Recent zbMATH articles in MSC 20https://zbmath.org/atom/cc/202023-09-22T14:21:46.120933ZUnknown authorWerkzeugLocally bi-2-transitive graphs and cycle-regular graphs, and the answer to a 2001 problem posed by Fouquet and Hahnhttps://zbmath.org/1517.050722023-09-22T14:21:46.120933Z"Conder, Marston"https://zbmath.org/authors/?q=ai:conder.marston-d-e"Zhou, Jin-Xin"https://zbmath.org/authors/?q=ai:zhou.jinxinSummary: A vertex-transitive but not edge-transitive graph \(\Gamma\) is called locally bi-2-transitive if the stabiliser \(S\) in the full automorphism group of \(\Gamma\) of every vertex \(v\) of \(\Gamma\) has two orbits of equal size on the neighbourhood of \(v\), and \(S\) acts 2-transitively on each of these two orbits. Also a graph is called cycle-regular if the number of cycles of a given length passing through a given edge in the graph is a constant, and a graph with girth \(g\) is called edge-girth-regular if the number of cycles of length \(g\) passing through any edge in the graph is a constant. In this paper, we prove that a graph of girth 3 is edge-girth-regular and locally bi-2-transitive if and only if \(\Gamma\) is the line graph of a semi-symmetric locally 3-transitive graph. Then as an application, we prove that every tetravalent edge-girth-regular locally bi-2-transitive graph of girth 3 is cycle-regular. This shows that vertex-transitive cycle-regular graphs need not to be edge-transitive, and hence resolves the problem posed by \textit{J.-L. Fouquet} and \textit{G. Hahn} [Discrete Appl. Math. 113, No. 2--3, 261--264 (2001; Zbl 0990.05086)] at the end of their paper.Binding number for coprime graph of groupshttps://zbmath.org/1517.050732023-09-22T14:21:46.120933Z"Mallika, A."https://zbmath.org/authors/?q=ai:mallika.a"Ahamed Thamimul Ansari, J."https://zbmath.org/authors/?q=ai:ahamed-thamimul-ansari.jSummary: Let \(G\) be a finite group with identity \(e\). The coprime of \(G\), \(\Gamma_G\) is a graph with \(G\) as the vertex set and two distinct vertices \(u\) and \(v\) are adjacent if and only if \((|u|, |v|)=1\). In this paper, we characterize the groups for which the binding number of coprime graph is 1 and investigate its bound range.Commuting graph of \(CA\)-groupshttps://zbmath.org/1517.050762023-09-22T14:21:46.120933Z"Torktaz, Mehdi"https://zbmath.org/authors/?q=ai:torktaz.mehdi"Ashrafi, Ali Reza"https://zbmath.org/authors/?q=ai:ashrafi.ali-rezaSummary: A group \(G\) is called a \(CA\)-group, if all the element centralizers of \(G\) are abelian and the commuting graph of \(G\) with respect to a subset \(A\) of \(G\), denoted by \(\Gamma (G, A)\), is a simple undirected graph with vertex set \(A\) and two distinct vertices \(a\) and \(b\) are adjacent if and only if \(ab = ba\). The aim of this paper is to generalize results of a recently published paper of \textit{F. Ali} et al. [Commun. Algebra 44, No. 6, 2389--2401 (2016; Zbl 1339.05176)] to the case that \(G\) is an \(CA\)-group.A note on the action of the Hecke group \(H(2)\) on subsets of the form \(\mathbb{Q}^{\ast}(\sqrt{n})\)https://zbmath.org/1517.051822023-09-22T14:21:46.120933Z"Cimpoeaş, Mircea"https://zbmath.org/authors/?q=ai:cimpoeas.mirceaSummary: Here, we study the action of the groups \(H(\lambda)\) generated by the linear fractional transformations \(x:z\mapsto -\frac{1}{z}\) and \(w:z\mapsto z+\lambda\) and \(\lambda\) is a positive integer, on the subsets \(\mathbb{Q}^{\ast}(\sqrt{n})=\{\frac{a+\sqrt{n}}{c}\mid a,b=\frac{a^2 -n}{c},c\in\mathbb{Z}\}\cup\{0,1,\infty\}\) and \(|n|\) is a square-free positive integer. We prove that this action has a finite number of orbits if and only if \(\lambda =1\) or \(\lambda =2\). Moreover, we give an upper bound for the number of orbits for \(\lambda =2\).Regular Cayley maps of elementary abelian \(p\)-groups: classification and enumerationhttps://zbmath.org/1517.051832023-09-22T14:21:46.120933Z"Du, Shaofei"https://zbmath.org/authors/?q=ai:du.shaofei"Yu, Hao"https://zbmath.org/authors/?q=ai:yu.hao.3|yu.hao.1|yu.hao.4|yu.hao.2|yu.hao"Luo, Wenjuan"https://zbmath.org/authors/?q=ai:luo.wenjuanSummary: Recently, regular Cayley maps of cyclic groups and dihedral groups have been classified in [\textit{M. D. E. Conder} and \textit{T. W. Tucker}, Trans. Am. Math. Soc. 366, No. 7, 3585--3609 (2014; Zbl 1290.05160)] and [\textit{I. Kovács} and \textit{Y. S. Kwon}, J. Comb. Theory, Ser. B 148, 84--124 (2021; Zbl 1459.05119)], respectively. A natural question is to classify regular Cayley maps of elementary abelian \(p\)-groups \(\mathbb{Z}_p^n\). In this paper, a complete classification of regular Cayley maps of \(\mathbb{Z}_p^n\) is given and moreover, the number of these maps and their genera are enumerated.Proofs of Chappelon and Ramírez Alfonsín conjectures on square Frobenius numbers and their relationship to simultaneous Pell equationshttps://zbmath.org/1517.110252023-09-22T14:21:46.120933Z"Binner, Damanvir Singh"https://zbmath.org/authors/?q=ai:binner.damanvir-singhSummary: Recently, \textit{J. Chappelon} and \textit{J. L. Ramírez Alfonsín} [Semigroup Forum 105, No. 1, 149--171 (2022; Zbl 07570869)] defined the square Frobenius number of coprime numbers \(m\) and \(n\) to be the largest perfect square that cannot be expressed in the form \(mx+ny\) for nonnegative integers \(x\) and \(y\). When \(m\) and \(n\) differ by 1 or 2, they found simple expressions for the square Frobenius number if neither \(m\) nor \(n\) is a perfect square. If either \(m\) or \(n\) is a perfect square, they formulated some interesting conjectures which have an unexpected close connection with a known recursive sequence, related to the denominators of Farey fraction approximations to \(\sqrt{2 }\). In this note, we prove these conjectures. Our methods involve solving Pell equations \(x^2 - 2y^2 = 1\) and \(x^2 - 2y^2 = -1\). Finally, to complete our proofs of these conjectures, we eliminate several cases using several results related to solutions of simultaneous Pell equations.On some arithmetic questions of reductive groups over algebraic extensions of local and global fieldshttps://zbmath.org/1517.110272023-09-22T14:21:46.120933Z"Thắng, Nguyễn Quốc"https://zbmath.org/authors/?q=ai:nguyen-quoc-thang.From the introduction: ``In the present paper, we are interested in answering the following question: which of the classical results in Galois cohomology theory of linear algebraic groups over local or global fields still hold in the case of infinite algebraic extensions of local and global fields? We investigate here some results which are related to the finiteness, the surjectivity (bijectivity) of a coboundary map in Galois cohomology which are important in arithmetic of algebraic groups over field. In particular, we extend Kneser's Theorem on the surjectivitiy of certain coboundary maps in Galois cohomology and Conrad's Theorem (thus partially also Borel-Serre's Theorem) on the finiteness of Galois cohomology of pseudo-reductive groups to the case of infinite algebraic extensions of local and global fields. As an application, we apply the finiteness of Galois cohomology to show the finiteness of the obstruction to weak approximation of connected reductive groups at finite set of places.''
A detailed version of the paper will be published elsewhere.
The following topics are discussed in sections:
\begin{itemize}
\item[--] Tate-Nakayama duality theory for algebraic group schemes of multiplicative type.
\item[--] Surjectivity of a coboundary map \(H^1_{\mathrm{fppf}}(S,\tilde G/Z)\to H^2_{\mathrm{fppf}}(S,Z(\tilde G))\) where \(Z\) is a subgroup \(S\)-scheme of the center \(Z(\tilde G)\) of a semisimple simply connected \(S\)-group scheme \(\tilde G\).
\item[--] Finiteness of Galois (flat) cohomology for pseudo-reductive groups.
\item[--] The obstruction to weak approximation.
\end{itemize}
Reviewer: Wilberd van der Kallen (Utrecht)Residual finiteness of extensions of arithmetic subgroups of \(\mathrm{SU}(d,1)\) with cuspshttps://zbmath.org/1517.110302023-09-22T14:21:46.120933Z"Hill, Richard M."https://zbmath.org/authors/?q=ai:hill.richard-michaelSummary: Let \(\Gamma\) be an arithmetic subgroup of \(\mathrm{SU}(d,1)\) with cusps, and let \(X_\Gamma\) be the associated locally symmetric space. In this paper we investigate the pre-image of \(\Gamma\) in the covering groups of \(\mathrm{SU}(d,1)\). Let \(H^\bullet_!(X_\Gamma ,\mathbb{C})\) be the inner cohomology, i.e. the image in \(H^\bullet (X_\Gamma ,\mathbb{C})\) of the compactly supported cohomology. We prove that if the first inner cohomology group \(H^1_!(X_\Gamma ,\mathbb{C})\) is non-zero then the pre-image of \(\Gamma\) in each connected cover of \(\mathrm{SU}(d,1)\) is residually finite. At the end of the paper we give an example of an arithmetic subgroup \(\Gamma\) satisfying the criterion \(H^1_!(X_\Gamma ,\mathbb{C}) \ne 0\).Error term concerning number of subgroups of group \(\mathbb{Z}_m \times \mathbb{Z}_n\) with \(m^2 + n^2 \le x\)https://zbmath.org/1517.111242023-09-22T14:21:46.120933Z"Sui, Yankun"https://zbmath.org/authors/?q=ai:sui.yankun"Liu, Dan"https://zbmath.org/authors/?q=ai:liu.danLet \(\mathbb{Z}_n\) be the additive group of residue classes modulo \(n\) (the cyclic group of order \(n\)). Let \(s(m, n)\) denote the number of subgroups of the group \(\mathbb{Z}_m \times \mathbb{Z}_n\), where \(m\) and \(n\) are arbitrary positive integers. Asymptotic formulas for the sum \(\sum_{m,n\le x} s(m,n)\) have been obtained by \textit{W. G. Nowak} and \textit{L. Tóth} [Int. J. Number Theory 10, No. 2, 363--374 (2014; Zbl 1307.11098)], \textit{L. Tóth} and \textit{W. Zhai} [Acta Arith. 183, No. 3, 285--299 (2018; Zbl 1435.11130)].
In this paper the authors deduce an asymptotic formula for the sum \(\sum_{m^2+n^2\le x} s(m,n)\) by using analytic arguments.
Reviewer: László Tóth (Pécs)An exact sequence and triviality of Bogomolov multiplier of groupshttps://zbmath.org/1517.130052023-09-22T14:21:46.120933Z"Hatui, Sumana"https://zbmath.org/authors/?q=ai:hatui.sumanaThe present paper is devoted to the Bogomolov multiplier \(B_0(G)\) of a finite group \(G\). Recall that \(B_0(G)\) is the subgroup of \(H^2(G,\mathbb Q/\mathbb Z)\), containing all the cohomology classes whose restriction to every abelian subgroup of \(G\) is 0. This group was introduced by \textit{F. A. Bogomolov} [Math. USSR, Izv. 30, No. 3, 455--485 (1988; Zbl 0679.14025); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 51, No. 3, 485--516 (1987)] and revisited by \textit{P. Moravec} in his seminal paper [Am. J. Math. 134, No. 6, 1679--1704 (2012; Zbl 1346.20072)]. Moravec suggested two isomorphisms which made possible to calculate explicitly the Bogomolov multiplier \(B_0(G)\). One of the isomorphisms is a Hopf-type formula: \[\tilde B_0(G)\simeq\frac{F'\cap R}{\langle K(F)\cap R\rangle},\] where \(B_0(G)\simeq\mathrm{Hom}(\tilde B_0(G),\mathbb Q/\mathbb Z)\) and \(F\) is a free group such that \(G\) is a quotient of \(F\) by the normal closure \(R\) of a given set of relations of \(F\). In the present paper the author gives another interpretation of this formula, by using the transgression map \(\mathrm{tra:Hom}(R/\langle K(F)\cap R\rangle,\mathbb Q/\mathbb Z)\to B_0(G)\). This formula is used to derive a long exact sequence (Theorem 1.2), which is then used to provide necessary and sufficient conditions for the corresponding inflation homomorphism to be an epimorphism and to be the zero map (Theorem 1.3). Finally, the author gives a complete list of groups of order \(p^6\), for odd prime \(p\), having trivial Bogomolov multiplier (Theorem 1.4), thus completing the investigation of \textit{Y. Chen} and \textit{R. Ma} [Commun. Algebra 49, No. 1, 242--255 (2021; Zbl 1459.13007)].
Reviewer: Ivo M. Michailov (Shumen)Cluster algebras and higher order generalizations of the \(q\)-Painleve equations of type \(A_7^{(1)}\) and \(A_6^{(1)}\)https://zbmath.org/1517.130192023-09-22T14:21:46.120933Z"Masuda, Tetsu"https://zbmath.org/authors/?q=ai:masuda.tetsu"Okubo, Naoto"https://zbmath.org/authors/?q=ai:okubo.naoto"Tsuda, Teruhisa"https://zbmath.org/authors/?q=ai:tsuda.teruhisaSummary: We construct higher order generalizations of the \(q\)-Painlevé equations of surface type \(A_7^{(1)}\) and \(A_6^{(1)}\) based on the cluster algebras corresponding to certain quivers. These equations possess the affine Weyl group symmetries of type \(A_1^{(1)}\) and \((A_1+A'_1)^{(1)}\), respectively. We show that these equations and symmetries can be realized as birational canonical transformations. A relationship between the quivers and the discrete KdV equation is also discussed.Derived categories of the Cayley plane and the coadjoint Grassmannian of type Fhttps://zbmath.org/1517.140312023-09-22T14:21:46.120933Z"Belmans, Pieter"https://zbmath.org/authors/?q=ai:belmans.pieter"Kuznetsov, Alexander"https://zbmath.org/authors/?q=ai:kuznetsov.alexander-gennadevich"Smirnov, Maxim"https://zbmath.org/authors/?q=ai:smirnov.maximA long standing conjecture states that the bounded derived category of coherent sheaves on a rational homogeneous variety admits a full exceptional collection. It is rather easy to see that the problem can be reduced to the study of generalized Grassmannians, i.e., quotients of simple algebraic groups by maximal parabolic subgroups.
While there is at least a general construction for groups of type \textrm{BCD}, little is known for exceptional groups. For instance, in types \textrm{E} and \textrm{F} the only previously known complete result belongs to \textit{D. Faenzi} and \textit{L. Manivel} [Proc. Am. Math. Soc. 143, No. 3, 1057--1074 (2015; Zbl 1318.14015)], who constructed a full exceptional collection in the bounded derived category of the so-called Cayley plane, i.e., the variety \(E_6/P_1\) (here and later the Bourbaki numbering of verticies in Dynkin diagrams is used). In the present paper the authors construct a full exceptional collection in the bounded derived category of \(F_4/P_4\), the coadjoint Grassmannian of Dynkin type \(F_4\). The collection that they construct is Lefschetz. It is known that small quantum cohomology of this Grassmannian is not generically semisimple. The authors show that the residual category of their Lefschetz exceptional collection is not generated by completely orthogonal exceptional objects, thus giving evidence to the generalization of Dubrovin's conjecture proposed by \textit{A. Kuznetsov} and \textit{M. Smirnov} [Proc. Lond. Math. Soc. (3) 120, No. 5, 617--641 (2020; Zbl 1493.14078)], the second and the third authors of the paper, respectively.
Exceptional collections, especially full ones, are notoriously difficult to construct. One approach is to look for a \textit{Lefschtz} exceptional collection. Namely, one can find a small number of objects forming an exceptional sequence and attempt to extend it by taking twists by a fixed (ample in most cases) line bundle \(\mathcal{L}\) of (some of) its elements. Thus one gets the second block and continues the process. If the given variety is Fano and its anti-canonical bundle is isomorphic to \(\omega\simeq \mathcal{L}^{-m}\), then each object from the initial sequence, the \textit{starting block}, may appear at most \(m\) times (twisted by \(\mathcal{L}^i\) for \(i=0,\ldots,m-1\)). Lefschetz exceptional collections behave particularly well with respect to taking hyperplane section. Assume that \(\mathcal{L}\) is very ample, and that \(Y\subset X\) is a smooth (for simplicity) hyperplane section of \(X\) with respect to \(\mathcal{L}\). Then the bounded derived category of coherent sheaves on \(Y\) has an exceptional collection: one drops the starting block of the Lefschetz collection on \(X\) and restricts the resulting collection to \(Y\). In general, the resulting collection is not full.
We have mentioned earlier that a full exceptional collection was known for the Cayley plane. Actually, Faenzi and Manivel [loc. cit.] constructed a collection which is Lefschetz with respect to the very ample generator of the Picard group. It so happens that the coadjoint Grassmannian is isomorphic to a generic hyperplane section of the Cayley plane. Thus, one gets an exceptional collection by dropping the starting block of the Manivel-Faenzi collection and restricting to the section. This is the collection that is in the center of the present paper. For istance, the authors (1) prove that it is full, (2) compute its residual category.
Fullness is usually a difficult question in this field. Fortunately for the authors, an argument previously used by \textit{A. Samokhin} [C. R., Math., Acad. Sci. Paris 340, No. 12, 889--893 (2005; Zbl 1078.14059)] is applicable in their case. Next, the authors show that the residual category of the Faenzi-Manivel collection is generated by three completely orthogonal objects (this is in favor of the modified Dubrovin's conjecture mentioned above since the small quantum cohomology of the Cayley plane is known to be generically semi-simple). Finally they compute the residual category for the collection on the adjoint Grassmannian (is is equivalent to the derived category of the quiver \(A_2\)). This, to our taste, is the nicest part of the paper: the authors establish a rather general statement about residual categories of hyperplane sections (it turns out that if the initial residual category is generated by a completely othogonal sequence, then the residual category for the hyperplane section is isomorphic to the product of derived categories of quivers of type \(A_i\), see Theorem 2.6).
Reviewer: Anton Fonarev (Moskva)Geometric structure of affine Deligne-Lusztig varieties for \(\mathrm{GL}_3\)https://zbmath.org/1517.140352023-09-22T14:21:46.120933Z"Shimada, Ryosuke"https://zbmath.org/authors/?q=ai:shimada.ryosukeLet \(k\) be the finite field with \(q\) elements, \(\overline{k}\) the algebraic closure of \(k\), \(L = \overline{k}((t))\) and \(\mathcal{O}=\overline{k}[[t]]\) the valuation ring of \(L\).
Let \(G\) be a split connected reductive group over \(k\) and let \(T\) be a split maximal torus of it. Let \(B\) be a Borel subgroup of \(G\) containing \(T\). For a cocharacter \(\lambda \in X_{\ast}(T)\), let \(t^{\lambda}\) be the image of \(t \in \mathbb{G}_{m}(F)\) under the homomorphism \(\lambda: \mathbb{G}_{m} \rightarrow T\). Then the affine Deligne-Lusztig variety \(X_{\lambda}(b)\) is the locally closed reduced \(\overline{k}\)-subscheme of the affine Grassmannian defined as
\[
X_{\lambda}(b)(\overline{k}) = \big \{ xG(\mathcal{O}) \in G(L)/G(\mathcal{O}) \; \big | \; x^{-1}b\sigma(x) \in G(\mathcal{O})t^{\lambda}G(\mathcal{O}\, \big \}
\]
where \(\sigma\) is the Frobenius morphism of \(\overline{k}/k\).
The affine Deligne-Lusztig variety \(X_{\lambda}(b)\) carries a natural action (by left multiplication) by the group \(J_{b}=J_{b}(F)=\{g \in G(L) \mid g^{-1}b\sigma(g)=b \}\) and this action induces an action of \(J_{b}\) on the set of irreducible components.
In this paper, the author studies the geometry of \(X_{\lambda}(b)\) for \(G=\mathrm{GL}_{3}\) and \(b\) basic. The main theorem is Theorem 1.1: The irreducible components of \(X_{\lambda}(b)\) are parameterized by the set \(\bigsqcup_{\mu \in M} J_{b}/K_{\mu}\), where \(M\) is a finite set consisting of certain dominant cocharacters \(\mu\) determined by \(\lambda\), and \(K_{\mu}\) is the stabilizer of a lattice depending on \(\mu\) under the action of \(J_{b}\). (The interested reader should see Theorem 6.1 and Theorem 6.3 for a more precise statement.)
As a corollary of this result, the author classifies the cases where all of the irreducible components of \(X_{\lambda}(b)\) are classical Deligne-Lusztig varieties times finite dimensional affine spaces (see Corollary 6.4). In such cases, \(X_{\lambda}(b)\) is a disjoint union of the irreducible components.
Reviewer: Enrico Jabara (Venezia)Two-term tilting complexes for preprojective algebras of non-Dynkin typehttps://zbmath.org/1517.160092023-09-22T14:21:46.120933Z"Kimura, Yuta"https://zbmath.org/authors/?q=ai:kimura.yuta"Mizuno, Yuya"https://zbmath.org/authors/?q=ai:mizuno.yuyaSummary: In this paper, we study two-term tilting complexes for preprojective algebras of non-Dynkin type. We show that there exist two families of two-term tilting complexes, which are respectively parameterized by the elements of the corresponding Coxeter group. Moreover, we provide the complete classification in the case of affine type by showing that any two-term silting complex belongs one of them. For this purpose, we also discuss the Krull-Schmidt property for the homotopy category of finitely generated projective modules over a complete ring.On the number of generators of an algebra over a commutative ringhttps://zbmath.org/1517.160142023-09-22T14:21:46.120933Z"First, Uriya A."https://zbmath.org/authors/?q=ai:first.uriya-a"Reichstein, Zinovy"https://zbmath.org/authors/?q=ai:reichstein.zinovy-b"Williams, Ben"https://zbmath.org/authors/?q=ai:williams.benLet \(k\) be an infinite field, and \(R\) a \(k\)-ring (meaning commutative associative unital \(k\)-algebra. \textit{O. Forster} [Math. Z. 84, 80--87 (1964; Zbl 0126.27303)] proved the following result: if \(R\) is noetherian with Krull dimension \(k\), then any projective \(R\)-module of rank \(n\) can be generated by \(k+n\) elements. The authors create the following setting for Forster type Theorems. An \(R\)-algebra is an \(R\)-module \(B\) with a \(B\)-bilinear map \(B^2\to B\). An \(R\)-form of a \(k\)-algebra \(A\) is an \(R\)-algebra \(B\) such that there exists a faithfully flat \(R\)-ring \(S\) such that \(A\otimes_k S\) and \(B\otimes_RS\) are isomorphic as \(S\)-algebras. In this way projective \(R\)-modules of rank \(n\) are forms of \(k^n\) with zero multiplication. Other examples of forms include finite étale \(R\)-algebras, Azumaya algebras, octonion algebras and Albert algebras. An even more general setting arises when one considers so-called multialgebras. An \(R\)-multialgebra is an \(R\)-module \(A\) together with a family of \(R\)-linear maps \(m_i:\ A^{n_i}\to A\). In fact an algebra with unit, or an algebra with an involution can be considered as a multialgebra.
For an \(R\)-(multi)algebra \(B\), let \(\mathrm{gen}_R(B)\) be the smallest cardinality of a set of generators of \(B\) as an \(R\)-algebra. The aim of the present paper is to find upper and lower bounds of \(\mathrm{gen}_R(B)\) in the situation where \(B\) is an \(R\)-form of a given \(R\)-algebra \(A\). There are three main result, Theorems 1.3, 1.4 and 1.5. Theorem 1.3 brings an upper bound in the case where \(A\) is an \(n\)-dimensional \(k\)-algebra, and \(R\) is a finite type \(k\)-ring. Theorem 1.4 gives a lower bound for certain rings \(R\) and certain forms of \(A\), under the assumption that \(\mathrm{Aut}_k(A)\) is not unipotent. Theorems 1.3 and 1.4 also hold for multialgebras. In Theorem 1.5, both the upper and lower bound can be sharpened in the situation where \(A\) is a matrix algebra (and \(B\) is an \(R\)-Azumaya algebra).
Reviewer: Stefaan Caenepeel (Brussels)UN rings and group ringshttps://zbmath.org/1517.160162023-09-22T14:21:46.120933Z"Jangra, Kanchan"https://zbmath.org/authors/?q=ai:jangra.kanchan"Udar, Dinesh"https://zbmath.org/authors/?q=ai:udar.dineshSummary: A ring \(R\) is called a UN ring if every non unit of it can be written as product of a unit and a nilpotent element. We obtain results about lifting of conjugate idempotents and unit regular elements modulo an ideal \(I\) of a UN ring \(R\). Matrix rings over UN rings are discussed and it is obtained that for a commutative ring \(R\), a matrix ring \(M_n (R)\) is UN if and only if \(R\) is UN. Lastly, UN group rings are investigated and we obtain the conditions on a group \(G\) and a field \(K\) for the group algebra \(KG\) to be UN. Then we extend the results obtained for \(KG\) to the group ring \(RG\) over a ring \(R\) (which may not necessarily be a field).A basis of a certain module for the hyperalgebra of \((\mathrm{SL}_2)_r\) and some applicationshttps://zbmath.org/1517.160172023-09-22T14:21:46.120933Z"Yoshii, Yutaka"https://zbmath.org/authors/?q=ai:yoshii.yutakaSummary: In the hyperalgebra \(\mathcal{U}_r\) of the \(r\) th Frobenius kernel \((\mathrm{SL}_2)_r\) of the algebraic group \(\mathrm{SL}_2\), we construct a basis of the \(\mathcal{U}_r\)-module generated by a certain element which was given by the author before. As its applications, we also prove some results on the \(\mathcal{U}_r\)-modules and the algebra \(\mathcal{U}_r\).Bound for the cocharacters of the identities of irreducible representations of \(\mathfrak{sl}_2(\mathbb{C})\)https://zbmath.org/1517.160232023-09-22T14:21:46.120933Z"Domokos, Mátyás"https://zbmath.org/authors/?q=ai:domokos.matyasLet \(L=\mathfrak{sl}_2(\mathbb{C})\) be the three dimensional simple Lie algebra over the complex numbers. The author of the paper under review studies numerical invariants of the polynomial identities of the irreducible f.d. representations of \(L\). The identities of representations of \(L\) were first studied by \textit{Yu. P. Razmyslov} [Transl., Ser. 2, Am. Math. Soc. 140, 101--109 (1988; Zbl 0658.17014); translation from Algebra, Work Collect., dedic. O. Yu. Shmidt, Moskva 1982, 139--150 (1982)], see also [\textit{Yu. P. Razmyslov}, Identities of algebras and their representations. Transl. from the Russian by A. M. Shtern. Transl. ed. by Simeon Ivanov. Providence, RI: American Mathematical Society (1994; Zbl 0827.17001)]. When studying polynomial identities in characteristic 0, one can restrict the considerations to the multilinear identities only. The multilinear identities of given degree \(n\) form a module over the symmetric group \(S_n\) and one applies the theory of representations of \(S_n\) in order to study ideals of identities. Another, equivalent, approach is often much better. It is well known that the representations of \(S_n\), and the polynomial representations of the general linear group \(\mathrm{GL}_m(\mathbb{C})\), are described in the same terms, and are ``dual'' to each other. Let \(\rho\) be an irreducible representation of \(L\), and let \(I(L,\rho)\) be the ideal of identities of \(\rho\). In other words, this is the ideal of weak identities of the pair \((V,L)\) (or \((V,\rho)\)) where \(\rho\colon L\to V\). The author studies the cocharacter sequence of \(I(L,\rho)\). The main theorem of the paper under review states that the only non-zero irreducible modules that appear in the decomposition of the multilinear part of the ``non-identities'' correspond to partitions of at most three parts. If \(\dim V=d\) then these multiplicities are bounded by \(3^{d-2}\).
It should be noted that if one considers the ordinary identities of the full matrix algebra of order 2, its cocharacter has no bounded multiplicities. (Though for every PI algebra \(A\) the multiplicities of the irreducible modules that appear in its cocharacter are polynomially bounded.) The author proves that in case \(\dim V=d\) then there exists a multiplicity which is \(\ge d-1\); thus one cannot expect any uniform (that is independent of \(d\)) bound.
Reviewer: Plamen Koshlukov (Campinas)Quasi-bialgebras from set-theoretic type solutions of the Yang-Baxter equationhttps://zbmath.org/1517.160282023-09-22T14:21:46.120933Z"Doikou, Anastasia"https://zbmath.org/authors/?q=ai:doikou.anastasia"Ghionis, Alexandros"https://zbmath.org/authors/?q=ai:ghionis.alexandros"Vlaar, Bart"https://zbmath.org/authors/?q=ai:vlaar.bartThere is an interest in studying classes of quantum algebras arising from set-theoretic solutions of the Yang-Baxter equation. The motivations for deepening this investigation lie in the paper [\textit{A. Doikou} and \textit{A. Smoktunowicz}, Lett. Math. Phys. 111, No. 4, Paper No. 105, 40 p. (2021; Zbl 1486.16039)].
In the paper under review, the authors investigate algebras coming from involutive and non-degenerate solutions and their \(q\)-analogues. It is shown that they are quasi-triangular quasi-bialgebras. To get this result, they provide some universal results on quasi-bialgebras and admissible Drinfeld twists. In fact, the property of being quasi-triangular (quasi-)bialgebra is preserved by twisting (see [\textit{V. G. Drinfel'd}, Sov. Math., Dokl. 32, 256--258 (1985; Zbl 0588.17015); translation from Dokl. Akad. Nauk SSSR 283, 1060--1064 (1985)]). Moreover, they make use of some first results on the admissible Drinfeld twist for involutive set-theoretic solution already derived in [\textit{A. Doikou}, J. Phys. A, Math. Theor. 54, No. 41, Article ID 415201, 21 p. (2021; Zbl 07654521)].
Reviewer: Marzia Mazzotta (Lecce)A wells type exact sequence for non-degenerate unitary solutions of the Yang-Baxter equationhttps://zbmath.org/1517.160292023-09-22T14:21:46.120933Z"Bardakov, Valeriy"https://zbmath.org/authors/?q=ai:bardakov.valerii-georgievich"Singh, Mahender"https://zbmath.org/authors/?q=ai:singh.mahenderThis paper is devoted to linear cycle sets which are closely related to set-theoretical solutions of the Yang-Baxter equation.
Let \(X\) be a set. Let \(r\colon X\times X\to X\times X\) be a map. Let \(r_{12},r_{23}\colon X\times X\times X\to X\times X\times X\) be the maps given respectively by \(r_{12}(x,y,z)=(r(x,y),z)\) and \(r_{23}(x,y,z)=(x,r(y,z))\). A pair \((X,r)\), where \(r\) is as above, is a \textit{set-theoretic solution} of the Yang-Baxter equation when \(r_{12}r_{23}r_{12}=r_{23}r_{12}r_{23}\). \(r\) then is \textit{non-degenerate} when the maps \(\pi_2\circ r(x,-)\colon X\to X\) and \(\pi_1\circ r(-,y)\colon X\to X\) are bijective for all \(x,y\in X\), where \(\pi_1,\pi_2\colon X\times X\to X\) are respectively the projections onto the first and the second factor.
A \textit{(left) cycle set} is a non-empty set \(X\) with a binary operation \(\cdot\) having bijective left translations \(x\mapsto y\cdot x\), and satisfying the equation \((x\cdot y)\cdot (x\cdot z)=(y\cdot x)\cdot (y\cdot z)\) for all \(x,y,z\in X\).
Cycle sets are in bijection with non-degenerate ``unitary'' set-theoretic solutions of the Yang-Baxter equation.
A \textit{(left) linear cycle set} is a cycle set \((X,\cdot)\) with an abelian group operation \(+\) such that \(\cdot\) is left distributive over \(+\) and such that \((x+y)\cdot z=(x\cdot y)\cdot (x\cdot z)\), \(x,y,z\in X\). E.g. any abelian group becomes a linear cycle set, called \textit{trivial}, when \((X,\cdot)\) is a right-zero band, that is, with \(x\cdot y=y\).
In this paper the authors, after recalling some notions about cohomology and about extensions of linear cycle sets, prove that there is a canonical group homomorphism between the second linear cycle set cohomology and the second symmetric cohomology of the underlying abelian group (Prop. 2.6), and, for trivial linear cycle sets, they show that this homomorphism is onto, and that in fact up to an isomorphism it is simply the projection onto the second factor (Prop. 2.7).
They also provide, for each central extension of linear cycle sets, a four term exact sequence in which occur group of \(1\)-cocyles, automorphism groups and second cohomology groups (Theorem 4.5).
Reviewer: Laurent Poinsot (Villetaneuse)Orders of units in integral group rings and blocks of defect 1https://zbmath.org/1517.160312023-09-22T14:21:46.120933Z"Caicedo, Mauricio"https://zbmath.org/authors/?q=ai:caicedo.mauricio"Margolis, Leo"https://zbmath.org/authors/?q=ai:margolis.leoLet \(\mathbb Z G\) be the integral group ring of a group \(G\) and \(V(\mathbb Z G)\) be the normalized unit group of \(\mathbb Z G\). The set of the orders of a finite group \(G\) is called the spectrum of \(G\). The so-called Spectrum Problem is the following. Whether the spectra of \(V(\mathbb Z G)\) and \(G\) coincide. The Spectrum Problem has a positive solution for many classes of groups, in particular for the solvable groups [\textit{M. Hertweck}, Commun. Algebra 36, No. 10, 3585--3588 (2008; Zbl 1157.16010)]. The weaker version of the Spectrum Problem is so-called Prime Graph Question: whether \(V(\mathbb Z G)\) and \(G\) have the same prime graph?
The authors prove the following basic result. If \(p\) is a prime and the Sylow \(p\)-subgroup of \(G\) is of order \(p\), then for any prime \(q\), there is an element of order \(pq\) in \(V(\mathbb Z G)\) if and only if there is an element of order \(pq\) in \(G\) (Theorem 1.1).
From this result they answer Prime Graph Question for most sporadic simple groups and some groups of Lie type, including seven new infinite series of such groups, as well as the corresponding almost simple groups. The methods of the indicated results use blocks of cyclic defect, in particular the theory of blocks of defect \(1\) as developed by Brauer, and Young tableaux combinatorics. These serve to restrict the possible actions of a critical unit on \(G\)-modules over fields of positive characteristic. To apply the results to infinitely many groups of Lie type, it is also proven that a product of cyclotomic polynomials of degree at most \(3\) evaluated at primes takes square-free values infinitely many times (this is due to R. Heath-Brown).
The arguments involved can be considered a generalization of the methods involved in earlier papers on the topic, e.g. [\textit{A. Bächle} and \textit{L. Margolis}, Proc. Am. Math. Soc. 147, No. 10, 4221--4231 (2019; Zbl 1444.16052)].
Reviewer: Todor Mollov (Plovdiv)Rota-Baxter operators on Clifford semigroups and the Yang-Baxter equationhttps://zbmath.org/1517.170152023-09-22T14:21:46.120933Z"Catino, Francesco"https://zbmath.org/authors/?q=ai:catino.francesco"Mazzotta, Marzia"https://zbmath.org/authors/?q=ai:mazzotta.marzia"Stefanelli, Paola"https://zbmath.org/authors/?q=ai:stefanelli.paolaThis paper aims to show how to obtain weak braces from Rota-Baxter operators defined on Clifford semigroups, for which the authors introduce the notion of Rota-Baxter operator on a Clifford semigroup \(\left(S,+\right)\), i.e., a map
\[
\mathfrak{R}:S\rightarrow S
\]
abiding by the relations
\begin{align*}
\mathfrak{R}\left(a\right)+\mathfrak{R}\left(b\right) & =\left(a+\mathfrak{R}\left(a\right)+b-\mathfrak{R}\left(a\right)\right)\\
a+\mathfrak{R}\left(a\right)-\mathfrak{R}\left(a\right) & =a
\end{align*}
for all \(a,b\in S\).
The synopsis of the paper goes as follows.
\begin{itemize}
\item[\S 1] gives essential results on the structures of weak braces introduced in [\textit{F. Catino} et al., Semigroup Forum 104, No. 2, 228--255 (2022; Zbl 07533946)] to find solutions of the Yang-Baxter equation. Some basics on Clifford semigroups are recalled [\textit{A. H. Clifford} and \textit{G. B. Preston}, The algebraic theory of semigroups. Vol. I. Providence, RI: American Mathematical Society (AMS) (1961; Zbl 0111.03403); \textit{G. Cooperman} and \textit{L. Finkelstein}, J. Symb. Comput. 17, No. 6, 513--528 (1994; Zbl 0835.20007); \textit{M. Petrich}, Inverse semigroups. Pure and Applied Mathematics. A Wiley-Interscience Publication. New York etc.: John Wiley \& Sons. (1984; Zbl 0546.20053)].
\item[\S 2] presents and investigates the Rota-Baxter operators on a Clifford semigroup, consistently with that introduced for groups in [\textit{L. Guo} et al., Adv. Math. 387, Article ID 107834, 34 p. (2021; Zbl 1468.17026)].
\item[\S 3] focuses on Rota-Baxter endomorphisms, giving a description of such maps \(f\) for which \(\mathrm{Im}\,f\) is commutative. The authors characterize Rota-Baxter endomorphisms of groups that are also idempotents.
\item[\S 4] illustrates a method for obtaining a dual weak brace \(S\) starting from a given Rota-Baxter operator on a Clifford semigroup \(\left(S,+\right) \). The construction is inspired by that of skew braces due to \textit{V. G. Bardakov} and \textit{V. Gubarev} [J. Algebra 596, 328--351 (2022; Zbl 1492.17019), Proposition 3.1]. It is shown that every commutative Rota-Baxter endormorphism determines a bi-weak brace.
\item[\S 5] makes explicit the solutions associated to dual weak braces obtained through Rota-Baxter operators.
\item[\S 6] deals with the ideals of a dual weak brace, by extending the already known theory for skew braces. In particular, the notion of the socle of a dual weak brace is introduced. The ideals of dual weak braces associated to a given Rota-Baxter operator are finally described.
\end{itemize}
Reviewer: Hirokazu Nishimura (Tsukuba)Automorphism orbits and element orders in finite groups: almost-solubility and the Monsterhttps://zbmath.org/1517.200012023-09-22T14:21:46.120933Z"Bors, Alexander"https://zbmath.org/authors/?q=ai:bors.alexander"Giudici, Michael"https://zbmath.org/authors/?q=ai:giudici.michael"Praeger, Cheryl E."https://zbmath.org/authors/?q=ai:praeger.cheryl-eLet \(G\) be a finite group, let \(\Gamma_{n}=\{ g \in G \mid o(g)=n \}\) be the set of all elements of order \(n\) in \(G\). If \(A=\Aut(G)\), then an \(A\)-orbit of an element \(g \in G\) is the set \(g^{A}=\{g^{\alpha} \mid \alpha \in A \}\). The group \(G\) is called an \(\mathsf{AT}\)-group if \(\Aut(G)\) is ``as transitive as possible'', that is for every \(n\) such that \(\Gamma_{n} \not = \emptyset\) we have \(\Gamma_{n}=g^{A}\) for every \(g \in \Gamma_{n}\). \textit{J. Zhang} in [J. Algebra 153, No. 1, 22--36 (1992; Zbl 0767.20009)] has studied \(\mathsf{AT}\)-groups extensively proving that a simple \(\mathsf{AT}\)-group is isomorphic to either \(\mathrm{PSL}(3,4)\) or to \(\mathrm{PSL}(2,q)\) for suitable \(q \in \{5,7,8,9\}\).
The aim of the paper under review is to study notions of finite groups that are ``close to being \(\mathsf{AT}\)-groups''. That is, the authors see \(\mathsf{AT}\)-groups as extremal structures, lying at one end of a quantitative spectrum of homogeneity conditions. They will do this by comparing, for a given group \(G\), the numbers of \(A\)-orbits on \(G\) and of distinct element orders in \(G\), observing that \(G\) is an \(\mathsf{AT}\)-group if and only if these two numbers are equal.
Let \(\omega(G)\) be the number of \(A\)-orbits on \(G\), \(\mathrm{Ord}(G)\) be the set of element orders in \(G\), \(o(G)=| \mathrm{Ord}(G) |\). For \(n \in \mathrm{Ord}(G)\) let \(\omega_{n}(G)\) be the number of \(A\)-orbits on \(\Gamma_{n}\) and define \(\mathfrak{d}(G)=\omega(G)-o(G)\) and \(\mathfrak{q}(G)=\omega(G)/o(G)\).
Let \(\mathrm{Rad}(G)\) be the soluble radical of \(G\). The first main result of this paper is Theorem 1.1.2 which states that there are monotonically increasing (in each component) functions \(f_{i}: [0, \infty)^{i} \rightarrow [1,\infty)\) for \(i \in \{ 1, 2\}\) such that if \(G\) is a finite group, then \(|G: \mathrm{Rad}(G)| \leq f_{1}(\mathfrak{d}(G))\) and \(|G: \mathrm{Rad}(G)| \leq f_{2}(\mathfrak{q}(G),o(\mathrm{Rad}(G)))\).
The second main result is Theorem 1.1.3 which, among other things, states that if \(S\) is a nonabelian finite simple group, then
\[
\lim \inf_{|S| \rightarrow \infty} \frac{\log \log \omega(S)}{\log \log |S|}=\frac{1}{2} \quad\text{ and } \quad \lim_{|S| \rightarrow \infty}\frac{\log o(G)}{\log \omega(S)}=0.
\]
Furthermore, Theorems 1.1.2 and 1.1.3 give a curious quantitative characterisation of the Fischer-Griess Monster group.
Reviewer: Enrico Jabara (Venezia)Topics in infinite group theory. Nielsen methods, covering spaces, and hyperbolic groupshttps://zbmath.org/1517.200022023-09-22T14:21:46.120933Z"Fine, Benjamin"https://zbmath.org/authors/?q=ai:fine.benjamin-l"Moldenhauer, Anja"https://zbmath.org/authors/?q=ai:moldenhauer.anja-i-s"Rosenberger, Gerhard"https://zbmath.org/authors/?q=ai:rosenberger.gerhard"Wienke, Leonard"https://zbmath.org/authors/?q=ai:wienke.leonardAs it is known the first development of group theory, concerning the ideas of Galois, was limited to finite groups. But the idea of an abstract infinite group is due to Cayley who formulated the axioms for a group. The recognition, notably by Klein, the role of groups, many of them infinite, in geometry, as well as the development of continuous initiated by Lie, was the motivation for the development of infinite group theory. Another stimulus to the study of infinite discontinuous groups was the development of topology. Here we mention particularly the work of Poincaré, Dehn and Nielsen. Dehn posed the well-known problems, which gave impetus to the development of group theory.
Important contributions to the development of the ideas initiated by Dehn were made by Magnus, who has in turn been one of the strongest influences on contemporary research. The book [Combinatorial group theory. Presentations of groups in terms of generators and relations. New York-London-Sydney: Interscience Publishers, a division of John Wiley and Sons, Inc. (1966; Zbl 0138.25604)], by \textit{W. Magnus} et al., appeared in 1966, immediately became the classic in its field. In the meantime, new results were obtained and gave furthering bloom in the area of infinite groups. In 1977, an other book (with the same title) [Combinatorial group theory. Springer-Verlag, Berlin (1977; Zbl 0368.20023)], by \textit{R. C. Lyndon} and \textit{P. E. Schupp} was issued. This book, in one sense, could be considered as a continuation/completion of the previous book. Since then there was an explosion and a storm of new results. A main principle in the study of geometric objects is the analysis of their isometry groups. Gromov \textit{reverses the order} making groups geometric objects and studying the algebraic properties of the groups by means of geometric properties of their Cayley graphs.
The book under review by B. Fine, A. Moldenhauer, G. Rosenberger, and L. Wienke, is coming to give an other contribution in the area of infinite group theory. There are two broad methods running through their treatment. The first is the `linear' cancellation method of Nielsen which is concerned with the formal expression of an element of a group in terms of a given set of generators for the group. The second is the geometric method, initiated by Poincaré and Dehn, based on the the theory of covering spaces and fundamental groups of spaces.
Nevertheless, great importance is given to the interconnectedness of these methods and for many theorems are given different proofs. Also, emphasis on applications is given, in particular in number theory.
As it is indicated by the title, only some topics from this huge area of mathematics are presented.
The book consists of three chapters.
The first chapter has the umbrella title ``Nielsen methods'' and is divided in seven paragraphs.
1. Free groups and group presentations.
Here, the definition of a free group over a set \(X\) is given. It is proved that there exist free groups and that every group is a quotient of a free group.
By the definition of the free groups it is easy to see that the three problems of Dehn have positive solution on free groups.
The notions of residually finite, residually free and Hopfian groups are introduced and it is proved that free groups satisfy these properties.
A group \(G\) is commutative transitive (CT), provided the relation of commutativity is transitive on the non-identity elements of \(G\).
A group \(G\) is conjugately separated abelian (CSA) provided the maximal abelian subgroups are malnormal.
Here, the authors prove that in non-abelian residually free groups CT and CSA are equivalent properties.
2. The Nielsen method in free groups.
For a finite subset \(U\) of a group \(G\), the elementary Nielsen transformation on \(U\) is defined and it is proved that every finite subset of a free group can be transformed into a Nielsen reduced set by a finite number of elementary Nielsen transformations.
From this result, main results concerning free groups are derived.
\begin{itemize}
\item[(i)] Every subgroup of a free group is free (a result which can be proved easily using the action of groups on trees).
\item[(ii)] The automorphism group of a finitely generated free group is finitely generated.
\end{itemize}
The elementary Nielsen transformations give the possibility to test if an element of a free group is a primitive element.
The Whitehead automorphisms of a free groups are defined and it is proved that the automorphism group of a finitely generated free group is not only finitely generated, but finitely presented.
For a finitely generated free group \(F\) and an automorphism \(\vartheta \) of \(F\), the fixed point subgroup \(\mathrm{Fix}(\vartheta )=\{u\in F\mid \vartheta (u)=u\}\) is finitely generated. The proof of this result is easier using the theory of covering spaces.
We note that many results, which are proved here using combinatorial methods, are developed and proved
in the light of hyperbolic groups. See, for example, Section 3.6 in the book.
3. Generators and relators.
Using the projective property of free groups, for each group \(G\) a presentation \(\langle X\mid R \rangle\) with generators and relators is obtained. Conversely, for a set \(X\) and a subset \(R\) of the free group \(F(X)\) generated by \(X\), a group with presentation \(\langle X\mid R \rangle\) can be defined. For a group, there are many presentations. The problem is to decide if two presentations define isomorphic groups. The Tietze transformations give an answer if two presentations define isomorphic groups. The Reidemeister-Schreier method is developed to obtain representatives and presentations of a subgroup of a group with a given presentation. This method it is used here to obtain results for subgroups of free groups. Many of these results can be proved using the theory of group actions on spaces. The Reidemeister-Schreier method is very complicated and it is not easily applicable for every presentation.
Here, another method (the method of Todd-Coxeter) is developed which leads to a Schreier set of representatives for a finitely generated subgroup of a finitely presented group.
4. Free products.
The free product of a family of groups is defined. Using Nielsen transformations on sets of elements of a free product, the structure of a subgroup of free product is studied and special generating sets of a free product are obtained.
Also, it is proved that some properties of the factor groups of a free product are inherited to the free product. For example:
\begin{itemize}
\item[(i)] The free product of finitely many residually finite groups is residually finite.
\item[(ii)] The free product of finitely many residually free groups is residually free.
\item[(iii)] The free product of commutative transitive groups is commutative transitive.
\end{itemize}
As an application of free products, it is proved that extensions of finitely presented groups are finitely presented.
5. Free products with amalgamation.
The free product of two groups with an amalgamated subgroup is defined and their university property is proved. The normal form of an element in a free product with amalgamation is defined and it is proved that the normal form of an element is unique. From this fact, many properties of a free product with amalgamation are proved and some interesting equations in free product with amalgamation are studied.
6. HNN extensions.
An other construction of groups, closely related with the free product with amalgamation, are HNN extensions. The universal property of HNN extensions is proved and the normal form of an element in an HNN extension is defined and it is proved that it is unique. Many properties of HNN extensions are proved and some interesting equations in HNN extensions are studied.
HNN extensions are used to obtain remarkable results concerning embedding groups in other groups.
For example:
\begin{itemize}
\item[(i)] Each countable group \(C\) is embeddable in a two generated group \(G\). If \(C\) is finitely presented, then \(G\) is also finitely presented.
\item[(ii)] Every countable group \(C\) can be embedded into a countable group \(G\) in which all elements with the same order are conjugate.
\item[(iii)] Every countable group \(C\) can be embedded into a countable simple and divisible group.
\end{itemize}
A group \(G\) is called \(SQ\) universal if every countable group is embeddable in some factor group.
This property has been extensively studied using HNN extensions. Here, some results are discussed.
7. One-relator groups.
After the free groups (which have not relators), one-relator groups are considered. One-relator groups have attracted the interest of many researchers and play a fundamental role in infinite group theory.
The first fundamental result, which motivated the thorough study of one-relator groups is \textit{Magnus's Freiheitssatz}. Here, the main results in this area are developed. Later (in Chapter 3), the study of one-relators groups is given in the light of hyperbolic groups. See, for example, Theorems 3.5.8 and 3.5.28.
In the second chapter with the title ``Covering spaces'', a brief introduction is given to the concepts of paths, homotopy equivalence and the fundamental group of a space. Properties of spaces are studied with the help of fundamental groups and conversely properties of fundamental groups are studied with the help of properties of spaces. The concept of covering spaces is introduced. The theorem of Seifert and van Kampen has a dominant position here. Short and elegant proofs of theorems concerning free groups, free products, amalgamated free products and HNN-extensions are given as applications. For example, we refer to Corollary 2.3.19, Theorems 2.3.20 and 2.3.39, and to the proofs of Theorems 1.2.52 and 1.3.23 (p. 222 in the book).
A reference is made to how the residual finiteness of groups and more generally the subgroup separability of groups is connected with the theory of covering spaces referring to [\textit{P. Scott}, J. Lond. Math. Soc., II. Ser. 17, 555--565 (1978; Zbl 0412.57006); \textit{M. Tretkoff}, Contemp. Math. 109, 179--191 (1990; Zbl 0742.20028)].
Α particular mention is also made of surface groups.
Finally, reference is made (if possible in two pages) to the graph of groups and to the fundamental group of a graph of groups, presenting them as applications of the general theory of covering spaces and lifting maps.
As mentioned at the beginning of this chapter, the reader should be familiar with basic concepts of topology and is referred to relevant literature. In addition to these references, we also mention the book [\textit{J. R. Munkres}, Topology. A first course. Englewood Cliffs, N.J.: Prentice-Hall, Inc. (1975; Zbl 0306.54001)].
In his seminal paper [in: Essays in group theory, Publ., Math. Sci. Res. Inst. 8, 75--263 (1987; Zbl 0634.20015)], \textit{M. Gromov} gave three equivalent definitions of hyperbolic groups and founded the theory of hyperbolic groups. Since then, on one hand, new research results have emerged in this area, on the other hand, many attempts have been made to present these concepts and results in a comprehensible manner.
In the third chapter, titled ``Hyperbolic groups'', an attempt is made for a systematic presentation of the most important results emphasizing how we can see topological and group-theoretic results in the light of this theory.
Given a presentation \(\langle S\mid R \rangle\) of a group \(\Gamma\), it ``becomes'' a geometric object (a metric space) \((\Gamma , S)\) (the Cayley graph of the group \(\Gamma\) related to the generating system \(S\)). For two generating systems \(S, S^{\prime }\) for the group \(\Gamma\), the spaces \((\Gamma ,S)\) and \((\Gamma ,S^{\prime})\), although they seem different, in reality are not very different (they are quasi-isometric).
If we have two geodesic metric spaces and quasi-isometric spaces, one of which is hyperbolic, then the other is also hyperbolic. Therefore, whether a group is hyperbolic does not depend on the choice of its generating system (Theorem 3.7.15 and Corollary 3.7.17).
More precisely, the Gromov product in hyperbolic metric spaces is defined and equivalent characterizations of the \(\delta\)-hyperbolic spaces are obtained, which lead to equivalent characterizations of hyperbolic groups. Here, we refer to Theorems 3.3.3 and 3.3.9.
In the study of the properties of hyperbolic groups, the notion of Rips complex (Def. 3.4.2), plays an important role. The Rips complex, in one sense, is a generalization of the Cayley graph of a presentation of a group. A connection between the Cayley graph and the Rips complex is given in Theorem 3.4.10.
The Dehn algorithm and the linear isoperimetric inequality for a group are defined. The connection of these two notions and the hyperbolicity of groups is studied, this implies an equivalence (see Theorem 3.4.30).
A connection between groups of type \(F\), hyperbolic groups and faithful representations in \(\mathrm{PSL}(2, \mathbb{R})\) is given (see Corollary 3.5.25 and Theorem 3.5.26).
To summarize, in this book the authors collect and successfully present results in infinite group theory which are interconnected to other areas of mathematics (algebraic topology, geometry, metric spaces etc.). This book can be useful to both beginners and advanced researchers in infinite group theory, as they can have quick access to important results. In addition, the book is supplied by reach bibliography which, in combination with the references inside the book, can be the starting point of a more comprehensive study of this mathematical area.
Reviewer: Dimitrios Varsos (Athína)Computing normalisers of intransitive groupshttps://zbmath.org/1517.200032023-09-22T14:21:46.120933Z"Chang, Mun See"https://zbmath.org/authors/?q=ai:chang.mun-see"Jefferson, Christopher"https://zbmath.org/authors/?q=ai:jefferson.christopher"Roney-Dougal, Colva M."https://zbmath.org/authors/?q=ai:roney-dougal.colva-mThe paper under review is dedicated to the study of the so-called normalizer problem: given two subgroup \(G\) and \(H\) of the symmetric group \(\mathrm{S}_{n}\) compute \(N_{G}(H)\). As a consequence of quasipolynomial solution to the string isomorphism problem [\textit{Babai}, Association for Computing Machinery (ACM), New York, 684-697 (2016; Zbl 1376.68058)], the intersection of permutation groups can be computed in quasipolynomial time. So, with a quasipolynomial cost, it suffices to compute \(N_{\mathrm{S}_{n}}(H)\), then \(N_{G}(H)=N_{\mathrm{S}_{n}}(H)\cap G\). Furthermore, in practice, computing intersections is much faster than computing normalisers. The fastest known algorithm for the normalizer problem is simply exponential, whilst more efficient algorithms are known for restricted classes of groups.
In this paper, the authors present new algorithms for the special case where \(H\) has many orbits and improve the state of the art by several orders of magnitude, at least for implementations in the computer algebra system \textsf{GAP}.
The authors also prove that the normaliser problem for \(G =\mathrm{S}_{n}\) is at least as hard as computing the group of monomial automorphisms of a linear code over any field of fixed prime order.
Reviewer: Egle Bettio (Venezia)On the height and relational complexity of a finite permutation grouphttps://zbmath.org/1517.200042023-09-22T14:21:46.120933Z"Gill, Nick"https://zbmath.org/authors/?q=ai:gill.nick"Lodà, Bianca"https://zbmath.org/authors/?q=ai:loda.bianca"Spiga, Pablo"https://zbmath.org/authors/?q=ai:spiga.pabloLet \(G\) be a finite permutation group on a set \(\Omega\) of size \(t\). A subset \(\Lambda \subseteq \Omega\) is an independent set if its pointwise stabilizer is not equal to the pointwise stabilizer of any proper subset of \(\Lambda\), the height \(\mathsf{H}(G)\) of \(G\) is the maximum size of an independent set.
The group \(G\) is a primitive large-base group if \(G\) is a primitive subgroup of \(\mathrm{Sym}(m) \wr \mathrm{Sym}(n)\) containing \((\mathrm{Alt}(m))^{r}\), where the action of \(\mathrm{Sym}(m)\) is on \(k\)-subsets of \(\{1,2, \dots, m\}\) and the wreath product has the product action of degree \(t = \binom{m}{k}^{r}\).
Let \(G\) be a primitive group, the main result in this paper asserts that either \(\mathsf{H}(G) \leq 9 \cdot \log_{2}t\) or else \(G\) is a primitive large-base group.
A corollary of this result is a characterization of primitive permutation groups with large relational complexity, a quantity introduced by \textit{G. Cherlin} [J. Algebr. Comb. 43, No. 2, 339-374 (2016; Zbl 1378.20001)] in his study of the model theory of permutation groups.
Reviewer: Egle Bettio (Venezia)Saxl graphs of primitive affine groups with sporadic point stabilizershttps://zbmath.org/1517.200052023-09-22T14:21:46.120933Z"Lee, Melissa"https://zbmath.org/authors/?q=ai:lee.melissa"Popiel, Tomasz"https://zbmath.org/authors/?q=ai:popiel.tomaszA base for a permutation group \(G \le \mathrm{Sym}(\Omega)\) is a subset \(B\subseteq\Omega\) with the property that the pointwise stabilizer of \(B\) in \(G\) is trivial. The \textit{base size} \(b(G)\) of \(G\) is the minimal cardinality of a base for \(G\). Bases have been studied since the late 19-th century [\textit{A. Bochert}, Math. Ann. 33, 584--590 (1889; JFM 21.0141.01)] with particular emphasis on \textit{primitive} groups, namely transitive groups that preserve no non-trivial partition of \(\Omega\), and on groups with small bases. \par If \(G\) has base size \(2\), then the corresponding \textit{Saxl graph} \(\Sigma(G)\) has vertex set \(\Omega\) and two vertices are adjacent if and only if they form a base for \(G\). A recent conjecture of \textit{T. C. Burness} and \textit{M. Giudici} [Math. Proc. Camb. Philos. Soc. 168, No. 2, 219--248 (2020; Zbl 1479.20006)] states that if \(G\) is a finite primitive permutation group with base size \(2\), then \(\Sigma(G)\) has the property that every two vertices have a common neighbour. \textit{T. C. Burness} and \textit{H. Y. Huang} [Algebr. Comb. 5, No. 5, 1053--1087 (2022; Zbl 1511.20012)] have verified that conjecture for almost simple groups with socle \(\mathrm{PSL}(2, q)\) and for almost simple groups with soluble point stabilizers. \par The authors investigate Burness-Giudici conjecture in the case where \(G\) is an affine group and a point stabilizer is an almost quasisimple group whose central quotient is either \(S\) or \(\Aut(S)\) for some sporadic simple group \(S\). Then, the conjecture is verified for all but \(16\) of the groups \(G\).
Reviewer: Marek Golasiński (Olsztyn)Finite simple automorphism groups of edge-transitive mapshttps://zbmath.org/1517.200062023-09-22T14:21:46.120933Z"Jones, Gareth A."https://zbmath.org/authors/?q=ai:jones.gareth-aLet \(\mathcal{G}(T)=\{G\mid G\cong\Aut\mathcal{M}\text{ for some map }\mathcal{M}\in T\}\), where \(T\) is one of the \(14\) Graver-Watkins classes of edge-transitive maps (see [\textit{J. E. Graver} and \textit{M. E. Watkins}, Locally finite, planar, edge-transitive graphs. Providence, RI: American Mathematical Society (AMS) (1997; Zbl 0901.05087)]). Building on earlier results for regular maps and for orientably regular chiral maps [\textit{D. Leemans} and \textit{M. W. Liebeck}, Bull. Lond. Math. Soc. 49, No. 4, 581--592 (2017; Zbl 1378.52013)], it is shown that the non-abelian finite simple groups arising as automorphism groups of maps are those which are not isomorphic to one of the groups listed in the corresponding class:
\begin{itemize}
\item Class \(1\): \(L_{3}(q)\), \(U_{3}(q)\), \(L_{4}(2^{e})\), \(U_{4}(2^{e})\), \(U_{4}(3)\), \(U_{5}(2)\), \(A_{6}\), \(A_{7}\), \(M_{11}\), \(M_{22}\), \(M_{23}\), \(McL\);
\item Classes \(2\), \(2^{\ast}\), \(2P\): \(U_{3}(3)\);
\item Classes \(2\mathrm{ex}\), \(2^{\ast}\mathrm{ex}\), \(2^{P}\mathrm{ex}\): \(L_{2}(q)\), \(L_{3}(q)\), \(U_{3}(q)\), \(A_{7}\);
\item Classes \(3\), \(4\), \(4^{\ast}\), \(4P\): none;
\item Classes \(5\), \(5^{\ast}\), \(5P\): \(L\mathrm{2}(q)\).
\end{itemize}
Reviewer: Wen-Fong Ke (Tainan)On the Weiss conjecture. Ihttps://zbmath.org/1517.200072023-09-22T14:21:46.120933Z"Trofimov, V. I."https://zbmath.org/authors/?q=ai:trofimov.vladimir-iThis is the first article by the author on the Weiss conjecture, going back to work by \textit{R. Weiss} from the late 1970s [Math. Proc. Camb. Philos. Soc. 85, 43--48 (1979; Zbl 0392.20002)]. Suppose that \(\Gamma\) is a connected graph with finite vertex set and that \(G\) is a subgroup of the automorphism group of \(\Gamma\) that acts transitively on the set of vertices. Moreover, suppose that, for all vertices \(x\), the action of the stabiliser \(G_x\) on the set \(\Gamma(x)\) of neighbours of \(x\) in the graph is primitive. Then Weiss' conjecture states that \(|G_x|\) is bounded from above by a number depending only on the number of neighbours of \(x\).
Some parts of the article are fairly general, which makes it a good introduction into the background of Weiss' conjecture. Then the author develops general group theoretic arguments for the structure of the point stabilisers, and later he uses results by \textit{U. Meierfrankenfeld} and \textit{B. Stellmacher} [J. Algebra 351, No. 1, 1--63 (2012; Zbl 1262.20003)], together with the O'Nan-Scott theorem, in order to investigate specific cases of the conjecture. In fact, the author confirms Weiss' conjecture except for the types AS and PA from O'Nan-Scott (as discussed in [\textit{M. W. Liebeck} et al., J. Aust. Math. Soc., Ser. A 44, No. 3, 389--396 (1988; Zbl 0647.20005)]).
Reviewer: Rebecca Waldecker (Halle)Generating wreath products of symmetric and alternating groupshttps://zbmath.org/1517.200082023-09-22T14:21:46.120933Z"East, James"https://zbmath.org/authors/?q=ai:east.james"Mitchell, James"https://zbmath.org/authors/?q=ai:mitchell.james-dSummary: We show that the wreath product of two finite symmetric or alternating groups is 2-generated.Inverse Satake isomorphism and change of weighthttps://zbmath.org/1517.200092023-09-22T14:21:46.120933Z"Abe, N."https://zbmath.org/authors/?q=ai:abe.norihiro|abe.nobukado|abe.natsuroh|abe.narishige|abe.noriyuki|abe.naoto|abe.naohito|abe.naohiko|abe.naoki|abe.nobuhisa"Herzig, F."https://zbmath.org/authors/?q=ai:herzig.florian"Vignéras, M. F."https://zbmath.org/authors/?q=ai:vigneras.marie-franceLet \(F\) be a local non-Archimedean field with finite residue field \(k\) of characteristic \(p\). Let \(\mathbf{G}\) be a connected reductive \(F\)-group, and \(C\) be an algebraically closed field of characteristic \(p\). In a previous paper of the authors and \textit{G. Henniart} [J. Am. Math. Soc. 30, No. 2, 495--559 (2017; Zbl 1372.22014)], a classification of irreducible admissible smooth \(C\)-representations of \(G=\mathbf{G}(F)\) was given in terms of supercuspidal representations of Levi subgroups of \(G\). A fundamental ingredient of that paper is the so-called ``change of weight theorem'', which we deduced from the existence of certain elements in the image of the \(\bmod\ {p}\) Satake transform. This follows from the existence of certain elements in the image of the \(\bmod\ {p}\) Satake transform.
In the paper under review, the authors make an intensive study of the \(\bmod\ {p}\) Satake transform. They determine the image of the \(\bmod\ {p}\) Satake transform and give an explicit give an explicit formula for the inverse of the \(\bmod\ {p}\) Satake transform, which they call the inverse Satake theorem, from which the change of weight is an immediate consequence.
Reviewer: Enrico Jabara (Venezia)Non-solvable groups whose character degree graph has a cut-vertex. IIhttps://zbmath.org/1517.200102023-09-22T14:21:46.120933Z"Dolfi, Silvio"https://zbmath.org/authors/?q=ai:dolfi.silvio"Pacifici, Emanuele"https://zbmath.org/authors/?q=ai:pacifici.emanuele"Sanus, Lucia"https://zbmath.org/authors/?q=ai:sanus.luciaLet \(G\) be a finite group and \(\mathrm{cd}(G)\) the set of degrees of the irreducible complex characters of \(G\). The character degree graph \(\Delta(G)\) is the (simple undirected) graph whose vertices consist of those primes dividing some element of \(\mathrm{cd}(G)\), and two distinct vertices \(p\) and \(q\) are connected by an edge whenever \(pq\) divides some number in \(\mathrm{cd}(G)\). This graph is a tool that has been devised in order to investigate the arithmetical structure of \(\mathrm{cd}(G)\).
The structure of the finite groups \(G\) such that \(\Delta(G)\) is disconnected is known since 2003. Indeed, \(\Delta(G)\) may have at most \(2\) connected components when \(G\) is solvable, and at most three if \(G\) is non-solvable. A further step in the study of the connectivity properties of \(\Delta(G)\) is to consider the existence of cut-vertices. A cut-vertex is a vertex whose removal yields a resulting graph with more connected components than the original one, and a connected graph that has a cut-vertex is said to have connectivity degree 1. The finite solvable groups \(G\) such that \(\Delta(G)\) has connectivity degree 1 have already been investigated in [\textit{M. L. Lewis} and \textit{Q. Meng}, Monatsh. Math. 190, No. 3, 541--548 (2019; Zbl 1475.20015)], where, among other things, it is shown that in that case \(\Delta(G)\) has exactly one cut-vertex. On the other hand, the first author et al. [``Non-solvable groups whose character degree graph has a cut-vertex. I'', Vietnam J. Math. (to appear; \url{doi:10.1007/s10013-023-00627-1})] have started a research work with the aim of characterizing those finite non-solvable groups \(G\) such that \(\Delta(G)\) has a cut-vertex. Concretely, in that paper they deal with the (non-solvable) groups having no composition factor isomorphic to PSL\((2,t^a)\) for a prime power \(t^a\geq 4\). The paper under review continues this investigation by addressing the complementary situation in the case when \(t\) is odd and \(t^a>5\). The proof of this classification is completed in a third and last paper [the first author et al., ``Non-solvable groups whose character degree graph has a cut vertex. III'', Ann. Mat. Pura Appl. (4) (to appear; \url{doi:10.1007/s10231-023-01328-9})].
Reviewer: Antonio Beltrán Felip (Castellón)On \(G\)-character tables for normal subgroupshttps://zbmath.org/1517.200112023-09-22T14:21:46.120933Z"Felipe, M. J."https://zbmath.org/authors/?q=ai:felipe.maria-jose"Pérez-Ramos, M. D."https://zbmath.org/authors/?q=ai:perez-ramos.maria-dolores"Sotomayor, V."https://zbmath.org/authors/?q=ai:sotomayor.victorLet \(G\) be a finite group and \(N\) a normal subgroup of \(G\). Since \(N\) is a union of \(G\)-conjugacy classes, it is natural to wonder whether those columns of the character table of \(G\) provide structural information of \(N\) (for an overview of this topic see [\textit{A. Beltran} et al., Int. J. Group Theory 7, No. 1, 23--36 (2018; Zbl 1446.20047)]).
From a result of \textit{R. Brauer} [Lect. Modern Math. 1, 133--175 (1963; Zbl 0124.26504)], it can be derived that the character table of \(G\) contains square submatrices which are induced by the \(G\)-conjugacy classes of elements in \(N\) and the \(G\)-orbits of irreducible characters of \(N\). In the paper under review, the authors provide an alternative approach to this fact through the structure of the group algebra. They also show that such matrices are non-singular and become a useful tool to obtain information of \(N\) from the character table of \(G\).
Reviewer: Enrico Jabara (Venezia)\(\mathcal{P}\)-characters of \(\mathrm{PSL}(2,q)\)https://zbmath.org/1517.200122023-09-22T14:21:46.120933Z"Ghorbani, M."https://zbmath.org/authors/?q=ai:ghorbani.maryam|ghorbani.morteza|ghorbani.morteza.1|ghorbani.mohammad.1|ghorbani.mazaher|ghorbani.mehrzad|ghorbani.majid|ghorbani.mohammad|ghorbani.mojtaba|ghorbani.modjtaba"Iranmanesh, A."https://zbmath.org/authors/?q=ai:iranmanesh.ali"Tehranian, A."https://zbmath.org/authors/?q=ai:tehranian.abolfazl"Ghasemabadi, M. Foroudi"https://zbmath.org/authors/?q=ai:ghasemabadi.m-foroudiLet \(H\) be a subgroup of a group \(G\), the transitive permutation character \((1_{H})^{G}\) is said to be multiplicity-free if all of its irreducible constituents are distinct. A \(\mathcal{P}\)-character of \(G\) is an irreducible character, which is a constituent of the permutation character \((1_{M})^{G}\) for some maximal subgroup \(M\) of \(G\).
In the paper under review, the authors determine the \(\mathcal{P}\)-characters of \(P=\mathrm{PSL}(2, q)\) and check which of the permutation characters \((1_{M})^{P}\) are multiplicity-free.
Reviewer: Enrico Jabara (Venezia)The uniqueness of vertex pairs in \(\pi\)-separable groupshttps://zbmath.org/1517.200132023-09-22T14:21:46.120933Z"Wang, Lei"https://zbmath.org/authors/?q=ai:wang.lei.76"Jin, Ping"https://zbmath.org/authors/?q=ai:jin.pingLet \(\pi\) be a set of primes and \(G\) a finite group. \(G\) is said to be \(\pi\)-separable if there exists a normal \(\pi\)-series (or \(\pi\)-chain) of normal subgroups \(N_i\) of \(G\)
\[
\mathcal{N} = \{1 = N_0 \leq N_1 \leq \dots \leq N_{k-1} \leq N_k = G\}
\]
such that each \(N_i/N_{i-1}\) is either a \(\pi\)-group or a \(\pi'\)-group. Given such a chain \(\mathcal{N}\), an irreducible character \(\chi\) of \(G\) is an \(\mathcal{N}\)-lift if every irreducible constituent of the restriction \(\chi_N\) is a \(\pi\)-lift for all \(N \in \mathcal{N}\). The authors define the twisted vertex for any \(\chi \in \operatorname{Irr}(G)\) to be \((Q, \epsilon \delta)\) where \((Q,\delta)\) is a vertex pair for \(\chi\) and \(\epsilon\) is an appropriate sign character (taking only the values \(\pm 1\)). A twisted vertex is said to be linear if \(\delta\) is linear (degree 1).
It is known due to \textit{J. P. Cossey} and \textit{M. L. Lewis} [J. Group Theory 14, No. 2, 201--212 (2011; Zbl 1230.20009)] that if \(2 \in \pi\), then all vertex pairs for \(\chi\) are linear and conjugate in \(G\), but this may fail if \(2 \notin \pi\). In this work, the authors investigate these twisted vertices when \(2 \notin \pi\) and study their uniqueness. In particular, they show that if \(2 \notin \pi\) and \(\chi \in \operatorname{Irr}(G)\) is an \(\mathcal{N}\)-lift for some \(\pi\)-chain \(\mathcal{N}\) of \(G\), then all linear twisted vertices for \(\chi\) are conjugate in \(G\). They also show that if \(2 \notin \pi\) and \(\chi \in \operatorname{Irr}(G)\) is a \(\pi\)-lift with a linear \emph{Navarro vertex}, then again all linear twisted vertices for \(\chi\) are conjugate in \(G\).
The authors go on to provide examples of characters \(\chi\) which satisfy these conditions.
Reviewer: Jack Saunders (Perth)The orthogonal unit group of the trivial source ringhttps://zbmath.org/1517.200142023-09-22T14:21:46.120933Z"Boltje, Robert"https://zbmath.org/authors/?q=ai:boltje.robert"Carman, Rob"https://zbmath.org/authors/?q=ai:carman.robLet \((K,\mathcal{O},F)\) be a \(p\)-modular system. The trivial source ring \(T(\mathcal{O}G)\) of a finite group \(G\) is the Grothendieck ring of the isomorphism classes of \(p\)-permutation \(\mathcal{O}G\)-modules, that is, the modules whose restrictions to a \(p\)-subgroup \(P\) of \(G\) is a permutation \(\mathcal{O}P\)-module. It is a crucial tool in modular representation theory of finite groups, for example its two sided bifree analogue provides an equivalence between Rickard equivalences and isotypies.
The authors find a characterization of \(T(\mathcal{O}G)\) in terms of coherent characters: the virtual characters of \(KN_G(P)/P\) satisfying certain equality conditions on their values. In particular, they show that \(T(\mathcal{O}G)\) can be embedded into a ring of coherent \(G\)-stable tuples (Theorem A).
As an application of their chracterization, the authors describe the group of orthogonal units of \(T(\mathcal{O}G)\) as a product of the unit group of the Burnside ring of \(G\) and the group of certain coherent tuples of \(\mathrm{Hom}(N_G(P)/P,F^\times)\) (Theorem C).
Reviewer: İsmail Alperen Öğüt (Ankara)Degree divisibility in Alperin-McKay correspondenceshttps://zbmath.org/1517.200152023-09-22T14:21:46.120933Z"Miquel Martínez, J."https://zbmath.org/authors/?q=ai:martinez.j-miquel"Rossi, Damiano"https://zbmath.org/authors/?q=ai:rossi.damianoThe McKay conjecture asserts that if \(G\) is a finite group, \(p\) is a prime and \(P\) is a Sylow \(p\)-subgroup of \(G\), then \(|\mathrm{Irr}_{p'}(G)|=|\mathrm{Irr}_{p'}(N_G(P))|\), where \(\mathrm{Irr}_{p'}(G)\) is the set of complex irreducible character of degree not divisible by \(p\) of \(G\). Alperin proposed the following generalization of McKay's conjecture, which is known as the Alperin-McKay conjecture: the number of \(p'\)-degree irreducible characters among Brauer correspondent \(p\)-blocks coincides.
It was proved in [\textit{A. Turull}, J. Algebra 307, No. 1, 300--305 (2007; Zbl 1118.20011)] that if \(G\) is solvable then there is a bijection \(f:\mathrm{Irr}_{p'}(G)\longrightarrow\mathrm{Irr}_{p'}(N_G(P))\) such that \(f(\chi)(1)\) divides \(\chi(1)\) for every \(\chi\in\mathrm{Irr}_{p'}(G)\). It was recently shown in [\textit{N. Rizo}, Arch. Math. 112, No. 1, 5--11 (2019; Zbl 1502.20007)] that this holds for \(p\)-solvable groups if we assume the degree-divisibility between Glauberman correspondent characters. This was subsequently proved in [\textit{M. Geck}, Ann. Math. (2) 192, No. 1, 229--249 (2020; Zbl 1491.20026)].
In this paper, the authors obtain a result analogous to the theorems of Turull and Rizo for Brauer correspondent \(p\)-blocks of \(p\)-solvable groups, both for ordinary characters and Brauer characters. They also show that if \(B\) is a \(p\)-block of a \(p\)-solvable group with defect group \(D\) and \(b\) is the Brauer correspondent block, then the dimension of \(b\) divides the dimension of \(B\).
Reviewer: Alexander Moretó (València)On the largest Brauer and \(p^\prime\)-character degreeshttps://zbmath.org/1517.200162023-09-22T14:21:46.120933Z"Moretó, Alexander"https://zbmath.org/authors/?q=ai:moreto.alexanderLet \(K\) be a field of characteristic \(p > 0\). \textit{J. F. Humphreys} [J. Lond. Math. Soc., II. Ser. 5, 233--234 (1972; Zbl 0238.20017)] proved that if \(G\) is a finite group and all irreducible representations over \(K\) of \(G\) have degree at most \(n\), then the order of \(G\) is bounded in terms of \(n\) and the order of a Sylow \(p\)-subgroup of \(G\).
In the paper under review, the author improves Humphreys' theorem (see Theorem A) by proving that there exists a function \(f\) such that if \(G\) is a finite group and all irreducible representations of \(G\) over \(K\) have degree at most \(n\), then \(G\) has a characteristic \(p\)-abelian subgroup \(A\) such that \(|G : A| \leq f(n)\). (A group \(A\) is \(p\)-abelian if its derived subgroup \(A'\) is a \(p\)-group.)
In Theorem B, the author proves that there exists a function \(f\) such that if \(G\) is a finite group and all characters in \(\mathrm{Irr}_{p'}(G)\) have degree at most \(n\), then \(G\) has a characteristic solvable \(p\)-nilpotent subgroup \(N\) such that \(|G : N| \leq f(n)\). Unlike Theorem A, Theorem B relies on the classification of finite simple groups.
Reviewer: Enrico Jabara (Venezia)On Héthelyi-Külshammer's conjecture for principal blockshttps://zbmath.org/1517.200172023-09-22T14:21:46.120933Z"Nguyen, Ngoc Hung"https://zbmath.org/authors/?q=ai:nguyen.ngoc-hung"Schaeffer Fry, A. A."https://zbmath.org/authors/?q=ai:schaeffer-fry.amanda-aLet \(p\) be a prime and let \(B\) be a \(p\)-block of positive defect in a finite group \(G\). \textit{L. Héthelyi} and the reviewer [Bull. Lond. Math. Soc. 32, No. 6, 668--672 (2000; Zbl 1024.20016)] conjectured that \(k(B)\), the number of irreducible complex characters of \(G\) in \(B\), satisfies the inequality \(k(B) \geq 2 \sqrt{p-1}\). In the paper under review, the authors verify this lower bound in the special case where \(B\) is the principal \(p\)-block \(B_0(G)\). They also classify the groups \(G\) with \(k(B_0(G)) = 2 \sqrt{p-1}\). Their results extend earlier results by Maróti and others. In their proof, which is based on the classification of the finite simple groups, the authors have to overcome the difficulty that the Alperin-McKay conjecture has not yet been proved completely.
Reviewer: Burkhard Külshammer (Jena)Lie elements and the matrix-tree theoremhttps://zbmath.org/1517.200182023-09-22T14:21:46.120933Z"Burman, Yurii"https://zbmath.org/authors/?q=ai:burman.yurii-m"Kulishov, Valeriy"https://zbmath.org/authors/?q=ai:kulishov.valeriySummary: For a finite-dimensional representation \(V\) of a group \(G\) we introduce and study the notion of a Lie element in the group algebra \(k[G]\). The set \(\mathcal{L}(V) \subset k[G]\) of Lie elements is a Lie algebra and a \(G\)-module acting on the original representation \(V\).
Lie elements often exhibit nice combinatorial properties. In particular, we prove a formula, similar to the classical matrix-tree theorem, for the characteristic polynomial of a Lie element in the permutation representation \(V\) of the group \(G = S_n\).Saturated Majorana representations of \(A_{12}\)https://zbmath.org/1517.200192023-09-22T14:21:46.120933Z"Franchi, Clara"https://zbmath.org/authors/?q=ai:franchi.clara"Ivanov, Alexander A."https://zbmath.org/authors/?q=ai:ivanov.alexander-a"Mainardis, Mario"https://zbmath.org/authors/?q=ai:mainardis.marioThe main ingredients of a Majorana representation, as defined by Ivanov, are a finite group \(G\), a \(G\)-stable set \(T\) of involutions in \(G\) and an action of \(G\) on a commutative non-associative algebra \(V\) called Majorana algebra. The motivating example of a Majorana representation is the Monster simple group \(M\), together with its set of Fischer involutions and its action on the Griess algebra.
In the paper under review, the authors investigate the case where \(G\) is the alternating group \(A_{12}\) and \(T\) is the set of involutions of cycle type \(2^2\) or \(2^6\). The authors prove that there is a unique (up to equivalence) corresponding Majorana representation. The relevant Majorana algebra has dimension 3960, and the decomposition of \(V\), viewed as an \(\mathbb{R}A_{12}\)-module, into simple submodules is determined. Thus this Majorana representation comes from an embedding of \(A_{12}\) into the Monster \(M\).
As a consequence of these results, the authors show that the Harada-Norton simple group \(HN\) also has a unique (up to equivalence) Majorana representation. Thus, this similarly comes from an embedding of \(HN\) into \(M\). Some results on Majorana representations of smaller alternating groups are also derived.
Reviewer: Burkhard Külshammer (Jena)Commuting involutions and elementary abelian subgroups of simple groupshttps://zbmath.org/1517.200202023-09-22T14:21:46.120933Z"Guralnick, Robert M."https://zbmath.org/authors/?q=ai:guralnick.robert-m"Robinson, Geoffrey R."https://zbmath.org/authors/?q=ai:robinson.geoffrey-rLet \(G\) be a symmetric group of degree \(n\) and \(E\) be the maximal elementary abelian group generated by pairwise commuting transpositions. Then any involution in the alternating group \(G^\prime\) can be conjugate within \(G^\prime\) into \(E \cap G^\prime\). Elementary abelian groups with this property will be called broad in this paper. The authors prove that such broad groups exist in all finite simple groups, moreover they also exist in all finite quasi simple groups. The proof split into two parts, the case of a simple group \(G\) and the case that \(2\) divides \(|Z(G)|\). In the later, there is some \(E\) in \(G/Z(G)\) but the preimage of \(E\) might be non abelian. Furthermore, preimages of involutions in \(G/Z(G)\) sometimes become elements of order four like \((1,2)((3,4)\) in the alternating group becomes an element of order four in the covering group. This is the main problem one has to deal with when considering quasi simple groups \(G\) with \(2 \mid |Z(G)|\).
The authors also show that the direct analogue of this result for elements of odd prime order \(p\) is not true. Furthermore, even for \(p =2\) one cannot extend it to almost simple groups as \(G = O^+_{4m}(2^n)\) shows.
The motivation for this comes from representation theory in particular a result due to \textit{R. Knörr} [Proc. Lond. Math. Soc. (3) 59, No. 1, 99--132 (1989; Zbl 0676.20006)]. In fact, a corollary of the result above reads as follows. Let \(G\) be a finite group with \(O(G) = 1\). Then there is some involution \(t \in F^\ast(G)\) such that the total number of irreducible characters of \(G\) which do not lie over 2-blocks of defect zero of \(F^\ast(G)\) is at most \(|C_G(t)|\).
The proof for the alternating groups is easy, for the sporadic group it goes by inspection. For the groups of Lie type, the authors use the corresponding algebraic groups \(X\) over the algebraically closed field. In fact, if \(H = X_{\sigma}\) where \(\sigma\) is a Steinberg endomorphism of \(X\) they show that for a field of odd characteristic there is a toral subgroup of \(H\), i.e. a group which is contained in a torus of \(X\), which intersect any conjugacy class of involutions of \(H\). This then yields in the split type that there is a maximal torus contained in a Borel subgroup, which intersects every conjugacy class of involutions non trivially.
Reviewer: Gernot Stroth (Halle)The isomorphism of generalized Cayley graphs on finite non-abelian simple groupshttps://zbmath.org/1517.200212023-09-22T14:21:46.120933Z"Zhu, Xiao-Min"https://zbmath.org/authors/?q=ai:zhu.xiaomin"Liu, Weijun"https://zbmath.org/authors/?q=ai:liu.weijun"Yang, Xu"https://zbmath.org/authors/?q=ai:yang.xuSuppose that \(G\) is a finite group, that \(S\) is a subset of \(G\) and that \(\alpha\) is an automorphism of \(G\) of order at most 2 such that the following holds for all \(g,h \in G\):
\(g (g^{-1})^\alpha \notin S\), and if \(g (h^{-1})^\alpha \in S\), then also \(h (g^{-1})^\alpha \in S\).
Then the generalised Cayley graph with respect to \(G\), \(S\) and \(\alpha\), denoted by \(GC(G, S, \alpha)\), has vertex set \(G\) and edge set \(\{\{g, sg^\alpha\}\mid g \in G, s \in S\}\).
In the special case where \(\alpha\) is the identity map and \(S\) is a generating set, this is the well-known Cayley graph, but the definition is more general and allows for \(G\) not to be generated by \(S\).
The paper focuses on a generalisation of so-called \(m\)-CI-groups, where \(m \in \mathbb{N}\). \(G\) is an \(m\)-CI-group if and only if, for all inverse-closed subsets \(S\) of \(G\) of size at most \(m\), the resulting Cayley graphs are isomorphic.
In the present article, this notion is extended for generalised Cayley graphs. The authors investigate, for finite simple non-abelian groups \(G\), how if \(S_1\) and \(S_2\) are small subsets and if the graphs \(GC(G, S_1, \alpha_1)\) and \(GC(G, S_2, \alpha_2)\) are isomorphic, this gives information about the subsets and the automorphisms. The article also gives some examples, discusses related questions and gives an overview over previous results in this area.
Reviewer: Rebecca Waldecker (Halle)\text{M}, \text{B} and \(\mathrm{Co}_1\) are recognisable by their prime graphshttps://zbmath.org/1517.200232023-09-22T14:21:46.120933Z"Lee, Melissa"https://zbmath.org/authors/?q=ai:lee.melissa"Popiel, Tomasz"https://zbmath.org/authors/?q=ai:popiel.tomaszThe prime graph or Kegel-Gruenberg graph \(\Gamma(G)\) of a finite group \(G\) is the graph whose vertices are the prime divisors of \(\lvert G\rvert\) and whose edges are the pairs \(\{p,q\}\) for which \(G\) possesses an element of order \(pq\). We say that a group is \textit{recognisable} by its prime graph if every group \(H\) such that \(\Gamma(H)=\Gamma(G)\) is isomorphic to \(G\).
The problem of recognisability by the prime graph of the sporadic finite simple groups has been addressed by several authors, and there were only three cases left: the Monster \(M\), the Baby Monster \(B\), and the first Conway group \(\operatorname{Co}_1\). In this paper, the authors prove that these three groups are recognisable by their prime graphs.
Reviewer: Ramón Esteban-Romero (València)On the soluble graph of a finite grouphttps://zbmath.org/1517.200242023-09-22T14:21:46.120933Z"Burness, Timothy C."https://zbmath.org/authors/?q=ai:burness.timothy-c"Lucchini, Andrea"https://zbmath.org/authors/?q=ai:lucchini.andrea"Nemmi, Daniele"https://zbmath.org/authors/?q=ai:nemmi.danieleGiven an insoluble group \(G\) with soluble radical \(R(G)\), we can consider a graph whose vertices are the elements of \(G\setminus R(G)\) and in which two vertices \(x\) and \(y\) are adjacent if and only if they generate a soluble subgroup of \(G\). This graph is called the \textit{soluble graph} of \(G\). This construction is a generalisation of the commutativity graph of \(G\), in which two elements are adjacent if, and only if, they commute, that is, they generate an abelian subgroup. The main result of this paper states that this graph is always connected and its diameter \(\delta_{\mathcal{S}}(G)\) is at most~\(5\). The authors prove some more accurate results in Theorem 2:
\begin{itemize}
\item[1.] If \(G\) is not almost simple, then \(\delta_{\mathcal{S}}(G)\le 3\).
\item[2.] If \(G\) is almost simple with socle \(G_0\), then \(\delta_{\mathcal{S}}(G)\le 5\). In addition:
\begin{itemize}
\item[(a)] If \(G_0=\operatorname{L}_2(q)\) and \(q\ge 8\), then \(\delta_{\mathcal{S}}(G)=2\) if \(\operatorname{PGL}_2(q)\le G\), otherwise, \(\delta_{\mathcal{S}}(G)=3\).
\item[(b)] If \(G_0=A_n\) and \(n\ge 7\), then either \(\delta_{\mathcal{S}}(G)=3\) or \(G=A_n\) and \(n\in \{p, p+1\}\), where \(p\) is a prime with \(p\equiv 3\pmod{4}\).
\item[(c)] If
\begin{align*}
\mathcal{A}&=\{A_{11}, A_{12}, \operatorname{L}_5^\epsilon(2), \operatorname{M}_{12}, \operatorname{M}_{22}, \operatorname{M}_{23}, \operatorname{M}_{24}, \operatorname{HS}, \operatorname{J}_3\},\\
\mathcal{B}&=\{A_n, \operatorname{L}_7^\epsilon(2), E_6(2), \operatorname{Co}_2, \operatorname{Co}_3, \operatorname{McL}, \mathbb{B}\},
\end{align*}
and \(G\in\mathcal{A}\cup \mathcal{B}\), then \(\delta_{\mathcal{S}}(G)\ge 4\), with equality if \(G\in\mathcal{A}\).
\item[(d)] If \(G=G_0\) is not isomorphic to a classical group, then \(\delta_{\mathcal{S}}(G)\ge 3\).
\end{itemize}
\end{itemize}
Recall that a natural number \(p\) is a \emph{Sophie Germain prime} if both \(p\) and \(2p+1\) are prime numbers. Theorem~4 states that if \(p\ge 5\) is a Sophie Germain prime, then \(\delta_{\mathcal{S}}(A_{2p+1})\in\{4,5\}\). For finite insoluble groups \(G\) with \(R(G)=1\) and \(x\in G\) nontrivial, Theorem~5 says that we have that either there exists an involution \(y\) with \(\delta(x,y)\le 2\), or \(G\) is the Mathieu group \(\operatorname{M}_{23}\) and \(x\) is of order~\(23\).
Some other properties of this graph and other related graphs, like the ones obtained by replacing soluble by nilpotent, metabelian or metacyclic, are shown in this paper.
The techniques used to prove the results include a reduction to almost simple groups and the classification of almost simple groups. Some \textsc{Magma} algorithms have also been used.
Section~8 of the paper includes some open problems and conjectures. The existence of groups \(G\) with \(\delta_{\mathcal{S}}(G)=5\) is left as an open problem, as well as the existence of infinitely many groups with \(\delta_{\mathcal{S}}(G)\ge 4\). The existence of infinitely many Sophie Germain primes would give a positive answer to this last problem, as well as if for all primes \(p\ge 11\) with \(p\equiv 3\pmod{4}\) we had \(\delta_{\mathcal{S}}(A_p)\ge 4\). The determination of all simple groups with \(\delta_{\mathcal{S}}(G)=2\) is also left as an open problem, as well as to know whether \(\delta_{\mathcal{S}}(G)\le 3\) for every non-simple group~\(G\). Another open problem are to understand the relation between \(\delta_{\mathcal{S}}(G)\) and \(\delta_{\mathcal{S}}(G_0)\) if \(G\) is a finite almost simple group with socle \(G_0\) and to understand the relation between \(\delta_{\mathcal{S}}(G)\) and \(\delta_{\mathcal{S}}(T)\) when \(G\) is a monolithic group with socle \(T^k\), where \(T\) is a non-abelian finite simple group and \(k\ge 2\).
Reviewer: Ramón Esteban-Romero (València)The \(IC\)-property of primary subgroupshttps://zbmath.org/1517.200252023-09-22T14:21:46.120933Z"Gao, Yaxin"https://zbmath.org/authors/?q=ai:gao.yaxin"Li, Xianhua"https://zbmath.org/authors/?q=ai:li.xianhuaA subgroup \(H\) of a finite group \(G\) has the \(IC\)-property if the intersection of \(H\) and \([H, G]\) satisfy some conditions (see Definitions 1.1 and 1.2).
In the paper under review, the authors investigate the structure of a finite group \(G\) under the assumption that some primary subgroups of \(G\) have \(IC\)-property. They obtain some new results.
Reviewer: Egle Bettio (Venezia)On finite factorized groups with \({\mathbb{T}}X\)-subnormal subgroupshttps://zbmath.org/1517.200262023-09-22T14:21:46.120933Z"Monakhov, V. S."https://zbmath.org/authors/?q=ai:monakhov.victor-stepanovich"Trofimuk, A. A."https://zbmath.org/authors/?q=ai:trofimuk.aleksandr-aleksandrovichLet \(G\) be a finite group, and \(\mathbb{T}\) a non-empty subset of the natural numbers, which is closed under taking divisors. A subgroup \(H\) of \(G\) is said to be \(\mathbb{T}\)-subnormal if there is a chain \(H = H_{0} \le H_{1} \le \dots \le H_{n} = G\) such that all indices \(\lvert H_{i+1} : H_{i} \rvert\) are in \(\mathbb{T}\). Let \(X\) be a normal subgroup of \(G\). A subgroup \(H\) of \(G\) is said to be \(\mathbb{T} X\)-subnormal if it is \(\mathbb{T}\)-subnormal in \(H X\). \par In the paper under review, the authors study groups \(G\) of the form \(A B\), where \(A, B \le G\) are \(\mathbb{T} X\)-subnormal for some normal subgroup \(X\) of \(G\). Under suitable assumptions on \(A\), \(B\), \(\mathbb{T}\), \(X\), they obtain conditions for \(G\) to be soluble or supersoluble.
Reviewer: Andrea Caranti (Trento)On the Schur multiplier of finite \(p\)-groups of maximal classhttps://zbmath.org/1517.200272023-09-22T14:21:46.120933Z"Joshi, Renu"https://zbmath.org/authors/?q=ai:joshi.renu"Sarkar, Siddhartha"https://zbmath.org/authors/?q=ai:sarkar.siddharthaFor a finite group \(G\), the Schur multiplier of \(G\) is the second homology group \(H_2(G,\mathbb{Z})\), the homology group with integer coefficients. It is well known that for a finite group \(G\), the Schur multiplier of \(G\) is a finite abelian group. It is also known that for a finite \(p\)-group, the Schur multiplier is a finite abelian \(p\)-group. Describing the structure of the Schur multiplier of a finite group is a difficult problem. The structure of the Schur multiplier is known for some \(2\)-groups such as dihedral and quaternion groups of order \(2^n\), \(n\geq 3\). It is also known for extraspecial \(p\)-groups of order \(p^3\), where \(p\) is odd. In this paper, the author considers the Schur multiplier of a \(p\)-group of maximal class, where \(p>2\). More specifically, the author proves that for an odd prime \(p\) and a finite \(p\)-group \(G\) of maximal class and order \(p^n\) (\(4\leq n \leq p+1\)), the Schur multiplier of \(G\) is elementary abelian.
Reviewer: Fatma Altunbulak Aksu (İstanbul)Groups with 2-generated Sylow subgroups and their character tableshttps://zbmath.org/1517.200282023-09-22T14:21:46.120933Z"Moretó, Alexander"https://zbmath.org/authors/?q=ai:moreto.alexander"Sambale, Benjamin"https://zbmath.org/authors/?q=ai:sambale.benjaminThe aim of the paper under review is to advance on the study of Brauer's Problem 12 that asks which properties of a Sylow \(p\)-subgroup \(P\) of a finite group \(G\) are determined by \(X(G)\) the character table of \(G\).
It is known that \(X(G)\) determines whether \(P\) is abelian [\textit{G. Malle} and \textit{G. Navarro}, J. Reine Angew. Math. 778, 119--125 (2021; Zbl 07392170)], and whether the order of \(P/[P, P]\), the abelianization of \(P\), or of \(P/\mathbf{Z}(P)\), the quotient of \(P\) by its center, is \(p^2\) [\textit{G. Navarro} and \textit{B. Sambale}, ``Characters, commutators and centers of Sylow subgroups'', Preprint, \url{arXiv:2204.04407}]. Besides, it can be proven in an elementary way that \(X(G)\) determines whether \(P\) is cyclic, that is, whether \(|P:\Phi(P)|\), the size of the quotient of \(P\) by its Frattini subgroup, is smaller or equal than \(p\).
One of the main goals of this article is to study whether \(X(G)\) determines if \(|P:\Phi(P)|=p^2\), that is, if \(P\) is 2-generated but not cyclic. This problem was positively solved in [\textit{G. Navarro} et al., Represent. Theory 25, 142--165 (2021; Zbl 1480.20029)] for the prime \(p=2\). For odd primes, the authors provide a positive solution whenever \(G\) is a \(p\)-constrained group (so in particular for \(p\)-solvable groups) in Corollary 5.
In Theorem A, the authors prove that \(X(G)\) determines whether \(G\) has a Sylow \(p\)-subgroup of maximal nilpotency class. Their proof does not depend on the classification of finite simple groups but on recent results on fusion systems [\textit{V. Grazian} and \textit{C. Parker}, ``Saturated fusion systems on \(p\)-groups of maximal class'', Preprint, \url{arXiv:2011.05011}].
Moreover, the authors show that \(X(G)\) does not determine \(X(P)\), the character table of one of its Sylow \(p\)-subgroups. For \(p=3\), counterexamples arise as as semidirect products of nonequivalent faithful actions of \(\mathrm{SL}(2, 3)\) on \(C_9 \times C_9\).
Reviewer: Carolina Vallejo Rodríguez (Firenze)Skew product groups for monolithic groupshttps://zbmath.org/1517.200292023-09-22T14:21:46.120933Z"Bachratý, Martin"https://zbmath.org/authors/?q=ai:bachraty.martin"Conder, Marston"https://zbmath.org/authors/?q=ai:conder.marston-d-e"Verret, Gabriel"https://zbmath.org/authors/?q=ai:verret.gabrielThe main result of the paper is a classification of the finite groups \(G\) having a complementary factorization \(G=B\cdot C\) (i.e. \(B\), \(C\) are subgroups of \(G\) with \(B\cap C=\{1\}\)), where \(B\) is monolithic (i.e. \(B\) has a unique minimal normal subgroup and it is non-abelian), \(C\) is non-trivial cyclic and core-free in \(G\) (i.e. \(C\) contains no non-trivial normal subgroup of \(G\)). As a consequence, it turns out that finite nonabelian simple groups rarely have skew morphisms that are not automorphisms. Recall that a \textit{skew morphism} of a group \(B\) is a bijection \(\varphi:B\to B\), \(\varphi(1)=1\), for which there exists a map \(\pi:B\to \mathbb{N}\) such that for all \(a,b\in B\) we have \(\varphi(a\cdot b)=\varphi(a)\cdot \varphi^{\pi(a)}(b)\). Note that skew morphisms of \(B\) are closely related to complementary factorizations of the form \(G=B\cdot C\) where \(C\) is cyclic and core-free in \(G\). Skew morphisms were introduced in the study of regular Cayley maps (certain embeddings of Cayley graphs into surfaces).
Reviewer: Matyas Domokos (Budapest)On some products of finite groupshttps://zbmath.org/1517.200302023-09-22T14:21:46.120933Z"Ballester-Bolinches, A."https://zbmath.org/authors/?q=ai:ballester-bolinches.adolfo"Madanha, S. Y."https://zbmath.org/authors/?q=ai:madanha.sesuai-yash"Pedraza-Aguilera, M. C."https://zbmath.org/authors/?q=ai:pedraza-aguilera.maria-carmen"Wu, X."https://zbmath.org/authors/?q=ai:wu.xinweiA result of \textit{R. Baer} [Ill. J. Math. 1, 115--187 (1957; Zbl 0077.03003)] states that a finite group \(G=AB\) which is the product of two normal supersoluble subgroups \(A\) and \(B\) is supersoluble if and only if \(G'\) is nilpotent.
Let \(G = AB\) be a product of subgroups \(A\) and \(B\), if \(A\) is normal in \(G\) and \(B\) permutes with all the maximal subgroups of Sylow subgroups of \(A\), then \(G\) is called a weak normal product of \(A\) and \(B\).
In the paper under review, the authors show that if \(G = AB\) is a weak normal product of supersoluble subgroups \(A\) and \(B\), then \(G\) is supersoluble, provided that \(G'\) is nilpotent. This provides an elegant generalization of Baer's theorem.
The authors also investigate weak normal products of subgroups when \(A \cap B = 1\) and in this case they obtain more general results.
Reviewer: Enrico Jabara (Venezia)A direct proof of Müller's result on automorphisms of finite \(p\)-groupshttps://zbmath.org/1517.200322023-09-22T14:21:46.120933Z"Singh, Mandeep"https://zbmath.org/authors/?q=ai:singh.mandeep"Garg, Rohit"https://zbmath.org/authors/?q=ai:garg.rohitThe authors give a concise, cohomology-free proof of \textit{O. Müller}'s result [Arch. Math. 32, 533--538 (1979; Zbl 0417.20025), Theorem p. 533] that a finite \(p\)-group which is neither elementary abelian nor extra-special admits a non-inner automorphism centralizing the Frattini quotient. A key ingredient in this is a new direct proof of an auxiliary proposition of Müller [loc. cit., Proposition 3.1] or Proposition 2.2 in the paper under review, which characterizes the nonabelian finite \(p\)-groups \(G\) that have no non-inner automorphisms centralizing both \(Z(G)\) and \(G/\Phi(G)\). Once this is established, one can infer that a nonabelian finite \(p\)-group \(G\) that does not admit a non-inner automorphism centralizing \(G/\Phi(G)\) satisfies \(Z(G)=\Phi(G)\), whence \(G\) also has no non-inner central automorphisms. From there, \textit{M. J. Curran} and \textit{D. J. McCaughan}'s characterization [Commun. Algebra 29, No. 5, 2081--2087 (2001; Zbl 0991.20019), Theorem p. 2081] yields the desired conclusion that \(G\) is extra-special.
Reviewer: Alexander Bors (Linz)On the sum of the inverses of the element orders in finite groupshttps://zbmath.org/1517.200332023-09-22T14:21:46.120933Z"Azad, Morteza Baniasad"https://zbmath.org/authors/?q=ai:baniasad-azad.morteza"Khosravi, Behrooz"https://zbmath.org/authors/?q=ai:khosravi.behrooz"Rashidi, Hamideh"https://zbmath.org/authors/?q=ai:rashidi.hamidehLet \(G\) be a finite group. Related to the set of order of elements of \(G\), various functions can be defined and using these functions, some criteria for solvability, nilpotency, supersolvability etc., are obtained. In this paper, for a finite group \(G\), the function \(m(G)=\sum\limits_{g\in G}1/o(g)\), where \(o(g)\) denotes the order of \(g\in G\), is considered. The following results are obtained.
{Theorem 2.7}. If \(m(G)<m(A_{5})=599/30\), then \(G\) is solvable.
{Theorem 2.8}. If \(m(G)<m(A_{4})=31/6\), then \(G\) is supersolvable.
{Theorem 2.9}. If \(m(G)<m(S_{3})=19/6\), then \(G\) is a cyclic group of order \(n\in \{p,p^{2},2p,8,15,16,21,27\}\), where \(p\) is prime or \(G\cong C_{2}\times C_{2}\) or \(G\cong Q_{8}\).
{Corollary 2.10}. If \(m(G)<m(C_{2}\times C_{2})\), \(m(G)<m(Q_{8})\) or \( m(G)<m(S_{3})\), then \(G\) is cyclic, abelian or nilpotent, respectively.
Reviewer: Grigore Călugăreanu (Cluj-Napoca)Extending results of Morgan and Parker about commuting graphshttps://zbmath.org/1517.200342023-09-22T14:21:46.120933Z"Beike, Nicolas F."https://zbmath.org/authors/?q=ai:beike.nicolas-f"Carleton, Rachel"https://zbmath.org/authors/?q=ai:carleton.rachel"Costanzo, David G."https://zbmath.org/authors/?q=ai:costanzo.david-g"Heath, Colin"https://zbmath.org/authors/?q=ai:heath.colin"Lewis, Mark L."https://zbmath.org/authors/?q=ai:lewis.mark-l"Lu, Kaiwen"https://zbmath.org/authors/?q=ai:lu.kaiwen"Pearce, Jamie D."https://zbmath.org/authors/?q=ai:pearce.jamie-dLet \(G\) be a finite group; the commuting graph of \(G\) is the graph \(\Gamma (G)\) whose vertex set is \(G\setminus Z(G)\), that is the set of all non central elements of \(G\), and two vertices \(x\) and \(y\) are adjacent if and only if \([x,y]=1\), that is if and only if they commute.
Commuting graphs were studied by many authors; much of the research regarding this subject is related to simple groups. For instance, \textit{R. Solomon} and \textit{A. Woldar} [J. Group Theory 16, 793--824 (2013; Zbl 1293.20024)] proved that simple groups are characterized by their commuting graph.
\textit{A. Iranmanesh} and \textit{A. Jafarzadeh} [Acta Math. Acad. Paedagog. Nyházi. (N.S.) 23, No. 1, 7--13 (2007; Zbl 1135.20014)] conjectured that there is a universal bound on the diameter of commuting graphs but \textit{M. Giudici} and \textit{C. Parker} [J. Comb. Theory, Ser. A 120, No. 7, 1600--1603 (2013; Zbl 1314.05055)] constructed a family of 2-groups, nilpotent of class 2, for which there is no bound on the diameter of the commuting graphs.
On the other hand, \textit{C. Parker} [Bull. Lond. Math. Soc. 45, No. 4, 839--848 (2013; Zbl 1278.20017)] proved that the commuting graph of a solvable group \(G\) with trivial center is disconnected if and only if \(G\) is a Frobenius group or it has normal subgroups \(K\le L\) such that \(L\) and \(G/K\) are Frobenius groups with Frobenius kernels \(K\) and \(L/K\), respectively (that is it is a 2-Frobenius group). Moreover, when \(\Gamma (G)\) is connected it has diameter at most 8.
Afterwards \textit{G. L. Morgan} and \textit{C. W. Parker} [J. Algebra 393, 41--59 (2013; Zbl 1294.20033)] removed the solvability hypothesis on \(G\) and proved that if \(G\) is any group with trivial center, then all the connected components of \(\Gamma(G)\) have diameter at most 10.
In this paper, the authors try to extend results of Parker [loc. cit,] and Parker and Morgan [loc. cit,] and they show that it is possible to replace the hypothesis that \(Z(G)=1\) with the hypothesis that \(G'\cap Z(G)=1\). The main result is the following
Theorem 1.1 Let \(G\) be a group and suppose that \(G'\cap Z(G)=1\); then
1) \(\Gamma(G)\) is connected if and only if \(\Gamma (G/Z(G))\) is connected;
2) every connected component of \(\Gamma(G)\) has diameter at most \(10\);
3) if \(G\) is solvable and \(\Gamma (G)\) is connected, then \(\Gamma (G)\) has diameter at most \(8\);
4) if \(G\) is solvable, then \(\Gamma (G)\) is disconnected if and only if \(G/Z(G)\) is either a Frobenius group or a \(2\)-Frobenius group.
The hypothesis that \(G'\cap Z(G)=1\) can be relaxed, in fact, it suffices to assume that, for all \(x,y\in G\), the commutator \([x,y]\in Z(G)\) if and only if \([x,y]=1\).
A class of finite groups satisfying the hypothesis of Theorem 1.1 is, for instance, the class of \(A\)-groups. In fact, if every Sylow subgroup of a group \(G\) is abelian (that is \(G\) is an \(A\)-group), then it is possible to prove that \(G'\cap Z(G)=1\)
Finally, the authors present some examples to clarify various points.
Reviewer: Chiara Nicotera (Salerno)Subindices and subfactors of finite groupshttps://zbmath.org/1517.200352023-09-22T14:21:46.120933Z"Hooshmand, Mohammad Hadi"https://zbmath.org/authors/?q=ai:hooshmand.mohammad-hadiThis paper studies the structure of subsets (not subgroups) of groups and the ways in which they interact. In particular, it develops a variety of results and questions about differences, products, and indexes (suitably defined) of subsets.
For example, let \(A\) be a subset of a group \(G\). Another subset \(B\) is a (right) sub-factor of \(G\) related to \(A\) if \(B\) is a maximal subset with the property that \(AB = A \cdot B\), i.e. everything in the product \(AB\) can be expressed uniquely. Then the right upper index of \(A\) in \(G\), denoted \(|G:A|^+\), is defined to be \(\sup|B|\) over all right sub-factors \(B\) of \(G\) related to \(A\). We can also analogously define the right lower index and the left upper and lower indexes. All four of these degenerate to the usual index \(|G:A|\) in the case that \(A\) is a subgroup, so the idea here is to study how we can use these more general definitions.
Again let \(A\) be a subset and define the (left) difference \(\mathrm{Dif}_\ell(A) = AA^{-1}\). Note this definition can be iterated to obtain \(\mathrm{Dif}_\ell^\infty(A)\).
The main part of the paper is a collection of results about indexes (and related metrics) and difference sets. For example, it starts with Theorem 3.1: \(|G:A|^+ + |\mathrm{Dif}_\ell(A)| \le |G|+1\) (plus a more extensive inequality under the assumption that \(A\ne \emptyset\)).
Because the objects of study are just groups and subsets with only a small amount of background (most of which is provided in the paper), anyone interested in groups should find reading the paper worthwhile.
Reviewer: Alden Walker (Chicago)Finite groups whose commuting conjugacy class graphs have isolated verticeshttps://zbmath.org/1517.200362023-09-22T14:21:46.120933Z"Saeidi, Amin"https://zbmath.org/authors/?q=ai:saeidi.aminIn this paper, the author studies the so-called \textit{commuting conjugacy class graph} of a finite group \(G\), denoted \(\Gamma(G)\), introduced by \textit{M. Herzog} et al. in [Commun. Algebra 37, No. 10, 3369--3387 (2009; Zbl 1187.20027)]. The vertex set of this graph is the set of non-identity conjugacy classes of \(G\), and two conjugacy classes \( C_1\) and \(C_2\) are adjacent if and only if we can find \(x \in C_1\) and \(y \in C_2\) such that \(xy =yx\). The vertex of the graph corresponding to the conjugacy class of an element \(x \in G\) is denoted by \([x]\).
Groups whose corresponding graphs contain isolated vertices are investigated in the paper under review. It is shown that if \(\Gamma(G)\) has an isolated vertex \([x]\), then the order of \(x\) must be a prime number \(p\), and in such case it is said that the group \(G\) \textit{satisfies the \(p\)-isolation property}. Moreover, the Sylow \(p\)-subgroups of such group \(G\) are \(CC\)-subgroups, i.e., for every non-trivial \(x \in P\), \(P\) a Sylow \(p\)-subgroup of \(G\), it holds \(C_G(x) \leq P\).
A complete classification of solvable groups with the \(p\)-isolation property, \(p\) a prime, is given in the paper (Theorem 1). It is also proved that if \(G\) is nonsolvable, then either \(G/F(G)\) is almost simple, or \(G\) is sharply \(2\)-transitive (Theorem 2), where \(F(G)\) denotes the Fitting subgroup of \(G\).
Since the structure of finite non-solvable sharply \(2\)-transitive groups is well-known (they are Frobenius groups with a restricted Frobenius complement), the next problem is to determine almost simple groups satisfying the \(p\)-isolation property. This problem has been solved in the paper for almost simple groups with sporadic socles (Theorem 3), as well as for certain families of finite simple groups (Theorems 3.3 and 3.7).
Moreover, a complete classification of non-solvable groups satisfying the \(p\)-isolation property is given for the primes \(p = 2\) and \(p = 3\) (Propositions 3.6 and 3.10).
Reviewer: Ana Martínez-Pastor (València)On \(B_8\)- and \(B_9\)-groupshttps://zbmath.org/1517.200372023-09-22T14:21:46.120933Z"Tian, Kaiyu"https://zbmath.org/authors/?q=ai:tian.kaiyu"Tan, Yilan"https://zbmath.org/authors/?q=ai:tan.yilan\textit{G. A. Freiman} introduced the notion of \(B_n\)-group in [Aequationes Math. 22, 140--152 (1981; Zbl 0489.20020)]; if \(n\) is a positive integer and \(G\) is a finite group, we say that \(G\) is a \(B_n\)-group if for any \(n\)-element subset \(A =\{a_1 , a_2 , .\ldots, a_n \}\) of \(G\) the condition \(|A^2| = |\{a_i a_j \mid 1 \le i, j \le n \}| \le n(n+1)/2\) is satisfied. Later, \textit{Ya. G. Berkovich} et al. [Bull. Aust. Math. Soc. 44, No. 3, 429--450 (1991; Zbl 0728.20020)] introduced the notion of the small squaring property: A finite group \(G\) has the small squaring property if \(|A^2 | < n^2\) for any \(n\)-subset \(A\) of \(G\). Weakening a bit more the upper bound on \(|A^2|\), \textit{T. Eddy} and \textit{M. M. Parmenter} [Ars Comb. 104, 321--331 (2012; Zbl 1258.20020)] introduced the notion of \(B(n,k)\)-group; a finite group \(G\) is a \(B(n,k)\)-group if \(|A^2 | \le k\) for any \(n\)-subset \(A\), where \(k\) is a positive integer such that \(n \le k \le n^2-1\). Of course, if \(k=n(n+1)/2\), then the \(B(n,k)\)-groups are exactly \(B_n\)-groups and if \(k=n^2-1\) we find the groups with the small squaring property. Classifications of \(B(5,18)\)-groups and \(B(5,19)\)-groups are available (see Lemmas 1 and 3) and the main results of the paper under review characterize \(B_8\)-groups and \(B_9\)-groups via \(B(5,18)\)-groups and \(B(5,19)\)-groups; for instance, Proposition 1 shows that \(B_8\)-groups of order \(>252\) should be \(B(5,18)\)-groups. Similarly, Proposition 3 shows that \(B_9\)-groups of order \(>360\) should be \(B(5,19)\)-groups. Consequently, structural results are provided in Theorems 1 and 2.
Reviewer: Francesco G. Russo (Rondebosch)Cogrowth series for free products of finite groupshttps://zbmath.org/1517.200382023-09-22T14:21:46.120933Z"Bell, Jason"https://zbmath.org/authors/?q=ai:bell.jason-p"Liu, Haggai"https://zbmath.org/authors/?q=ai:liu.haggai"Mishna, Marni"https://zbmath.org/authors/?q=ai:mishna.marniLet \(G\) be a group with finite generating set \(S\). Let \(\mathcal{L}(G,S)\) be the set of elements in the free monoid \(S^{*}\) of \(S\) whose image in \(G\) is the identity. For each \(n\geq 0\), let \(CL(n,G,S)\) be the number of words of length \(n\) which are equal to the identity of \(G\). This is the cogrowth function of \(G\) and generating set \(S\). The generating function for this sequences is the cogrowth series and is given by \(F_{G,S}(t)=\sum_{n\geq 0}CL(n,G,S)t^{n}\). This paper studies these series for virtually free groups. The authors provide explicit formulae and several examples. The main results are as follows: Consider \(G_{1}, \ldots ,G_{r}\) finite groups with generating sets \(S_{1},\ldots, S_{r}\), respectively. For each \(i=1,\dots,r\), let \(G_{i}^{*m_{i}}\) denote the \(m_{i}\)-times free product of \(G_{i}\) and let \(S_{i}^{(j)}\), \(j=1,\ldots ,m_{i}\) be copies of \(S_{i}\) in the corresponding copies of \(G_{i}\). Moreover, let \(G=G_{1}^{*m_{1}}*\cdots *G_{r}^{*m_{r}}\) and \(S=\bigcup_{i=1}^{r}\bigcup_{j=1}^{m_{i}}S_{i}^{(j)}\). The authors prove:
Theorem. Let \(G\) and \(S\) be as above. Then the cogrowth series \(F(t)=F_{G,S}(t)\) is algebraic and satisfies \(\Lambda(t,F(t))=0\), where \(\Lambda(t,z)\) is a non-zero polynomial with rational coefficients satisfying
\[
\deg_{t}(\Lambda), \deg_{z}(\Lambda)\leq (\prod_{i=1}^{r}\Delta_{i})(1+\sum_{i=1}^{r}\frac{1}{\Delta_{i}}),
\]
where \(\Delta_{i}\) is the sum of the degrees of the irreducible representations of \(G_{i}\), \(i=1,\ldots , r\).
In the second result, the authors provide explicit expressions for the cogrowth series for the following groups with specific generating sets: \(\mathbb{Z}/d*\cdots *\mathbb{Z}/d\), the \(m\)-times free product of the finite cyclic group \(\mathbb{Z}/d\) for \(d,m\geq 2\), the \(m\)-times free product of \(\mathbb{Z}/2\) free product with a free group of rank \(s\), \(m,s\geq 0\) and \(\mathbb{Z}/2*\mathbb{Z}/n\), \(n\geq 2\).
Lastly, the authors find a gap in the radius of convergence of the generating series of the cogrowth series for a group \(G\) with generating set \(S\).
Reviewer: Daniel Juan Pineda (Michoacán)Intersection of centralizers in a partially commutative metabelian grouphttps://zbmath.org/1517.200392023-09-22T14:21:46.120933Z"Timoshenko, E. I."https://zbmath.org/authors/?q=ai:timoshenko.evgenii-iLet \(\Gamma=\langle X;E\rangle\) be a finite undirected graph without loops with vertex set \(X=\{x_1,\dots, x_n\}\) and edge set \(E\subseteq X\times X\). A partially commutative group \(F(\Gamma, \mathfrak{M})\) of the variety \(\mathfrak{M}\) has presentation \(F(\Gamma, \mathfrak{M})=\langle X\mid x_ix_j=x_jx_i \text{ if }\{x_i,x_j\}\in E, \mathfrak{M}\rangle\). Let \(\mathfrak{A}^2\) denote the variety of all metabelian groups and recall that a clique of a graph is any of its complete subgraphs. The main aim of the paper is to prove the following result: Let \(G=F(\Gamma, \mathfrak{A}^2)\) and \(X=\{x_1,\dots, x_n\}\) the vertex set of the graph \(\Gamma\). Then for distinct vertices \(x, y\) of the graph \(\Gamma\), \(C(x)\cap C(y)\cap G'\) is trivial if and only if whenever \(x_{i_1},\dots, x_{i_m},x_{i_1}\) is a cycle, then \(\{x_{i_1},\dots, x_{i_m}\}\) is a clique. (Here \(C(x)\) denotes the centralizer of \(x\) in \(G\) and \(G'\) is the commutator subgroup.)
Reviewer: Martyn Dixon (Tuscaloosa)Profinite groups with restricted centralizers of \(\pi\)-elementshttps://zbmath.org/1517.200402023-09-22T14:21:46.120933Z"Acciarri, Cristina"https://zbmath.org/authors/?q=ai:acciarri.cristina"Shumyatsky, Pavel"https://zbmath.org/authors/?q=ai:shumyatsky.pavelWe say that a group has restricted centralisers if for each \(g\in G\) the centraliser \(C_G(g)\) is finite or has finite index in \(G\). We say that a profinite group satisfies a property virtually if it has an open subgroup with that property. \textit{A. Shalev} [Proc. Am. Math. Soc. 122, No. 4, 1279--1284 (1994; Zbl 0822.20031)] proved that a profinite group with restricted centralisers is virutally abelian. A previous result of the authors [Isr. J. Math. 242, No. 1, 269--278 (2021; Zbl 1476.20036)], presented here as Theorem~1.1, states that if \(p\) is a prime and \(G\) is a profinite group in which the centraliser of each \(p\)-element is either finite or open, then \(G\) has a normal abelian pro-\(p\)-subgroup \(N\) such that \(G/N\) is virtually pro-\(p'\). The aim of this paper is to given an extension of this result to a set of primes~\(\pi\). As usual, an element \(x\) of a profinite group \(G\) is a \(\pi\)-element if the images of \(x\) in every finite continous homomorphic image of \(G\) is only divisible by primes in~\(\pi\). The authors develop new techniques and prove the following result (Theorem~1.2): Let \(n\) be a positive integer, \(\pi\) a set of primes, and \(G\) a finite group such that \(\lvert g^G\rvert \le n\) for each \(\pi\)-element \(g\in G\). Let \(H=\operatorname{O}^{\pi'}(G)\). Then \(G\) has normal subgroup \(N\) such that
\begin{itemize}
\item[1.] the index \([G,N]\) is bounded by a function of~\(n\);
\item[2.] \([H,N]=[H,H]\);
\item[3.] the order of \([H,N]\) is bounded by a function of~\(n\).
\end{itemize}
This result is stronger than Theorem~1.1 even if \(\pi\) consists of a single prime. They also prove (Theorem~1.3) that if \(\pi\) is a set of primes and \(G\) is a profinite group in which the centraliser of each \(\pi\)-element is either finite or open, then \(G\) has an open subgroup of the form \(P\times Q\), where \(P\) is an abelian pro-\(\pi\) subgroup and \(Q\) is a pro-\(\pi'\) subgroup.
Reviewer: Ramón Esteban-Romero (València)Commutators in \(\mathrm{SL}_2\) and Markoff surfaces. Ihttps://zbmath.org/1517.200412023-09-22T14:21:46.120933Z"Ghosh, Amit"https://zbmath.org/authors/?q=ai:ghosh.amit"Meiri, Chen"https://zbmath.org/authors/?q=ai:meiri.chen"Sarnak, Peter"https://zbmath.org/authors/?q=ai:sarnak.peter-cSummary: We show that the commutator equation over \(\mathrm{SL}_2\mathbb{Z}\) satisfies a profinite local to global principle, while it can fail with infinitely many exceptions for \(\mathrm{SL}_2(\mathbb{Z}[\frac{1}{p}])\). The source of the failure is a reciprocity obstruction to the Hasse Principle for cubic Markoff surfaces.Constructing uncountably many groups with the same profinite completionhttps://zbmath.org/1517.200422023-09-22T14:21:46.120933Z"Nikolov, Nikolay"https://zbmath.org/authors/?q=ai:nikolov.nikolay-i|nikolov.nikolay-m|nikolov.nikolay"Segal, Dan"https://zbmath.org/authors/?q=ai:segal.dan.1|segal.danSummary: Two constructions are described: one gives soluble groups of derived length 4, the other uses groups acting on a rooted tree.On some series of a group related to the non-abelian tensor square of groupshttps://zbmath.org/1517.200432023-09-22T14:21:46.120933Z"Bastos, R."https://zbmath.org/authors/?q=ai:bastos.raimundo"de Oliveira, R."https://zbmath.org/authors/?q=ai:de-oliveira.roberto-t-g|de-oliveira.ronaldo-junio|de-oliveira.rudinei-martins|de-oliveira.rosevaldo|de-oliveira.reinaldo-resende|de-oliveira.ricardo-t-a|de-oliveira.renata-z-g|de-oliveira.rafael-mendes|de-oliveira.rodrigo-m-s|de-oliveira.rodolfo-alves|de-oliveira.ricardo-puziol|de-oliveira.rogerio-luiz-quintino-jun|de-oliveira.roberson-assis|de-oliveira.r-i-jun|de-oliveira.rafael-massambone"Monetta, C."https://zbmath.org/authors/?q=ai:monetta.carmine"Rocco, N. R."https://zbmath.org/authors/?q=ai:rocco.norai-romeuThe non-abelian tensor square \(G\otimes G\) of a group \(G\) is the group generated by all symbols \(g\otimes h\) with \(g,h\in G\), satisfying the following relations
\[
gg_1\otimes h=(g^{g_1}\otimes h^{g_1})(g_1\otimes h)\>\>\hbox{and}\>\> g\otimes hh_1=(g\otimes h_1)(g^{h_1}\otimes h^{h_1})
\]
where, for all \(g,h\in G\), the cojugate of \(g\) by \(h\), denoted by \(g^h\), is the product \(h^{-1}gh\).
In [Bol. Soc. Bras. Mat. 22, 63--79 (1991; Zbl 0791.20020)], \textit{N. R. Rocco} proved that it is possible to obtain the non-abelian tensor square of a group \(G\) by the following construction.
Let \(G^{\varphi}=\{g^{\varphi}\mid g\in G\}\) be an isomorphic copy of a group \(G\); put
\[
\nu(g):=\langle G\cup G^{\varphi}| [g_1,g_2^{\varphi}]^{g_3}=[g_1^{g_3}, (g_2^{g_3})^{\varphi}]=[g_1,g_2^{\varphi}]^{g_3^{\varphi}}, g_1,g_2,g_3\in G\rangle,
\]
it is possible to show that its subgroup \({\Upsilon}_1(G):=[G,G^{\varphi}]\) is isomorphic to the non-abelian tensor square \(G\otimes G\) and this fact motivates the study of the group \(\nu(G)\).
In this paper, the authors apply the biderivation technique to investigate the structure of the terms of the derived series and of the lower central series of the group \(\nu(G)\). First of all, they denote by \(\Theta (G)\) the kernel of the epimorphism \(\rho: \nu(G)\rightarrow G\) defined by \(\rho(g)=g=\rho(g^{\varphi})\) for every \(g\in G\), then they define \(\Upsilon_2(G):=[\Theta(G), G]\) and \(\Upsilon_3(G):=[\Theta(G), G^{\varphi}]\). These two subgroups are normal in \(\nu(G)\), they are contained in \(\Theta(G)\), moreover they permute with each other and their intersection is a central subgroup of \(\nu(G)\).
They prove the following results:
Theorem A. Let \(G\) be a group. Then
(a) the subgroups \(\Upsilon_2(G)\) and \(\Upsilon_3(G)\) are both isomorphic to \(G\otimes G\);
(b) the derived subgroup \(\nu(G)'\) is a central product of the subgroups \(\Upsilon_1(G)\), \(\Upsilon_2(G)\) and \(\Upsilon_3(G)\).
Moreover, the group \(\nu(G)'\) is isomorphic to \(G'\times G'\times G'\) modulo \(\mu(G)\), where \(\mu(G)\) is the kernel of the map \(\rho':\Upsilon_1(G)\rightarrow G'\), induced by \([g,h^{\varphi}]\rightarrow [g,h]\) for all \(g,h\in G\)
For \(h\ge 1\), they define \(A_h(G):=[\Upsilon_1(G),_{h-1} G]\), \(B_h(G):=[\Upsilon_2(G),_{h-1} G]\) and \(C_h(G):=[\Upsilon_3(G),_{h-1} G^{\varphi}]\).
Theorem B. Let \(G\) be a group and let \(k\ge 0\). Then
\[
\nu(G)^{(k+1)}=\Upsilon_1(G)^{(k)}\Upsilon_2(G)^{(k)}\Upsilon_3(G)^{(k)}
\]
Theorem C. Let \(G\) be a group; if \(k\ge 2\), then
\[
\gamma_k(\nu(G))=A_{k-1}(G)B_{k-1}(G)C_{k-1}(G).
\]
In the last section of the paper, the authors provide some applications of Theorems A and B, computing the derived subgroups \(\nu(G)'\) for some groups. Moreover, they obtain some bounds for the exponent of \(\nu(G)\) and its subgroups.
Reviewer: Chiara Nicotera (Salerno)Classification of Thompson related groups arising from Jones technology. Ihttps://zbmath.org/1517.200442023-09-22T14:21:46.120933Z"Brothier, Arnaud"https://zbmath.org/authors/?q=ai:brothier.arnaudAuthor's abstract: In the quest in constructing conformal field theories (CFTs), Jones has discovered a beautiful and deep connection between CFT, Richard Thompson's groups, and knot theory. This led to a powerful functorial framework for constructing actions of particular groups arising from categories such as Thompson's groups and braid groups. In particular, given a group and two of its endomorphisms one can construct a semidirect product where the largest Thompson's group \(V\) is acting. These semidirect products have remarkable diagrammatic descriptions that were previously used to provide new examples of groups having the Haagerup property. They naturally appear in certain field theories as being generated by local and global symmetries. Moreover, these groups occur in a construction of Tanushevski and can be realised using Brin-Zappa-Szep's products with the technology of cloning systems of Witzel and Zaremsky. We consider in this article the class of groups obtained in that way where one of the endomorphism is trivial leaving the case of two nontrivial endomorphisms to a 2nd article [Bull. Soc. Math. Fr. 149, No. 4, 663--725 (2021; Zbl 07513439)]. We provide an explicit description of all these groups as permutational restricted twisted wreath products where \(V\) is the group acting and the twist depends on the endomorphism chosen. We classify this class of groups up to isomorphisms and provide a thin description of their automorphism group thanks to an unexpected rigidity phenomena.
Reviewer: Francesco G. Russo (Rondebosch)Extensions of linear cycle sets and cohomologyhttps://zbmath.org/1517.200452023-09-22T14:21:46.120933Z"Guccione, Jorge A."https://zbmath.org/authors/?q=ai:guccione.jorge-alberto"Guccione, Juan J."https://zbmath.org/authors/?q=ai:guccione.juan-jose"Valqui, Christian"https://zbmath.org/authors/?q=ai:valqui.christianAfter the pioneering paper by \textit{V. G. Drinfel'd} [Lect. Notes Math. 1510, 1--8 (1992; Zbl 0765.17014)], which first mentioned the possibility to study set-theoretic solution of the Yang-Baxter equation, many novel algebraic structures have been introduced for this purpose. One such example is linear cyclic sets, which were introduced by \textit{W. Rump} [J. Algebra 307, No. 1, 153--170 (2007; Zbl 1115.16022)] where Rump also defined the equivalent structure of braces.
A linear cyclic set is an abelian group with an additional binary operation that satisfies certain conditions. There are natural notions of ideals and of triviality (where the additional operation is trivial in a suitable sense, reducing the linear cyclic set to just an abelian group).
The paper under review explores general notions of extensions of linear cyclic sets, focusing specifically on providing a cohomological description when the involved ideals are trivial. In doing so, the authors propose a generalisation of the results of [\textit{V. Lebed} and \textit{L. Vendramin}, Pac. J. Math. 284, No. 1, 191--212 (2016; Zbl 1357.20009)], where a more specific class of ideals is considered, and they also reinterpret the work of \textit{D. Bachiller} [J. Pure Appl. Algebra 222, No. 7, 1670--1691 (2018; Zbl 1437.20031)], developed in the setting of braces without involving cohomology. The introduction features a helpful diagram that clearly illustrates the connections with these two related papers, providing an overview of the relationships between different sections.
After a section of preliminaries, the paper presents a definition for an extension of linear cyclic sets, motivated by the traditional notion for abelian groups. It introduces properties of all the involved operations and defines specific maps and actions to control these properties. Throughout the subsequent sections, the authors narrow their focus to more manageable cases, specially in the trivial ideal case, resulting in simplified statements that were initially technical. Finally, in these favorable situations, the authors introduce a double cochain complex, so that the related second cohomology group bijectively describes extensions of linear cyclic sets (with respect to a natural notion of equivalence).
Despite the necessary technicalities of certain results, the authors have done an excellent job of consistently highlighting the main findings of the paper, resulting in a smoother reading experience.
Reviewer: Lorenzo Stefanello (Pisa)Aritmethic lattices of \(\operatorname{SO}(1,n)\) and units of group ringshttps://zbmath.org/1517.200462023-09-22T14:21:46.120933Z"Chagas, Sheila"https://zbmath.org/authors/?q=ai:chagas.sheila-c"del Rio, Ángel"https://zbmath.org/authors/?q=ai:del-rio.angel"Zalesskii, Pavel A."https://zbmath.org/authors/?q=ai:zalesskij.pavel-aLet \(G\) be a group. \(G\) is conjugacy separable if given any two elements \(x\) and \(y\) that are non-conjugate in \(G\), there exists some finite quotient of \(G\) in which the images of \(x\) and \(y\) are not conjugate. \(G\) is virtually compact special if it has a finite index subgroup isomorphic to the fundamental group of a compact special cube complex and \(G\) is toral relatively hyperbolic if \(G\) is torsion-free, and hyperbolic relative to a finite set of finitely generated abelian subgroups.
The main result of the paper under review is Theorem 1.1: Let \(G\) be a group having compact special and toral relatively hyperbolic subgroup of finite index. Then \(G\) is conjugacy separable.
As a consequence of Theorem 1.1, the authors also prove Theorem 1.2: Standard arithmetic lattices of special orthogonal groups \(\mathrm{SO}(1, n)\) are conjugacy separable.
The results above allow the authors to prove conjugacy separability for the group of units \(\mathcal{U}(\mathbb{Z}(G)\) of the integral group rings of some finite groups \(G\). These groups of units have nice residual properties that allows to approach the question to what extent a finitely generated group is determined by its finite quotients or equivalently by its profinite completion for \(U(\mathbb{Z}G)\).
Reviewer: Enrico Jabara (Venezia)Bruhat-Tits theory from Berkovich's point of view. Analytic filtrationshttps://zbmath.org/1517.200472023-09-22T14:21:46.120933Z"Mayeux, Arnaud"https://zbmath.org/authors/?q=ai:mayeux.arnaudAuthor's abstract: We define filtrations by affinoid groups, in the Berkovich analytification of a connected reductive group, related to Moy-Prasad filtrations. They are parametrized by a cone, whose basis is the Bruhat-Tits building and whose vertex is the neutral element, via the notions of Shilov boundary and holomorphically convex envelope.
Reviewer: Egle Bettio (Venezia)Correction to: ``Conjugacy classes and union of cosets of normal subgroups''https://zbmath.org/1517.200492023-09-22T14:21:46.120933Z"Beltrán, Antonio"https://zbmath.org/authors/?q=ai:beltran.antonioCorrection to the author's paper [ibid. 201, No. 2, 349--358 (2023; Zbl 1514.20106)].Twisted conjugacy in linear algebraic groupshttps://zbmath.org/1517.200502023-09-22T14:21:46.120933Z"Bhunia, S."https://zbmath.org/authors/?q=ai:bhunia.swarup|bhunia.sushil|bhunia.subodh-c|bhunia.supriya|bhunia.santanu"Bose, A."https://zbmath.org/authors/?q=ai:bose.anjan|bose.abhijit|bose.arup|bose.abhigyan|bose.amarnath|bose.arijit|bose.anil-k|bose.akash|bose.amitabha|bose.arindam|bose.anirbanLet \(G\) be a group and \(\phi\) an endomorphism of \(G\). Two elements \(x,y \in G\) are said to be \(\phi\)-twisted conjugate, if \(y = gx\phi(g)^{-1}\) for some \(g \in G\). The equivalence classes with respect to this relation are called \(\phi\)-twisted conjugacy classes or Reidemeister classes of \(\phi\). The cardinality of the \(\phi\)-twisted conjugacy class containing \(x \in G\) is called the Reidemeister number of \(\phi\).
Let \(k\) be an algebraically closed field and \(G\) a linear algebraic group defined over \(k\). It is said that \(G\) has the algebraic \(R_\infty\)-property if \(R(\phi) = \infty\) for all \(\phi \in \Aut(G)\). The authors pose the natural question: under which conditions such \(G\) has the \(R_\infty\)-property? The main result of this article is the following: Let \(G\) be a connected non-solvable linear algebraic group over an algebraically closed field \(k\). Then \(G\) possesses the \(R_\infty\)-property. The condition here is not necessary (see Example 5 in Section 4 of this article). For the case of solvable groups \(G\), the authors prove the following: Let \(G\) be a connected solvable algebraic group and \(T\) a maximal torus of \(G\). Suppose that the condition \(\phi(T ) = T\) implies \(R(\phi|_T ) = \infty\) for all \(\phi \in \Aut(G)\). Then the group \(G\) has the \(R_\infty\)-property.
Reviewer: V. V. Gorbatsevich (Moskva)On characterization of a finite group with non-simple socle by the set of conjugacy class sizeshttps://zbmath.org/1517.200512023-09-22T14:21:46.120933Z"Gorshkov, Ilya"https://zbmath.org/authors/?q=ai:gorshkov.ilya-borisovichLet \(G\) be a finite group and \(N(G)\) be the set of its conjugacy class sizes excluding \(1\). In 1987, John G. Thompson conjectured that if \(S\) is a finite non-abelian simple group and \(G\) is a finite group with trivial center such that \(N(G)=N(S)\), then \(G\cong L\) (briefly, \(S\) is said to be recognizable by class sizes). Since then, many authors have made much progress for many families of non-abelian simple groups until in 2019 the proof of the validity of Thompson's conjecture was completed.
In this paper, the author proposes a question that goes further: If \(S\) is a finite non-abelian simple group, is it true that \(S^n\) is recognizable by class sizes for every \(n\in {\mathbb{N}}\)? The author gives an affirmative answer for the case \(L_2(q)\times L_2(q)\) where \(3< q\) is an odd prime-power and the numbers \(q-1\) and \(q+1\) are not powers of \(2\). The author himself has previously solved this problem for the particular case \(\mathrm{Alt}(5)\times\mathrm{Alt}(5)\) [J. Algebra Appl. 21, No. 11, Article ID 2250226, 8 p. (2022; Zbl 07596149)].
Reviewer: Antonio Beltrán Felip (Castellón)Free polynilpotent groups and the Magnus propertyhttps://zbmath.org/1517.200522023-09-22T14:21:46.120933Z"Klopsch, Benjamin"https://zbmath.org/authors/?q=ai:klopsch.benjamin"Mendonça, Luis"https://zbmath.org/authors/?q=ai:de-mendonca.luis-augusto"Petschick, Jan Moritz"https://zbmath.org/authors/?q=ai:petschick.jan-moritzIn this article, the authors deal with the Magnus property in relatively free groups, more specifically, in free polynilpotent groups. The Magnus property for free groups is a classic result: ``If two elements \(g\) and \(h\) of a free group \(F\) generate the same normal subgroup of \(F\), then \(g\) is conjugated to \(h^\epsilon\), \(\epsilon = 1\) or \(\epsilon = -1\)''. Based on this classic result, a general problem posed by the authors is: ``for a given variety \(\mathcal{V}\) of groups, which \(\mathcal{V}\)-free groups have the Magnus property (MP, for short)?''
The results mainly concern the variety \(\mathcal{N}_c\) of all polynilpotent groups of class-row \(c\), for any given length \(l \in \mathbb{N}\) and class tuple \(c = (c_1 , \ldots, c_l) \in \mathbb{N}^l\): a group \(G\) belongs to \(\mathcal{N}_c\) if the term \(\gamma_{(c_1 +1,\ldots, \, c_l +1)}(G)\) of its iterated lower central series vanishes. Here, \(\gamma_{(1)}(G) = \gamma_1(G) = G\), and inductively \(\gamma_{(c_1 +1,\ldots, \, c_l +1)}(G) = \gamma_{(c_l +1)}(\gamma_{(c_1 +1,\ldots, c_{{l} - {1}} + 1)}(G))\) for \(l > 1\), where \(\gamma_{(c_1 +1)}(G) = \gamma_{c_1+1}(G) = [\gamma_{c_1}(G), G]\) is the \((c_1 + 1)\)-st term of the ordinary lower central series of G. As a main result the authors prove:
(Theorem 1.1) Let \(G\) be an \(\mathcal{N}_c\)-free group of rank \(d\), i.e., a free polynilpotent group of class row \(c\) that is freely generated by \(d\) elements, where \(d, l \in \mathbb{N}\) and \(c \in \mathbb{N}^l\). Then \(G\) has the MP if and only if \(G\) is nilpotent of class at most 2; equivalently, if and only if \(d = 1\) or \(c \in \{(1), (2)\}\).
While \(\mathcal{N}_c\)-free group are torsion-free, the authors also prove a similar result for center-by-\(\mathcal{N}_c\)-free groups (which can involve central torsion of exponent 2):
(Theorem 1.2) Let \(G\) be a center-by-\(\mathcal{N}_c\)-free group of rank \(d\), where \(d, l \in \mathbb{N}\) and \(c \in \mathbb{N}^l\). Then \(G\) has the MP if and only if \(G\) is nilpotent of class at most 2; equivalently, if and only if \(d = 1\) or \(c = (1)\).
An interesting tool for proving a group \(G\) does not have MP is the concept of a \textit{basic} \(\neg\)(MP)-\textit{witness pair} for \(G\), that is, a pair of elements \((g, \nu)\), \(g \in G\) and \(\nu \in [G, G] \setminus \{[g, w] | w \in G \}\) such that \(g^2 \not\in [G, G]\) and \(\langle g \rangle^{G} = \langle g\nu \rangle^{G}\) (cf. Lemma 2.6). Use of such a pair has been made, for instance, to prove that the restricted wreath product \(C_{\infty} \wr C_{\infty}\) does not have MP, a starting point in proving Theorem 1.1.
In proving Theorems 1.1 and 1.2, the authors explore MP in more general groups, producing new techniques for establishing or denying MP.
The authors provide (Example 3.8) an explicit family of finitely generated, nilpotent groups (with non-trivial 3-torsion) of any prescribed nilpotency class that possess MP and, in (Example 3.11), they explicitly construct a 4-generated, torsion-free, class-3 nilpotent group of Hirsch length 9 with MP.
Finally, using an ultraproduct construction they establish:
(Theorem 1.3) For every \(c \in \mathbb{N}\), there exists a countable, metabelian, torsion-free, nilpotent group with the Magnus property that has nilpotency class precisely \(c\).
Reviewer: Noraí Romeu Rocco (Brasília)On the domino problem of the Baumslag-Solitar groupshttps://zbmath.org/1517.200532023-09-22T14:21:46.120933Z"Aubrun, Nathalie"https://zbmath.org/authors/?q=ai:aubrun.nathalie"Kari, Jarkko"https://zbmath.org/authors/?q=ai:kari.jarkkoSummary: In [Electron. Proc. Theor. Comput. Sci. (EPTCS) 128, 35--46 (2013; Zbl 1469.20028)] we construct aperiodic tile sets on the Baumslag-Solitar groups \(B S(m, n)\). Aperiodicity plays a central role in the undecidability of the classical domino problem on \(\mathbb{Z}^2\), and analogously to this we state as a corollary of the main construction that the Domino problem is undecidable on all Baumslag-Solitar groups. In the present work we elaborate on the claim and provide a full proof of this fact. We also provide details of another result reported in [loc. cit.]: there are tiles that tile the Baumslag-Solitar group \(B S(m, n)\) but none of the valid tilings is recursive. The proofs are based on simulating piecewise affine functions by tiles on \(B S(m, n)\).On generalized conjugacy and some related problemshttps://zbmath.org/1517.200542023-09-22T14:21:46.120933Z"Carvalho, André"https://zbmath.org/authors/?q=ai:ponce-de-leon-ferreira-de-carvalho.andre-carlos|carvalho.andre-r-rThe conjugacy problem \(\mathsf{CP}(G)\) on \(G\) consists on deciding whether two elements, given as input, are conjugate or not in \(G\). The twisted conjugacy problem \(\mathsf{TCP}(G)\) on \(G\) consists of deciding whether, taking two elements \(g, h \in G\) and an automorphism \(\varphi \in \Aut(G)\), there is some \(x \in G\) such that \(h = x^{-1}gx^{\varphi}\) (that is if \(g\) and \(h\) are \(\varphi\)-twisted conjugate in \(G\)).
In [J. Algebra 324, No. 5, 1083--1097 (2010; Zbl 1209.20023)], \textit{P. Brinkmann} proved that, in case \(G\) is a free group of finite rank, there is an algorithm to decide whether, on input \(g, h \in G\) and \(\varphi \in \Aut(G)\), there is some \(k \in \mathbb{Z}\) such that \(g^{\varphi^{k}}\) is conjugate to \(h\). Such a problem is called \(\mathsf{BrCP}(G)\) (Brinkmann's conjugacy problem). In this paper, the author proposes several generalizations of the previous problems. In particular, if \(\mathcal{C}\) is a class of subsets of \(G\) he defines
\begin{itemize}
\item \(\mathsf{GTCP}_{\mathcal{C}}(G)\) (the \(\mathcal{C}\)-generalized twisted conjugacy problem): taking as input a subset \(K \in \mathcal{C}\), an automorphism \(\varphi \in \Aut(G)\) and an element \(x \in G\), decide whether \(x\) has a \(\varphi\)-twisted conjugate in \(K\);
\item \(\mathsf{GBrCP}_{\mathcal{C}}(G)\) (the \(\mathcal{C}\)-generalized Brinkmann conjugacy problem): taking as input a subset \(K \in \mathcal{C}\), an automorphism \(\varphi \in \Aut\)(G) and an element \(x \in G\), decide whether there is some \(k\in \mathbb{Z}\) such that \(x^{\varphi^{k}}\) has a conjugate in \(K\).
\end{itemize}
In the article under review, the author establishes connections between the generalized conjugacy problem for \(G \rtimes \mathbb{Z}\), \(\mathsf{GTCP}_{\mathcal{C}}(G)\) and \(\mathsf{GBrCP}_{\mathcal{C}}(G)\). Using such results, he is able to prove that \(\mathsf{GBrCP}_{\mathcal{C}}(G)\) is decidable when \(G\) is a virtually polycyclic group. Furthermore, he proves that if \(G\) is a finitely generated virtually free group, then \(\mathsf{TCP}(G)\) is decidable.
Reviewer: Egle Bettio (Venezia)Lamplighter groups and automatahttps://zbmath.org/1517.200552023-09-22T14:21:46.120933Z"Jain, Sanjay"https://zbmath.org/authors/?q=ai:jain.sanjay"Moldagaliyev, Birzhan"https://zbmath.org/authors/?q=ai:moldagaliyev.birzhan"Stephan, Frank"https://zbmath.org/authors/?q=ai:stephan.frank"Tien Dat Tran"https://zbmath.org/authors/?q=ai:tien-dat-tran.\noindent A \textit{transducer} \(M=(Q,\Sigma,\delta,s_0,F)\) is a non-deterministic finite automaton, where~\(Q\) is a finite set of states, \(\Sigma\) is a finite alphabet, \(s_0\) is the starting state, \(F\) is a set of accepting states and \(\delta\) is a function from \(Q\times\big(\Sigma^*\big)^k\) to the powerset of \(Q\) (here, \(\Sigma^*\) is the set of words on the alphabet \(\Sigma\)).
The \(k\)-tuple \((w_1,\dots,w_k)\in\Sigma^*\) is \textit{accepted} by \(M\) if there are \(q_0,q_1,\dots,q_r\in Q\), and \(w_{i,j}\in\Sigma^*\) (with \(i=1,\dots,k\) and \(j=0,1,\dots,r-1\)) such that:
\begin{itemize}
\item[(1)] \(q_{j+1}\in\delta(q_j,w_{1,j},\dots,w_{k,j})\) for \(j<r\);
\item[(2)] \(q_0=s_0\) and \(q_r\in F\);
\item[(3)]\(w_i=w_{i,0}\dots w_{i,r-1}\) for \(i=1,\dots,k\).
\end{itemize}
A relation is \textit{transducer recognisable} if there is a transducer accepting every element of the relation. A function is \textit{transducer recognisable} if its graph is transducer recognisable.
A group \((G,\circ)\) is said to be \textit{transducer one-to-one presented} if it admits a \textit{transducer presentation}, that is, a bijective map \(\phi:D\rightarrow G\) such that
\begin{itemize}
\item \(D\) is a transducer recognisable language;
\item The relation \(\{(x,y)\in D\times D\,:\, \phi(x)=\phi(y)\}\) is transducer recognisable;
\item Some transducer recognisable function computes for inputs \(x,y\in D\), a \(z\in D\) such that \(\phi(x)\circ\phi(y)=\phi(z)\).
\end{itemize}
The main aim of the paper under review is to show that given a transducer one-to-one presented group \(G\), then also the wreath product \(G\wr\mathbb{Z}\) is transducer one-to-one presented (see Theorem 14). The paper contains other results on the argument. For example, it is shown that the Baumslag-Solitar group \(BS(1,n)\) is also transducer one-to-one presented (see Theorem 16). Other types of presentations (that is, Cayley transducer presentations and tree automata presentations) for certain types of wreath products are also dealt with.
Reviewer: Marco Trombetti (Napoli)A Deligne complex for Artin monoidshttps://zbmath.org/1517.200562023-09-22T14:21:46.120933Z"Boyd, Rachael"https://zbmath.org/authors/?q=ai:boyd.rachael"Charney, Ruth"https://zbmath.org/authors/?q=ai:charney.ruth-m"Morris-Wright, Rose"https://zbmath.org/authors/?q=ai:morris-wright.roseA \textit{Deligne complex} for Artin monoids is defined. It is a cube complex, and the authors show that it is contractible. An embedding of the monoid Deligne complex into the Deligne complex for the corresponding Artin group is constructed. This embedding is shown to be a locally isometric embedding. In the case of FC-type Artin groups, this result is strengthened to a globally isometric embedding, and it follows that the monoid Deligne complex is CAT(0) and its image in the Deligne complex is convex.
At last, it is shown that for a finite-type Artin group, the monoid Cayley graph embeds isometrically, but not quasi-convexly, into the group Cayley graph.
Reviewer: Stephan Rosebrock (Karlsruhe)Euclidean Artin-Tits groups are acylindrically hyperbolichttps://zbmath.org/1517.200572023-09-22T14:21:46.120933Z"Calvez, Matthieu"https://zbmath.org/authors/?q=ai:calvez.matthieuAn \textit{Artin-Tits group} is a group \(G\) which admits a presentation \(\langle S \mid R \rangle\) with a finite set of generators \(S\) and the relations in \(R\) are all of the form \( abab\ldots = baba\ldots\) for distinct \(a, b\) in \(S\), where both sides have equal lengths and there exists at most one relation for each pair of distinct generators. This presentation can be encoded by a Coxeter graph \(\Gamma\), which defines a Coxeter group associated with \(G\). An Artin-Tits groups is said to be of \textit{Euclidean type} if the associated Coxeter group is Euclidean.
The main result of the paper establishes that all irreducible Artin-Tits groups of Euclidean type are acylindrically hyperbolic, i.e. admit an acylindrical isometric action with unbounded orbits on a hyperbolic metric space.
The proof is based on previous work by the author and \textit{B. Wiest} [Geom. Dedicata 191, 199--215 (2017; Zbl 1423.20028)], where spherical Artin-Tits groups were considered, and on a result by \textit{J. McCammond} and \textit{R. Sulway} [Invent. Math. 210, No. 1, 231--282 (2017; Zbl 1423.20032)], who showed that a Euclidean Artin-Tits group embeds into an infinite-type Garside group. The proof of acylindrical hyperbolicity uses \textit{D. Osin}'s characterization from [Trans. Am. Math. Soc. 368, No. 2, 851--888 (2016; Zbl 1380.20048)].
Reviewer: Mikhail Belolipetsky (Rio de Janeiro)The commutator subgroup of the braid group is generated by two elementshttps://zbmath.org/1517.200592023-09-22T14:21:46.120933Z"Kordek, Kevin"https://zbmath.org/authors/?q=ai:kordek.kevinSimilar to the symmetric group, the braid group \(B_n\) (\(n \geq 3\)) is not cyclic and can be generated by only two elements, say \(\sigma_1\) and \(\sigma_1\cdots\sigma_{n-1}\). Here, the author shows that the commutator subgroup \(B_n'\) of \(B_n\) (\(n \geq 3\)) is also not cyclic and that it can be generated by two elements when \(n=5\) or \(n\geq7\), but requires at least three elements when \(n=4\) or \(6\).
Reviewer: Diego Arcis (Talca)Invariants for metabelian groups of prime power exponent, colorings, and stairshttps://zbmath.org/1517.200602023-09-22T14:21:46.120933Z"Barmak, Jonathan Ariel"https://zbmath.org/authors/?q=ai:barmak.jonathan-arielIn this paper, the authors study the free metabelian group \(M(2,n)\) of prime power exponent \(n\) in two generators, that is, the quotient of the Burnside group \(B(2,n)\) by its second derived subgroup. Contrary to Burnside groups, the groups \(M(2, n)\) are known to be finite for every \(n\). However, their order has been determined in very few cases if \(n\) is not prime. The authors define invariants (group homomorphisms) \(M(2,n)' \rightarrow \mathbb{Z}_{n}\) (that they construct from colorings of the squares in the integer grid \(\mathbb{R} \times \mathbb{Z} \cup \mathbb{Z} \times \mathbb{R}\)) and use them to prove that certain identities do not hold in \(M(2, n)\) and to improve bounds for its order obtained by \textit{M. F. Newman} [Lect. Notes Math. 1098, 87--98 (1984; Zbl 0566.20016)]. They study identities in \(M(2,n)\), which give information about identities in the Burnside group \(B(2,n)\) and the restricted Burnside group \(R(2,n)\).
Reviewer: M. Carmen Pedraza-Aguilera (València)On periodic Shunkov's groups with almost layer-finite normalizers of finite subgroupshttps://zbmath.org/1517.200612023-09-22T14:21:46.120933Z"Senashov, Vladimir Iavnovich"https://zbmath.org/authors/?q=ai:senashov.vladimir-iavnovichSummary: Layer-finite groups first appeared in the work by \textit{S. N. Chernikov} [C. R. (Dokl.) Acad. Sci. URSS, n. Ser. 50, 71--74 (1945; Zbl 0061.02905)]. Almost layer-finite groups are extensions of layer-finite groups by finite groups. The class of almost layer-finite groups is wider than the class of layer-finite groups; it includes all Chernikov groups, while it is easy to give examples of Chernikov groups that are not layer-finite. The author develops the direction of characterizing well-known and well-studied classes of groups in other classes of groups with some additional (rather weak) finiteness conditions. A Shunkov group is a group \(G\) in which for any of its finite subgroups \(K\) in the quotient group \(N_G (K) / K\) any two conjugate elements of prime order generate a finite subgroup. In this paper, we prove the properties of periodic not almost layer-finite Shunkov groups with condition: the normalizer of any finite nontrivial subgroup is almost layer-finite. Earlier, these properties were proved in various articles of the author, as necessary, sometimes under some conditions, then under others (the minimality conditions for not almost layer-finite subgroups, the absence of second-order elements in the group, the presence of subgroups with certain properties in the group). At the same time, it was necessary to make remarks that this property is proved in almost the same way as in the previous work, but under different conditions. This eliminates the shortcomings in the proofs of many articles by the author, in which these properties are used without proof.Canonical universal locally finite groupshttps://zbmath.org/1517.200622023-09-22T14:21:46.120933Z"Shelah, Saharon"https://zbmath.org/authors/?q=ai:shelah.saharonIn this paper, the author studies the class \(\mathbf{K}^{\mathrm{lf}}\) and the subclasses \(\mathbf{K}^{\mathrm{lf}}_{\lambda}\) of members of \(\mathbf{K}^{\mathrm{lf}}\) of cardinality \(\lambda\). By a result of \textit{R. Grossberg} and the author [Isr. J. Math. 44, 289--302 (1983; Zbl 0525.20025)], if \(\lambda=\lambda^{\aleph_{0}}\), then there is no universal member for \(\mathbf{K}^{\mathrm{lf}}_{\lambda}\). However, if \(\lambda\) is a strong limit cardinal of cofinality \(\aleph_{0}\) above a compact cardinal \(\kappa\), then there is \(G \in \mathbf{K}^{\mathrm{lf}}_{\lambda}\) which is universal.
The main result of this article is the proof that the existence of universally locally finite groups implies that there is a canonical one in any strong limit singular cardinality of countable cofinality. For showing the existence, the author relies on the existence of enough indecomposable such groups which he proved in [Rend. Semin. Mat. Univ. Padova 144, 253--270 (2020; Zbl 1511.20007)].
Reviewer: Enrico Jabara (Venezia)On automorphisms of undirected Bruhat graphshttps://zbmath.org/1517.200632023-09-22T14:21:46.120933Z"Gaetz, Christian"https://zbmath.org/authors/?q=ai:gaetz.christian"Gao, Yibo"https://zbmath.org/authors/?q=ai:gao.yiboIn this paper, the authors study automorphisms of the undirected Bruhat graph and classify when the undirected Bruhat graph is vertex-transitive.
Reviewer: Chen Sheng (Harbin)Salem numbers, spectral radii and growth rates of hyperbolic Coxeter groupshttps://zbmath.org/1517.200642023-09-22T14:21:46.120933Z"Kellerhals, R."https://zbmath.org/authors/?q=ai:kellerhals.ruth"Liechti, L."https://zbmath.org/authors/?q=ai:liechti.livioA Coxeter system \((W, S)\) is a pair with \(W\) a group and \(S\) a set of generators that are reflections and the only relations arise from the angles of reflecting hyperplanes. A Coxeter group admits a faithful representation where the generators act as reflections on \(\mathbb{R}^N\). A product of all the generators of \(W\) is called a Coxeter element of \(W\), and the corresponding element in \(\mathrm{GL}_N(\mathbb{R})\) is called a Coxeter transformation. For a Coxeter group, the growth series is defined as
\[
f_S(t) = 1 + \sum_{k\ge 1}a_kt^k
\]
where \(a_k\) is the number of elements of \(S\)-length \(k\) and it is known to be a rational function that depends of the finite subgroups of \(W\). The inverse \(\tau = 1/R = \limsup_{k\to \infty}\sqrt[n]{a_k} \) of the radius of convergence of the series is called the growth rate of \(W\).
A Coxeter polyhedron \(P \subset \mathbb{H}^n\) is a convex polyhedron where all the dihedral angles are \({\pi}/m\) where \(m \in \{2, \dots \infty\}\). It is assumed that the polyhedron is of finite volume and thus it is bounded by finitely many hyperplanes. For \(n = 2\), \(P = (p_1, \dots p_k)\) is a hyperbolic polygon where \(p_1, \dots p_k \ge 2\) are integers and \(1/p_1 + \dots + 1/p_k < k - 2\). Reflections on \(\mathbb{H}^n\) with respect to the hyperplanes define a Coxeter group which is called a hyperbolic Coxeter group. It acts on \(\mathbb{H}^n\) with hyperbolic isometries. If \(n\) is 2 or 3, the growth rate is either a quadratic unit or a Salem number, that is a real algebraic integer \(\tau > 1\), whose conjugates have absolute value \(\le 1\) and at least one of them has absolute value 1.
The first main result of the paper states that not all Salem numbers appear as growth rates of hyperbolic Coxeter groups. For \(n = 2\), it is proved that the smallest growth rate is \(\tau_{[3, 7]}\) which is Lehmer's number \(\alpha_L\) and after that is \(\tau_{[3,8]}\) which is the seventh smallest number in the known list of Salem numbers. So the numbers in between in the list can not be realised as growth rates of hyperbolic Coxeter groups. For \(n = 3\), it is known that the smallest growth rate is the Salem number \(\tau_{[3,4,3]}\). So the first 47 known numbers are not realised. For \(n \ge 4\), it is proved that the growth rate is either larger that \(\tau_{[3,8]}\) or it is not a Salem number. There is no example of a hyperbolic Coxeter group with \(\tau > \tau_{[3,8]}\).
Let \((p_1, \dots p_k)\) be a sequence of integers, \(p_i \ge 2\). We construct the star graph \(\mathrm{Star}(p_1, \dots p_k)\) to be a tree with a vertex with degree \(k\) and \(k\) paths emanating from the vertex of length \(p_i -1\), respectively. Let \((p_1, \dots p_k)\) denote a convex polygon in \(\mathbb{H}^2\) and \(W\) the Coxeter group with Coxeter graph \(\mathrm{Star}(p_1, \dots p_k)\), i.e. with generators corresponding to the vertices of \(\mathrm{Star}(p_1, \dots p_k)\) and relations \((st)^3 = 1\) if two vertices are joined by an edge and \((st)^2 = 1\) otherwise. Then the growth rate of \(W\) equals to the spectral radius of the Coxeter transformation of \(W\).
The tree \(H(i,j,k)\) is defined as follows: we start with two vertices \(v_1\) and \(V_2\) of degree 3 and a path of length \(j\) joining them. To \(v_1\) we attach two paths, one having length \(1\) and the other of length \(i - 1\). Similarly, to \(v_2\) we attach a vertex and a path of length \(k-1\). To this graph, we associate a Coxeter group as before. The authors prove that to Coxeter group \([4, 3, 5]\) we associate the graph \(H(2, 8, 3)\) and the Coxeter group \(W\). They prove that \(\tau_{[4, 3, 5]}\) (which is a Salem number) equal to the spectral radius of the Coxeter transformation of \(W\). Also, they prove that \(\tau_{[3,5,3]}\) is not the spectral radius of a Coxeter element of a Coxeter group associated to a graph as above.
Reviewer: Stratos Prassidis (Karlovasi)On some Garsideness properties of structure groups of set-theoretic solutions of the Yang-Baxter equationhttps://zbmath.org/1517.200652023-09-22T14:21:46.120933Z"Chouraqui, Fabienne"https://zbmath.org/authors/?q=ai:chouraqui.fabienneA set-theoretic solution of the Yang-Baxter equation is a pair \((X, r)\), where \(X\) is a set, \(r: X \times X \rightarrow X \times X\) and \(r(x, y) = (\sigma_{x}(y), \tau_{y}(x))\) is a bijective map satisfying \(r^{12}r^{23}r^{12} = r^{23}r^{12}r^{23}\), where \(r^{12} = r \times \mathrm{Id}_{X}\) and \(r^{23} = \mathrm{Id}_{X} \times r\). If \(r\) is a non-degenerate and involutive set-theoretic solution of the Yang-Baxter equation its structure group is defined by \(G(X, r)=\langle X \mid x_{i}x_{j}=x_{k}x_{\ell} \mbox{ iff } r(x_{i},x_{j})=(x_{k},x_{\ell}) \rangle\).
There exists a multiplicative homomorphism from the structure group \(G(X, r)\) with \(| X |= n\) to the matrix algebra \(M_{n+1}(\mathbb{R})\). In this paper, the author constructs a finite basis of the underlying vector space of the image of \(G(X, r)\) using the correspondence between the structure groups and Garside groups with a particular presentation.
Reviewer: Enrico Jabara (Venezia)The surface group conjectures for groups with two generatorshttps://zbmath.org/1517.200662023-09-22T14:21:46.120933Z"Gardam, Giles"https://zbmath.org/authors/?q=ai:gardam.giles"Kielak, Dawid"https://zbmath.org/authors/?q=ai:kielak.dawid"Logan, Alan D."https://zbmath.org/authors/?q=ai:logan.alan-dA Mel'nikov group is a non-free infinite one-relator group with every subgroup of finite index also a one-relator group. Residually finite Mel'nikov groups are not necessarily surface groups, with the Baumslag-Solitar groups \(\mathrm{BS}(1, n)\) forming a family of counterexamples.
The main results of the paper under review are summarized in Theorem 1.5:
Let \(G\) be a two-generated group.
\begin{itemize}
\item[(1)] If \(G\) is a residually finite Mel'nikov group, then \(G\) is a surface group or \(\mathrm{BS}(1, n)\) for some nonzero integer \(n\).
\item[(2)] If \(G\) is a Mel'nikov group with every subgroup of infinite index free, then \(G\) is a surface group.
\item[(3)] If \(G\) is an infinite (non-free) one-relator group with every subgroup of infinite index free, then \(G\) is a surface group.
\end{itemize}
Reviewer: Egle Bettio (Venezia)Corrigendum to: ``Quasi-isometry invariants of weakly special square complexes''https://zbmath.org/1517.200672023-09-22T14:21:46.120933Z"Oh, Sangrok"https://zbmath.org/authors/?q=ai:oh.sangrokCorrigendum to the author's paper [ibid. 307, Article ID 107945, 46 p. (2022; Zbl 1508.20051)].Bi-exactness of relatively hyperbolic groupshttps://zbmath.org/1517.200682023-09-22T14:21:46.120933Z"Oyakawa, Koichi"https://zbmath.org/authors/?q=ai:oyakawa.koichiA group \(G\) is exact if there is a compact Hausdorff space on which \(G\) acts topologically amenably and \(G\) is bi-exact if it is exact and there is a map \(\mu: G\rightarrow \mathrm{Prob}(G)\) such that for all \(s,t\in G\) we have \(\lim_{x\to\infty}\|\mu(sxt)-s\cdot\mu(x)\|_1=0\). The main result of the paper is: Suppose that \(G\) is a finitely generated group hyperbolic relative to a collection \(\{H_{\lambda}\}_{\lambda\in\Lambda}\) of \(G\). Then, \(G\) is bi-exact if and only if all subgroups \(H_{\lambda}\) are bi-exact.
Reviewer: Martyn Dixon (Tuscaloosa)Convexity in hierarchically hyperbolic spaceshttps://zbmath.org/1517.200692023-09-22T14:21:46.120933Z"Russell, Jacob"https://zbmath.org/authors/?q=ai:russell.jacob"Spriano, Davide"https://zbmath.org/authors/?q=ai:spriano.davide"Tran, Hung Cong"https://zbmath.org/authors/?q=ai:tran-cong-hung.Summary: Hierarchically hyperbolic spaces (HHSs) are a large class of spaces that provide a unified framework for studying the mapping class group, right-angled Artin and Coxeter groups, and many \(3\)-manifold groups. We investigate strongly quasiconvex subsets in this class and characterize them in terms of their contracting properties, relative divergence, the coarse median structure, and the hierarchical structure itself. Along the way, we obtain new tools to study HHSs, including two new equivalent definitions of hierarchical quasiconvexity and a version of the bounded geodesic image property for strongly quasiconvex subsets. Utilizing our characterization, we prove that the hyperbolically embedded subgroups of hierarchically hyperbolic groups are precisely those that are almost malnormal and strongly quasiconvex, producing a new result in the case of the mapping class group. We also apply our characterization to study strongly quasiconvex subsets in several specific examples of HHSs. We show that while many commonly studied HHSs have the property that every strongly quasiconvex subset is either hyperbolic or coarsely covers the entire space, right-angled Coxeter groups exhibit a wide variety of strongly quasiconvex subsets.Strong Gelfand pairs of \(\mathrm{SL} (2, p)\)https://zbmath.org/1517.200702023-09-22T14:21:46.120933Z"Barton, Andrea"https://zbmath.org/authors/?q=ai:barton.andrea"Humphries, Stephen P."https://zbmath.org/authors/?q=ai:humphries.stephen-pFor a group \(G\) and a subgroup \(H\subseteq G\), consider the coset action \(\sigma : G \longrightarrow S_{G/H}\), on the coset space \(G/H\). This produces a complex representation \(\widetilde{\sigma}\) of \(G\). The pair \((G,H)\) is called to be a \textit{Gelfand pair} if the centralizer algebra \(C(\widetilde{\sigma})\) is commutative. This is equivalent to saying that the induction of the trivial character of \(H\) gives a multiplicity-free character of \(G\). A pair \((G,H)\) of finite groups \(G\), \(H\) with \(H\subseteq G\) is called a \textit{strong Gelfand pair} if every irreducible character of \(H\) induces a multiplicity-free character of \(G\). Note that a strong Gelfand pair is also referred to as ``multiplicity one property'' or ``multiplicity one theorem''.
This article determines the strong Gelfand pairs when \(G=\mathrm{SL}(2,p)\), the special linear group of \(2\times 2\) matrices over the field of prime order \(p\). Consider the following two subgroups of \(\mathrm{SL}(2,p)\). The first one is \(\mathrm{U}(2,p)\), the subgroup of upper triangular matrices in \(\mathrm{SL}(2,p)\) and the second one being \(T\), the unique subgroup of index \(2\), of \(\mathrm{U}(2,p)\). Then the main result of the paper says that for a prime \(p>11\),
\begin{itemize}
\item if \(p\equiv 1\pmod{4}\), then the only strong Gelfand pair is given by \(\left(\mathrm{SL}(2,p),\mathrm{U}(2,p)\right)\),
\item if \(p\equiv 3\pmod{4}\), then there are exactly two strong Gelfand pairs: \(\left(\mathrm{SL}(2,p),\mathrm{U}(2,p)\right)\) and \(\left(\mathrm{SL}(2,p),T\right)\).
\end{itemize}
The scenario for \(p\leq 11\) is a little different. We have the following cases.
\begin{itemize}
\item For \(p=2\), \(\mathrm{SL}(2,2)\) is isomorphic to the symmetric group on \(3\) letters. The strong Gelfand pairs are \((\mathrm{SL}(2,2), C_2)\) and \((\mathrm{SL}(2,2), C_3)\).
\item For \(p=3\), the strong Gelfand pairs are \((\mathrm{SL}(2,3),C_3)\), \((\mathrm{SL}(2,3),C_6)\) and \((\mathrm{SL}(2,3),\mathrm{U}(2,3))\).
\item For \(p=5\), the strong Gelfand pairs are \((\mathrm{SL}(2,5),\mathrm{SL}(2,3))\) and \((\mathrm{SL}(2,5),\mathrm{U}(2,5))\).
\item For \(p=7\), the strong Gelfand pairs are \((\mathrm{SL}(2,7),\mathrm{U}(2,7))\), \((\mathrm{SL}(2,7), T)\) and \((\mathrm{SL}(2,7), K)\) where \(K\) is the preimage of \(\Sigma_4\subseteq\mathrm{PSL}(2,7)\) (the symmetric group on \(4\) symbols) under the canonical projection map.
\item For \(p=11\), the strong Gelfand pairs are \((\mathrm{SL}(2,11),\mathrm{U}(2,11))\), \((\mathrm{SL}(2,11), T)\) and \((\mathrm{SL}(2,11), 2I)\) where \(2I=\langle r,s,t|r^2=s^3=t^5=rst \rangle\).
\end{itemize}
The cases \(p\leq 11\) have been computed using Magma. For the rest of the cases, the approach has been uniform, as follows. The authors start with the description of character tables of the groups \(\mathrm{SL}(2,p)\), \(\mathrm{U}(2,p)\) and the projective special linear group \(\mathrm{PSL}(2,p)\). Then the problem has been reduced to considering the cases \((\mathrm{SL}(2,p),H)\) where \(H\) is a maximal subgroup of \(\mathrm{SL}(2,p)\). Since there is a one-one correspondence between the maximal subgroups of \(\mathrm{SL}(2,p)\) and those of \(\mathrm{PSL}(2,p)\), the finding has been further reduced to considering the maximal subgroups of \(\mathrm{PSL}(2,p)\) up to conjugacy. Next, the authors show that if \((\mathrm{SL}(2,p),H)\) is a strong Gelfand pair, then \(H\) must be a subgroup of \(\mathrm{U}(2,p)\). The proof is then completed after showing that if \(K\subseteq\mathrm{U}(2,p)\) is any proper subgroup and \(K\neq T\), then \((\mathrm{SL}(2,p), K)\) is not a strong Gelfand pair.
Reviewer: Saikat Panja (Prayāgrāj)Infinite characters of type II on \(\mathrm{SL}_n(\mathbb{Z})\)https://zbmath.org/1517.200712023-09-22T14:21:46.120933Z"Boutonnet, Rémi"https://zbmath.org/authors/?q=ai:boutonnet.remiBy definition, a character (finite or infinite) on a group \(\Gamma\) is a tracial weight on the universal \(C^*\)-algebra \(C^*(\Gamma)\) of the form \(\mathrm{Tr} \circ \pi\), where \(\pi : \Gamma \rightarrow \mathcal{U}(M)\) is a generating unitary representation into a semi-finite factor \(M\) admitting a normal faithful semi-simple trace \(\mathrm{Tr}\) such that \(\pi(C^*(\Gamma))\) contains a non-zero positive operator with finite trace. The type of a character is the von Neumann type of the factor \(M\). An unitary representation \(\pi\) is called factorial if the von Neumann algebra \(\pi(\Gamma)''\) is a factor.
The following is the main result of the paper (Theorem 1.3). For every \(n \geq 2\), \(\mathrm{SL}(n, \mathbb{Z})\) admits uncountably many factorial representations of type \(\mathrm{II}_{\infty}\) which are traceable, none of which weakly contains any other.
This work was motivated by the problem posed by \textit{B. Bekka} in [``Infinite characters on \(\mathrm{GL}_n(\mathbb{Q})\), on \(\mathrm{SL}_n(\mathbb{Z})\), and on groups acting on trees'', Preprint, \url{arXiv:1806.10110}].
Reviewer: Alla Detinko (Hull)Character bounds for regular semisimple elements and asymptotic results on Thompson's conjecturehttps://zbmath.org/1517.200722023-09-22T14:21:46.120933Z"Larsen, Michael"https://zbmath.org/authors/?q=ai:larsen.michael-j"Taylor, Jay"https://zbmath.org/authors/?q=ai:taylor.jay"Pham Huu Tiep"https://zbmath.org/authors/?q=ai:tiep.pham-huuA conjecture of J. Thompson states that each finite non-abelian simple group \(G\) contains a conjugacy class \(C \subset G\) such that \(C^{2} = G\). The main result of [\textit{E. W. Ellers} and \textit{N. Gordeev}, Trans. Am. Math. Soc. 350, No. 9, 3657--3671 (1998; Zbl 0910.20007)] shows that this conjecture holds, unless \(G\) is a finite simple group of Lie type over a field of at most 8 elements. Recently the first and third authors proved that Thompson's conjecture holds for simple symplectic, unitary, or orthogonal groups (with some condition on the center of the Schur cover in the two latter cases) of sufficiently large rank.
In this paper, the authors study an asymptotic version of Thompson's conjecture for \(\mathrm{SL}_{n}(q)\), \(\mathrm{SU}_{n}(q)\), \(\mathrm{Sp}_{2n}(q)\), \(\mathrm{SO}_{2n+1}(q)\) and \(\mathrm{SO}^{\pm}_{2n}(q)\), which are all closely related to simple groups.
If \(G\) is a finite classical group with natural module \(V = \mathbb{F}_{q}^{n}\), the support \(\mathsf{supp}(x)\) of an element \(x \in G\) is the codimension of the largest eigenspace of \(x\) on \(V \otimes_{\mathbb{F}_{q}} \overline{\mathbb{F}_{q}}\).
The main result of the paper under review is Theorem 1: For all positive integers \(k\) there exists an explicit constant \(B = B(k) > 0\) such that for all \(n \geq 1\) and all prime powers \(q\) the following statement holds. Suppose \(G\) is one of \(\mathrm{SL}_{n}(q)\), \(\mathrm{SU}_{n}(q)\), \(\mathrm{Sp}_{2n}(q)\), \(\mathrm{SO}_{2n+1}(q)\), and \(\mathrm{SO}^{\pm}_{2n}(q)\), and \(g \in G\) is a regular semisimple element whose characteristic polynomial on the natural module is a product of \(k\) pairwise distinct irreducible polynomials, of pairwise distinct degrees if \(G\) is of type \(\mathrm{Sp}\) or \(\mathrm{SO}\). Then \(g^{G}\cdot g^{G}\) contains every element \(x \in [G,G]\) with \(\mathsf{supp}(x) \geq B\).
Reviewer: Egle Bettio (Venezia)The second lowest two-sided cell in the affine Weyl group \(\widetilde{B}_n\)https://zbmath.org/1517.200732023-09-22T14:21:46.120933Z"Shi, Jian-yi"https://zbmath.org/authors/?q=ai:shi.jianyiLet \((W_a, S_a)\) be an irreducible affine Weyl group with \(W_0\) the associated Weyl group. The author assumes that \(W_a=\tilde B_n\), the affine Weyl group, and proves that any left cell of \(\tilde{B_n}\) in \(\Omega_{qr}\) is left-connected, verifying a conjecture of Lusztig in this case; he shows that \(\Omega_{qr}\) consists of all the extensions of \(w_J\) in the set \(\tilde{B}_n - W_{(\nu)}\), where \(W_{(\nu)}\) is the lowest two-sided cell of \(\tilde{B}_n\) and \(J \subset S_a\) is such that \(w_J\) is the longest element in the subgroup \(W_J\) of \(\tilde{B}_n\) of type \(D_n\).
Reviewer: Erich W. Ellers (Toronto)Correction to: ``Proper affine actions: a sufficient criterion''https://zbmath.org/1517.200742023-09-22T14:21:46.120933Z"Smilga, Ilia"https://zbmath.org/authors/?q=ai:smilga.iliaCorrection to the author's paper [ibid. 382, No. 2, 513--605 (2022; Zbl 1510.20043)].Functions whose orbital integrals and those of their Fourier transforms are at topologically nilpotent supporthttps://zbmath.org/1517.200752023-09-22T14:21:46.120933Z"Waldspurger, J.-L."https://zbmath.org/authors/?q=ai:waldspurger.jean-loupSummary: Let \(F\) be a \(p\)-adic field and let \(G\) be a connected reductive group defined over \(F\). We assume \(p\) is large. Denote \(\mathfrak{g}\) the Lie algebra of \(G\). To each vertex \(s\) of the reduced Bruhat-Tits' building of \(G\), we associate as usual a reductive Lie algebra \({\mathfrak{g}_s}\) defined over the residual field \({\mathbb{F}_q} \). We normalize suitably a Fourier-transform \(f\mapsto \hat{f}\) on \({C_c^{\infty }}(\mathfrak{g}(F))\). We study the subspace of functions \(f\in{C_c^{\infty }}(\mathfrak{g}(F))\) such that the orbital integrals of \(f\) and of \(\hat{f}\) are 0 for each element of \(\mathfrak{g}(F)\) which is not topologically nilpotent. This space is related to the characteristic functions of the character-sheaves on the spaces \({\mathfrak{g}_s}({\mathbb{F}_q})\), for each vertex \(s\), which are cuspidal and with nilpotent support. We prove that our subspace behave well under endoscopy.Unipotent generators for arithmetic groupshttps://zbmath.org/1517.200762023-09-22T14:21:46.120933Z"Venkataramana, T. N."https://zbmath.org/authors/?q=ai:venkataramana.tyakal-nanjundiahLet \(G \subset \mathrm{SL}_n\) be a semi-simple \(\mathbb{Q}\)-simple algebraic group defined over \(\mathbb{Q}\) and let \(Q\subset G\) be a proper parabolic \(\mathbb{Q}\)-subgroup with unipotent radical \(U^+\). Let \(U^-\) be the opposite unipotent radical and for any integer \(k\geq 1\) denote by \(E_Q(k)\) the subgroup generated by \(U^+\cap \mathrm{SL}_n(k\mathbb{Z})\) and \(U^-\cap \mathrm{SL}_n(k\mathbb{Z})\). The focus of attention in this paper is the following result.
{Theorem A.} If \(\mathbb{R}-\mathrm{rank}(G) \geq 2\) then the group \(E_Q(k)\) is an arithmetic subgroup of \(G(\mathbb{Q})\), i.e. has finite index in \(G(\mathbb{Z})=G\cap \mathrm{SL}_n(\mathbb{Z})\).
This is very much a ``higher rank'' result. For \(G=\mathrm{SL}_2\) the corresponding subgroups \(E_Q(k)\) are, for all but finitely many \(k\), of infinite index in \(\mathrm{SL}_2(\mathbb{Z})\). The first version of this result, for the special case where \(G=\mathrm{SL}_n\) and \(n \geq 3\), \(k\geq 1\), is due to \textit{J. Tits} [C. R. Acad. Sci., Paris, Sér. A 283, 693--695 (1976; Zbl 0381.14005)]. Various contributions, including one from the author [Pac. J. Math. 166, No. 1, 193--212 (1994; Zbl 0822.22005)], have led to a proof of the general case. The purpose of this paper is to provide a more uniform approach to this result.
The proof presented here is based on the completions of \(G(\mathbb{Q})\) with respect to two topologies. Let \(G\) be a group and let \(\mathcal{C}\) be a collection of subgroups of \(G\). Subject to some restrictions it can be shown that \(G\) is a topological group with respect to that topology whose open sets are \(gW\), where \(g \in G\) and \(W \in \mathcal{C}\). In a standard way by means of Cauchy sequences, \(G\) can be extended to \(\widehat{G(\mathbb{Q})}\), a \textit{completion} of \(G(\mathbb{Q})\) with respect to this topology. Here, \(\widehat{G(\mathbb{Q})}\) is determined by \(\mathcal{C}=\{F(k):k\geq1\}\), where \(F(k) \subset G(k\mathbb{Z})=G(\mathbb{Q})\cap \mathrm{SL}_n(k\mathbb{Z})\) is subgroup derived from the Levi-decomposition of a \textit{maximal} parabolic \(\mathbb{Q}\)-subgroup \(P\subset G\). A second completion, the \textit{congruence completion} \(\overline{G(\mathbb{Q})}\) is determined by \(W=\{G(k\mathbb{Z}):k \geq1\}\). Since one topology refines the other, there is an open map \(\widehat{G(\mathbb{Q})}\rightarrow \overline{G(\mathbb{Q})}\) which is also \textit{surjective}. The \textit{kernel} is denoted by \(C\). If \(\widehat{G(\mathbb{Q})}\) is replaced by the usual profinite completion of \(G(\mathbb{Q})\), then \(C\) becomes the well-known \textit{congruence kernel} associated with arithmetic groups which plays a central role in the \textit{congruence subgroup problem}. In contrast to \(C\), the congruence kernel is a compact (profinite) group. By a complicated route, the author shows that Theorem A follows from the following.
{Theorem B.} If \(\mathbb{R}-\mathrm{rank}(G)\geq 2\), then the kernel \(C\) is central in \(\widehat{G(\mathbb{Q})}\).
Most (affirmative) solutions for congruence subgroup problems involve a result of this type. As with Theorem A, the higher rank restriction is in general necessary. The proof of Theorem B is intricate and involves an interesting new technique introduced by the author [Mich. Math. J. 72, 599--619 (2022; Zbl 07599284)] concerning the centrality of the congruence kernel. This paper is one of many impressive contributions made by the author to the study of arithmetic groups.
For the entire collection see [Zbl 1511.20004].
Reviewer: Alexander W. Mason (Glasgow)Profinite rigidity, Kleinian groups, and the cofinite Hopf propertyhttps://zbmath.org/1517.200772023-09-22T14:21:46.120933Z"Bridson, M. R."https://zbmath.org/authors/?q=ai:bridson.martin-r"Reid, A. W."https://zbmath.org/authors/?q=ai:reid.alan-wSummary: Let \(\Gamma\) be a nonelementary Kleinian group and \(H < \Gamma\) be a finitely generated, proper subgroup. We prove that if \(\Gamma\) has finite covolume, then the profinite completions of \(H\) and \(\Gamma\) are not isomorphic. If \(H\) has finite index in \(\Gamma \), then there is a finite group onto which \(H\) maps but \(\Gamma\) does not. These results streamline the existing proofs that there exist full-sized groups that are profinitely rigid in the absolute sense. They build on a circle of ideas that can be used to distinguish among the profinite completions of subgroups of finite index in other contexts, for example, limit groups. We construct new examples of profinitely rigid groups, including the fundamental group of the hyperbolic 3-manifold \(\mathrm{Vol}(3)\) and that of the 4-fold cyclic branched cover of the figure-eight knot. We also prove that if a lattice in \(\mathrm{PSL}(2,\mathbb{C})\) is profinitely rigid, then so is its normalizer in \(\mathrm{PSL}(2,\mathbb{C})\).Cohomology of Fuchsian groups and non-Euclidean crystallographic groupshttps://zbmath.org/1517.200782023-09-22T14:21:46.120933Z"Hughes, Sam"https://zbmath.org/authors/?q=ai:hughes.samA non-Euclidean crystallographic group (NEC group) \(\Gamma \) is a discrete subgroup of isometries of the hyperbolic plane \(\mathbb{R}\mathbf{H}^2\), including orientation-reversing elements, with compact quotient \(X=\mathbb{R}\mathbf{H}^2/\Gamma \). Since their introduction in the late 1960s, these groups have been an important tool for the study of Klein surfaces and their automorphism groups, albeit usually restricted to the co-compact case. In the present paper, parabolic transformations are allowed. Thus, the quotient is an orbifold that can have punctures.
The aim of the paper is to calculate the cohomology groups of NEC groups. In Section 2, the author gives the preliminaries about these groups. The signature of \(\Gamma\) is the symbol:
\[
(g, s, \epsilon, [m_1, \dots, m_r], \{(n_{1,1}, \dots, n_{1,s_1}), \dots, (n_{k,1}, \dots, n_{k,s_k}), (), \dots, ()\}),
\]
where \(g\) is the topological genus of the quotient, \(s\) is the number of punctures, \(\epsilon = +\) if the quotient is orientable and \(\epsilon = -\) otherwise. The number of empty period cycles \(()\) is denoted by \(d\). Let \(C_E\) denote the number of even \(n_{i,j}\) and \(C_O\) the number of period cycles for which every \(n_{i,j}\) is odd. The presentation of an NEC group, its canonical fundamental region and the rational characteristic are also provided in this section. Observe that there are a couple of misprints in the notation of the surface symbol and another on in the exponent of \((c_{i,l-1}c_{i,l})\) in the presentation of the group.
In Section 3, the author introduces the Cartan-Leray spectral sequences for a \(\Gamma\)-space. In Section 4, the cohomology groups of NEC groups are obtained. The study is divided in four cases, according to the sign of \(\epsilon\) and according to whether \(k+d+s=0\) or not. The results are presented in Corollaries 4.1, 4.2, 4.4 and 4.5. As an example of these results consider the orientable case with \(k+d+s>0\), (Corollary 4.4). The groups of cohomology of \(\Gamma\) are
\[
H^q(\Gamma)=
\begin{cases}
\begin{alignedat}{2}
&\mathbb{Z} & &q=0,\\
&\mathbb{Z}^{2g+s+k+d-1}& &q=1,\\
&\mathbb{Z}^{\frac{1}{2}qC_E+C_O+d} \oplus (\bigoplus_{j=1}^r \mathbb{Z}_{m_j}) & &q\equiv 2 \pmod{4},\\
&\mathbb{Z}^{\frac{1}{2}(q-1)C_E+C_O+d} & &q=2p+1\ \text{where}\ p\geq 1,\\
&\mathbb{Z}^{\frac{1}{2}qC_E+C_O+d} \oplus (\bigoplus_{j=1}^r \mathbb{Z}_{m_j}) \oplus (\bigoplus_{i=1}^k \bigoplus_{l=1}^{s_i} \mathbb{Z}_{n_{i,j}})& \qquad &q>0 \text{ and } q\equiv 0 \pmod{4}.\\
\end{alignedat}
\end{cases}
\]
In the case where \(\Gamma\) is a Fuchsian group (\(\epsilon = +\) and \(k+d+s=0\)), the cohomology ring is also calculated in Theorem 1.4. The author comments that some of the results in the paper have appeared in the literature, but the methods in those earlier results are different to the ones used here. The paper ends with several closing remarks. The author points out that Fuchsian groups are not determined by their cohomology and presents two non-isomorphic Fuchsian groups with isomorphic cohomology rings.
Reviewer: Ernesto Martínez (Madrid)Bounded cohomology of classifying spaces for families of subgroupshttps://zbmath.org/1517.200792023-09-22T14:21:46.120933Z"Li, Kevin"https://zbmath.org/authors/?q=ai:li.kevin-x|li.kevin-wThe author introduces a generalization of bounded cohomology as bounded version of Bredon cohomology for groups relative to a family of subgroups, i.e. a collection of subgroups of a group G such that it is closed under conjugation and under taking subgroups. This generalization of bounded cohomology, like the Bredon cohomology, is well-behaved with respect to normal subgroups and admits a topological interpretation in terms of classifying spaces for families.
Analogous to \textit{B. E. Johnson}'s characterization of amenability [Cohomology in Banach algebras. Providence, RI: American Mathematical Society (AMS) (1972; Zbl 0256.18014)], the author proves a characterization of relatively amenable groups in terms of bounded Bredon cohomology. He also provides a characterization of relative amenability in terms of relatively injective modules. Analogous to \textit{I. Mineyev}'s characterization of hyperbolicity [Q. J. Math. 53, No. 1, 59--73 (2002; Zbl 1013.20048)], the relative hyperbolicity is characterized in terms of this generalized bounded cohomology.
Reviewer: Mădălina Buneci (Targu-Jiu)On \(p\)-adic modules with isomorphic endomorphism algebrashttps://zbmath.org/1517.200802023-09-22T14:21:46.120933Z"Goldsmith, Brendan"https://zbmath.org/authors/?q=ai:goldsmith.brendan"White, Noel"https://zbmath.org/authors/?q=ai:white.noelA well-known fact that two (finite-dimensional) modules over a field (i.e. vector spaces) are isomorphic, if and only if their endomorphism algebras are isomorphic, has attracted many attempts to generalize the result, one way or the other, to modules over various rings. It seems that the question in its full generality is intractable, judging by examples dealing with pathologies in abelian group theory (see, e.g. a paper by \textit{A. L. S. Corner} et al. [in: Models, modules and abelian groups. In memory of A. L. S. Corner. Berlin: Walter de Gruyter. 315--323 (2008; Zbl 1233.20051)]). Nevertheless, there are a number of papers attacking the problem for some specialized rings. The reviewer's paper [Arch. Math. 51, No. 5, 419--424 (1988; Zbl 0635.13002)] for modules over valuation domains is one of many attempts of this kind. The paper under review considers the category of modules over the ring of \(p\)-adic integers \(J_p\) (\(p\) a fixed prime), which is fairly close to a field, considering its properties.
A few of the results proved in the paper are as follows: Assume that the endomorphism algebras of two modules are isomorphic: \(E(G)\cong E(H)\). This will imply \(G\cong H\) in either of the following cases: a) \(G\) and \(H\) are bounded modules b) \(G\) and \(H\) are reduced modules and \(G\) has a direct summand isomorphic to \(R\). In particular, the conclusion follows, if \(G, H\) are reduced, torsion-free modules.
In the case when \(G=D\oplus G_1\), \(D\neq 0\) divisible and \(G_1\neq 0\) reduced, the endo-algebras isomorphism implies \(H=D^\prime\oplus H_1\), where \(D^\prime\cong D\) and \(H_1\) is reduced with \(E(H_1)\cong E(G_1)\).
Combined with some forthcoming and already published results of the authors, the main theorem is as follows: If \(G\) and \(H\) are \(J_p\)-adic modules with isomorphic endomorphism algebras \(E(G)\cong E(H)\), then \(G\cong H\), except in the following two cases: (i) \(G=\oplus_\mu Q/R\) and \(H\) is a completion of \(\oplus_\mu R\) in the \(p\)-adic topology (\(\mu\) a cardinal) or (ii) \(G\) and \(H\) have isomorphic torsion basic submodules \(B\) and at least one of \(G, H\) is a non-torsion extension of \(B\), by a divisible module.
Reviewer: Radoslav M. Dimitrić (New York)Presentations for wreath products involving symmetric inverse monoids and categorieshttps://zbmath.org/1517.200812023-09-22T14:21:46.120933Z"Clark, Chad"https://zbmath.org/authors/?q=ai:clark.chad"East, James"https://zbmath.org/authors/?q=ai:east.jamesWreath products involving symmetric inverse monoids/semigroups/categories arise in many areas of algebra and science, and presentations by generators and relations are crucial tools in such studies. Let \(M\) be an arbitrary monoid. The symmetric inverse monoid \(\mathcal{I}_n\) is the set of all partial permutations of the set \(\textbf{n}=\{1, \ldots , n\}\), under the operation of relational composition, Sing(\(\mathcal{I}_n\)) is the singular ideal of \(\mathcal{I}_n\), consisting of all strictly partial permutations and \(\mathcal{I}\) is the category of all partial bijections \(\textbf{m} \rightarrow \textbf{n}\) where \(m\) and \(n\) range over \(\{0,1, \ldots , n\}\). The authors begin with some preliminary material in Section 2, which includes the definition of the wreath products in Subsection 2.3. In Section 3, the authors give presentations for the monoid \(M \wr \mathcal{I}_n\) (Theorems 3.22 and 3.29). In Section 4, the authors apply the results of Section 3 to obtain a category presentation (Theorem 4.11) of \(M \wr \mathcal{I}\), which is then used to obtain a tensor presentation (Theorem 4.17) for \(M \wr \mathcal{I}\). In Section 5, the authors give a presentation for the singular wreath product \(M\) \(\wr\) Sing\((\mathcal{I}_n)\) (Theorem 5.37).
Reviewer: Ronnason Chinram (Hat Yai)An explicit algorithm for normal forms in small overlap monoidshttps://zbmath.org/1517.200822023-09-22T14:21:46.120933Z"Mitchell, James D."https://zbmath.org/authors/?q=ai:mitchell.james-d"Tsalakou, Maria"https://zbmath.org/authors/?q=ai:tsalakou.mariaSummary: We describe a practical algorithm for computing normal forms for semigroups and monoids with finite presentations satisfying so-called small overlap conditions. Small overlap conditions are natural conditions on the relations in a presentation, which were introduced by J. H. Remmers and subsequently studied extensively by M. Kambites. Presentations satisfying these conditions are ubiquitous; Kambites showed that a randomly chosen finite presentation satisfies the \(C(4)\) condition with probability tending to 1 as the sum of the lengths of relation words tends to infinity. Kambites also showed that several key problems for finitely presented semigroups and monoids are tractable in \(C(4)\) monoids: the word problem is solvable in \(O(\min \{| u |, | v | \})\) time in the size of the input words \(u\) and \(v\); the uniform word problem for \(\langle A \mid R \rangle\) is solvable in \(O( N^2 \min \{| u |, | v | \})\) where \(N\) is the sum of the lengths of the words in \(R\); and a normal form for any given word \(u\) can be found in \(O(| u |)\) time. Although Kambites' algorithm for solving the word problem in \(C(4)\) monoids is highly practical, it appears that the coefficients in the linear time algorithm for computing normal forms are too large in practice. In this paper, we present an algorithm for computing normal forms in \(C(4)\) monoids that has time complexity \(O(| u |^2)\) for input word \(u\), but where the coefficients are sufficiently small to allow for practical computation. Additionally, we show that the uniform word problem for small overlap monoids can be solved in \(O(N \min \{| u |, | v | \})\) time.On the word problem for free products of semigroups and monoidshttps://zbmath.org/1517.200832023-09-22T14:21:46.120933Z"Nyberg-Brodda, Carl-Fredrik"https://zbmath.org/authors/?q=ai:nyberg-brodda.carl-fredrikSummary: We study the language-theoretic aspects of the word problem, in the sense of \textit{A. Duncan} and \textit{R. H. Gilman} [Math. Proc. Camb. Philos. Soc. 136, No. 3, 513--524 (2004; Zbl 1064.20055)], of free products of semigroups and monoids. First, we provide algebraic tools for studying classes of languages known as super-AFLs, which generalise e.g. the context-free or the indexed languages. When \(\mathcal{C}\) is a super-AFL closed under reversal, we prove that the semigroup (monoid) free product of two semigroups (resp. monoids) with word problem in \(\mathcal{C}\) also has word problem in \(\mathcal{C}\). This recovers and generalises a recent result by \textit{T. Brough} et al. [Lect. Notes Comput. Sci. 11647, 292--305 (2019; Zbl 07117554)] that the class of context-free semigroups (monoids) is closed under taking free products. As a group-theoretic corollary, we deduce that the word problem of the (group) free product of two groups with word problem in \(\mathcal{C}\) is also in \(\mathcal{C}\). As a particular case, we find that the free product of two groups with indexed word problem has indexed word problem.On d-semigroups, r-semigroups, dr-semigroups and their subclasseshttps://zbmath.org/1517.200842023-09-22T14:21:46.120933Z"Wang, Shoufeng"https://zbmath.org/authors/?q=ai:wang.shoufengA \textit{d-semigroup} (resp. \textit{r-semigroup}) is a semigroup \((S,\cdot)\) with a unary operation \(x\mapsto x^+\) (resp. \(x\mapsto x^*\)) satisfying the identities
\begin{align*}
x^+\cdot x &= x,\ (x\cdot y)^+ = (x\cdot y^+)^+.\\
(\text{resp. }x\cdot x^* &= x,\ (y\cdot x)^* = (y^*\cdot x)^*.)
\end{align*}
\textit{dr-semigroups} are two-sided versions of d-semigroups and r-semigroups. These are semigroups \((S,\cdot)\) with two unary operations \(x\mapsto x^+\) and \(x\mapsto x^*\), such that \((S,\cdot,{}^+)\) is a d-semigroup, \((S,\cdot,{}^*)\) is a r-semigroup and
\begin{align*}
(x^+)^*=x^+,\ (x^*)^+=x^*
\end{align*}
for all \(x\in S\).
In the first part of the paper, the author studies the varieties of d-semigroups, r-semigroups and dr-semigroups and their relationships with other known varieties of semigroups with unary operation(s). In the second part, he proves several Munn-type representation theorems for these classes of semigroups. Finally, an Ehresmann-Schein-Nambooripad-type theorem for dr-semigroups is provided.
Reviewer: Mykola Khrypchenko (Florianópolis)Arf numerical semigroups with prime multiplicityhttps://zbmath.org/1517.200852023-09-22T14:21:46.120933Z"Karakaş, Halil İbrahim"https://zbmath.org/authors/?q=ai:karakas.halil-ibrahimAn \textit{Arf numerical semigroup} \(S\) is a numerical semigroup satisfying additionally
\[
x, y, z\in S;\ x\geq y\geq z\Rightarrow x+y-z\in S.
\]
Let \(\mathcal{S}_{\mathrm{ARF}}(m, c)\) denote the set of all Arf numerical semigroups with multiplicity \(m\) (the smallest positive element of \(S\)) and conductor \(c\) (the smallest element of \(S\) for which all subsequent natural numbers belong to \(S\)). It is proved that if \(p\) is prime and \(c > 2p\), then \(\mathcal{S}_{\mathrm{ARF}}(p, c+p)=\{(p+S)\cup\{0\}\colon S\in \mathcal{S}_{\mathrm{ARF}}(p, c)\}\) (and, consequently, \(\vert \mathcal{S}_{\mathrm{ARF}}(p, c+p)\vert=\vert\mathcal{S}_{\mathrm{ARF}}(p, c)\vert\)).
Reviewer: Peeter Normak (Tallinn)Blocks of epigroupshttps://zbmath.org/1517.200862023-09-22T14:21:46.120933Z"Liu, Jingguo"https://zbmath.org/authors/?q=ai:liu.jingguoA semigroup \(S\) is called: 1) an \textit{epigroup} if some power \(x^n\) of any element \(x\in S\) lies in some (maximal) subgroup of \(S\) (the identity of this subgroup is denoted by \(x^{\omega}\)); 2) \(E\)-\textit{solid} if for all \(e,f,g\in E_S\) with \(e\mathcal{L}g\mathcal{R} f\), there exists \(h\in E_S\) such that \(e\mathcal{R}h\mathcal{L}f\). Let \(D\) be a regular \(\mathcal{D}\)-class and \(T\) the subsemigroup of its trace \(D^0\) generated by the union of all maximal subgroups of \(D\). Then the regular \(\mathcal{D}\)-classes of \(T\) are called the \textit{blocks} of \(D\) (if \(D\) is a \(\mathcal{D}\)-class of an epigroup, then the multiplication \(*\) on its trace \(D^0\) can be defined by \(a*b=ab\) if \(ab\in D\) and \(0\) otherwise). To each block \(B\), a semigroup \(B'\) is associated (which, depending on some conditions, equals either \(B\) or \(B\cup 0\) or \(0\)). For a subclass \(\mathbf{V}\) of the class \(\mathbf{E}\) of all epigroups, the following subclass is defined: \(B(\mathbf{V})=\{ S\in \mathbf{E}\mid \text{for any block }B\text{ of }S, B'\in \mathbf{V}\}\). Some properties of blocks are derived. The main theorem proves the equivalence of twelve properties of an epigroup \(S\), including the following:
\begin{itemize}
\item[1)] \(S\) is \(E\)-solid;
\item[2)] for any block \(B\) of \(S\), \(B'\) is completely simple;
\item[3)] for any \(e,f,g\in E_S\), \(ef\vert g\Rightarrow (ef)^2\vert g\);
\item[4)] \(S\) satisfies the identity \((x^{\omega}(y^{\omega}zx^{\omega})^{\omega}y^{\omega})^{\omega+1}=x^{\omega}(y^{\omega}zx^{\omega})^{\omega}y^{\omega}\) (\(a^{\omega+1}\) means \(a^{\omega}a\)).
\end{itemize}
The following subclasses of epigroups are also described, by several equivalent conditions in each case: \(B(\mathbf{R}_e\mathbf{G})\) (\(\mathbf{R}_e\mathbf{G}\) -- the class of rectangular groups), \(B(\mathbf{LG})\) (\(\mathbf{LG}\) -- the class of left groups), \(B(\mathbf{G})\) (\(\mathbf{R}\) -- the class of \(\mathcal{R}\)-trivial epigroups).
Reviewer: Peeter Normak (Tallinn)On automorphism groups of endomorphism semigroups of finite elementary abelian groupshttps://zbmath.org/1517.200872023-09-22T14:21:46.120933Z"Bayramyan, A. A."https://zbmath.org/authors/?q=ai:bayramyan.a-aConsider a monoid \(M\) and its group of invertible elements \(I(M)\). Every automorphism of \(M\) stabilizes \(I(M)\), so that one has a group homomorphism \(\tau = \tau_M: \Aut(M) \to \Aut(I(M))\) given by restriction of maps: \(\tau(\varphi) = \varphi|_{I(M)}\).
Assume now that \(M\) is the multiplicative monoid of matrices \(\operatorname{M}_n(\mathbb{Z}/p\mathbb{Z})\), where \(p\) is a prime number and \(n\geq 2\). The group \(I(M)\) is then \(\operatorname{GL}_n(\mathbb{Z}/p\mathbb{Z})\). In this setting, the main theorems of the article are:
\begin{itemize}
\item[1.] The morphism \(\tau_{M}\) is injective.
\item[2.] Every automorphism of the monoid \(M = \operatorname{M}_n(\mathbb{Z}/p\mathbb{Z})\) is inner.
\end{itemize}
The proofs are elementary, based on matrix operations, but rely on a particular generating set of \(\operatorname{M}_n(\mathbb{Z}/p\mathbb{Z})\) for Theorem 1, and on a classification of the automorphisms of \(\operatorname{GL}_n(\mathbb{Z}/p\mathbb{Z})\) for Theorem 2.
Another case of interest is when \(M\) is the monoid \(\operatorname{End}(G)\) of endomorphisms of a group \(G\); in this case, \(I(M)\) is the group \(\Aut(G)\) of automorphisms of \(G\). If \(G\) is the finite elementary abelian group \((\mathbb{Z}/p\mathbb{Z})^n = (\mathbb{Z}/p\mathbb{Z}) \oplus \cdots \oplus (\mathbb{Z}/p\mathbb{Z})\), we recover the above setting.
The author was motivated by articles where the relationship between \(\Aut(\operatorname{End}(G))\) and \(\Aut(\Aut(G))\) is studied for other groups \(G\), namely:
\begin{itemize}
\item free groups of finite rank in [\textit{E. Formanek}, Proc. Am. Math. Soc. 130, No. 4, 935--937 (2002; Zbl 0993.20020)];
\item free Burnside groups in [\textit{V. S. Atabekyan}, Int. J. Algebra Comput. 25, No. 4, 669--674 (2015; Zbl 1318.20038)];
\item relatively free groups in [\textit{V. S. Atabekyan} and \textit{H. T. Aslanyan}, Int. J. Algebra Comput. 28, No. 2, 207--215 (2018; Zbl 1384.20033)].
\end{itemize}
Reviewer: Yves Stalder (Aubière)The orbits of generalized orthogonal monoids under the action of their unit groupshttps://zbmath.org/1517.200882023-09-22T14:21:46.120933Z"Feng, Jianqiang"https://zbmath.org/authors/?q=ai:feng.jianqiang"Li, Zhenheng"https://zbmath.org/authors/?q=ai:li.zhenhengIn this paper, the authors introduce a class of new monoids termed as the generalised orthogonal monoids containing the traditional orthogonal monoids as proper submonoids and, in fact, they are the examples of dilating monoids. The orbit structure of this type of monoids denoted by \(M\) under the two-sided actions of their unit groups are also studied. Introducing the concept of standard generalised orthogonal matrices, they determine the complete set of all orbits and deduce that the set of all standard generalised orthogonal matrices is not a submonoid of \(M\). Moreover, the orbits in \(M\) are described in the paper under the flavour of the set \(Y\) of standard generalised orthogonal matrices of the second kind and with this notion the authors prove \(Y\) to be a submonoid of \(M\). They also impose the partial order on \(Y\) and sketch the corresponding Hasse diagram. As a result, the total number of orbits in \(M\) are determined. Furthermore, the results in the paper are supported by sharper methodologies. Also, a resourceful reference list is given at the end of the paper.
Reviewer: Sanjib Kumar Datta (Kalyani)On the monoid of partial isometries of a finite star graphhttps://zbmath.org/1517.200892023-09-22T14:21:46.120933Z"Fernandes, Vítor H."https://zbmath.org/authors/?q=ai:fernandes.vitor-h"Paulista, Tânia"https://zbmath.org/authors/?q=ai:paulista.taniaLet \(S_n =(\{0, 1,\ldots, n-1\},\{\{0,i\}\mid i=1, \ldots, n-1\})\) be the \textit{star graph} with \(n\) vertices and \(\mathcal{DP}S_n\) the monoid of all partial isometries of \(S_n\) (distance preserving partial transformations of \(S_n\)). In this paper, the cardinal of \(\mathcal{DP}S_n\) is calculated, Green's relations and generating sets of \(\mathcal{DP}S_n\) are described and it is proved that the rank (the minimum size of a generating set) of \(\mathcal{DP}S_n\) is 3, for \(n=3\), and 5, for \(n\geq 4\). The main theorem states that for \(n\geq 4\) the presentation of a monoid \(\mathcal{DP}S_n\) is defined by \(3n+9\) relations, all explicitly described.
Reviewer: Peeter Normak (Tallinn)On certain semigroups of transformations with restricted rangehttps://zbmath.org/1517.200902023-09-22T14:21:46.120933Z"Li, De Biao"https://zbmath.org/authors/?q=ai:li.debiao"Zhang, Wen Ting"https://zbmath.org/authors/?q=ai:zhang.wen-ting.1"Luo, Yan Feng"https://zbmath.org/authors/?q=ai:luo.yan-fengThe authors consider a finite chain \(X=\{1,\ldots,n\}\) where \(n\) is a positive integer. For any nonempty subset \(Y\) of \(X\), the semigroup \(\mathcal{POI}(X, Y )\) of all injective order-preserving partial transformations with range contained in \(Y\) with respect to the usual composition is mainly focused. Particularly, Green's relations and regularity of \(\mathcal{POI}(X, Y )\) are characterized in Section 2. The authors further give a sufficient and necessary condition under which \(\mathcal{POI}(X, Y )\) is a regular semigroup. In Section 3, Green's \(*\)-relations \(\mathcal{L}^*\) and \(\mathcal{R}^*\) on the semigroup \(\mathcal{POI}(X, Y )\) are described. Moreover, for a proper nonempty subset \(Y\) of \(X\), the authors prove in Theorem 3.4 that the semigroup \(\mathcal{POI}(X, Y )\) is left abundant but not right abundant. The authors continue their investigation in Section 4 with counting the cardinality of the semigroup \(\mathcal{POI}(X, Y )\) where \(Y\) is a subset of \(X\) containing \(r \) elements, \(1\leq r \leq n\). An isomorphism theorem for \(\mathcal{POI}(X, Y )\) is provided in Theorem 4.3. In Section 5, the authors determine the rank of \(\mathcal{POI}(X, Y )\) and give a conclusion in Theorem 5.16.
Reviewer: Thodsaporn Kumduang (Nakhon Pathom)Degree 2 transformation semigroups as continuous maps on graphs: complexity and exampleshttps://zbmath.org/1517.200912023-09-22T14:21:46.120933Z"Margolis, Stuart"https://zbmath.org/authors/?q=ai:margolis.stuart-w"Rhodes, John"https://zbmath.org/authors/?q=ai:rhodes.john-lSummary: In this paper, we give a number of illuminating examples of transformation semigroups of degree 2 acting on graphs by functions that preserve vertices and edges by inverse image. It is known that the complexity of such a transformation semigroup is at most 2. We give examples that use sophisticated lower bounds to complexity to distinguish between complexity 1 and complexity 2.Acts over semigroupshttps://zbmath.org/1517.200922023-09-22T14:21:46.120933Z"Kozhukhov, I. B."https://zbmath.org/authors/?q=ai:kozhukhov.igor-borisovich"Mikhalev, A. V."https://zbmath.org/authors/?q=ai:mikhalev.aleksandr-vThe article surveys the main results in the last two decades in the area of structural theory of acts over semigroups. Several results in the acts over completely (0-)simple semigroups, subdirectly irreducible and uniform acts, acts with conditions on the congruence lattice, diagonal acts, biacts and multiacts, partial acts etc. are also discussed.
Reviewer: Azeef Muhammed Parayil Ajmal (Ekaterinburg)A monoid structure on the set of all binary operations over a fixed sethttps://zbmath.org/1517.200932023-09-22T14:21:46.120933Z"López-Permouth, Sergio R."https://zbmath.org/authors/?q=ai:lopez-permouth.sergio-r"Owusu-Mensah, Isaac"https://zbmath.org/authors/?q=ai:mensah.isaac-owusu"Rafieipour, Asiyeh"https://zbmath.org/authors/?q=ai:rafieipour.asiyehLet \(M(S)\) be the set of all binary operations on \(S\). In this paper, the monoid structure \((M(S),\triangleleft)\) satisfying that each outset
\[
\mathrm{out}(*) = \{\circ\in M(S)\,|\, *\text{ distributes over }\circ\},
\]
and several properties of \((M(S),\triangleleft)\) are considered, including a complete characterization of its group of units and of a subgroup of its group of automorphisms, induced by permutations, which is a retraction. In addition, various submonoids and ideals are studied. A generic decomposition, called the kernel-cokernel decomposition, of arbitrary magmas and ideals are characterized.
Reviewer: Wiesław A. Dudek (Wrocław)Extensions and tangent prolongations of differentiable loopshttps://zbmath.org/1517.200942023-09-22T14:21:46.120933Z"Figula, Ágota"https://zbmath.org/authors/?q=ai:figula.agota"Nagy, Péter T."https://zbmath.org/authors/?q=ai:nagy.peter-tiborThe aim of the authors' research project is to study the tangent prolongation of differentiable loops, giving a natural generalization of the tangent prolongation of Lie groups. The initial research of the authors showed that the prolonged loop structure to the tangent bundle belongs to a special category of loop extensions [Commentat. Math. Univ. Carol. 61, No. 4, 501--511 (2020; Zbl 07332724)]. The authors have previously studied the abstract construction of abelian extensions of loops that have some weak inverse property, and they showed that the tangent prolongation of loops inherits the classically weak associative properties of the base loop [\textit{P. T. Nagy} and \textit{Á. Figula}, Publ. Math. Debr. 97, No. 1--2, 241--252 (2020; Zbl 1474.22007)]. The authors' purpose in this paper is to give a systematic investigation of abelian extensions of differentiable loops and the corresponding extension theory of the tangent algebras of degree 3 and find the algebraic characterization of these tangent algebras of the tangent prolongation of differentiable loops. The authors address in this paper: In Section 3 -- Extension of binary-ternary algebras. In Section 4 -- Akivis algebra and Sabinin algebra of degree 3 of abelian extensions of local loops. In Section 5 -- Abelian loop extensions are associated with tangent algebra extensions. In Section 6 -- Tangent algebras of the tangent prolongation of loops.
Reviewer: C. Pereira da Silva (Curitiba)Free Bol loops of exponent twohttps://zbmath.org/1517.200952023-09-22T14:21:46.120933Z"Grishkov, A."https://zbmath.org/authors/?q=ai:grishkov.alexander-n"Rasskazova, M."https://zbmath.org/authors/?q=ai:rasskazova.m-n|rasskazova.marina"Souza dos Anjos, G."https://zbmath.org/authors/?q=ai:souza-dos-anjos.gA \textit{loop} consists of a nonempty set \(L\) with a binary operation \(\ast\) such that, for each \(a, b \in L\), the equations \(a \ast x = b\) and \(y \ast a = b\) have unique solutions for \(x, y \in L\), and there exists an identity element \(1\in L\) satisfying \(1 \ast x = x = x \ast 1\), for any \(x \in L\). A (\textit{right}) \textit{Bol loop} \(L\) is a loop that satisfies the (right) Bol identity \(x((yz)y) = ((xy)z)y\) for \(x,y,z\in L\). One of the most interesting subvarieties of Bol loops is the variety \(B_2\) of Bol loops of exponent two. Many constructions of non-associative loops of \(B_2\) can be found in the literature (see [\textit{H. Kiechle}, Theory of K-loops. Berlin: Springer (2002; Zbl 0997.20059)], for example). In [\textit{G. P. Nagy}, Trans. Am. Math. Soc. 361, No. 10, 5331--5343 (2009; Zbl 1179.20061)], a class of non-associative simple Bol loops of exponent \(2\) was constructed.
The authors give a construction of free objects in the variety \(B_2\). Let \(B(X)\) be the free Bol loop of exponent two with free set of generators \(X\). A subset \(R(X)\subseteq B(X)\) such that every element \(b \in B(X) \backslash \{1\}\) has the canonical form \(b = (\dots(b_1b_2)b_3\dots)b_m)b_{m-1})\dots)b_2)b_1\) is constructed, where \(b_i \in R(X)\) and \(b_i \not= b_{i+1}\), for all \(i\). Then, the multiplication law of \(B(X)\) based on this form is described. Furthermore, it is proved that the nuclei and the center of \(B(X)\) are trivial.
Reviewer: Marek Golasiński (Olsztyn)On the commuting probability of \(p\)-elements in a finite grouphttps://zbmath.org/1517.200962023-09-22T14:21:46.120933Z"Burness, Timothy C."https://zbmath.org/authors/?q=ai:burness.timothy-c"Guralnick, Robert"https://zbmath.org/authors/?q=ai:guralnick.robert-m"Moretó, Alexander"https://zbmath.org/authors/?q=ai:moreto.alexander"Navarro, Gabriel"https://zbmath.org/authors/?q=ai:navarro.gabriel.1A fairly well-known result due to \textit{W. H. Gustafson} [Am. Math. Mon. 80, 1031--1034 (1973; Zbl 0276.60013)] says: The probability that two random elements in a finite group commute is greater than \(\frac{5}{8}\) this is equivalent to \(G\) being abelian, in which case the probability is 1. The paper under review deals with a local version of this result. Let \(p\) be a prime which divides \(|G|\). Then the probability \(\operatorname{Pr}_p(G)\) that two random \(p\)-elements commute is defined by \(\frac{|\{(x,y) \in G_p \times G_p: xy=yx\}|}{|G_p|^2}\), where \(G_p\) is the set of \(p\)-elements in \(G\). Set \(f(p) = \frac{p^2+p-1}{p^3}\). Recall that \(f(2) =\frac{5}{8}\), the Gustafson bound. The main result of the paper is that \(G\) has an abelian normal Sylow \(p\)-subgroup if and only if \(\operatorname{Pr}_p(G) > f(p)\). The authors also give a complete classification in the boundary case that \(\operatorname{Pr}_p(G) = f(p)\) and \(G = O^{p^\prime}(G)\). In fact, if \(p \geq 5\), \(\operatorname{Pr}_p(G) = f(p)\) and \(G\) is non abelian simple, then \(G \cong \mathrm{PSL}_2(p)\). There are no non solvable examples \(G = O^{p^\prime}(G)\) for \(p = 2\) or \(3\). There is a further result in the paper of independent interest. Let \(G\) be a finite group, \(p\) a prime and \(x \in G \setminus O_p(G)\) be a \(p\)-element. Then \(\frac{|C_G(x)_p|}{|G_p|} \leq \frac{1}{p}\).
The proof of the main result depends on the classification of the finite simple groups. In fact, it uses a result on fixed point ratios due to \textit{T. C. Burness} and \textit{R. M. Guralnick} [Adv. Math. 411 A, Article ID 108778, 90 p. (2022; Zbl 1511.20010)] which depends on the classification of the finite simple groups. But in case of \(F^\ast(G)\) is a \(p^\prime\)-group, the classification is not needed.
The authors also claim that for \(G\) simple, \(\operatorname{Pr}_p(G)\) goes to zero with \(|G|\) to infinity. They prove this for \(G\) an alternating group. They also give a sketch of a proof of this result for groups of Lie type. There are many further results in this worthwhile reading paper.
Reviewer: Gernot Stroth (Halle)Some new results about left ideals of \(\beta S\)https://zbmath.org/1517.220012023-09-22T14:21:46.120933Z"Hindman, Neil"https://zbmath.org/authors/?q=ai:hindman.neil"Strauss, Dona"https://zbmath.org/authors/?q=ai:strauss.donaGiven a semigroup \(S\) with the discrete topology, there is a compact, right topological semigroup structure on its Stone-Čech compactification \(\beta S\). The minimal left ideals of \(\beta S\), which are also universal minimal \(S\)-flows with respect to the natural action of \(S\) on \(\beta S\), play a foundational part in the theory of \(\beta S\). In this paper the authors investigate (and answer) several natural questions concerning these minimal left ideals.
First is the question of when the minimal left ideals of \(\beta S\) are finite. If \(S\) already contains a finite left ideal, then this remains true (for relatively simple reasons) in \(\beta S\). The situation becomes more subtle, and more interesting, when all the minimal left ideals of \(\beta S\) are contained in \(\beta S \setminus S\). In this case the authors formulate some necessary conditions for the existence of finite minimal left ideals. They also show that these necessary conditions are not sufficient.
Next, the authors investigate how the taking of Cartesian products interacts with the taking of minimal left ideals. Are the minimal left ideals of a product of semigroups each equal (identifiable via a topological isomorphism) with the product of a minimal left ideal of each of the factors? Several results are proved in this vein.
Lastly, the authors show that, if \(S\) is any countably infinite cancellative semigroup, then every non-minimal semiprincipal left ideal in \(\beta S\) contains many semiprincipal left ideals defined by right cancelable elements of \(\beta S\). Some of the properties of these left ideals are explored.
Reviewer: Will Brian (Charlotte)The product of lattice covolume and discrete series formal dimension: \(\mathfrak{p}\)-adic \(\mathrm{GL}(2)\)https://zbmath.org/1517.220092023-09-22T14:21:46.120933Z"Ruth, L. C."https://zbmath.org/authors/?q=ai:ruth.lauren-cIn 1930, J. von Neumann introduced the concept of von Neumann algebras as one special type of \(C^*\)-algebras. In fact, if \(\mathcal{M}\) is a self-adjoint unital subalgebra of bounded linear operators on complex Hilbert space \(\mathcal{H}\) that is closed in the strong operator topology then \(\mathcal{M}\) is called a von Neumann algebra. Seven years later, \textit{F. J. Murray} and \textit{J. von Neumann} [Ann. Math. (2) 37, 116--229 (1936; Zbl 0014.16101; JFM 62.0449.03)] introduced the concept of the von Neumann dimension \(\dim _{\mathcal{M}} \mathcal{H}\) of a representation of the von Neumann algebra \(\mathcal{M}\) on a Hilbert space \(\mathcal{H}\). Let \(G\) be a connected semisimple Lie group, a discrete series representation of \(G\) is an irreducible unitary representation \((\pi, \mathcal{H})\) of \(G\) if one (hence all) its matrix coefficients are square-integrable, i.e. \(\pi\) is equivalent to a subrepresentation of the right regular representation of \(G\) on \(L^2 (G)\) (see Theorem 16.2 in [\textit{A. Robert}, Introduction to the representation theory of compact and locally compact groups. Cambridge: Cambridge University Press. London: London Mathematical Society (1983; Zbl 0498.22006)]). The formal dimension is a constant \(d_{\pi} > 0\) satisfied the Schur's relation:
\[
\int\limits^{}_{G}(\pi (g) u, v) \overline{(\pi (g) u', v')}\,dg = \frac{1}{d_{\pi}} (u, u') \overline{(v, v')}; \, u, u', v, v'\in \mathcal{H}.
\]
By definition, a von Neumann algebra with trivial center is called a factor. If a factor that has a unique faithful normal (weakly and strongly continuous) tracial state, then it is called a finite factor. The finite factors acting (irreducibly) on finite-dimensional complex vector spaces are the complex matrix algebras \(M_n (\mathbb{C})\) and they are called \(I_n\) factors. Of course, the trace on projections in a \(I_n\) factor attains all values in \(\{0, \frac{1}{n}, \frac{2}{n}, \dots, \frac{n}{n} = 1\}\) (normalized to equal 1 on the identity). Finite factors acting on infinite-dimensional Hilbert spaces are called \(II_1\) factors; the trace on projections in a \(II_1\) factor attains all values in [0, 1] (normalized to equal 1 on the identity). Let \(G\) be a locally compact unimodular group, then a discrete subgroup \(\Gamma\) of \(G\) is called a lattice if \(G / \Gamma\) supports a finite Haar measure.
In 2018, the author introduced two new settings for examples of von Neumann dimension in his dissertation [\textit{L. C. Ruth}, Two new settings for examples of von Neumann dimension, Preprint, \url{arxiv:1811.11749}] in the case \(G\) is the projective linear group \(\mathrm{PGL}(2, F)\) with \(F\) is a nonarchimedean local field of characteristic 0 and residue field of order not divisible by 2. In this paper, the author introduce a method to calculate the product of the covolume of a torsion-free lattice in \(\mathrm{PGL}(2, F)\) and the formal dimension of a discrete series representation of \(\mathrm{GL}(2, F)\). Let \(F_n\) be a free group contained as a lattice in \(\mathrm{PGL}(2, F)\) and \(RF_n\) is the closure with respect to the strong operator topology of the right regular representation of \(F_n\) on the Hilbert space \(l^2 (F_n)\). The main result of the paper is Theorem 1.1 in which give the list of the von Neumann dimension \(\dim _{RF_n}\mathcal{H}\) for any discrete series representation \((\pi, \mathcal{H})\) of the \(II_1\) factor \(RF _n\) in each \(\mathcal{H}\). This list extends the one of von Neumann dimensions in the setting of \(\mathrm{PGL}(2, F)\) of the author's dissertation [loc. cit.].
Reviewer: Le Anh Vu (Ho Chi Minh City)Endomorphism algebras and Hecke algebras for reductive \(p\)-adic groupshttps://zbmath.org/1517.220102023-09-22T14:21:46.120933Z"Solleveld, Maarten"https://zbmath.org/authors/?q=ai:solleveld.maartenAccording to Harish-Chandra's philosophy of cusp forms, the representation theory of a connected reductive \(p\)-adic group \(G\) is built via parabolic induction from the supercuspidal representations of its Levi subgroups. Moreover, according to a celebrated result of Bernstein, this manifests on the level of categories as the decomposition
\[
\mathrm{Rep}(G)=\prod_{\mathfrak{s}}\mathrm{Rep}(G)^\mathfrak{s},\tag{1}
\]
where the blocks \(\mathrm{Rep}(G)^{\mathfrak{s}}\) are the categories of admissible representations of \(G\) with supercuspidal support \(\mathfrak{s}=[M,\sigma]\), for \(M\) a Levi subgroup of \(G\) and \(\sigma\) a supercuspidal representation of \(M\). While the decomposition (1) is classical, it has been an ongoing effort to obtain a general and computationally-tractable understanding of the blocks themselves, which is important to understanding the local Langlands correspondence. The paper under review is one part of the recent flurry of activity in understanding individual blocks.
The entire category \(\mathrm{Rep}(G)\) is equivalent to the category of nondegenerate modules over the Hecke algebra of \(G\), but individual blocks can be equivalent to categories of modules over much smaller and more tractable algebras. Most famously, if \(\mathfrak{s}=[T, \mathrm{triv}]\) where \(T\) is a maximal torus of \(G\), then \(\mathrm{Rep}(G)^{\mathfrak{s}}\) is the category of representations with nonzero vector fixed by the Iwahori subgroup \(I\) of \(G\), and is equivalent to the category of finite-dimensional modules over the Iwahori-Hecke algebra \(C_c^\infty(I\backslash G/I)\) of \(G\), an example of an affine Hecke algebra. These algebras have a very convenient and explicit presentation by generators and relations. As we will partially recall below, there is reason to hope that affine Hecke algebras are in fact almost the entire story for all blocks. The present paper is a major advance in this direction.
There are two main ways via which a block \(\mathrm{Rep}(G)^\mathfrak{s}\) may be equivalent to modules over some algebra:
\begin{itemize}
\item Firstly, one can try to find a type, in the sense of Bushnell-Kutzko, for \(\mathfrak{s}\). This is the data of an open compact subgroup \(K\) of \(G\) and an irreducible finite-dimensional representation \(\lambda\) of \(K\) such that a representation \(\pi\) lies in \(\mathrm{Rep}(G)^{\mathfrak{s}}\) if and only if \(\pi|_K\) contains \(\lambda\). In this case, \(\mathrm{Rep}(G)^\mathfrak{s}\) is equivalent to the category of modules over the algebra \(H(G,K,\lambda)=\mathrm{End}\left(c-\mathrm{Ind}_K^G(\lambda)\right)\) of functions \(G\to\mathrm{End}(\lambda)\) transforming under \(K\times K\) according to \(\lambda\).
Thanks to celebrated work of \textit{J. Fintzen} [Ann. Math. (2) 193, No. 1, 303--346 (2021; Zbl 1492.22013)], \(\mathfrak{s}\)-types are known to exist in very wide generality, but it is only for certain types that one knows a presentation of \(H(G,K,\lambda)\) by generators and relations. Namely, when \((K,\lambda)\) is a depth-zero type in the sense of Moy-Prasad, \textit{L. Morris} [Invent. Math. 114, No. 1, 1--54 (1993; Zbl 0854.22022)] proved that \(H(G,K,\lambda)\) is isomorphic to an extension of an affine Hecke algebra by an abelian group.
\item Alternatively, there is a much more general way for any abelian category \(\mathcal{C}\) to be equivalent to the module category of an algebra: if one has a projective generator \(\Pi\) of \(\mathcal{C}\), then \(\mathcal{C}\) is equivalent to the category of \(\mathrm{End}_\mathcal{C}(\Pi)\)-modules. Such projective generators \(\Pi^{\mathfrak{s}}\) of \(\mathrm{Rep}(G)^\mathfrak{s}\) were given by Bernstein for every \(\mathfrak{s}\) in the early 1990s. For classical groups \(G\) and inner forms of \(\mathrm{GL}_n\), Heiermann showed that \(\mathrm{End}_G(\Pi)\) is isomorphic to an affine Hecke algebra extended by a finite group [\textit{V. Heiermann}, Math. Z. 287, No. 3--4, 1029--1052 (2017; Zbl 1381.22014)].
\end{itemize}
The present paper proves a description of \(\mathrm{End}_G(\Pi)\) similar to that of Heiermann for general \(G\). The ring \(\mathrm{End}_G(\Pi)\) is an algebra over functions \(\mathbb{C}[X_{\mathrm{nr}}(M)]\) on the unramified characters of \(M\). The first result is that, over the generic point of \(\mathbb{C}[X_{\mathrm{nr}}(M)]\), \(\mathrm{End}_G(\Pi)\) is isomorphic to the twisted group algebra of a group built by extension from \(\mathrm{Stab}_{X_{\mathrm{nr}}(M)}(\sigma)\), a Weyl group, and an \(R\)-group (Theorem A). Under an assumption on the cocyle \(\natural\) from Theorem A that holds in many natural cases, the author then extracts a crossed product \(\tilde{\mathcal{H}}\) of an affine Hecke and the twisted group algebra of the \(R\)-group above, obtaining equivalences
\[
\mathrm{End}_G(\Pi^{\mathfrak{s}})-\mathrm{mod}_{\mathrm{finite}~\mathrm{length}}\simeq\tilde{\mathcal{H}}-\mathrm{mod}_{\mathrm{finite}~\mathrm{length}}\simeq\mathrm{Rep}(G)^\mathfrak{s}_{\mathrm{finite}~\mathrm{length}}.\text{2}
\]
Absent the assumption on \(\natural\), a description in terms of a family of graded Hecke algebras is given (Theorem B). As the author points out, this is almost as good as (2) because any affine Hecke algebra appearing in (2) will have unequal parameters in general; the study of such affine Hecke algebras naturally begets their graded versions. Moreover, Theorem B is proven without recourse to any construction of supercuspidals, and holds for an arbitrary reductive \(p\)-adic group; in particular it holds without the tameness hypothesis of [\textit{J. Fintzen}, Ann. Math. (2) 193, No. 1, 303--346 (2021; Zbl 1492.22013)].
These equivalences also interact well with temperedness and square-integrability (Theorem C) and give a matching of irreducibles (Theorem D), which induces a bijection (Theorem E)
\[
\mathrm{Irr}(G)\simeq\coprod_M\left(\mathrm{Irr}_{\mathrm{Cuspidal}}(M)//N_G(M)/M\right)_\natural,\tag{3}
\]
where the sets on the right hand side of (3) are twisted extended quotients by \(N_G(M)/M\), itself an extension of a Weyl group by an \(R\)-group. It is very surprising that such a simple bijection exists.
Projective generators are highly non-unique. In the case of \(\mathrm{Rep}(G)^{\mathfrak{s}}\), another choice of (smaller) projective generator is \(\Pi_1^\mathfrak{s}=\mathrm{Ind}_P^G\left(\mathrm{c-Ind}_{M^1}^M\left(\sigma|_{M^1}\right)\right)\), where \(M^1\subset M\) is generated by all open compact subgroups of \(M\). The inclusion
\[
\mathrm{End}_G(\Pi_1^{\mathfrak{s}})\hookrightarrow\mathrm{End}_G(\Pi^{\mathfrak{s}})
\]
is a Morita equivalence, but considering \(\Pi_1^\mathfrak{s}\) gets one closer to affine Hecke algebras: If \(\Pi_1^\mathfrak{s}|_{M^1}\) decomposes with multiplicity one, then \(\mathrm{End}_G(\Pi_1^{\mathfrak{s}})\) is isomorphic to an extension of an affine Hecke algebra by the twisted group algebra of an \(R\)-group (Theorem F). The multiplicity one hypothesis is of course equivalent to this \(R\)-group being abelian, and the author shows in an example that this does not always hold.
However, if the assumption does hold and \(\mathfrak{s}\) has a depth-zero type in the sense of Moy-Prasad, then it makes sense to consider the relationship between the algebra in Theorem F and Morris' results recalled above. Indeed, a major motivation of the present paper is to give a generators-and-relations understanding of blocks, and the depth-zero case is the main example of this coming about via the theory of types. In [\textit{K. Ohara}, ``A comparison of endomorphism algebras'', Preprint, \url{arXiv:2301.09182}] Ohara proves that the Solleveld and Morris endomorphism algebras are actually isomorphic under the multiplicity one hypothesis. (As Morris' extension is by an abelian group, this hypothesis is certainly necessary.) In fact, Ohara shows that the two projective generators are themselves isomorphic.
With Theorems A--F in hand, the author observes that a classification of \(\mathrm{End}_G(\Pi^\mathfrak{s})\)-modules is in principal possible, thanks to the well-developed representation theory of graded Hecke algebras. The latter depends on certain parameter functions, which can be read off from an \(\mathfrak{s}\)-type when one is available. The author defers computing these parameters without using types to [\textit{M. Solleveld}, ``Parameters of Hecke algebras for Bernstein components of \(p\)-adic groups'', Preprint, \url{arXiv:2103.13113}].
Finally, we point out that a series of works of \textit{A.-M. Aubert} et al. [``Affine Hecke algebras for Langlands parameters'', Preprint, \url{arXiv:1701.03593}] constructs certain (twisted) affine and graded Hecke algebras attached to every Bernstein component in the space of enhanced Langlands parameters. Hence, as the author remarks, it could be possible to establish new instances of the local Langlands correspondence by comparing these algebras. Indeed, the present paper has since served as an input to the works [\textit{A.-M. Aubert} et al., ``Affine Hecke algebras for Langlands parameters'', Preprint, \url{arXiv:2202.01305}] and [\textit{A.-M. Aubert} et al., ``The explicit local Langlands correspondence for \(G_2\)'', Preprint, \url{arXiv:2208.12391}] of Aubert-Xu.
Reviewer: Stefan Dawydiak (Bonn)Connections between vector-valued and highest weight Jack and Macdonald polynomialshttps://zbmath.org/1517.330072023-09-22T14:21:46.120933Z"Colmenarejo, Laura"https://zbmath.org/authors/?q=ai:colmenarejo.laura"Dunkl, Charles F."https://zbmath.org/authors/?q=ai:dunkl.charles-f"Luque, Jean-Gabriel"https://zbmath.org/authors/?q=ai:luque.jean-gabrielSummary: We analyze conditions under which a projection from the vector-valued Jack or Macdonald polynomials to scalar polynomials has useful properties, specially commuting with the actions of the symmetric group or Hecke algebra, respectively, and with the Cherednik operators for which these polynomials are eigenfunctions. In the framework of representation theory of the symmetric group and the Hecke algebra, we study the relation between singular nonsymmetric Jack and Macdonald polynomials and highest weight symmetric Jack and Macdonald polynomials. Moreover, we study the quasistaircase partition as a continuation of our study on the conjectures of Bernevig and Haldane on clustering properties of symmetric Jack polynomials.On certain inverse semigroups associated with one-sided topological Markov shiftshttps://zbmath.org/1517.370132023-09-22T14:21:46.120933Z"Matsumoto, Kengo"https://zbmath.org/authors/?q=ai:matsumoto.kengoSummary: We will introduce an inverse semigroup written \({\mathcal{S}}_A\) associated with a one-sided topological Markov shift \((X_A,\sigma_A)\) for an irreducible matrix \(A\) with entries in \(\{0,1\}\). We will show that two inverse semigroups \({\mathcal{S}}_A\) and \({\mathcal{S}}_B\) are isomorphic if and only if the one-sided topological Markov shifts \((X_A,\sigma_A)\) and \((X_B, \sigma_B)\) are continuous orbit equivalent. As a result, the isomorphism class of the inverse semigroup \({\mathcal{S}}_A\) is described in terms of an étale groupoid \(G_A\), Cuntz-Krieger algebra \({\mathcal{O}}_A\), and so on.Force recurrence of semigroup actionshttps://zbmath.org/1517.370222023-09-22T14:21:46.120933Z"Yan, Kesong"https://zbmath.org/authors/?q=ai:yan.kesong"Zeng, Fanping"https://zbmath.org/authors/?q=ai:zeng.fanping"Tian, Rong"https://zbmath.org/authors/?q=ai:tian.rongSummary: We investigate the sets of countable discrete semigroups that force recurrence, that is, the recurrent properties of a point along a subset of a countable semigroup action. We show that a subset of a monoid forces recurrence (resp., forces minimality) if and only if it contains a broken \textit{IP}-set (resp., broken syndetic set), and forces infinite recurrence implies it is contains a broken infinite \textit{IP}-sets. As an example, we show that every subset with positive upper Banach density of infinite countable amenable groups forces infinite recurrence.Measure rigidity of Anosov flows via the factorization methodhttps://zbmath.org/1517.370422023-09-22T14:21:46.120933Z"Katz, Asaf"https://zbmath.org/authors/?q=ai:katz.asafSummary: Using the factorization method of \textit{A. Eskin} and \textit{M. Mirzakhani} [Publ. Math., Inst. Hautes Étud. Sci. 127, 95--324 (2018; Zbl 1478.37002)], we show that generalized \(u\)-Gibbs states over quantitatively non-integrable partially hyperbolic systems have absolutely continuous disintegrations on unstable manifolds. As an application, we show a pointwise equidistribution theorem analogous to the equidistribution results of \textit{D. Kleinbock} et al. [Math. Ann. 367, No. 1--2, 857--879 (2017; Zbl 1417.37056)] and \textit{A. Eskin} and \textit{J. Chaika} [J. Mod. Dyn. 9, 1--23 (2015; Zbl 1358.37008)].Isolations of geodesic planes in the frame bundle of a hyperbolic 3-manifoldhttps://zbmath.org/1517.370432023-09-22T14:21:46.120933Z"Mohammadi, Amir"https://zbmath.org/authors/?q=ai:mohammadi.amir-amjad|mohammadi.amir|mohammadi.amir-hossein-mousavi"Oh, Hee"https://zbmath.org/authors/?q=ai:oh.hee-keun|oh.heeSummary: We present a quantitative isolation property of the lifts of properly immersed geodesic planes in the frame bundle of a geometrically finite hyperbolic \(3\)-manifold. Our estimates are polynomials in the tight areas and Bowen-Margulis-Sullivan densities of geodesic planes, with degree given by the modified critical exponents.Infinite quantum permutationshttps://zbmath.org/1517.460522023-09-22T14:21:46.120933Z"Voigt, Christian"https://zbmath.org/authors/?q=ai:voigt.christianSummary: We define and study quantum permutations of infinite sets. This leads to discrete quantum groups which can be viewed as infinite variants of the quantum permutation groups introduced by \textit{S.-Z. Wang} [Commun. Math. Phys. 195, No.~1, 195--211 (1998; Zbl 1013.17008)]. More precisely, the resulting quantum groups encode universal quantum symmetries of the underlying sets among all discrete quantum groups. We also discuss quantum automorphisms of infinite graphs, including some examples and open problems regarding both the existence and non-existence of quantum symmetries in this setting.On the rank 5 polytopes of the Higman-Sims simple grouphttps://zbmath.org/1517.520072023-09-22T14:21:46.120933Z"Kelsey, Veronica"https://zbmath.org/authors/?q=ai:kelsey.veronica"Nicolaides, Robert"https://zbmath.org/authors/?q=ai:nicolaides.robert"Rowley, Peter"https://zbmath.org/authors/?q=ai:rowley.peter-jIn a paper by \textit{M. I. Hartley} and \textit{A. Hulpke} [Contrib. Discrete Math. 5, No. 2, 106--118 (2010; Zbl 1320.51021)] a computer-led census revealed that the Higman-Sims simple group \(\mathsf{HS}\) is the automorphism group of 520 abstract regular polytopes with the highest rank being 5. Just four of these have rank 5 and they form two dual pairs. In [\textit{D. G. Higman} and \textit{C. C. Sims}, Math. Z. 105, 110--113 (1968; Zbl 0186.04002)] \(\mathsf{HS}\) was constructed as an index 2 subgroup of the automorphism group of a certain strongly regular graph with 100 vertices, now called the Higman-Sims graph.
In the paper under review, the authors describe the rank 5 polytopes of \(\mathsf{HS}\) using the Higman-Sims graph and the decomposition of this graph into five double covers of the Petersen graph.
Reviewer: Enrico Jabara (Venezia)Notices of the international congress of Chinese mathematicians, Vol. 10, No. 2 (December 2022)https://zbmath.org/1517.530032023-09-22T14:21:46.120933ZPublisher's description: This is the twentieth issue (Vol. 10, No. 2, December 2022) of the Notices of the International Consortium of Chinese Mathematicians, the organization's official periodical.
Formerly entitled Notices of the International Congress of Chinese Mathematicians, this journal brings research, news, and the presentation of various perspectives relevant to Chinese mathematics development and education.
Readers of the Notices will find research papers on various topics by prominent experts from around the world, interesting and timely articles on current applications and trends, biographical and historical essays, profiles of important institutions of research and learning, and more.
The articles of this volume were reviewed individually within the journal [ICCM Not. 10, No. 2 (2022)].The first-countability in the quotient spaces of topological gyrogroupshttps://zbmath.org/1517.540112023-09-22T14:21:46.120933Z"Liu, Limin"https://zbmath.org/authors/?q=ai:liu.limin"Zhang, Jing"https://zbmath.org/authors/?q=ai:zhang.jing.5The authors prove that if \(H\) is a closed strong subgyrogroup of a strongly topological gyrogroup \(G\) and \(H\) is neutral, then the following are true: (1) \(G/H\) is completely regular; (2) \(G/H\) is metrizable if and only if it is first-countable; (3) \(G/H\) is submetrizable if and only if it has countable pseudocharacter; (4) \(G/H\) is metrizable if and only if it is a bisequential space. Finally, when \(H\) is a compact L-subgyrogroup of a topological gyrogroup \(G\), each compact subspace of \(G/H\) is first-countable if and only if every compact subspace of \(G/H\) is metrizable.
Reviewer: Watchareepan Atiponrat (Chiang Mai)The bridge number of arborescent links with many twigshttps://zbmath.org/1517.570022023-09-22T14:21:46.120933Z"Baader, Sebastian"https://zbmath.org/authors/?q=ai:baader.sebastian"Blair, Ryan"https://zbmath.org/authors/?q=ai:blair.ryan-c"Kjuchukova, Alexandra"https://zbmath.org/authors/?q=ai:kjuchukova.alexandra"Misev, Filip"https://zbmath.org/authors/?q=ai:misev.filipNote: This paper is written succinctly, without detailed definitions or explanations. A reader who is not already familiar with the topic will need to refer to other sources, such as those referenced in the text.
An arborescent link can be described (non-uniquely) by a planar tree with an integer weighting on each vertex. The links of particular interest in this article are those where this tree `has many twigs' -- that is, where every vertex with valence higher than \(1\) has at least three adjacent vertices with valence \(1\).
The main result presented is that the meridional rank conjecture holds for links arising from trees of this form where all vertex weights have absolute value at least \(2\). That is, the meridional rank, which is a lower bound for the bridge number, is in fact equal to the bridge number. The proof works by establishing inequality relationships between several different values, giving a circle of inequalities for the links in question.
One of the values considered is the maximal number of link components among arborescent links with the same underlying tree (i.e. by changing the vertex weightings). Another is the `flattening number', which relates to the location of vertices of valence at least \(3\) in the tree. Other ideas used are Coxeter quotients, and the Wirtinger number, which has a combinatorial description in terms of link diagrams, giving a bound on the bridge number.
Reviewer: Jessica Banks (Liverpool)New families of hyperbolic twisted torus knots with generalized torsionhttps://zbmath.org/1517.570062023-09-22T14:21:46.120933Z"Himeno, Keisuke"https://zbmath.org/authors/?q=ai:himeno.keisuke"Teragaito, Masakazu"https://zbmath.org/authors/?q=ai:teragaito.masakazuThe existence of an element of finite order precludes a group from being bi-ordered, but so does the existence of a set of conjugates of a non-trivial element whose product in some order is the identity. Finding such a product in the groups of various classes of hyperbolic twisted torus knots is the point of this work. Such an element is called a generalized torsion element. (It is known that all knot groups can be right-ordered.)
A twisted torus knot is, roughly speaking, a torus knot \(T(p, q)\) on which \(r\) adjacent strands have been cut, given \(s\) full right-handed twists and then reattached. The quadruple \(T(p, q; r, s)\) denotes such a twisted torus knot. Here \(p > q> 1\), \(r < q\) and \(s \geq 1\). The new families of hyperbolic twisted torus knots which the authors find whose groups contain generalized torsion elements are the following:
1. \(T(5m+2, 5; 3, 1)\) for \(m\geq 1\),
2. \(T(5m-2, 5; 3, 1)\) for \(m \geq 2\),
3. \(T(5m+2, 5; 4, 1)\) for \(m \geq 1\),
4. \(T(5m-2, 5; 4, 1)\) for \(m \geq 2\),
and for \(r\geq 2\),
1. \(T(r+2, r+1; r, s)\) for \(s \geq 2\),
2. \(T(m(r+1)+1, r+1; r, 2)\) for \(m \geq 2\),
3. \(T(2r+1, r+1; r, s)\) for \(s \geq 1\),
4. \(T(m(r+1)-1, r+1; r, 1)\) for \(m \geq 3\),
(Thus in these last \(4\) cases \(r\) can be arbitrarily large.)
As a consequence, none of these \(8\) families of knots have a group which can be bi-ordered.
The authors achieve their results by careful selection of certain presentations of the knot groups in question, and application of the following general lemma:
Lemma 3.2. Let \(G\) be a group generated by two elements \(x\) and \(y\), and let \(w\) consist of \(x^{\pm 1}\) and \(y\). Suppose that \(G\) is not abelian. If \([x, w] = 1\), then the commutator \([x, y]\) is a generalized torsion element in \(G\).
Reviewer: Lee P. Neuwirth (Princeton)The torsion generating set of the extended mapping class groups in low genus caseshttps://zbmath.org/1517.570112023-09-22T14:21:46.120933Z"Du, Xiaoming"https://zbmath.org/authors/?q=ai:du.xiaomingSummary: We prove that for genus \(g=3,4\), the extended mapping class group \(\text{Mod}^{\pm}(S_g)\) can be generated by two elements of finite orders. But for \(g=1\), \(\text{Mod}^{\pm}(S_1)\) cannot be generated by two elements of finite orders.Fuglede-Kadison determinants over free groups and Lehmer's constantshttps://zbmath.org/1517.570142023-09-22T14:21:46.120933Z"Ben Aribi, Fathi"https://zbmath.org/authors/?q=ai:ben-aribi.fathiSummary: Lehmer's famous problem asks whether the set of Mahler measures of polynomials with integer coefficients admits a gap at \(1\). In [J. Topol. Anal. 14, No. 4, 901--932 (2022; Zbl 07635915)], \textit{W. Lück} extended this question to Fuglede-Kadison determinants of a general group, and he defined the Lehmer's constants of the group to measure such a gap.
In this paper, we compute new values for Fuglede-Kadison determinants over non-cyclic free groups, which yields the new upper bound \(\frac{2}{\sqrt{3}}\) for Lehmer's constants of all torsion-free groups which have non-cyclic free subgroups.
Our proofs use relations between Fuglede-Kadison determinants and random walks on Cayley graphs, as well as works of \textit{L. Bartholdi} [Enseign. Math. (2) 45, No. 1--2, 83--131 (1999; Zbl 0961.05032)] and \textit{O. T. Dasbach} and \textit{M. N. Lalin} [Forum Math. 21, No. 4, 621--637 (2009; Zbl 1225.11131)].
Furthermore, via the gluing formula for \(L^2\)-torsions, we show that the Lehmer's constants of an infinite number of fundamental groups of hyperbolic 3-manifolds are bounded above by even smaller values than \(\frac{2}{\sqrt{3}}\).A group theoretic description of the \(\kappa\)-Poincaré Hopf algebrahttps://zbmath.org/1517.810632023-09-22T14:21:46.120933Z"Arzano, Michele"https://zbmath.org/authors/?q=ai:arzano.michele"Kowalski-Glikman, Jerzy"https://zbmath.org/authors/?q=ai:kowalski-glikman.jerzySummary: It is well known in the literature that the momentum space associated to the \(\kappa\)-Poincaré algebra is described by the Lie group \(\mathsf{AN}(3)\). In this letter we show that the full \(\kappa\)-Poincaré Hopf algebra structure can be obtained from rather straightforward group-theoretic manipulations starting from the Iwasawa decomposition of the \(\mathsf{SO}(1, 4)\) group.Twisted composition algebras and Arthur packets for triality Spin\(_8\)https://zbmath.org/1517.810692023-09-22T14:21:46.120933Z"Gan, Wee Teck"https://zbmath.org/authors/?q=ai:gan.wee-teck"Savin, Gordan"https://zbmath.org/authors/?q=ai:savin.gordanSummary: The purpose of this paper is to construct and analyze certain square-integrable automorphic forms on the quasi-split simply-connected groups \(\mathrm{Spin}_8\) of type \(D_4\) over a number field \(F\). Since the outer automorphism group of \(\mathrm{Spin}_8\) is \(S_3\), these quasi-split groups are parametrised by étale cubic \(F\)-algebras \(E\) and we denote them by \(\mathrm{Spin}^E_8\) (to indicate the dependence on \(E)\). We shall specialize to the case when \(E\) is a cubic field: this gives the so-called triality \(\mathrm{Spin}_8\).Noncommutative tensor triangular geometryhttps://zbmath.org/1517.810712023-09-22T14:21:46.120933Z"Nakano, Daniel K."https://zbmath.org/authors/?q=ai:nakano.daniel-k"Vashaw, Kent B."https://zbmath.org/authors/?q=ai:vashaw.kent-b"Yakimov, Milen T."https://zbmath.org/authors/?q=ai:yakimov.milen-tSummary: We develop a general noncommutative version of s tensor triangular geometry that is applicable to arbitrary monoidal triangulated categories M\(\Delta\)Cs). Insight from noncommutative ring theory is used to obtain a framework for prime, semiprime, and completely prime (thick) ideals of an M\(\Delta\)C, \textbf{K}, and then to associate to \textbf{K} a topological space-the Balmer spectrum \(\operatorname{Spc} \mathbf{K}\). We develop a general framework for (noncommutative) support data, coming in three different flavors, and show that \(\operatorname{Spc} \mathbf{K}\) is a universal terminal object for the first two notions (support and weak support). The first two types of support data are then used in a theorem that gives a method for the explicit classification of the thick (two-sided) ideals and the Balmer spectrum of an M\(\Delta\)C. The third type (quasi support) is used in another theorem that provides a method for the explicit classification of the thick right ideals of \textbf{K}, which in turn can be applied to classify the thick two-sided ideals and \(\operatorname{Spc} \mathbf{K}\).
As a special case, our approach can be applied to the stable module categories of arbitrary finite dimensional Hopf algebras that are not necessarily cocommutative (or quasitriangular). We illustrate the general theorems with classifications of the Balmer spectra and thick two-sided/right ideals for the stable module categories of all small quantum groups for Borel subalgebras, and classifications of the Balmer spectra and thick two-sided ideals of Hopf algebras studied by \textit{D. Benson} and \textit{S. Witherspoon} [Arch. Math. 102, No. 6, 513--520 (2014; Zbl 1310.16025)].Quantum statistics of identical particleshttps://zbmath.org/1517.820072023-09-22T14:21:46.120933Z"Garrison, J. C."https://zbmath.org/authors/?q=ai:garrison.john-cSummary: The empirical rule that systems of identical particles always obey either Bose or Fermi statistics is customarily imposed on the theory by adding it to the axioms of nonrelativistic quantum mechanics, with the result that other statistical behaviors are excluded a priori. A more general approach is to ask what other many-particle statistics are consistent with the indistinguishability of identical particles. This strategy offers a way to discuss possible violations of the Pauli Exclusion Principle, and it leads to some interesting issues related to preparation of states and a superselection rule arising from invariance under the permutation group.Commutative monoid dualityhttps://zbmath.org/1517.820302023-09-22T14:21:46.120933Z"Latz, Jan Niklas"https://zbmath.org/authors/?q=ai:latz.jan-niklas"Swart, Jan M."https://zbmath.org/authors/?q=ai:swart.jan-mSummary: We introduce two partially overlapping classes of pathwise dualities between interacting particle systems that are based on commutative monoids (semigroups with a neutral element) and semirings, respectively. For interacting particle systems whose local state space has two elements, this approach yields a unified treatment of the well-known additive and cancellative dualities. For local state spaces with three or more elements, we discover several new dualities.Geometrical trinity of unimodular gravityhttps://zbmath.org/1517.830062023-09-22T14:21:46.120933Z"Nakayama, Yu"https://zbmath.org/authors/?q=ai:nakayama.yuSummary: We construct a Weyl transverse diffeomorphism invariant theory of teleparallel gravity by employing the Weyl compensator formalism. The low-energy dynamics has a single spin two gravition without a scalar degree of freedom. By construction, it is equivalent to unimodular gravity (as well as Einstein's general relativity with an adjustable cosmological constant) at the non-linear level. Combined with our earlier construction of a Weyl transverse diffeomorphism invariant theory of symmetric teleparallel gravity, unimodular gravity is represented in three alternative ways.Higher derivative gravity's anti-Newtonian limit and the Caldirola-Kanai oscillatorhttps://zbmath.org/1517.830202023-09-22T14:21:46.120933Z"Niedermaier, M."https://zbmath.org/authors/?q=ai:niedermaier.max-rSummary: Fourth order gravity in \(1+d\) dimensions is investigated in its anti-Newtonian limit where lightcones shrink to lines. The limiting theory is fourth order in time derivatives, still fully diffeomorphism invariant, and retains the original number of physical degrees of freedom. In an unimodular Hamiltonian formulation the dynamics can, however, be reduced to a second order one by regarding the unimodular metric as a composite field. In a gauge where the logarithm of the volume element is `time' a Hamiltonian proper arises, which (for each spatial point) is a matrix generalization of a Caldirola-Kanai oscillator with inverted quartic potential. In an alternative `Wick-flipped' Hamiltonian formulation an analogous reduced system arises with non-inverted quartic potential.Cryptographic multilinear maps using pro-\(p\) groupshttps://zbmath.org/1517.941132023-09-22T14:21:46.120933Z"Kahrobaei, Delaram"https://zbmath.org/authors/?q=ai:kahrobaei.delaram"Stanojkovski, Mima"https://zbmath.org/authors/?q=ai:stanojkovski.mimaSummary: In [\textit{D. Kahrobaei} et al., in: Elementary theory of groups and group rings, and related topics. Proceedings of the conference held at Fairfield University and at the Graduate Center, CUNY, New York, NY, USA, November 1--2, 2018. Berlin: De Gruyter. 127--134 (2020; Zbl 1455.94168)], the authors show how, to any nilpotent group of class \(n\), one can associate a non-interactive key exchange protocol between \(n+1\) users. The multilinear commutator maps associated to nilpotent groups play a key role in this protocol. In the present paper, we explore some alternative platforms, such as pro-\(p\) groups.