Recent zbMATH articles in MSC 20Ehttps://zbmath.org/atom/cc/20E2024-02-28T19:32:02.718555ZWerkzeugAn introduction to \(p\)-modular representations of \(p\)-adic groupshttps://zbmath.org/1527.110402024-02-28T19:32:02.718555Z"Abdellatif, Ramla"https://zbmath.org/authors/?q=ai:abdellatif.ramlaThis paper is based on the lecture notes the author gave in the GABY summer school, Milan, summer 2022. It offers a self-contained introduction to \(p\)-modular representations of \(p\)-adic groups, focusing on the results of \(\mathrm{GL}_2(F)\) and \(\mathrm{SL}_2(F)\), where \(F\) is a \(p\)-adic field. A \(p\)-modular representation of \(G(F)\) for a reductive group \(G\) over \(F\) is an action of \(G(F)\) on a vector space defined over a field of characteristic \(p\), the characteristic of the residue field of \(F\). Such representations appear naturally in number theory, but its study was quite recent and only started in the mid-nineties after the seminal work of \textit{L. Barthel} and \textit{R. Livné} [Duke Math. J. 75, No. 2, 261--292 (1994; Zbl 0826.22019)].
As explained in the paper, the story of \(p\)-modular representations are quite different from the complex representation theory of \(p\)-adic groups. Even for the groups \(\mathrm{GL}_2(F)\) and \(\mathrm{SL}_2(F)\), the classification problem of \(p\)-modular representations is still not complete. In this well-written article, the author tries to explain how much we know towards this problem. In section 2, the author starts from some basic materials like \(p\)-adic groups, general representation theory and Bruhat-Tits buildings. Let \(\mathbf{C}\) be an algebraically closed field of characteristic \(p\). In section 3, parabolically induced representations \(\mathrm{Ind}_B^G(\sigma)\) of \(G\) (where \(G\) is \(\mathrm{GL}_2(F)\) and \(\mathrm{SL}_2(F)\) as usual) over \( \mathbf{C}\) were introduced. In particular, Theorem 3.4 covers the classical result about (ir)reducibility of \(\mathrm{Ind}_B^G(\sigma)\) which was due to Barthel and Ron Livné [loc. cit.; J. Number Theory 55, No. 1, 1--27 (1995; Zbl 0841.11026)] for \(\mathrm{GL}_2\) and to the author for \(\mathrm{SL}_2\) [Bull. Soc. Math. Fr. 142, No. 3, 537--589 (2014; Zbl 1319.11029)]. Section 4 is the technical heart of this paper, where the author introduced certain cokernel representation \(\pi(\sigma,\lambda)\) (Definition 4.7) of Laure Barthel and Ron Livné. These representations play a key role in the classification problem. For example, Theorem 4.8 says that any irreducible admissible representation is a quotient of certain \(\pi(\sigma,\lambda)\) and if \(\lambda\ne 0\), then the quotient of \(\pi(\sigma,\lambda)\) must be non supercuspidal. For \(F=\mathbb{Q}_p\), a classification of irreducible admissible representations of \(G\) is given in section 4.4. Some open questions were discussed in section 5.
This paper is nicely written and contains enough details so that a graduate student who wants to enter this beautiful subject should be able to follow.
Reviewer: Qing Zhang (Wuhan)Mild pro-\(p\) groups and the Koszulity conjectureshttps://zbmath.org/1527.160302024-02-28T19:32:02.718555Z"Mináč, J."https://zbmath.org/authors/?q=ai:minac.jan"Pasini, F. W."https://zbmath.org/authors/?q=ai:pasini.federico-william"Quadrelli, C."https://zbmath.org/authors/?q=ai:quadrelli.claudio"Tân, N. D."https://zbmath.org/authors/?q=ai:nguyen-duy-tan.Summary: Let \(p\) be a prime, and \(\mathbb{F}_p\) the field with \(p\) elements. We prove that if \(G\) is a mild pro-\(p\) group with quadratic \(\mathbb{F}_p\)-cohomology algebra \(H^\bullet (G, \mathbb{F}_p)\), then the algebras \(H^\bullet (G, \mathbb{F}_p)\) and \(\operatorname{gr} \mathbb{F}_p [[G]]\) -- the latter being induced by the quotients of consecutive terms of the \(p\)-Zassenhaus filtration of \(G\) -- are both Koszul, and they are quadratically dual to each other. Consequently, if the maximal pro-\(p\) Galois group of a field is mild, then Positselski's and Weigel's Koszulity conjectures hold true for such a field.On products of groups and indices not divisible by a given primehttps://zbmath.org/1527.200302024-02-28T19:32:02.718555Z"Felipe, María José"https://zbmath.org/authors/?q=ai:felipe.maria-jose"Kazarin, Lev S."https://zbmath.org/authors/?q=ai:kazarin.lev-sergeevich"Martínez-Pastor, Ana"https://zbmath.org/authors/?q=ai:martinez-pastor.a"Sotomayor, Víctor"https://zbmath.org/authors/?q=ai:sotomayor.victorSummary: Let the group \(G = AB\) be the product of subgroups \(A\) and \(B\), and let \(p\) be a prime. We prove that \(p\) does not divide the conjugacy class size (index) of each \(p\)-regular element of prime power order \(x \in A \cup B\) if and only if \(G\) is \(p\)-decomposable, i.e. \(G = O_p(G) \times O_{p'}(G)\).Subgroups with a small sum of element ordershttps://zbmath.org/1527.200322024-02-28T19:32:02.718555Z"Lazorec, Mihai-Silviu"https://zbmath.org/authors/?q=ai:lazorec.mihai-silviuA finite group \(G\) is said to be a \(\mathcal {B}_\psi\)-group if \(\psi(H ) < |G|\) for any proper subgroup \(H\) of \(G\), where \(\psi : H \in \mathrm{L}(G) \mapsto \psi(H) \in \mathbb{N}\) is a function from the lattice of subgroups \(\mathrm{L}(G) \) of \(G\) to the natural numbers \(\mathbb{N}\) and \(\psi(H)\) denotes the sum of element orders of \(H\).
Of course, if \(o(x)\) denotes the order of an element \(x \in G\), the definition shows that
\[
\psi: G \in \mathrm{L}(G) \mapsto \psi(G) = \sum_{x \in G} o(x) \in \mathbb{N}
\]
and so \(\mathcal {B}_\psi\)-groups are finite groups, possessing a uniform upper bound for the sum of element orders of the subgroups.
It turns out that \(\mathcal {B}_\psi\)-groups have an interesting structure. For instance:
\begin{itemize}
\item[1.] Abelian \(\mathcal {B}_\psi\)-groups either are elementary abelian \(p\)-groups (\(p\) prime) or cyclic \(p\)-groups of order \(p^2\), see Theorem 1.2.
\item[2.] Nilpotent \(\mathcal {B}_\psi\)-groups either have exponent \(p\) or again are cyclic \(p\)-groups of order \(p^2\), see Theorem 2.5.
\end{itemize}
Among minimal non-nilpotent groups, studied originally by \textit{L. Rédei} [Publ. Math. Debr. 4, 303--324 (1956; Zbl 0075.24003)] and also by \textit{R. Schmidt} [Subgroup lattices of groups. Berlin: Walter de Gruyter (1994; Zbl 0843.20003)], \(\mathcal {B}_\psi\)-groups can be characterized very well as per Theorem 2.7. Finally, there are open questions in order to explore similar behaviours for different classes of groups, see Questions 2, 4 and 5.
Reviewer: Francesco G. Russo (Rondebosch)Weak potency and cyclic subgroup separability of certain free products and tree productshttps://zbmath.org/1527.200342024-02-28T19:32:02.718555Z"Asri, Muhammad Sufi Mohd"https://zbmath.org/authors/?q=ai:asri.muhammad-sufi-mohd"Othman, Wan Ainun Mior"https://zbmath.org/authors/?q=ai:othman.wan-ainun-mior"Wong, Kok Bin"https://zbmath.org/authors/?q=ai:wong.kok-bin"Wong, Peng Choon"https://zbmath.org/authors/?q=ai:wong.peng-choonA group \(G\) is called weakly potent if for any element \(x \in G\) of infinite order there is a positive integer \(r\) with the property that for each positive integer \(n\), there exists a normal subgroup \(M_{n}\) of finite index in \(G\) such that \(xM_{n} \in G/M_{n}\) has order \(rn\).
The main result proved in this paper is Theorem 4.7: Let \(G = A\ast_{H} B\), where \(A\) and \(B\) are finite extensions of finitely generated torsion-free nilpotent groups and \(H\) is a finitely generated normal subgroup of \(A\) and \(B\). Then \(G\) is weakly potent.
Reviewer: Egle Bettio (Venezia)Train track maps on graphs of groupshttps://zbmath.org/1527.200352024-02-28T19:32:02.718555Z"Lyman, Rylee Alanza"https://zbmath.org/authors/?q=ai:lyman.rylee-alanzaLet \(G\) be a connected graph. A homotopy equivalence \(f:G\rightarrow G\) is a train track map if \(f\) maps vertices to vertices and if the restriction of any multiple of \(f\) to an edge of \(G\) produces an immersion. This train track map notion was introduced by \textit{M. Bestvina} and \textit{M. Handel} [Ann. Math. (2) 135, No. 1, 1--51 (1992; Zbl 0757.57004)] and is mainly used for studying the outer automorphisms of free groups.
In the paper, the author develops train track map theory on graphs of groups. Following from a definition by \textit{H. Bass} [J. Pure Appl. Algebra 89, No. 1--2, 3--47 (1993; Zbl 0805.57001)], the map for graphs of groups and the homotopy equivalence notions are defined. It is proved that under one of two technical assumptions, any homotopy equivalence of a graph of groups can be represented by a relative train track map. The first of these can be used for graphs of groups with finite edge groups while the other one can be applied to certain generalized Baumslag-Solitar groups which are examples of two-generator, one- relator groups.
Reviewer: Müge Saadetoğlu (Gazimağusa)Spectra of groupshttps://zbmath.org/1527.200362024-02-28T19:32:02.718555Z"Facchini, Alberto"https://zbmath.org/authors/?q=ai:facchini.alberto"de Giovanni, Francesco"https://zbmath.org/authors/?q=ai:de-giovanni.francesco"Trombetti, Marco"https://zbmath.org/authors/?q=ai:trombetti.marcoThe first author et al. [Algebra Univers. 83, No. 1, Paper No. 8, 38 p. (2022; Zbl 1485.18007)] proved that it is possible to associate a Zariski spectrum with any complete multiplicative lattice. In this paper, the authors associate with any group \(G\) the Zariski spectrum \(\mathrm{Spec}(G)\) of the multiplicative lattice \(\mathcal{N}(G)\) of all normal subgroups of \(G\). The multiplication in this lattice is given by the commutator operation, that is, the product of two normal subgroups \(A\) and \(B\) of \(G\) is the normal subgroup \([A,B]\). The points of the spectrum \(\mathrm{Spec}(G)\) of G are the prime subgroups of \(G\), i.e. the prime elements of the multiplicative lattice \(\mathcal{N}(G)\).
The aim of the paper under review is to investigate the behaviour of prime and semiprime subgroups of groups and their relation with the existence of abelian normal subgroups. In particular, the authors study the set \(\mathrm{Spec}(G)\) of all prime subgroups of a group \(G\) endowed with the Zariski topology and among other examples, they construct an infinite group whose proper normal subgroups are prime and form a descending chain of type \(\omega+1\).
Reviewer: Egle Bettio (Venezia)On the cohomology of pro-fusion systemshttps://zbmath.org/1527.200372024-02-28T19:32:02.718555Z"Díaz Ramos, Antonio"https://zbmath.org/authors/?q=ai:diaz-ramos.antonio"Ocaña, Oihana Garaialde"https://zbmath.org/authors/?q=ai:garaialde-ocana.oihana"Mazza, Nadia"https://zbmath.org/authors/?q=ai:mazza.nadia"Park, Sejong"https://zbmath.org/authors/?q=ai:park.sejongLet \(p\) be a prime number. Fusion systems for finite groups and compact Lie groups have been successfully defined as algebraic models for their \(p\)-completed classifying spaces. For profinite groups, fusion was first studied by \textit{A. L. Gilotti} et al. [Ann. Mat. Pura Appl., IV. Ser. 177, 349--362 (1999; Zbl 0960.20015)]. More recently, fusion systems have been defined over pro-\(p\) groups and are termed pro-fusion systems (see [\textit{R. Stancu} and \textit{P. Symonds}, J. Lond. Math. Soc., II. Ser. 89, No. 2, 461--481 (2014; Zbl 1329.20037)]).
In this paper, the authors prove the Cartan-Eilenberg stable elements theorem for pro-fusion systems. Furthermore, they construct a Lyndon-Hochschild-Serre-type spectral sequence for such systems. As an application, they determine the continuous mod-\(p\) cohomology ring of \(\mathrm{GL}_{2}(\mathbb{Z}_{p})\) for any odd prime \(p\).
Reviewer: Enrico Jabara (Venezia)Amenability and profinite completions of finitely generated groupshttps://zbmath.org/1527.200382024-02-28T19:32:02.718555Z"Kionke, Steffen"https://zbmath.org/authors/?q=ai:kionke.steffen"Schesler, Eduard"https://zbmath.org/authors/?q=ai:schesler.eduardA residually finite group \(G\) embeds into its profinite completion \(\widehat{G}\) and it is natural to wonder what properties of \(G\) can be detected from \(\widehat{G}\). In particular, \textit{A. Grothendieck} [Manuscr. Math. 2, 375--396 (1970; Zbl 0239.20065)] posed the following question: Is an embedding \(\iota: H \rightarrow G\) of finitely presented, residually finite groups an isomorphism if it induces an isomorphism \(\widehat{\iota}: \widehat{H} \rightarrow \widehat{G}\) of profinite completions? The answer is negative. Finitely generated counterexamples were constructed by \textit{V. P. Platonov} and \textit{O. I. Tavgen'} [Sov. Math., Dokl. 33, 822--825 (1986; Zbl 0614.20016)] and finitely presented counterexamples was settled later by \textit{M. R. Bridson} and \textit{F. J. Grunewald} [Ann. Math. (2) 160, No. 1, 359--373 (2004; Zbl 1083.20023)].
In the paper under review, the authors construct a finitely generated, residually finite, amenable group \(A\) and an uncountable family of finitely generated, residually finite nonamenable groups, all of which are profinitely isomorphic to \(A\). All of these groups are branch groups. Moreover, picking up Grothendieck's problem, the group \(A\) embeds in these groups such that the inclusion induces an isomorphism of profinite completions.
In addition, the authors review the concept of uniform amenability (as defined in [\textit{G. Keller}, Ill. J. Math. 16, 257--269 (1972; Zbl 0231.22006)]) and they prove the interesting result that uniform amenability is indeed detectable from the profinite completion.
Reviewer: Enrico Jabara (Venezia)The Bernstein centre in natural characteristichttps://zbmath.org/1527.200392024-02-28T19:32:02.718555Z"Ardakov, Konstantin"https://zbmath.org/authors/?q=ai:ardakov.konstantin"Schneider, Peter"https://zbmath.org/authors/?q=ai:schneider.peter-r|schneider.peter.5|schneider.peter.1|schneider.peter.4|schneider.peter.2|schneider.peter.3|schneider.peter.6Summary: Let \(G\) be a locally profinite group and let \(k\) be a field of positive characteristic \(p\). Let \(Z(G)\) denote the centre of \(G\) and let \(\mathfrak Z(G)\) denote the Bernstein centre of \(G\), that is, the \(k\)-algebra of natural endomorphisms of the identity functor on the category of smooth \(k\)-linear representations of \(G\). We show that if \(G\) contains an open pro-\(p\) subgroup but no proper open centralisers, then there is a natural isomorphism of \(k\)-algebras \(\mathfrak{Z}(Z(G)) \xrightarrow{\cong} \mathfrak{Z}(G)\). We also describe \(\mathfrak{Z}(Z(G))\) explicitly as a particular completion of the abstract group ring \(k[Z(G)]\). Both conditions on \(G\) are satisfied whenever \(G\) is the group of points of any connected smooth algebraic group defined over a local field of residue characteristic \(p\). In particular, when the algebraic group is semisimple, we show that \(\mathfrak{Z}(G) = k[Z(G)]\).Edifices: building-like spaces associated to linear algebraic groupshttps://zbmath.org/1527.200402024-02-28T19:32:02.718555Z"Bate, Michael"https://zbmath.org/authors/?q=ai:bate.michael"Martin, Benjamin"https://zbmath.org/authors/?q=ai:martin.benjamin-m-s"Röhrle, Gerhard"https://zbmath.org/authors/?q=ai:rohrle.gerhard-eSummary: Given a semisimple linear algebraic \(k\)-group \(G\), one has a spherical building \(\Delta_G\), and one can interpret the geometric realisation \(\Delta_G(\mathbb{R})\) of \(\Delta_G\) in terms of cocharacters of \(G\). The aim of this paper is to extend this construction to the case when \(G\) is an arbitrary connected linear algebraic group; we call the resulting object \(\Delta_G(\mathbb{R})\) the spherical edifice of \(G\). We also define an object \(V_G(\mathbb{R})\) which is an analogue of the vector building for a semisimple group; we call \(V_G(\mathbb{R})\) the vector edifice. The notions of linear map and of isomorphism between edifices are introduced; we construct some linear maps arising from natural group-theoretic operations. We also devise a family of metrics on \(V_G(\mathbb{R})\) and show they are all bi-Lipschitz equivalent to each other; with this extra structure, \(V_G(\mathbb{R})\) becomes a complete metric space. Finally, we present some motivation in terms of geometric invariant theory and variations on the Tits centre conjecture.A new axiomatics for masureshttps://zbmath.org/1527.200412024-02-28T19:32:02.718555Z"Hébert, Auguste"https://zbmath.org/authors/?q=ai:hebert.augusteSummary: Masures are generalizations of Bruhat-Tits buildings. They were introduced by Gaussent and Rousseau to study Kac-Moody groups over ultrametric fields that generalize reductive groups. Rousseau gave an axiomatic definition of these spaces. We propose an equivalent axiomatic definition, which is shorter, more practical, and closer to the axiom of Bruhat-Tits buildings. Our main tool to prove the equivalence of the axioms is the study of the convexity properties in masures.The classical Tits quadrangleshttps://zbmath.org/1527.200422024-02-28T19:32:02.718555Z"Mühlherr, Bernhard"https://zbmath.org/authors/?q=ai:muhlherr.bernhard-matthias"Weiss, Richard M."https://zbmath.org/authors/?q=ai:weiss.richard-mMoufang polygons were classified by \textit{J. Tits} and the second author [Moufang polygons. Berlin: Springer (2002; Zbl 1010.20017)]. Tits polygons, as defined by the authors and \textit{H. P. Petersson} [Tits polygons. Providence, RI: American Mathematical Society (2022; Zbl 1498.51001)] generalize Moufang polygons. The authors, in a series of papers classified the dagger-sharp Tits triangles, hexagons and octagons that are \(7\)-plump.
In the paper under review, the authors prove that a genuine Tits quadrangle that is \(5\)-sturdy and laser-sharp but not exceptional is uniquely determined by either a quadratic space over a field or a pseudo-quadratic module over a simple associative ring with involution. This result completes the classification of \(7\)-sturdy laser-sharp Tits polygons.
Reviewer: Egle Bettio (Venezia)On conjugacy classes in groupshttps://zbmath.org/1527.200432024-02-28T19:32:02.718555Z"Herzog, Marcel"https://zbmath.org/authors/?q=ai:herzog.marcel"Longobardi, Patrizia"https://zbmath.org/authors/?q=ai:longobardi.patrizia"Maj, Mercede"https://zbmath.org/authors/?q=ai:maj.mercedeThe authors first define the deficient and non-deficient elements of groups as follows: A non-identity element \(x\) of a group \(G\) is called deficient if \(\langle x \rangle < C_{G}(x)\) and non-deficient if \(\langle x \rangle =C_{G}(x)\). The conjugacy class of a deficient (respectively non-deficient) element is called a deficient (respectively non-deficient) conjugacy class. Then they prove the following consequence of these definitions:
Theorem. Let \(G\) be a non-trivial finite group with no deficient conjugacy classes. Then either \(G\) is either cyclic of prime order or a Frobenius group \(C_{p}\rtimes C_{q}\) with \(p, q\) distinct primes.
Then they also characterize finite groups with one deficient conjugacy class. Moreover, they prove that if \(G\) is a locally finite or (periodic) locally graded or residually finite group with no (or one) deficient conjugacy class, then \(G\) is finite. Since the Tarski monster groups constructed by \textit{A. Yu. Ol'shanskiĭ} [Izv. Akad. Nauk SSSR, Ser. Mat. 43, 1328--1393 (1979; Zbl 0431.20027)] constitute examples of infinite simple periodic groups with no deficient conjugacy glasses, the locally graded or residually finite assumptions in these theorems are crucial.
The results in this paper, together with the authors' earlier results [J. Algebra Appl. 22, No. 10, Article ID 2350217, 7 p. (2023; Zbl 1527.20056)], can be seen as important discussions related to the following problem by Avinoam Mann published in [\textit{V. D. Mazurov} (ed.) and \textit{E. I. Khukhro} (ed.), The Kourovka notebook. Unsolved problems in group theory. 18th edition. Novosibirsk: Institute of Mathematics, Russian Academy of Sciences, Siberian Div. (2014; Zbl 1372.20001), Problem 11.56(a)].
Problem. Does there exist an infinite residually finite group such that all whose centralizers are finite?
Reviewer: Kıvanç Ersoy (Berlin)Corrigendum to: ``Growth of unbounded subsets in nilpotent groups, random mapping statistics, and geometry of group laws''https://zbmath.org/1527.200442024-02-28T19:32:02.718555Z"Greenfeld, Be'eri"https://zbmath.org/authors/?q=ai:greenfeld.beeri"Lavner, Hagai"https://zbmath.org/authors/?q=ai:lavner.hagaiThe authors correct Lemma 6.2 in their recent paper [ibid. 2023, No. 6, 5046--5086 (2023; Zbl 1516.20065)] and correct the constants derived from it in Theorems 1.1, 1.3, 1.4 and Corollary 6.5.
Reviewer: Alexander Ivanovich Budkin (Barnaul)On the transition monoid of the Stallings automaton of a subgroup of a free grouphttps://zbmath.org/1527.200462024-02-28T19:32:02.718555Z"Guimarães, Inês F."https://zbmath.org/authors/?q=ai:guimaraes.ines-fSummary: \textit{J. C. Birget} et al. [Theor. Comput. Sci. 242, No. 1--2, 247--281 (2000; Zbl 0944.68100)] proved that a finitely generated subgroup \(K\) of a free group is pure if and only if the transition monoid \(M(K)\) of its Stallings automaton is aperiodic. In this paper, we establish further connections between algebraic properties of \(K\) and algebraic properties of \(M(K)\). We mainly focus on the cases where \(M(K)\) belongs to the pseudovariety of finite monoids all of whose subgroups lie in a given pseudovariety of finite groups. We also discuss normal, malnormal and cyclonormal subgroups of \(F_A\) using the transition monoid of the corresponding Stallings automaton.Groups with some restrictions on non-Baer subgroupshttps://zbmath.org/1527.200472024-02-28T19:32:02.718555Z"Badis, Abdelhafid"https://zbmath.org/authors/?q=ai:badis.abdelhafid"Trabelsi, Nadir"https://zbmath.org/authors/?q=ai:trabelsi.nadirSummary: It is proved that if \(G\) is an \(\mathfrak{X}\)-group of infinite rank whose proper subgroups of infinite rank are Baer groups, then so are all proper subgroups of \(G\), where \(\mathfrak{X}\) is the class defined by \textit{N. S. Chernikov} [Ukr. Math. J. 42, No. 7, 855--861 (1990; Zbl 0751.20030); translation from Ukr. Mat. Zh. 42, No. 7, 962--970 (1990)] as the closure of the class of periodic locally graded groups by the closure operations \(\boldsymbol{\acute{P}}\), \(\boldsymbol{\grave{P}}\) and \(\boldsymbol{L}\). We prove also that if a locally graded group, which is neither Baer nor Černikov, satisfies the minimal condition on non-Baer subgroups, then it is a Baer-by-Černikov group which is a direct product of a \(p\)-subgroup containing a minimal non-Baer subgroup of infinite rank, by a Černikov nilpotent \(p^{\prime}\)-subgroup, for some prime \(p\). Our last result states that a group is locally graded and has only finitely many conjugacy classes of non-Baer subgroups if, and only if, it is Baer-by-finite and has only finitely many non-Baer subgroups.Groups with iterated restrictions on conjugacy classeshttps://zbmath.org/1527.200482024-02-28T19:32:02.718555Z"Ferrara, Maria"https://zbmath.org/authors/?q=ai:ferrara.maria"Trombetti, Marco"https://zbmath.org/authors/?q=ai:trombetti.marcoAuthors' abstract: Let \(\mathfrak{X}\) be a group class (such as the class \(\mathfrak{F}\) of all finite groups). Starting from \(\mathfrak{X}\), we can define the class \(\mathfrak{X}C\) of all groups \(G\) such that, for any \(g\in G\), the co-centralizer \(G/C_{G}(\langle g \rangle^{G})\) of \(g\) in \(G\) is an \(\mathfrak{X}\)-group; of course, if \(\mathfrak{X}=\mathfrak{F}\), these are the well-known FC-groups. Iterating this request, we define the class \(\mathfrak{X}C^{2}\) of groups whose co-centralizers are \(\mathfrak{X}C\)-groups, and so on. We generically refer to these groups as groups with \(\mathfrak{X}\)-iterated conjugacy classes. Of course, if \(\mathfrak{X}\) is quotient closed, then any group \(G\) such that \(G/\zeta_{k}(G) \in \mathfrak{X}\), for some \(k \geq 0\), has \(\mathfrak{X}\)-iterated conjugacy classes, and actually these concepts are almost always equivalent in the universe of linear groups. For \(\mathfrak{X} =\mathfrak{F}\), this type of restrictions have recently been investigated, and the aim of this paper is to study the general theory of groups with \(\mathfrak{X}\)-iterated conjugacy classes, paying particular attention to the case in which \(\mathfrak{X}\) is the class \(\mathfrak{C}\) of Černikov groups: we extend (and improve) results concerning groups with \(\mathfrak{F}\)-iterated conjugacy classes. The main focus is on Sylow theory, serial subgroups and groups with many proper subgroups having \(\mathfrak{C}\)-iterated conjugacy classes.
Reviewer: Egle Bettio (Venezia)Linearity and nonlinearity of groups of polynomial automorphisms of the planehttps://zbmath.org/1527.200502024-02-28T19:32:02.718555Z"Mathieu, Olivier"https://zbmath.org/authors/?q=ai:mathieu.olivierAuthor's abstract: Given a field \(K\), we investigate which subgroups of the group \(\Aut \mathbb{A}_K^2\) of polynomial automorphisms of the plane are linear or not. The results are contrasted. The group \(\Aut \mathbb{A}_K^2\) itself is nonlinear, except if \(K\) is finite, but it contains some large subgroups, of ``codimension-five'' or more, which are linear. This phenomenon is specific to dimension two: it is easy to prove that any natural ``finite-codimensional'' subgroup of \(\Aut \mathbb{A}_K^3\) is nonlinear, even for a finite field \(K\). When \(\operatorname{ch} K = 0\), we also look at a similar questions for finitely generated subgroups, and the results are again disparate. The group \(\Aut \mathbb{A}_K^2\) has a one-related finitely generated subgroup which is not linear. However, there is a large subgroup, of ``codimension-three'', which is locally linear but not linear.
Reviewer: Francesco G. Russo (Rondebosch)The automorphism groups of Artin groups of edge-separated CLTTF graphshttps://zbmath.org/1527.200512024-02-28T19:32:02.718555Z"An, Byung Hee"https://zbmath.org/authors/?q=ai:an.byung-hee"Cho, Youngjin"https://zbmath.org/authors/?q=ai:cho.youngjinLet \(A_{\Gamma}\) be the Artin group with graph \(\Gamma\). In [Geom. Topol. 9, 1381--1441 (2005; Zbl 1135.20027)], \textit{J. Crisp} considered the \textsf{CLTTF} Artin groups defined by graphs \(\Gamma\) that are connected, have edge labels \(\geq 3\) (Large Type) and are triangle free. The \textsf{CLTTF} Artin groups form a somewhat manageable family of non-rigid Artin groups in studying their automorphism groups.
The paper under review is a continuation of Crisp's work. More precisely, the authors provide an explicit presentation of the automorphism group of an edge-separated \textsf{CLTTF} Artin group \(A_{\Gamma}\) in which \(\Gamma\) has no separating vertices.
Reviewer: Egle Bettio (Venezia)On groups in which every element has a prime power order and which satisfy some boundedness conditionhttps://zbmath.org/1527.200562024-02-28T19:32:02.718555Z"Herzog, Marcel"https://zbmath.org/authors/?q=ai:herzog.marcel"Longobardi, Patrizia"https://zbmath.org/authors/?q=ai:longobardi.patrizia"Maj, Mercede"https://zbmath.org/authors/?q=ai:maj.mercedeIn this paper, the authors prove many interesting results about periodic groups in which every element has prime power order and and for each prime \(p \in \pi(G)\) there is a a natural number \(u_{p}\) such that the order of any \(p\)-element in \(G\) is less than \(p^{u_{p}}\), which the authors call BCP-groups. Groups whose finite \(p\)-subgroups have order less than \(p^{u_{p}}\) with the given \(u_{p}\), they call BSP groups. They prove that finitely generated BCP-groups have only finitely many subgroups of finite index. Furthermore, they prove that locally graded BCP groups are BSP and locally finite BSP groups are finite.
This paper together with [\textit{M. Herzog} et al., J. Algebra 637, 112--131 (2024; Zbl 1527.20043)] brings attention to problems related to periodic groups with conditions on centralizers or conjugacy classes, subject to being locally finite.
Reviewer: Kıvanç Ersoy (Berlin)Random subcomplexes of finite buildings, and fibering of commutator subgroups of right-angled Coxeter groupshttps://zbmath.org/1527.200682024-02-28T19:32:02.718555Z"Schesler, Eduard"https://zbmath.org/authors/?q=ai:schesler.eduard"Zaremsky, Matthew C. B."https://zbmath.org/authors/?q=ai:zaremsky.matthew-curtis-burkholderLet \(\mathcal{P}\) be a group theoretic property. A group \(G\) is said to algebraically \(\mathcal{P}\)-fiber if it admits an epimorphism \(G \rightarrow \mathbb{Z}\) whose kernel has property \(\mathcal{P}\). The question raised by this paper is to find which right-angled Coxeter groups \(\mathrm{F}_{n}\)-fibers or \(\mathrm{FP}_{n}\)-fibers, and which do not.
Let \(G\) be a right-angled Coxeter group with underlying graph \(\mathcal{G}=(V,E)\). In the paper under review, the authors determine a sufficient condition on \(\mathcal{G}\) under which \(G\) algebraically \(\mathrm{F}_{n}\)-fibers. Then they prove that this condition holds with high probability when \(\mathcal{G}\) is the 1-skeleton of a finite building. Furthermore, they use their techniques to present examples where the kernel is of type \(\mathrm{F}_{2}\) but not \(\mathrm{FP}_{3}\) and examples where the right-angled Coxeter group is hyperbolic and the kernel is finitely generated and non-hyperbolic.
Reviewer: Egle Bettio (Venezia)Equations over solvable groupshttps://zbmath.org/1527.200722024-02-28T19:32:02.718555Z"Klyachko, Anton A."https://zbmath.org/authors/?q=ai:klyachko.anton-a"Mikheenko, Mikhail A."https://zbmath.org/authors/?q=ai:mikheenko.mikhail-a"Roman'kov, Vitaly A."https://zbmath.org/authors/?q=ai:romankov.vitaly-aLet \(G\) be a group, \(X\) a set disjoint of \(G\) and let \(F(X)\) be the free group on \(X\). A system of equations \(\{w_{i} = 1 \mid i \in I \}\) with coefficients in \(G\), where \(w_{i}\) are words in the alphabet \(G \sqcup X^{\pm 1}\) is solvable over \(G\), if there exists a group \(\widetilde{G}\) containing \(G\) as a subgroup and a retraction of the free product \(\widetilde{G} \ast F(X)\) onto \(\widetilde{G}\) onto \(\widetilde{G}\) containing all elements \(w_{i}\) in its kernel. If the solution group \(G\) can be chosen from a class \(\mathcal{K}\), then we say that the system is solvable in \(\mathcal{K}\).
A system of equations (possibly infinite and with possibly infinitely many unknowns) over a group is nonsingular if the rows composed of the exponent-sums of unknowns in each equation are linearly independent over \(\mathbb{Q}\). If these rows are linearly independent over \(\mathbb{F}_{p}\) for each prime \(p\), then the system of equations is unimodular. In particular, one equation with one unknown is (a) nonsingular if the exponent sum of the unknown in this equation is nonzero, (b) unimodular if this sum is \(\pm 1\).
Unimodular equations behave better than arbitrary nonsingular ones: the first author proved in [Commun. Algebra 21, No. 7, 2555--2575 (1993; Zbl 0788.20017)] that any unimodular equation over a torsion-free group is solvable over this group. Section 1 of this paper contains examples showing that there exists a unimodular equation with one unknown over a metabelian group (that can be chosen finite or, on the contrary, torsion-free), which is not solvable in any larger metabelian group.
In [\textit{J. Howie}, J. Reine Angew. Math. 324, 165--174 (1981; Zbl 0447.20032)], the following conjecture was proposed: Any nonsingular system of equations over any group is solvable over this group.
In this paper, the authors call a class \(\mathcal{K}\) of groups a Howie class if any nonsingular system of equations over each group from \(\mathcal{K}\) has a solution in a larger group from \(\mathcal{K}\). The quasivariety generated by a Howie class is a Howie class too (see Section 2).
The main result of the paper under review establishes that the class consisting of all extensions of groups from a Howie quasivariety by abelian torsion-free groups is a Howie quasivariety too. Since abelian groups form an Howie quasivariety, the authors obtain as a corollary that for each positive integer \(n\), the class of groups \(G\) admitting a subnormal series \(G= G \triangleright G_{1} \triangleright \dots \triangleright G_{n}=\{1\}\) whose quotients \(G_{i}/G_{i+1}\) are abelian and all, except maybe the last one, are torsion-free is a Howie quasivariety. The authors point out that the minimum length of such a series can be larger than the derived length.
Reviewer: Enrico Jabara (Venezia)Affine Deligne-Lusztig varieties and folded galleries governed by chimneyshttps://zbmath.org/1527.200782024-02-28T19:32:02.718555Z"Milićević, Elizabeth"https://zbmath.org/authors/?q=ai:milicevic.elizabeth-t"Schwer, Petra"https://zbmath.org/authors/?q=ai:schwer.petra-n"Thomas, Anne"https://zbmath.org/authors/?q=ai:thomas.anne|thomas.anne.1Summary: ``We characterize the nonemptiness and dimension problems for an affine Deligne-Lusztig variety \(X_x(b)\) in the affine flag variety in terms of galleries that are positively folded with respect to a chimney. If the parabolic subgroup associated to the Newton point of \(b\) has rank 1, we then prove nonemptiness for a certain class of Iwahori-Weyl group elements \(x\) by explicitly constructing such galleries.''
Reviewer's remarks: It is the intention of the authors to present only those results for which they have a logically independent proof, using the distinct method of positively folded galleries.
More about the setting:
Let \(G\) be a split connected reductive group over \(\mathbb{F}_q\) with \(q\) a prime power and let \(T\) be a split maximal torus of \(G\). Let \(F = \overline{\mathbb{F}}q((t))\) be the discretely valued field with ring of integers \(O = \overline{\mathbb{F}}_q[[t]]\). The choice of \(T\) corresponds to fixing an apartment \(\mathcal{A}\) in the Bruhat-Tits building for \(G(F)\).
The Iwahori subgroup \(I\) of \(G(F)\) is the preimage of the opposite Borel under the projection \(G(O) \to G(\mathbb{F}_q)\) defined by \(t \mapsto 0\).
The Frobenius automorphism \(a \mapsto a^q\) on \(\mathbb{F}_q\) can be extended to \(\sigma : F \to F\) by defining \(\sigma\) to raise each coefficient to the \(q\)th power.
Given \(x\) in the extended affine Weyl group \(\widetilde W\cong X_*(T) \rtimes W\) and \(b\in G(F)\), the \textit{affine Deligne-Lusztig variety} associated to this pair of elements is defined as
\[
X_x(b)=\{\;g\in G(F)/I\;\mid\; g^{-1}b\sigma (g)\in IxI\;\}.
\]
One wants to know when \(X_x(b)\) is non-empty and if it is non-empty, what its dimension is.
The set \(B(G)\) of \(\sigma\)-conjugacy classes in \(G(F)\) is characterized by a pair of invariants: the Newton point and the Kottwitz point.
Every choice of a sub-root system and hence choice of a standard spherical parabolic subgroup \(P\), determines a \(P\)-chimney in \(\mathcal{A}\).
Let \(\Lambda_G\) be the quotient of the coweight lattice \(X_*(T)\) by the coroot lattice.
The folded galleries now consist of ``extended alcoves''. The extended alcoves lie in sheets, indexed locally by \(\Lambda_G\), above ordinary alcoves. There are also signs that determine if a fold is positive with respect to a chimney.
Reviewer: Wilberd van der Kallen (Utrecht)Nilpotent probability of compact groupshttps://zbmath.org/1527.220072024-02-28T19:32:02.718555Z"Abdollahi, Alireza"https://zbmath.org/authors/?q=ai:abdollahi.alireza"Soleimani Malekan, Meisam"https://zbmath.org/authors/?q=ai:soleimani-malekan.meisamLet \(G\) be a compact (Hausdorff) group and put
\[
\mathcal{N}_k=\{(x_1,\ldots,x_{k+1})\in G^{k+1}\mid[x_1,\ldots,x_{k+1}]=1\}.
\]
The Haar measure of \(\mathcal{N}_k\) in \(G^{k+1}\) is denoted by \textbf{np}\(_k(G)\). The paper deals with the case in which \textbf{np}\(_k(G)>0\) or, in other words, with the case in which the probability that \(k+1\) randomly chosen elements \(x_1,\ldots,x_{k+1}\) of \(G\) satisfy \([x_1,\ldots,x_{k+1}]=1\) is strictly greater than \(0\).
The main result of the paper is the following.
{Theorem 1.2}. Let \(G\) be a compact group with \textbf{np}\(_k(G)>0\). Then the connected component \(G^0\) of \(G\) is abelian and there exists a closed normal nilpotent subgroup \(N\) of class at most \(k\) such that \(G^0N\) is open in \(G\).
Reviewer: Mattia Brescia (Napoli)Topological groups with a compact open subgroup, relative hyperbolicity and coherencehttps://zbmath.org/1527.220082024-02-28T19:32:02.718555Z"Arora, Shivam"https://zbmath.org/authors/?q=ai:arora.shivam"Martínez-Pedroza, Eduardo"https://zbmath.org/authors/?q=ai:martinez-pedroza.eduardoIn this article, the pairs \((\mathcal G, H)\) are studied where \(\mathcal G\) is a topological group with a compact open subgroup, and \(\mathcal H\) is a finite collection of open subgroups.
Let \(G\) be a topological group. \(G\) is compactly generated if it admits a compact generating set. It is compactly presented if it admits a standard presentation \(\langle S\mid R\rangle\) where \(S\) is a compact subset of \(G\) and \(R\) is a set of words in \(S\) of uniformly bounded length. A topological group is said to be coherent if every closed compactly generated subgroup is compactly presented. There is also a notion of \(\mathcal G\) being compactly generated and compactly presented relative to \(\mathcal H\). In this case the authors show if \(\mathcal G\) is compactly generated, then each subgroup \(H\in\mathcal H\) is compactly generated and if each subgroup \(H\in\mathcal H\) is compactly presented, then \(G\) is compactly presented.
An approach is presented to relative hyperbolicity for pairs \((\mathcal G, H)\) based on Bowditch's work using discrete actions on hyperbolic fine graphs. For example, it is shown that if \(\mathcal G\) is hyperbolic relative to \(\mathcal H\) then \(\mathcal G\) is compactly presented relative to \(\mathcal H\).
As an application combination results for coherent topological groups with a compact open subgroup are shown, and the McCammond-Wise perimeter method is applied to this general framework.
Reviewer: Stephan Rosebrock (Karlsruhe)A type I conjecture and boundary representations of hyperbolic groupshttps://zbmath.org/1527.220092024-02-28T19:32:02.718555Z"Caprace, Pierre-Emmanuel"https://zbmath.org/authors/?q=ai:caprace.pierre-emmanuel"Kalantar, Mehrdad"https://zbmath.org/authors/?q=ai:kalantar.mehrdad"Monod, Nicolas"https://zbmath.org/authors/?q=ai:monod.nicolasThe main result of this paper is the beautiful Theorem A: Let \(G\) be a hyperbolic locally compact group admitting a uniform lattice. If \(G\) is of type I, then \(G\) has a cocompact amenable subgroup.
Thanks to Theorem D of the first author et al. [J. Eur. Math. Soc. (JEMS) 17, No. 11, 2903--2947 (2015; Zbl 1330.43002)], the above result leads to a rather precise description of \(G\): see Corollary C. Specializing Theorem B and Corollary C, the authors obtain Corollary D: Let \(\mathcal{T}\) be a locally finite tree and \(G \leq \Aut(\mathcal{T})\) be a closed nonamenable subgroup acting minimally on \(\mathcal{T}\). If \(G\) is type \(\mathrm{I}\), then the \(G\)-action on the set of ends \(\partial \mathcal{T}\) is 2-transitive.
Another important result is Theorem E: Any two boundary representations of a nonamenable hyperbolic locally compact group are weakly equivalent. This answers a question of \textit{C. Houdayer} and \textit{S. Raum} [Comment. Math. Helv. 94, No. 1, 185--219 (2019; Zbl 1468.22016)].
The paper under review contains several other interesting results that are too technical to be reported here.
Reviewer: Egle Bettio (Venezia)On absolutely profinitely solitary lattices in higher rank Lie groupshttps://zbmath.org/1527.220182024-02-28T19:32:02.718555Z"Kammeyer, Holger"https://zbmath.org/authors/?q=ai:kammeyer.holgerSummary: We establish conditions under which lattices in certain simple Lie groups are profinitely solitary in the absolute sense, so that the commensurability class of the profinite completion determines the commensurability class of the group among finitely generated residually finite groups. While cocompact lattices are typically not absolutely solitary, we show that noncocompact lattices in \(\mathrm{Sp}(n,\mathbb{R})\), \(G_{2(2)}\), \(E_8(\mathbb{C})\), \(F_4(\mathbb{C})\), and \(G_2(\mathbb{C})\) are absolutely solitary if a well-known conjecture on Grothendieck rigidity is true.On the profinite rigidity of lattices in higher rank Lie groupshttps://zbmath.org/1527.220192024-02-28T19:32:02.718555Z"Kammeyer, Holger"https://zbmath.org/authors/?q=ai:kammeyer.holger"Kionke, Steffen"https://zbmath.org/authors/?q=ai:kionke.steffenSummary: We investigate which higher rank simple Lie groups admit profinitely but not abstractly commensurable lattices. We show that no such examples exist for the complex forms of type \(E_8,F_4\), and \(G_2\). In contrast, there are arbitrarily many such examples in all other higher rank Lie groups, except possibly \(\mathrm{SL}_{2n+1}(\mathbb{R})\), \(\mathrm{SL}_{2n+1}(\mathbb{C})\), \(\mathrm{SL}_n(\mathbb{H})\), or groups of type \(E_6\).Characterization of positive definite, radial functions on free groupshttps://zbmath.org/1527.430032024-02-28T19:32:02.718555Z"Chuah, Chian Yeong"https://zbmath.org/authors/?q=ai:chuah.chian-yeong"Liu, Zhenchuan"https://zbmath.org/authors/?q=ai:liu.zhenchuan"Mei, Tao"https://zbmath.org/authors/?q=ai:mei.taoSummary: This article studies the properties of positive definite, radial functions on free groups following the work of
\textit{U.~Haagerup} and \textit{S.~Knudby} [Proc. Am. Math. Soc. 143, No.~4, 1477--1489 (2015; Zbl 1311.43009)].
We obtain characterizations of radial functions with respect to the \(\ell^2\) length on the free groups with infinite generators and the characterization of the positive definite, radial functions with respect to the \(\ell^p\) length on the free real line with infinite generators for \(0<p\leq 2\). We obtain a Lévy-Khintchine formula for length-radial conditionally negative functions as well.