Recent zbMATH articles in MSC 20Fhttps://zbmath.org/atom/cc/20F2023-05-31T16:32:50.898670ZWerkzeugvon Staudt constructions for skew-linear and multilinear matroidshttps://zbmath.org/1508.050212023-05-31T16:32:50.898670Z"Kühne, Lukas"https://zbmath.org/authors/?q=ai:kuhne.lukas"Pendavingh, Rudi"https://zbmath.org/authors/?q=ai:pendavingh.rudi-a"Yashfe, Geva"https://zbmath.org/authors/?q=ai:yashfe.gevaSummary: This paper compares skew-linear and multilinear matroid representations. These are matroids that are representable over division rings and (roughly speaking) invertible matrices, respectively. The main tool is the von Staudt construction, by which we translate our problems to algebra. After giving an exposition of a simple variant of the von Staudt construction we present the following results:
\begin{itemize}
\item Undecidability of several matroid representation problems over division rings.
\item An example of a matroid with an infinite multilinear characteristic set, but which is not multilinear in characteristic 0.
\item An example of a skew-linear matroid that is not multilinear.
\end{itemize}Coarse geometry of the fire retaining property and group splittingshttps://zbmath.org/1508.051142023-05-31T16:32:50.898670Z"Martínez-Pedroza, Eduardo"https://zbmath.org/authors/?q=ai:martinez-pedroza.eduardo"Prytuła, Tomasz"https://zbmath.org/authors/?q=ai:prytula.tomaszSummary: Given a non-decreasing function \(f:\mathbb{N}\rightarrow \mathbb{N}\) we define a single player game on (infinite) connected graphs that we call fire retaining. If a graph \(G\) admits a winning strategy for any initial configuration (initial fire) then we say that \(G\) has the \(f\)-retaining property; in this case if \(f\) is a polynomial of degree \(d\), we say that \(G\) has the polynomial retaining property of degree \(d\). We prove that having the polynomial retaining property of degree \(d\) is a quasi-isometry invariant in the class of uniformly locally finite connected graphs. Henceforth, the retaining property defines a quasi-isometric invariant of finitely generated groups. We prove that if a finitely generated group \(G\) splits over a quasi-isometrically embedded subgroup of polynomial growth of degree \(d\), then \(G\) has polynomial retaining property of degree \(d-1\). Some connections to other work on quasi-isometry invariants of finitely generated groups are discussed and some questions are raised.Restricted rotation distance between \(k\)-ary treeshttps://zbmath.org/1508.051582023-05-31T16:32:50.898670Z"Cleary, Sean"https://zbmath.org/authors/?q=ai:cleary.seanSummary: We study restricted rotation distance between ternary and higher-valence trees using approaches based upon generalizations of Thompson's group \(F\). We obtain bounds and a method for computing these distances exactly in linear time, as well as a linear-time algorithm for computing rotations needed to realize these distances. Unlike the binary case, the higher-valence notions of rotation distance do not give Tamari lattices, so there are fewer tools for analysis in the higher-valence settings. Higher-valence trees arise in a range of database and filesystem applications where balance is important for efficient performance.Fully preorderable groupshttps://zbmath.org/1508.060112023-05-31T16:32:50.898670Z"Ok, Efe A."https://zbmath.org/authors/?q=ai:ok.efe-a"Riella, Gil"https://zbmath.org/authors/?q=ai:riella.gilSummary: We say that a group is \textit{fully preorderable} if every (left- and right-) translation invariant preorder on it can be extended to a translation invariant total preorder. Such groups arise naturally in applications, and relate closely to orderable and fully orderable groups (which were studied extensively since the seminal works of Philip Hall and A. I. Mal'cev in the 1950s). Our first main result provides a purely group-theoretic characterization of fully preorderable groups by means of a condition that goes back to \textit{M. Ohnishi} [Osaka Math. J. 2, 161--164 (1950; Zbl 0039.25202)]. In particular, this result implies that every fully orderable group is fully preorderable, but not conversely. Our second main result shows that every locally nilpotent group is fully preorderable, but a solvable group need not be fully preorderable. Several applications of these results concerning the inheritance of full preorderability, connections between full preorderability and full orderability, vector preordered groups, and total extensions of translation invariant binary relations on a group, are provided.On the action of the restricted Weyl group on the set of orbits a minimal parabolic subgrouphttps://zbmath.org/1508.140532023-05-31T16:32:50.898670Z"Zhgoon, V. S."https://zbmath.org/authors/?q=ai:zhgoon.vladimir-s"Knop, F."https://zbmath.org/authors/?q=ai:knop.felipe|knop.friedrichSummary: We construct the action of the restricted Weyl group on the set of principal families of orbits of a minimal parabolic subgroup over an algebraically nonclosed field. Additionally, we relate this action to the action on a polarized cotangent bundle. These results generalize the corresponding results of Knop on the action of the Weyl group on the families of Borel orbits of maximal complexity and rank.Bijective 1-cocycles, braces, and non-commutative prime factorizationhttps://zbmath.org/1508.160432023-05-31T16:32:50.898670Z"Rump, Wolfgang"https://zbmath.org/authors/?q=ai:rump.wolfgangThe algebraic structure of the \textit{brace}, a generalisation of the classical Jacobson radical rings, has been introduced by the author in [J. Algebra 307, No. 1, 153--170 (2007; Zbl 1115.16022)]. One of the motivations for studying braces is their interplay with non-degenerate involutive set-theoretic solutions of the Yang-Baxter equation, a fundamental equation of mathematical physics (see [\textit{V. G. Drinfel'd}, Lect. Notes Math. 1510, 1--8 (1992; Zbl 0765.17014)]).
In the paper under review, the author focuses on \textit{quasirings}, namely additive abelian groups \(\left(A,+\right)\) endowed with a multiplication (written as juxtaposition) satisfying the following identities
\begin{align*}
0a &=0\\
a\left(b + c\right) &= ab + ac\\
\left(ab + a + b\right)c &= a\left(bc\right) + ac + bc,
\end{align*}
for all \(a,b,c\in A\). By defining the operation \(\circ\) given by \(a\circ b:= ab + a + b\), one has that \(\left(A, \circ \right)\) is a monoid; if \(\left(A, \circ \right)\) is a group, then \(A\) is a brace. Consistently with braces, quasirings are equivalent to bijective \(1\)-cocycles \(M\to A\) from a monoid \(M\) onto an \(M\)-module \(A\).
The author shows that a specific class of lattice-ordered quasirings characterises the divisor groups of non-commutative smooth algebraic curves. Moreover, the adjoint monoid structure extends the multiplication of fractional ideals of a hereditary noetherian ring to the set of all divisors. Besides, he provides a description of the multiplication of divisors as an extension of the functional representation of fractional ideals given in [the author and \textit{Y. Yang}, J. Algebra 468, 214--252 (2016; Zbl 1400.20058)].
Reviewer: Paola Stefanelli (Lecce)On the (crossed) Burnside ring of profinite groupshttps://zbmath.org/1508.190012023-05-31T16:32:50.898670Z"Mazza, Nadia"https://zbmath.org/authors/?q=ai:mazza.nadiaThe author generalizes certain results on the crossed Burnside ring of a finite group to profinite groups.
A classic result of \textit{D. Gluck} [Ill. J. Math. 25, 63--67 (1981; Zbl 0424.16007)] states that a finite group is solvable if and only if 0 and 1 are the only idempotents in its Burnside ring. An analogue of this result for the profinite case is given by the author as a profinite group is prosoluble if and only if the only idempotents in its Burnside ring (Proposition 6.1).
Another important result regarding Burnside ring of a finite groups is Gluck's formula for its primitive idempotents. Tha author generalizes this formula to Burnside rings of profinite groups (Proposition 6.4).
Reviewer: İsmail Alperen Öğüt (Ankara)Elliptic double affine Hecke algebrashttps://zbmath.org/1508.200062023-05-31T16:32:50.898670Z"Rains, Eric M."https://zbmath.org/authors/?q=ai:rains.eric-mSummary: We give a construction of an affine Hecke algebra associated to any Coxeter group acting on an abelian variety by reflections; in the case of an affine Weyl group, the result is an elliptic analogue of the usual double affine Hecke algebra. As an application, we use a variant of the \(\tilde{C}_n\) version of the construction to construct a flat noncommutative deformation of the \(n\)th symmetric power of any rational surface with a smooth anticanonical curve, and give a further construction which conjecturally is a corresponding deformation of the Hilbert scheme of points.On finite simple images of triangle groupshttps://zbmath.org/1508.200182023-05-31T16:32:50.898670Z"Jambor, Sebastian"https://zbmath.org/authors/?q=ai:jambor.sebastian"Litterick, Alastair"https://zbmath.org/authors/?q=ai:litterick.alastair-j"Marion, Claude"https://zbmath.org/authors/?q=ai:marion.claudeSummary: For a simple algebraic group \(G\) in characteristic \(p\), a triple \((a, b, c)\) of positive integers is said to be rigid for \(G\) if the dimensions of the subvarieties of \(G\) of elements of order dividing \(a\), \(b\), \(c\) sum to \(2 \dim G\). In this paper we complete the proof of a conjecture of the third author, that for a rigid triple \((a, b, c)\) for \(G\) with \(p > 0\), the triangle group \(T_{a,b,c}\) has only finitely many simple images of the form \(G(p^r)\). We also obtain further results on the more general form of the conjecture, where the images \(G(p^r)\) can be arbitrary quasisimple groups of type \(G\).Worst-case approximability of functions on finite groups by endomorphisms and affine mapshttps://zbmath.org/1508.200292023-05-31T16:32:50.898670Z"Bors, Alexander"https://zbmath.org/authors/?q=ai:bors.alexanderSummary: We study the maximum Hamming distance (or rather, the complementary notion of ``minimum approximability'') of a general function on a finite group \(G\) to either of the sets \(\operatorname{End} (G)\) and \(\operatorname{Aff} (G)\), of group endomorphisms of \(G\) and affine maps on \(G\), respectively, the latter being a certain generalization of endomorphisms. We give general bounds on these two quantities and discuss an infinite class of extremal examples (where each of the two Hamming distances can be made as large as generally possible). Finally, we compute the precise values of the two quantities for all finite groups \(G\) with \(|G|\leq 15\).Failure of the finitely generated intersection property for ascending HNN extensions of free groupshttps://zbmath.org/1508.200302023-05-31T16:32:50.898670Z"Bamberger, Jacob"https://zbmath.org/authors/?q=ai:bamberger.jacob"Wise, Daniel T."https://zbmath.org/authors/?q=ai:wise.daniel-tA group \(G\) has the \textit{finitely generated intersection property} (FGIP for short) if, whenever \(H,K\) are finitely generated subgroups of \(G\), the intersection \(H\cap K\) is still finitely generated. This property is also known as the \textit{Howson property}, since Howson proved that free groups have the FGIP [\textit{A. G. Howson}, J. Lond. Math. Soc. 29, 428--434 (1954; Zbl 0056.02106)]. Other interesting examples of groups with the FGIP are given by the following results:
\begin{itemize}
\item the FGIP is stable under taking free products [\textit{B. Baumslag}, J. Lond. Math. Soc. 41, 673--679 (1966; Zbl 0145.02402)];
\item locally quasiconvex word-hyperbolic groups have the FGIP [\textit{H. Short}, in: Group theory from a geometrical viewpoint. Proceedings of a workshop, held at the International Centre for Theoretical Physics in Trieste, Italy, 26 March to 6 April 1990. Singapore: World Scientific. 168--176 (1991; Zbl 0869.20023)].
\end{itemize}
On the other hand, let us mention the following classical counter-examples to the FGIP.
\begin{itemize}
\item The direct product \(\mathbb{F}_2 \times \mathbb{Z}\), where \(\mathbb{F}_2\) denotes the rank \(2\) free group, fails to have the FGIP [\textit{D. I. Moldavanskiĭ}, Sib. Math. J. 9, 1066--1069 (1969; Zbl 0221.20050); translation from Sib. Mat. Zh. 9, 1422--1426 (1968)].
\item Any extension of a free group \(\mathbb{F}\) of finite rank \(k\geq 2\) by \(\mathbb{Z}\) fails to have the FGIP [\textit{R. G. Burns} and \textit{A. M. Brunner}, Algebra Logic 18, 319--325 (1980; Zbl 0448.20030); translation from Algebra Logika 18, 513--522 (1979)]. -- Note that every such extension is split, hence isomorphic to a semidirect product \(\mathbb{F} \rtimes \mathbb{Z}\).
\end{itemize}
In the article under review, the following generalization is proven.
Theorem 1.3. Any ascending HNN extension of a free group \(\mathbb{F}\) of finite rank \(k\geq 2\) fails to have the FGIP.
Let us recall that an ascending HNN extension of a group \(G\) is a group of the form \(\langle G, t \mid txt^{-1} = \phi(x) \rangle\), where \(\phi: G \to G\) is a monomorphism of groups. It is straightforward that semidirect products \(G \rtimes \mathbb{Z}\) are exactly ascending HNN extensions of \(G\) with a surjective \(\phi\). Hence, Theorem 1.3 indeed generalizes the result of Burns and Brunner [loc. cit.].
As noticed in the footnote p. 886, Theorem 1.3 also appears in [\textit{D. I. Moldavanskiĭ}, Chebyshevskiĭ Sb. 11, No. 3(35), 103--110 (2010; Zbl 1274.20026)]. In this article, it is a consequence of the following theorem:
Let \(G\) be a finitely generated group, let \(\phi\) be an injective but not surjective endomorphism of \(\phi\). If the subgroup \(\phi(G) \leq G\) is freely complemented, then the associated ascending HNN extension fails to have the FGIP.
In the article under review, the authors give a new and more geometrical proof of Theorem 1.3. This proof is divided in three cases:
\begin{itemize}
\item the ascending HNN extension is proper (that is, \(\phi\) is not surjective);
\item \(\phi\) is a polynomially growing automorphism of \(\mathbb{F}\);
\item \(\phi\) is an exponentially growing automorphism of \(\mathbb{F}\).
\end{itemize}
For the last two cases (which cover the former result of Burns and Brunner [loc. cit.]) the authors use ``relative train-tracks'' developed in [\textit{M. Bestvina} et al., Ann. Math. (2) 151, No. 2, 517--623 (2000; Zbl 0984.20025)] and relative hyperbolicity techniques.
About relative hyperbolicity, the authors establish the following result of independant insterest.
Theorem 5.2. Let \(G\) be hyperbolic relative to a collection of subgroups. Let \(N \subseteq G\) be a finitely generated subgroup containing a loxodromic element \(w\). Suppose \(tNt^{-1} \subseteq N\) for some infinite order element \(t\) with \(\langle t \rangle \cap N = \{1_G\}\). Then \(G\) fails to have the FGIP. In particular, there exists \(m\) such that \(\langle t^m , w^m \rangle \cap N\) is not finitely generated.
Reviewer: Yves Stalder (Aubière)Groups with finitely many non-isomorphic factor-groupshttps://zbmath.org/1508.200312023-05-31T16:32:50.898670Z"Kurdachenko, Leonid A."https://zbmath.org/authors/?q=ai:kurdachenko.leonid-a"Longobardi, Patrizia"https://zbmath.org/authors/?q=ai:longobardi.patrizia"Maj, Mercede"https://zbmath.org/authors/?q=ai:maj.mercedeSummary: We prove that if \(G\) is either a hypercentral-by-finite group or a soluble Baer group and if \(G\) has finitely many non-isomorphic factor-groups, then \(G\) is a Chernikov group. The converse is also true. Furthermore, we give some information on the structure of a metabelian group with finitely many non-isomorphic factor-groups.Quotients of buildings by groups acting freely on chambershttps://zbmath.org/1508.200342023-05-31T16:32:50.898670Z"Norledge, William"https://zbmath.org/authors/?q=ai:norledge.williamSummary: We introduce certain directed multigraphs with extra structure, called Weyl graphs, which model quotients of Tits buildings by type-preserving chamber-free group actions. Their advantage over complexes of groups, which are often used for the CAT(0) Davis realization of buildings, is that Weyl graphs exploit the ultimate combinatorial W-metric structure of buildings. Weyl graphs generalize Tits's chamber systems of type M by allowing rank two residues to be quotients of generalized polygons by flag-free group actions, and Weyl graphs are easily constructed by amalgamating such quotients. We develop covering theory of Weyl graphs, which can be used to construct buildings as universal covers. We describe a method for obtaining a group presentation of the fundamental group of a Weyl graph, which acts chamber-freely on the covering building. The theory developed here is part of a fully general theory which deals with not necessarily chamber-free actions and the stacky version of buildings.Word problems for finite nilpotent groupshttps://zbmath.org/1508.200362023-05-31T16:32:50.898670Z"Camina, Rachel D."https://zbmath.org/authors/?q=ai:camina.rachel-deborah"Iñiguez, Ainhoa"https://zbmath.org/authors/?q=ai:iniguez.ainhoa"Thillaisundaram, Anitha"https://zbmath.org/authors/?q=ai:thillaisundaram.anithaSummary: Let \(w\) be a word in \(k\) variables. For a finite nilpotent group \(G\), a conjecture of Amit states that \(N_w(1)\ge |G|^{k-1}\), where for \(g\in G\), the quantity \(N_w(g)\) is the number of \(k\)-tuples \((g_1,\dots,g_k)\in G^{(k)}\) such that \(w(g_1,\dots,g_k)={g}\). Currently, this conjecture is known to be true for groups of nilpotency class 2. Here we consider a generalized version of Amit's conjecture, which states that \(N_w(g)\ge |G|^{k-1}\) for \(g\) a \(w\)-value in \(G\), and prove that \(N_w(g)\ge |G|^{k-2}\) for finite groups \(G\) of odd order and nilpotency class 2. If \(w\) is a word in two variables, we further show that the generalized Amit conjecture holds for finite groups \(G\) of nilpotency class 2. In addition, we use character theory techniques to confirm the generalized Amit conjecture for finite \(p\)-groups \((p\) a prime) with two distinct irreducible character degrees and a particular family of words. Finally, we discuss the related group properties of being rational and chiral, and show that every finite group of nilpotency class 2 is rational.On 3-strand singular pure braid grouphttps://zbmath.org/1508.200372023-05-31T16:32:50.898670Z"Bardakov, Valeriy G."https://zbmath.org/authors/?q=ai:bardakov.valerii-georgievich"Kozlovskaya, Tatyana A."https://zbmath.org/authors/?q=ai:kozlovskaya.tatyana-anatolevnaSummary: In this paper, we study the singular pure braid group \(\mathrm{SP}_n\) for \(n = 2, 3\). We find generators, defining relations and the algebraical structure of these groups. In particular, we prove that \(\mathrm{SP}_3\) is a semi-direct product \(\mathrm{SP}_3= \widetilde{V}_3 \leftthreetimes \mathbb{Z}\), where \(\widetilde{V}_3\) is an HNN-extension with base group \(\mathbb{Z}^2 \ast \mathbb{Z}^2\) and cyclic associated subgroups. We prove that the center \(Z( \mathrm{SP}_3)\) of \(\mathrm{SP}_3\) is a direct factor in \(\mathrm{SP}_3\).On virtual cabling and a structure of 4-strand virtual pure braid grouphttps://zbmath.org/1508.200382023-05-31T16:32:50.898670Z"Bardakov, Valeriy G."https://zbmath.org/authors/?q=ai:bardakov.valerii-georgievich"Wu, Jie"https://zbmath.org/authors/?q=ai:wu.jie.2Summary: This paper is dedicated to cabling on virtual braids. This construction gives a new generating set for the virtual pure braid group \(\text{VP}_n\). We define simplicial group \(\text{VP}_\ast\) and its simplicial subgroup \(T_\ast\) which is generated by \(\text{VP}_2\). Consequently, we describe \(\text{VP}_4\) as HNN-extension and find presentation of \(T_2\) and \(T_3\). As an application to classical braids, we find a new presentation of the Artin pure braid group \(P_4\) in terms of the cabled generators.Unrestricted virtual braids and crystallographic braid groupshttps://zbmath.org/1508.200392023-05-31T16:32:50.898670Z"Bellingeri, Paolo"https://zbmath.org/authors/?q=ai:bellingeri.paolo"Guaschi, John"https://zbmath.org/authors/?q=ai:guaschi.john"Makri, Stavroula"https://zbmath.org/authors/?q=ai:makri.stavroulaThe group of unrestricted virtual braids, denoted by \(UVB_n\) for \(n\ge 1\), was introduced by \textit{L. H. Kauffman} and \textit{S. Lambropoulou} [J. Knot Theory Ramifications 15, No. 6, 773--811 (2006; Zbl 1105.57002)] as the analogue of fused links in the setting of braids. Such links are distinguished by their virtual linking number, are considered by \textit{T. Nasybullov} [J. Knot Theory Ramifications 25, No. 14, Article ID 1650076, 21 p. (2016; Zbl 1383.57012)] as the closure of unrestricted virtual braids and classified in terms of Gauss diagrams by \textit{B. Audoux} et al. [Mich. Math. J. 67, No. 3, 647--672 (2018; Zbl 1406.57003)]. Those groups may be decomposed as a semi-direct product of a right-angled Artin group, which is in fact, by \textit{V. G. Bardakov} et al. [J. Knot Theory Ramifications 24, No. 12, Article ID 1550063, 23 p. (2015; Zbl 1360.20026)], the pure subgroup \(UVP_n\) of \(UVB_n\), by the symmetric group \(S_n\). \par The main aim of this paper is to characterise the torsion elements of \(UVB_n\) using this decomposition, namely to show that any element of finite order is a conjugate of an element of \(S_n\) by an element of \(UVP_n\). The authors show that the crystallographic braid group \(B_n=B_n/[P_n,P_n]\) embeds naturally in the group of unrestricted virtual braids \(UVB_n\), Then, they give new proofs of known results by \textit{D. L. Gonçalves} et al. [J. Algebra 474, 393--423 (2017; Zbl 1367.20034)] about the torsion elements of \(B_n=B_n/[P_n,P_n]\), and characterise the torsion elements of \(UVB_n\), where \(B_n\) is the Artin braid group, \(P_n\) is the pure Artin braid group, and \([P_n,P_n]\) is its commutator subgroup.
Reviewer: Marek Golasiński (Olsztyn)A note on representations of welded braid groupshttps://zbmath.org/1508.200402023-05-31T16:32:50.898670Z"Bellingeri, Paolo"https://zbmath.org/authors/?q=ai:bellingeri.paolo"Soulié, Arthur"https://zbmath.org/authors/?q=ai:soulie.arthurSummary: In this paper, we adapt the procedure of the Long-Moody procedure to construct linear representations of welded braid groups. We exhibit the natural setting in this context and compute the first examples of representations we obtain thanks to this method. We take this way also the opportunity to review the few known linear representations of welded braid groups.Property \(R_\infty\) for some spherical and affine Artin-Tits groupshttps://zbmath.org/1508.200412023-05-31T16:32:50.898670Z"Calvez, Matthieu"https://zbmath.org/authors/?q=ai:calvez.matthieu"Soroko, Ignat"https://zbmath.org/authors/?q=ai:soroko.ignatThe notion of twisted conjugacy classes arises in Nielsen fixed point theory and appears naturally in Selberg theory. Indeed, for a group \(G\) and an automorphism \(\varphi\) of it, two elements \(g,h\in G\) are said to be \textit{\(\varphi\)-twisted conjugate} if and only if there is \(x\in G\) such that \(h=xg\varphi(x)^{-1}\). The group \(G\) is said to have the \textit{property \(R_\infty\)} if the number of twisted conjugacy classes is infinity for all automorphism \(\varphi\). In the present paper, the authors give a short proof of this property for the Artin-Tits groups of spherical type: \(A_n\), \(B_n\), \(D_4\), \(I_2(m)\) with \(m\geq3\), also for their pure subgroups, and for the Artin-Tits groups of affine types: \(\tilde{A}_{n-1}\) and \(\tilde{C}_{n-1}\). In particular, they provide an alternative proof of the fact that pure braid groups have the property \(R_{\infty}\), which was recently proved by \textit{K. Dekimpe} et al. [Monatsh. Math. 195, No. 1, 15--33 (2021; Zbl 1507.20017)].
Reviewer: Diego Arcis (Talca)Commensurability invariance for abelian splittings of right-angled Artin groups, braid groups and loop braid groupshttps://zbmath.org/1508.200422023-05-31T16:32:50.898670Z"Zaremsky, Matthew C. B."https://zbmath.org/authors/?q=ai:zaremsky.matthew-curtis-burkholderSummary: We prove that if a right-angled Artin group \(A_\Gamma\) is abstractly commensurable to a group splitting nontrivially as an amalgam or HNN extension over \(\mathbb{Z}^n\), then \(A_\Gamma\) must itself split nontrivially over \(\mathbb{Z}^k\) for some \(k\le n\). Consequently, if two right-angled Artin groups \(A_\Gamma\) and \(A_\Delta\) are commensurable and \(\Gamma\) has no separating \(k\)-cliques for any \(k\le n\), then neither does \(\Delta\), so ``smallest size of separating clique'' is a commensurability invariant. We also discuss some implications for issues of quasi-isometry. Using similar methods we also prove that for \(n\ge 4\) the braid group \(B_n\) is not abstractly commensurable to any group that splits nontrivially over a ``free group-free'' subgroup, and the same holds for \(n\ge 3\) for the loop braid group \(\operatorname{LB}_n\). Our approach makes heavy use of the Bieri-Neumann-Strebel invariant.The adjoint group of a Coxeter quandlehttps://zbmath.org/1508.200432023-05-31T16:32:50.898670Z"Akita, Toshiyuki"https://zbmath.org/authors/?q=ai:akita.toshiyukiSummary: We give explicit descriptions of the adjoint group \(\mathrm{Ad}(Q_W)\) of the Coxeter quandle \(Q_W\) associated with an arbitrary Coxeter group \(W\). The adjoint group \(\mathrm{Ad}(Q_W)\) turns out to be an intermediate group between \(W\) and the corresponding Artin group \(A_W\), and it fits into a central extension of \(W\) by a finitely generated free abelian group. We construct \(2\)-cocycles of \(W\) corresponding to the central extension. In addition, we prove that the commutator subgroup of the adjoint group \(\mathrm{Ad}(Q_W)\) is isomorphic to the commutator subgroup of \(W\). Finally, the root system \(\Phi_W\) associated with a Coxeter group \(W\) turns out to be a rack. We prove that the adjoint group \({\mathrm{Ad}}(\Phi_W)\) of \(\Phi_W\) is isomorphic to the adjoint group of \(Q_W\).The low-dimensional homology of finite-rank Coxeter groupshttps://zbmath.org/1508.200442023-05-31T16:32:50.898670Z"Boyd, Rachael"https://zbmath.org/authors/?q=ai:boyd.rachaelSummary: We give formulas for the second and third integral homology of an arbitrary finitely generated Coxeter group, solely in terms of the corresponding Coxeter diagram. The first of these calculations refines a theorem of Howlett, while the second is entirely new and is the first explicit formula for the third homology of an arbitrary Coxeter group.Virtually fibering right-angled Coxeter groupshttps://zbmath.org/1508.200452023-05-31T16:32:50.898670Z"Jankiewicz, Kasia"https://zbmath.org/authors/?q=ai:jankiewicz.kasia"Norin, Sergey"https://zbmath.org/authors/?q=ai:norine.serguei"Wise, Daniel T."https://zbmath.org/authors/?q=ai:wise.daniel-tSummary: We show that certain right-angled Coxeter groups have finite index subgroups that quotient to \(\mathbb{Z}\) with finitely generated kernels. The proof uses Bestvina-Brady Morse theory facilitated by combinatorial arguments. We describe a variety of examples where the plan succeeds or fails. Among the successful examples are the right-angled reflection groups in \(\mathbb{H}^4\) with fundamental domain the \(120\)-cell or the \(24\)-cell.Some remarks on twin groupshttps://zbmath.org/1508.200462023-05-31T16:32:50.898670Z"Naik, Tushar K."https://zbmath.org/authors/?q=ai:naik.tushar-kanta"Nanda, Neha"https://zbmath.org/authors/?q=ai:nanda.neha"Singh, Mahender"https://zbmath.org/authors/?q=ai:singh.mahenderSummary: The twin group \(T_n\) is a right angled Coxeter group generated by \(n-1\) involutions and having only far commutativity relations. These groups can be thought of as planar analogues of Artin braid groups. In this paper, we study some properties of twin groups whose analogues are well known for Artin braid groups. We give an algorithm for two twins to be equivalent under individual Markov moves. Further, we show that twin groups \(T_n\) have \(R_\infty\)-property and are not co-Hopfian for \(n \geq 3\).Cannon-Thurston maps for CAT(0) groups with isolated flatshttps://zbmath.org/1508.200472023-05-31T16:32:50.898670Z"Benjamin, Beeker"https://zbmath.org/authors/?q=ai:benjamin.beeker"Cordes, Matthew"https://zbmath.org/authors/?q=ai:cordes.matthew"Gardam, Giles"https://zbmath.org/authors/?q=ai:gardam.giles"Gupta, Radhika"https://zbmath.org/authors/?q=ai:gupta.radhika"Stark, Emily"https://zbmath.org/authors/?q=ai:stark.emilyIf $M$ is a closed hyperbolic 3-manifold fibering over the circle and if $F\subset M$ is a fiber of the fibration, Cannon and Thurston proved in the 80s that the natural map from the hyperbolic plane to hyperbolic 3-space lifting the inclusion of $F$ into $M$ can be extended to a continuous map from the circle (the boundary of the hyperbolic plane) to the $2$-sphere (the boundary of hyperbolic 3-space), which is equivariant with respect to the action of the fundamental group of $F$.
Mj later extended this result by considering a hyperbolic group $G$ containing an infinite normal hyperbolic subgroup $H$ and by proving that the inclusion of $H$ into $G$ gives rise to an $H$-equivariant continuous surjective map from the Gromov boundary of $H$ to the Gromov boundary of $G$.
In this work, the authors consider a group $G$ acting geometrically on a $\mathrm{CAT}(0)$ space with isolated flats and an infinite normal subgroup $H$. Assuming either that $H$ is hyperbolic or that it acts itself geometrically on a $\mathrm{CAT}(0)$ space with isolated flats (and is of infinite index in the latter case), the authors prove that there is no Cannon-Thurston map in this context. The boundaries under consideration here are the visual boundary of the model space on which $G$ acts and either the Gromov boundary or the visual boundary of the model space for $H$ (depending on the assumption made).
We observe that the existence of Cannon-Thurston maps in a non-hyperbolic context depends on which type of boundary is considered. Such maps sometime exist in a relatively hyperbolic context, when one picks the Bowditch boundaries. We refer to the article for more detailed statements.
The authors also prove some structural results for hyperbolic normal subgroups in groups acting geometrically on $\mathrm{CAT}(0)$ spaces with isolated flats (building, among other things, on work of Rips and Sela).
Reviewer: Pierre Py (Strasbourg)On the topological dimension of the Gromov boundaries of some hyperbolic \(\mathrm{Out}(F_N)\)-graphshttps://zbmath.org/1508.200482023-05-31T16:32:50.898670Z"Bestvina, Mladen"https://zbmath.org/authors/?q=ai:bestvina.mladen"Horbez, Camille"https://zbmath.org/authors/?q=ai:horbez.camille"Wade, Richard D."https://zbmath.org/authors/?q=ai:wade.richard-dSummary: We give upper bounds, linear in the rank, to the topological dimensions of the Gromov boundaries of the intersection graph, the free factor graph and the cyclic splitting graph of a finitely generated free group.Subdirect products of surfaces and residually free groupshttps://zbmath.org/1508.200492023-05-31T16:32:50.898670Z"Bridson, Martin"https://zbmath.org/authors/?q=ai:bridson.martin-rFor the entire collection see [Zbl 1477.00012].A mixed identity-free elementary amenable grouphttps://zbmath.org/1508.200502023-05-31T16:32:50.898670Z"Jacobson, B."https://zbmath.org/authors/?q=ai:jacobson.bengt|jacobson.bo-h|jacobson.brianSummary: A group \(G\) is called \textit{mixed identity-free} if for every \(n\in\mathbb{N}\) and every \(w\in G\ast F_n\), there exists a homomorphism \(\varphi :G\ast F_n\rightarrow G\) such that \(\varphi\) is the identity on \(G\) and \(\varphi(w)\) is nontrivial. In this paper, we make a modification to the construction of elementary amenable lacunary hyperbolic groups provided by Ol'shanskii et al. to produce finitely generated elementary amenable groups which are mixed identity-free. As a byproduct of this construction, we also obtain locally finite \(p\)-groups which are mixed identity-free.Quasi-isometry invariants of weakly special square complexeshttps://zbmath.org/1508.200512023-05-31T16:32:50.898670Z"Oh, Sangrok"https://zbmath.org/authors/?q=ai:oh.sangrokSummary: We define the intersection complex for the universal cover of a compact weakly special square complex and show that it is a quasi-isometry invariant. By using this quasi-isometry invariant, we study the quasi-isometric classification of \(2\)-dimensional right-angled Artin groups and planar graph \(2\)-braid groups. Our results cover two well-known cases of \(2\)-dimensional right-angled Artin groups: (1) those whose defining graphs are trees and (2) those whose outer automorphism groups are finite. Finally, we show that there are infinitely many graph \(2\)-braid groups which are quasi-isometric to right-angled Artin groups and infinitely many which are not.On the cohomology of integral \(p\)-adic unipotent radicalshttps://zbmath.org/1508.200572023-05-31T16:32:50.898670Z"Ronchetti, Niccolò"https://zbmath.org/authors/?q=ai:ronchetti.niccoloSummary: Let \(G\) be a reductive split \(p\)-adic group and let \( \mathrm U\) be the unipotent radical of a Borel subgroup. We study the cohomology with trivial \(\mathbb Z_p\)-coefficients of the profinite nilpotent group \(N = \mathrm U (\mathcal O_F)\) and its Lie algebra \(\mathfrak n\) by extending a classical result of Kostant to our integral \(p\)-adic setup. The techniques used are a combination of results from group theory, algebraic groups and homological algebra.Bounded cohomology via quasi-treeshttps://zbmath.org/1508.200612023-05-31T16:32:50.898670Z"Bestvina, Mladen"https://zbmath.org/authors/?q=ai:bestvina.mladenFor the entire collection see [Zbl 1477.00012].Tied monoidshttps://zbmath.org/1508.200632023-05-31T16:32:50.898670Z"Arcis, Diego"https://zbmath.org/authors/?q=ai:arcis.diego"Juyumaya, Jesús"https://zbmath.org/authors/?q=ai:juyumaya.jesusSummary: We construct certain monoids, called \textit{tied monoids}. These monoids result to be semidirect products finitely presented and commonly built from braid groups and their relatives acting on monoids of set partitions. The nature of our monoids indicate that they should give origin to new knot algebras; indeed, our tied monoids include the tied braid monoid and the tied singular braid monoid, which were used, respectively, to construct new polynomial invariants for classical links and singular links. Consequently, we provide a mechanism to attach an algebra to each tied monoid; this mechanism not only captures known generalizations of the bt-algebra, but also produces possible new knot algebras. To build the tied monoids it is necessary to have presentations of set partition monoids of types A, B and D, among others. For type A we use a presentation due to FitzGerald and for the other type it was necessary to built them.The word problem for one-relation monoids: a surveyhttps://zbmath.org/1508.200682023-05-31T16:32:50.898670Z"Nyberg-Brodda, Carl-Fredrik"https://zbmath.org/authors/?q=ai:nyberg-brodda.carl-fredrikSummary: This survey is intended to provide an overview of one of the oldest and most celebrated open problems in combinatorial algebra: the word problem for one-relation monoids. We provide a history of the problem starting in 1914, and give a detailed overview of the proofs of central results, especially those due to Adian and his student Oganesian. After showing how to reduce the problem to the left cancellative case, the second half of the survey focuses on various methods for solving partial cases in this family. We finish with some modern and very recent results pertaining to this problem, including a link to the Collatz conjecture. Along the way, we emphasise and address a number of incorrect and inaccurate statements that have appeared in the literature over the years. We also fill a gap in the proof of a theorem linking special inverse monoids to one-relation monoids, and slightly strengthen the statement of this theorem.Special idempotents and projectionshttps://zbmath.org/1508.201182023-05-31T16:32:50.898670Z"Sentinelli, Paolo"https://zbmath.org/authors/?q=ai:sentinelli.paoloSummary: We define, for any special matching of a finite graded poset, an idempotent, regressive and order preserving function. We consider the monoid generated by such functions. We call \textit{special idempotent} any idempotent element of this monoid. They are interval retracts. Some of them realize a kind of parabolic map and are called \textit{special projections}. We prove that, in Eulerian posets, the image of a special projection, and its complement, are graded induced subposets. In a finite Coxeter group, all projections on right and left parabolic quotients are special projections, and some projections on double quotients too. We extend our results to special partial matchings.Finite orbits of monodromies of rank two Fuchsian systemshttps://zbmath.org/1508.341202023-05-31T16:32:50.898670Z"Tykhyy, Yuriy"https://zbmath.org/authors/?q=ai:tykhyy.yuriySummary: We classified finite orbits of monodromies of the Fuchsian system for five \(2\times 2\) matrices. The explicit proof of this result is given. We have proposed a conjecture for a similar classification for 6 or more \(2\times 2\) matrices. Cases in which all monodromy matrices have a common eigenvector are excluded from the consideration. To classify the finite monodromies of the Fuchsian system we combined two methods developed in this paper. The first is an induction method: using finite orbits of smaller number of monodromy matrices the method allows the construction of such orbits for bigger numbers of matrices. The second method is a formalism for representing the tuple of monodromy matrices in a way that is invariant under common conjugation way, this transforms the problem into a form that allows one to work with rational numbers only. The classification developed in this paper can be considered as the first step to a classification of algebraic solutions of the Garnier system.Metric equivalences of Heintze groups and applications to classifications in low dimensionhttps://zbmath.org/1508.530612023-05-31T16:32:50.898670Z"Kivioja, Ville"https://zbmath.org/authors/?q=ai:kivioja.ville"Le Donne, Enrico"https://zbmath.org/authors/?q=ai:le-donne.enrico"Golo, Sebastiano Nicolussi"https://zbmath.org/authors/?q=ai:nicolussi-golo.sebastianoSummary: We approach the quasi-isometric classification questions on Lie groups by considering low dimensional cases and isometries alongside quasi-isometries. First, we present some new results related to quasi-isometries between Heintze groups. Then we will see how these results together with the existing tools related to isometries can be applied to groups of dimension 4 and 5 in particular. Thus, we take steps toward determining all the equivalence classes of groups up to isometry and quasi-isometry. We completely solve the classification up to isometry for simply connected solvable groups in dimension 4 and for the subclass of groups of polynomial growth in dimension 5.Higher topological complexity of hyperbolic groupshttps://zbmath.org/1508.550012023-05-31T16:32:50.898670Z"Hughes, Sam"https://zbmath.org/authors/?q=ai:hughes.sam"Li, Kevin"https://zbmath.org/authors/?q=ai:li.kevin-x|li.kevin-wThis paper is concerned with higher topological complexity of certain hyperbolic groups.
Let \(r \geq 2\) be an integer. Consider the path-fibration \(p : X^{[0,1]} \rightarrow X^{r}\) that maps a path \(\omega: [0, 1] \rightarrow X\) to the tuple \((\omega(0), \omega\frac{1}{r-1}, \dots ,\omega \frac{r-2}{r-1}, \omega(1))\). Then \(TC_{r} (X)\) is defined as the minimal integer \(n\) for which \(X^{r}\) can be covered by \(n + 1\) many open subsets \(U_{0}, \dots ,U_{n}\) such that \(p\) admits a local section over each \(U_{i}\). If no such \(n\) exists, one sets \(TC_{r}(X) :=\infty\). Note that \(TC_{2}(X)\) recovers the usual topological complexity.
Let \(\Gamma\) be a non-elementary torsion-free hyperbolic group. Since the higher topological complexities are homotopy invariants, one obtains interesting invariants of groups \(\Gamma\) by setting \(TC_{r} (\Gamma) := TC_{r}(K(\Gamma, 1))\), where \(K(\Gamma, 1)\) is an Eilenberg-MacLane space. In a celebrated result by \textit{A. Dranishnikov} [Proc. Am. Math. Soc. 148, No. 10, 4547--4556 (2020; Zbl 1447.55002)] (see also [\textit{M. Farber} and \textit{S. Mescher}, J. Topol. Anal. 12, No. 2, 293--319 (2020; Zbl 1455.55003)]), the topological complexity \(TC_{2}(\Gamma)\) of groups with cyclic centralisers, such as hyperbolic groups, is equal \(cd(\Gamma \times \Gamma)\). Here \(cd\) denotes the cohomological dimension. The authors generalise this result to all higher topological complexities \(TC_{r} (\Gamma)\) for \(r \geq 2\), as well as to a larger class of groups containing certain toral relatively hyperbolic groups.
Theorem 1.1. Let \(r\geq 2\) and let \(\Gamma\) be a torsion-free group with \(cd(\Gamma) \geq 2\). Suppose that \(\Gamma\) admits a malnormal collection of abelian subgroups \(\lbrace P_i \mid i \in I \rbrace\) satisfying \(cd(P^{r}_{ i }) < cd(\Gamma^{r})\) such that the centraliser \(C_{\Gamma}(g)\) is cyclic for every \(g \in \Gamma\) that is not conjugate into any of the \(P_{i}\). Then \(TC_{r} (\Gamma) = cd(\Gamma^{r})\).
For a space \(X\), the \(TC\)-generating function \(f_{X} (t)\) is defined as the formal power series
\[
f_{X} (t) := \sum^{\infty}_{r=1}TC_{r+1}(X) \cdot t^{r} .
\]
The \(TC\)-generating function of a group \(\Gamma\) is set to be \(f_{\Gamma}(t) := f_{K(\Gamma,1)}(t)\). Recall that a group \(\Gamma\) is said to be of type \(F\) (or geometrically finite) if it admits a finite model for \(K(\Gamma, 1)\).
Following [\textit{M. Farber} and \textit{J. Oprea}, Topology Appl. 258, 142--160 (2019; Zbl 1412.55003)], we say that a finite \(CW\)-complex \(X\) (resp. a group \(\Gamma\) of type \(F\)) satisfies the rationality conjecture if the \(TC\)-generating function \(f_{X} (t)\) (resp. \(f_{\Gamma}(t))\) is a rational function of the form \(\frac{P(t)}{(1-t)^{2}}\), where \(P(t)\) is an integer polynomial with \(P(1) = cat(X)\) (resp. \(P(1) = cd(\Gamma))\). Here cat denotes the Lusternik-Schnirelmann category. While a counter-example to the rationality conjecture for finite \(CW\)-complexes was found in [\textit{M. Farber} et al., Topology Appl. 278, Article ID 107235, 4 p. (2020; Zbl 1440.55001)], the rationality conjecture for groups of type \(F\) remains open. The result in the present paper extends the class of groups for which the rationality conjecture holds as follows:
Corollary 1.2. Let \(\Gamma\) be a group as in Theorem 1.1. If \(\Gamma\) is of type \(F\), then
\[
f_{\Gamma}(t) = cd(\Gamma) \frac{(2-t) t}{(1-t)^{2}}.
\]
In particular, the rationality conjecture holds for \(\Gamma\).
Corollary 1.3 The rationality conjecture holds for torsion-free hyperbolic groups.
Reviewer: Fezzeh Akhtarifar (Tabriz)The special rank of virtual knot groupshttps://zbmath.org/1508.570112023-05-31T16:32:50.898670Z"Mira-Albanés, Jhon Jader"https://zbmath.org/authors/?q=ai:mira-albanes.jhon-jader"Rodríguez-Nieto, José Gregorio"https://zbmath.org/authors/?q=ai:rodriguez-nieto.jose-gregorio"Salazar-Díaz, Olga Patricia"https://zbmath.org/authors/?q=ai:salazar-diaz.olga-patriciaSummary: In this paper we introduce the special rank for virtual knots and some properties of this number are studied. Although we do not know if it can be considered as a nontrivial extension of the meridional rank given by \textit{H. U. Boden} and \textit{A. I. Gaudreau} [J. Knot Theory Ramifications 24, No. 2, Article ID 1550008, 15 p. (2015; Zbl 1326.57005)] and by \textit{M. Boileau} and \textit{B. Zimmermann} [Math. Z. 200, No. 2, 187--208 (1989; Zbl 0663.57006)], we prove that classical knots with special rank \(2\) are \(2\)-bridge knots. Therefore, a modified version of the so called Cappell and Shaneson conjecture could be considered.A small generating set for the balanced superelliptic handlebody grouphttps://zbmath.org/1508.570212023-05-31T16:32:50.898670Z"Omori, Genki"https://zbmath.org/authors/?q=ai:omori.genkiFor \(g \geq 0\), let \(H_g\) be the oriented three-dimensional handlebody of genus \(g\) and \(\Sigma_{g,k}\) be the oriented surface of genus \(g\) with \(k\geq 0\) punctures. Let \(\mathcal{H}_g\) be the handlebody group of \(H_g\) and \(\mathrm{Mod}_{g,k}\) be the mapping class group of \(\Sigma_{g,k}\). (We denote \(S_{g,0}\) simply by \(S_g\).) For \(n \geq 1\) and \(k \geq 2\) with \(g = n(k-1)\), let \(p_{g,k}: H_g \to H_0\) be the \(k\)-fold balanced superelliptic branched covering space introduced by \textit{T. Ghaswala} and \textit{R. R. Winarski} [Mich. Math. J. 66, No. 4, 885--890 (2017; Zbl 1383.57001)] (who considered its restriction \(p_{g,k}\vert_{\partial H_g} : \Sigma_g \to \Sigma_0\) with \(2n+2\) branched points.) Let \(\mathbf{H}_{2n+2}\) be the \textit{Hilden group} [\textit{H. M. Hilden}, Pac. J. Math. 59, 475--486 (1975; Zbl 0317.57005)], \(\mathrm{LMod}_{2n+2,k}\) be the liftable mapping class group, and \(\mathrm{SMod}_{g,k}\) be the symmetric mapping class group [\textit{D. Margalit} and \textit{R. R. Winarski}, Bull. Lond. Math. Soc. 53, No. 3, 643--659 (2021; Zbl 1470.57045)] with respect to \(p_{g,k}\).
The main result of this paper establishes that for \(k \geq 2\), the \textit{liftable Hilden group} \(\mathbf{LH}_{2n+2,k} =\mathrm{LMod}_{2n+2,k} \cap \mathbf{H}_{2n+2}\), is generated by three elements. As an application of this result and a result due to \textit{S. Hirose} and \textit{E. Kin} [Q. J. Math. 68, No. 3, 1035--1069 (2017; Zbl 1393.57005)], it is shown that the \textit{balanced superelliptic handlebody group} \(\mathcal{SH}_{g,k} = \mathrm{SMod}_{g,k} \cap \mathcal{H}_g\) is generated by four elements. Furthermore, by considering known abelianizations of \(\mathbf{LH}_{2n+2,k}\) and \(\mathcal{SH}_{g,k}\) [\textit{S. Hirose} and \textit{E. Kin}, loc. cit.; \textit{G. Omori} and \textit{Y. Yoshida}, ``A finite presentation for the balanced superelliptic handlebody group'', Preprint, \url{arXiv:2206.13006}; \textit{S. Tawn}, Math. Res. Lett. 15, No. 5--6, 1277--1293 (2008; Zbl 1166.57001)], the author proves that the derived generating set for \(\mathbf{LH}_{2n+2,k}\) is minimal for \(k > 2\) and even \(n\), while the generating set for \(\mathcal{SH}_{g,k}\) is minimal for even \(k \geq 4\) and odd \(n\).
Reviewer: Kashyap Rajeevsarathy (Bhopal)The Kauffman bracket skein module of the handlebody of genus 2 via braidshttps://zbmath.org/1508.570222023-05-31T16:32:50.898670Z"Diamantis, Ioannis"https://zbmath.org/authors/?q=ai:diamantis.ioannisSummary: In this paper we present two new bases, \( B_{H_2}^\prime\) and \(\mathcal{B}_{H_2}\), for the Kauffman bracket skein module KBSM\(( H_2)\) of \( H_2\), the handlebody of genus 2. We start from the well-known Przytycki-basis of KBSM\((H_2), B_{H_2} \), and using the technique of parting we present elements in \(B_{H_2}\) in open braid form. We define an ordering relation on an augmented set \(L\) consisting of monomials of all different ``loopings'' in \(H_2\), that contains the sets \(B_{H_2}, B_{H_2}^\prime\) and \(\mathcal{B}_{H_2}\) as proper subsets. Using the Kauffman bracket skein relation we relate \(B_{H_2}\) to the sets \(B_{H_2}^\prime\) and \(\mathcal{B}_{H_2}\) via a lower triangular infinite matrix with invertible elements in the diagonal. The basis \(B_{H_2}^\prime\) is an intermediate step in order to reach at elements in \(\mathcal{B}_{H_2}\) that have no crossings on the level of braids, and in that sense, \(\mathcal{B}_{H_2}\) is a more natural basis of KBSM\(( H_2)\). Moreover, this basis is appropriate in order to compute Kauffman bracket skein modules of closed-connected-oriented (c.c.o.) 3-manifolds \(M\) that are obtained from \(H_2\) by surgery, since isotopy moves in \(M\) are naturally described by elements in \(\mathcal{B}_{H_2} \).Representation stability in the level 4 braid grouphttps://zbmath.org/1508.570272023-05-31T16:32:50.898670Z"Kordek, Kevin"https://zbmath.org/authors/?q=ai:kordek.kevin"Margalit, Dan"https://zbmath.org/authors/?q=ai:margalit.danLet \(k\) be the field of either \(\mathbb{Q}\) or \(\mathbb{C}\). Let \(\mathbf{B}_n\) and \(\mathbf{PB}_n\) be the braid and pure braid groups of \(n\)-strands, respectively. Then there is an exact sequence
\[
1 \to \mathbf{PB}_n \to \mathbf{B}_n \to \mathcal{S}_n \to 1,
\]
where \(\mathcal{S}_n\) is the symmetric group. There is a natural (conjugation) \(\mathcal{S}_n\)-action on \(\mathbf{PB}_n\) permuting the strands, giving rise to \(\mathcal{S}_n\)-representations \(H_q(\mathbf{PB}_n,k)\). There is a natural inclusion morphism \(\iota : \mathbf{PB}_n \to \mathbf{PB}_{n+1}\). From this, \textit{T. Church} and \textit{B. Farb} [Adv. Math. 245, 250--314 (2013; Zbl 1300.20051); Duke Math. J. 164, No. 9, 1833--1910 (2015; Zbl 1339.55004)] showed a remarkable relation between the \(H_q(\mathbf{PB}_n,\mathbb{Q})\) as one varies \(n\) for large \(n\). This is called ``representation stability''.
Consider the sequence
\[
\mathbf{B}_n \to \mathbf{GL}_n(\mathbb{Z}[t, t^{-1}]) \to \mathbf{GL}_n(\mathbb{Z}) \to \mathbf{GL}_n(\mathbb{Z}/m\mathbb{Z}),
\]
where the first map is the Burau representation, the second one is by setting \(t=-1\) and the last one being \(mod \ m\) and \(m\) here is known as the level in the tradition of modular forms. This induces the exact sequence
\[
1 \to \mathbf{B}_n[m] \to \mathbf{B}_n \to \mathcal{Z}_n \to 1,
\]
where \(\mathcal{Z}_n\) is the image of the composition in \(\mathbf{GL}_n(\mathbb{Z}/m\mathbb{Z})\). The group \(\mathcal{Z}_n\) has a quotient (compatible with first sequence above):
\[
1 \to \mathcal{PZ}_n \to \mathcal{Z}_n \to \mathcal{S}_n \to 1.
\]
Again, there is an analogous natural map \(\iota_n : \mathbf{B}_n[m] \to \mathbf{B}_{n+1}[m]\).
Set \(m = 4\) and \(q = 1\). This paper studies the natural \(\mathcal{Z}_n\)-representations \(H_1(\mathbf{B}_n[4];k)\) in the context of representation stability.
The first result (Thm 2.1) is that the dimension of \(H_1(\mathbf{B}_n[4];\mathbb{Q})\) is a degree 4 polynomial in \(n\). Along the line, the authors produce a basis in terms of the natural (Artin) generators of \(\mathbf{B}_n\). The main result (Thm 2.5) shows that \(H_1(\mathbf{B}_n[4];\mathbb{C})\) is a direct sum of four irreducible representations for \(n \ge 4\), inductively defined on \(n\).
For a closed orientable surface \(\Sigma_g\) of genus \(g \ge 1\), denote by \(\mathbf{Mod}_g\) its mapping class group. There is a double cover \(\psi : \Sigma_g \to S^2\) to the 2-sphere, ramified at the \(2g+2\) points on \(S^2\) together with an involution \(\tau\) which induces an element in \(\mathbf{Mod}_g\), preserving the fibres of \(\psi\). Denote by \(\mathbf{SMod}_g\) the mapping class group of \(\Sigma_g\) and the subgroup centralizing \(\tau\), respectively.
Denote by \(\mathbf{Mod}_g[m]\) the subgroup of \(\mathbf{Mod}_g\) fixing \(H_1(\Sigma_g, \mathbb{Z}/m\mathbb{Z})\) (in the spirit of Torelli) and
\[
\mathbf{SMod}_g[m] = \mathbf{SMod}_g \cap \mathbf{Mod}_g[m].
\]
From \(\psi, \tau\), we obtain morphisms
\[
\mathbf{SMod}_g[m] \to \mathbf{B}_{2g+1}[m] \to \mathbf{B}_{2g+3}[m] \to \mathbf{SMod}_{g+1}[m].
\]
From this, the paper shows a similar representation stability, namely that the \(\mathbf{SMod}_g[m]\)-representation \(H_1(\mathbf{SMog}_g[4]; \mathbb{C})\) decomposes into four irreducible representations defined inductively on \(g \ge 2\) (Cor 2.6). There is a quartic polynomial lower bound on the dimension of \(H_1(\mathbf{B}_{2g+1}[0]; \mathbb{Q})\) in terms of genus \(g\).
There are a few more negative results (corollaries) concerning the level 4 Albanese cohomology of \(\mathbf{SMod}_g[4]\) and the \(d\)-characteristic variety of the complement of the braid arrangement moduli space.
Reviewer: Eugene Xia (Tainan)Algorithmic aspects of branched coverings. III/V: Erasing maps, orbispaces, and the Birman exact sequencehttps://zbmath.org/1508.570282023-05-31T16:32:50.898670Z"Bartholdi, Laurent"https://zbmath.org/authors/?q=ai:bartholdi.laurent"Dudko, Dzmitry"https://zbmath.org/authors/?q=ai:dudko.dzmitryAuthors' abstract: Let \(\widetilde{f}:(S^2,\widetilde{A})\circlearrowleft\) be a Thurston map and let \(M(\widetilde{f})\) be its mapping class biset: isotopy classes rel \(\widetilde{A}\) of maps obtained by pre- and post-composing \(\widetilde{f}\) by the mapping class group of \((S^2,\widetilde{A})\). Let \(A\subseteq \widetilde{A}\) be an \(\widetilde{f}\)-invariant subset, and let \(f:(S^2,A)\circlearrowleft\) be the induced map. We give an analogue of the Birman short exact sequence: just as the mapping class group \textbf{Mod}\((S^2,\widetilde{A})\) is an iterated extension of \textbf{Mod}\((S^2,{A})\) by fundamental groups of punctured spheres, \(M(\widetilde{f})\) is an iterated extension of \(M({f})\) by the dynamical biset of \(f\).
Thurston equivalence of Thurston maps classically reduces to a conjugacy problem in mapping class bisets. Our short exact sequence of mapping class bisets allows us to reduce in polynomial time the conjugacy problem in \(M(\widetilde{f})\) to that in \(M({f})\). In case \(\widetilde{f}\) is geometric (either expanding or doubly covered by a hyperbolic torus endomorphism) we show that the dynamical biset \(B(f)\) together with a ``portrait of bisets'' induced by \(\widetilde{A}\) is a complete conjugacy invariant of \(\widetilde{f}\).
Along the way, we give a complete description of bisets of \((2,2,2,2)\)-maps as a crossed product of bisets of torus endomorphisms by the cyclic group of order 2, and we show that non-cyclic orbisphere bisets have no automorphism.
We finally give explicit, efficient algorithms that solve the conjugacy and centralizer problems for bisets of expanding or torus maps.
Reviewer: Samyon R. Nasyrov (Kazan)Cannon-Thurston maps in Kleinian groups and geometric group theoryhttps://zbmath.org/1508.570312023-05-31T16:32:50.898670Z"Mj, Mahan"https://zbmath.org/authors/?q=ai:mj.mahanSummary: We give a survey account of Cannon-Thurston maps, both in the original context of Kleinian groups, as well as in the more general context of geometric group theory. Some of the principal applications are mentioned.
For the entire collection see [Zbl 1502.57002].Polynomial time machines equipped with word problems over algebraic structures as their acceptance criteriahttps://zbmath.org/1508.681212023-05-31T16:32:50.898670Z"Hertrampf, Ulrich"https://zbmath.org/authors/?q=ai:hertrampf.ulrichSummary: We investigate the power of polynomial time machines whose acceptance mechanism is defined by a word problem over some finite semigroup, monoid, or group. For the case of non-solvable groups or monoids (semigroups, resp.) containing non-solvable groups it follows from
[the author et al., ``On the power of polynomial time bit-reductions'', in: Proceedings of the 8th annual structure in complexity theory conference, SCT'93. Los Alamitos, CA: IEEE Computer Society. 200--207 (1993; \url{doi:10.1109/SCT.1993.336526})]
that the according complexity class is PSPACE. For solvable monoids it was shown there that the according class is always a subclass of MOD-PH.
We obtain the following results for finite groups: Commutative groups with \(k\) elements exactly characterize \(\mathrm{co}\)-\(\mathrm{MOD}_k \mathrm{P}\), solvable groups with \(k\) elements, having a composition chain of length \(r\), characterize a class that contains \(\mathrm{co}\)-\(\mathrm{MOD}_k\mathrm{P}\) and is contained in (\(\mathrm{co}\)-\(\mathrm{MOD}_k)^r\mathrm{P}\), the class obtained by \(r\)-fold iterated application of the \(\mathrm{co}\)-\(\mathrm{MOD}_k\)-operator to P. Our results for finite monoids are the following: The classes characterized by commutative finite monoids are the eventually periodic counting classes (see Section 2 for definitions). If we restrict our attention to aperiodic commutative finite monoids, we obtain exactly the classes of bounded counting type, and if we consider idempotent commutative finite monoids, we obtain the classes of the Boolean Hierarchy over NP.
Finally, our results for finite semigroups are: The class characterized by a commutative finite semigroup is representable as a P-disjoint union of classes characterized by commutative finite monoids. Thus the aperiodic and idempotent commutative cases have similar solutions as for monoids.
For the entire collection see [Zbl 0921.00032].Proving group isomorphism theorems (extended abstract)https://zbmath.org/1508.684102023-05-31T16:32:50.898670Z"Zhang, Hantao"https://zbmath.org/authors/?q=ai:zhang.hantaoSummary: We report the first computer proof of the three isomorphism theorems in group theory. The first theorem, the easiest of the three, was considered by Larry Wos as one of challenging problems for theorem provers. The technique we used is conditional completion which consists of one simplification rule called \textit{contextual rewriting} and one inference rule called \textit{clausal superposition}. Conditional completion works on conditional equations made from clauses and is a powerful method for clause-based theorem proving with equality.
For the entire collection see [Zbl 0825.00125].