Recent zbMATH articles in MSC 20Fhttps://www.zbmath.org/atom/cc/20F2021-04-16T16:22:00+00:00WerkzeugAlgorithmic constructions of relative train track maps and CTs.https://www.zbmath.org/1456.200442021-04-16T16:22:00+00:00"Feighn, Mark"https://www.zbmath.org/authors/?q=ai:feighn.mark-e"Handel, Michael"https://www.zbmath.org/authors/?q=ai:handel.michaelSummary: Building on
[\textit{M. Bestvina} et al., Ann. Math. (2) 135, No. 1, 1--51 (1992; Zbl 0757.57004); Ann. Math. (2) 151, No. 2, 517--623 (2000; Zbl 0984.20025)], we proved in
[the authors, Groups Geom. Dyn. 5, No. 1, 39--106 (2011; Zbl 1239.20036)] that every element \(\psi\) of the outer automorphism group of a finite rank free group is represented by a particularly useful relative train track map. In the case that \(\psi\) is rotationless (every outer automorphism has a rotationless power), we showed that there is a type of relative train track map, called a CT, satisfying additional properties. The main result of this paper is that the constructions of these relative train tracks can be made algorithmic. A key step in our argument is proving that it is algorithmic to check if an inclusion \(\mathcal{F}\sqsubset\mathcal{F}'\) of \(\phi\)-invariant free factor systems is reduced. We also give applications of the main result.HNN extensions and stackable groups.https://www.zbmath.org/1456.200462021-04-16T16:22:00+00:00"Hermiller, Susan"https://www.zbmath.org/authors/?q=ai:hermiller.susan-m"Martínez-Pérez, Conchita"https://www.zbmath.org/authors/?q=ai:martinez-perez.conchitaSummary: Stackability for finitely presented groups consists of a dynamical system that iteratively moves paths into a maximal tree in the Cayley graph. Combining with formal language theoretic restrictions yields auto- or algorithmic stackability, which implies solvability of theword problem. In this paperwe give two newcharacterizations of the stackable property for groups, and use these to show that every HNN extension of a stackable group over finitely generated subgroups is stackable. We apply this to exhibit a wide range of Dehn functions that are admitted by stackable and autostackable groups, as well as an example of a stackable group with unsolvable word problem. We use similarmethods to show that there exist finitely presented metabelian groups that are non-constructible but admit an autostackable structure.Classifying spaces for families of subgroups for systolic groups.https://www.zbmath.org/1456.200472021-04-16T16:22:00+00:00"Osajda, Damian"https://www.zbmath.org/authors/?q=ai:osajda.damian"Prytuła, Tomasz"https://www.zbmath.org/authors/?q=ai:prytula.tomaszSummary: We determine the large scale geometry of the minimal displacement set of a hyperbolic isometry of a systolic complex. As a consequence, we describe the centraliser of such an isometry in a systolic group. Using these results, we construct a low-dimensional classifying space for the family of virtually cyclic subgroups of a group acting properly on a systolic complex. Its dimension coincides with the topological dimension of the complex if the latter is at least four. We show that graphical small cancellation complexes are classifying spaces for proper actions and that the groups acting on them properly admitthree-dimensional classifying spaces with virtually cyclic stabilisers. This is achieved by constructing a systolic complex equivariantly homotopy equivalent to a graphical small cancellation complex. On the way we develop a systematic approach to graphical small cancellation complexes. Finally, we construct low-dimensional models for the family of virtually abelian subgroups for systolic, graphical small cancellation, and some CAT(0) groups.Braids, orderings, and minimal volume cusped hyperbolic 3-manifolds.https://www.zbmath.org/1456.570082021-04-16T16:22:00+00:00"Kin, Eiko"https://www.zbmath.org/authors/?q=ai:kin.eiko"Rolfsen, Dale"https://www.zbmath.org/authors/?q=ai:rolfsen.daleSummary: It is well known that there is a faithful representation of braid groups on automorphism groups of free groups, and it is also well known that free groups are bi-orderable. We investigate which \(n\)-strand braids give rise to automorphisms which preserve some bi-ordering of the free group \(F_n\) of rank \(n\). As a consequence of our work we find that of the two minimal volume hyperbolic 2-cusped orientable 3-manifolds, one has bi-orderable fundamental groupwhereas the other does not. We prove a similar result for the 1-cusped case, and have further results for more cusps. In addition, we study pseudo-Anosov braids and find that typically those with minimal dilatation are not order-preserving.A characterization of relatively hyperbolic groups via bounded cohomology.https://www.zbmath.org/1456.200522021-04-16T16:22:00+00:00"Franceschini, Federico"https://www.zbmath.org/authors/?q=ai:franceschini.federicoSummary: It was proved by \textit{L. Mineyev} and \textit{A. Yaman} [``Relative hyperbolicity and bounded cohomology'', Preprint] that, if \((\Gamma,\Gamma')\) is a relatively hyperbolic pair, the comparison map
\[
H_b^k(\Gamma,\Gamma';V)\to H^k(\Gamma,\Gamma';V)
\]
is surjective for every \(k\geq2\), and any bounded \(\Gamma\)-module \(V\). By exploiting results of \textit{D. Groves} and \textit{J. F. Manning} [Isr. J. Math. 168, 317--429 (2008; Zbl 1211.20038)], we give another proof of this result. Moreover, we prove the opposite implication under weaker hypotheses than the ones required by Mineyev and Yaman [loc. cit.] .On the numbers of the form \(x^2 + 11y^2\).https://www.zbmath.org/1456.110422021-04-16T16:22:00+00:00"Kreuzer, Martin"https://www.zbmath.org/authors/?q=ai:kreuzer.martin"Rosenberger, Gerhard"https://www.zbmath.org/authors/?q=ai:rosenberger.gerhardPrimes expressible as \(x^2+ny^2\) for a positive integer \(n\) were
studied by Euler. For number like \(n=1,2,3\) etc., the quadratic
reciprocity law already gives characterization of these primes as
precisely those belonging to certain congruence classes. However,
there are only finitely many \(n\) for which such a characterization
can be given. Euler wrote down a list of 65 `Idoneus Numerus'
(roughly translated as `convenient numbers'), the largest of which
is 1848; this list is expected to be complete but this is not yet
proved. Here, \(n\) is said to be `convenient' if each positive
integer that is expressible as \(x^2+ny^2\) with \(\gcd(x^2,ny^2)=1\), is
uniquely so expressible if, and only if, \(n\) is the power of a prime
or twice such a number. The smallest ``inconvenient'' number is \(11\).
Ring class field theory enables us to characterize prime numbers
that are expressible as \(x^2+ny^2\). One considers the order
\(\mathbb{Z}[\sqrt{-n}]\) in the corresponding imaginary quadratic
field (so, this is the full ring of integers when \(n \equiv 1,2\) mod
\(4\)) and its ring class field. If \(p\) is an odd prime not dividing
\(n\), and also, not dividing the discriminant of a monic integer
polynomial \(f_n\) of degree \(h(-4n)\) whose root is an algebraic
integer that generates the ring class field, the necessary and
sufficient condition for \(p\) to be expressible as \(x^2+ny^2\) is that
\(-n\) is a quadratic residue mod \(p\) and \(f_n(a) \equiv 0\) mod \(p\)
has a solution \(a \in \mathbb{Z}\). For \(n \equiv 1\) or \(2\) mod \(4\),
this condition is in terms of the class field of
\(\mathbb{Q}(\sqrt{-n})\).
The first inconvenient number is \(11\) and, in particular, the
above-mentioned theorem characterizes primes expressible as
\(x^2+11y^2\); the polynomial \(f_{11}(x) = x^3-2x^2+2x-2\) in this
case. However, the authors of the paper address the more difficult
task of characterizing positive integers (not just primes) that are
expressible as \(x^2+ 11y^2\) with gcd\((x,11y)=1\). They use the
detailed structure of the class group \(G_{11}\) of level \(11\) to
obtain and phrase their results. The group \(G_{11}\) (is the image in
\(PSL_2(\mathbb{R})\) of the set of matrices of the form \(\begin{pmatrix} a
\sqrt{11} & b \\ c & d \sqrt{11}\end{pmatrix}\) or \(\begin{pmatrix} a & b \sqrt{11} \cr
c \sqrt{11} & d\end{pmatrix}\) where \(a,b,c,d\) are integers and the matrix has
determinant \(1\).
In particular, they show that \(G_{11}\) has four conjugacy classes of
elliptic elements of order \(2\) represented by \(t_1,t_2,t_3,t_4\) say,
where \(t_1 = \begin{pmatrix} 0 & 1 \cr -1 & 0\end{pmatrix}\). A conjugate of \(t_1\) in
\(G_{11}\) gives a matrix in \(G_{11}\) whose \((1,2)\)-th entry is
expressible as \(x^2+11y^2\) with gcd\((x,11y)=1\). Conversely,
corresponding to any positive integer \(n\) for which \(-11\) is a
quadratic residue modulo \(n\), the authors construct an elliptic
element \(A_n\) of order \(2\) in \(G_{11}\) which is conjugate to exactly
one of the \(t_i\)'s, and this is \(t_1\) if and only if \(n\) is
expressible as \(x^2+11y^2\). Thus, there are four sets \(S_i\)
consisting of those \(n\) for which the \((1,2)\)-th entry of \(A_n\) is
\(n\) and \(A_n\) is conjugate to \(t_i\) (\(1 \leq i \leq 4\)). The authors
prove quite easily that \(S_2=S_3\) and \(S_4=2S_2\) consists precisely
the positive integers \(\equiv 2\) modulo \(4\). The set of interest is
\(S_1\) and the difficult task they accomplish is to distinguish
between \(S_1\) and \(S_2\).
For the entire collection see [Zbl 1435.20002].
Reviewer: Balasubramanian Sury (Bangalore)From Schritte and Wechsel to Coxeter groups.https://www.zbmath.org/1456.000972021-04-16T16:22:00+00:00"Schmidmeier, Markus"https://www.zbmath.org/authors/?q=ai:schmidmeier.markusSummary: The PLR-moves of neo-Riemannian theory, when considered as reflections on the edges of an equilateral triangle, define the Coxeter group \(\widetilde{S}_3\). The elements are in a natural one-to-one correspondence with the triangles in the infinite Tonnetz. The left action of \(\widetilde{S}_3\) on the Tonnetz gives rise to interesting chord sequences. We compare the system of transformations in \(\widetilde{S}_3\) with the system of Schritte and Wechsel introduced by Hugo Riemann in 1880. Finally, we consider the point reflection group as it captures well the transition from Riemann's infinite Tonnetz to the finite Tonnetz of neo-Riemannian theory.
For the entire collection see [Zbl 1425.00082].Some linear groups with \(G\)-core-free subspaces.https://www.zbmath.org/1456.200582021-04-16T16:22:00+00:00"Dixon, Martyn R."https://www.zbmath.org/authors/?q=ai:dixon.martyn-russell"Kurdachenko, Leonid A."https://www.zbmath.org/authors/?q=ai:kurdachenko.leonid-a"Subbotin, Igor Ya."https://www.zbmath.org/authors/?q=ai:subbotin.igor-yaLet $V$ be a vector space over a field $F$ and consider some subgroup $G$ of $\Aut_FV$. The authors discuss the situation where each subspace $W$ of $V$ satisfies either $WG = W$ or $\bigcap_{g\in G}Wg = \langle 1\rangle$. For example we clearly have this situation whenever $V$ is $FG$-irreducible. The authors prove a substantial number of lemmas studying various special cases. These are then put together to produce their main theorem, which lists 9 properties of the situation where $G$ is hypercentral and the intersection $M$ of all the non-zero $FG$-submodules of $V$ is neither $V$ nor $\langle 0\rangle$. Many of these are not easy to summarize briefly, but for example Property 6 is that $G$ is abelian and Properties 7, 8 and 9 describe $G$ if $C_G(M)$ is $G$, is neither $G$ nor $\langle 1\rangle$ and is $\langle 1\rangle$, respectively. In [Rocky Mt. J. Math. 50, No. 6, 2023--2034 (2020; Zbl 07297910)], the authors consider to the dual situation where each subspace $W$ of $V$ satisfies $WG = W$ or $WG = V$. Finally, the paper under review presents an example illustrating the situation studied in their main theorem.
Reviewer: B. A. F. Wehrfritz (London)Quasi-invariants in characteristic \(p\) and twisted quasi-invariants.https://www.zbmath.org/1456.812522021-04-16T16:22:00+00:00"Ren, Michael"https://www.zbmath.org/authors/?q=ai:ren.michael-s"Xu, Xiaomeng"https://www.zbmath.org/authors/?q=ai:xu.xiaomengSummary: The spaces of quasi-invariant polynomials were introduced by \textit{O. A. Chalykh} and \textit{A. P. Veselov} [Commun. Math. Phys.126, No. 3, 597-611 (1990; Zbl 0746.47025)]. Their Hilbert series over fields of characteristic 0 were computed by \textit{M. Feigin} and \textit{A. P. Veselov} [Int. Math. Res. Not. 2002, No. 10, 521-545 (2002; Zbl 1009.20044)]. In this paper, we show some partial results and make two conjectures on the Hilbert series of these spaces over fields of positive characteristic. On the other hand, \textit{A. Braverman, P. Etingof}, and \textit{M. Finkelberg} [preprint (2020; \url{arxiv:1611.10216})] introduced the spaces of quasi-invariant polynomials twisted by a monomial. We extend some of their results to the spaces twisted by a smooth function.Stability in outer space.https://www.zbmath.org/1456.200452021-04-16T16:22:00+00:00"Hamenstädt, Ursula"https://www.zbmath.org/authors/?q=ai:hamenstadt.ursula"Hensel, Sebastian"https://www.zbmath.org/authors/?q=ai:hensel.sebastian-wolfgangSummary: We characterize strongly Morse quasi-geodesics in Outer space as quasi-geodesics which project to quasi-geodesics in the free factor graph. We define convex cocompact subgroups of \(\mathrm{Out}(F_n)\) as subgroups such that an orbit map in the free factor graph is a quasi-isometric embedding, and we characterize such groups via their action on Outer space in a way which resembles the characterization of convex cocompact subgroups of mapping class groups.Thompson groups for systems of groups, and their finiteness properties.https://www.zbmath.org/1456.200502021-04-16T16:22:00+00:00"Witzel, Stefan"https://www.zbmath.org/authors/?q=ai:witzel.stefan"Zaremsky, Matthew C. B."https://www.zbmath.org/authors/?q=ai:zaremsky.matthew-curtis-burkholderSummary: We describe a procedure for constructing a generalized Thompson group out of a family of groups that is equipped with what we call a cloning system. The previously known Thompson groups \(F\), \(V\), \(V_{\mathrm{br}}\) and \(F_{\mathrm{br}}\) arise from this procedure using, respectively, the systems of trivial groups, symmetric groups, braid groups and pure braid groups.
We give new examples of families of groups that admit a cloning system and study how the finiteness properties of the resulting generalized Thompson group depend on those of the original groups. The main new examples here include upper triangular matrix groups, mock reflection groups, and loop braid groups. For generalized Thompson groups of upper triangular matrix groups over rings of \(S\)-integers of global function fields, we develop new methods for (dis-)proving finiteness properties, and show that the finiteness length of the generalized Thompson group is exactly the limit inferior of the finiteness lengths of the groups in the family.Subspace arrangements, BNS invariants, and pure symmetric outer automorphisms of right-angled Artin groups.https://www.zbmath.org/1456.200432021-04-16T16:22:00+00:00"Day, Matthew B."https://www.zbmath.org/authors/?q=ai:day.matthew-b"Wade, Richard D."https://www.zbmath.org/authors/?q=ai:wade.richard-dSummary: We introduce a homology theory for subspace arrangements, and use it to extract a new system of numerical invariants from the Bieri-Neumann-Strebel invariant of a group. We use these to characterize when the set of basis conjugating outer automorphisms (a.k.a. the pure symmetric outer automorphism group) of a right-angled Artin group is itself a right-angled Artin group.Automorphisms of the cube \(n^d\).https://www.zbmath.org/1456.051212021-04-16T16:22:00+00:00"Dvořák, Pavel"https://www.zbmath.org/authors/?q=ai:dvorak.pavel"Valla, Tomáš"https://www.zbmath.org/authors/?q=ai:valla.tomasSummary: Consider a hypergraph \(n^d\) where the vertices are points of the \(d\)-dimensional cube \([ n ]^d\) and the edges are all sets of \(n\) points such that they are in one line. We study the structure of the group of automorphisms of \(n^d\), i.e., permutations of points of \([ n ]^d\) preserving the edges. In this paper we provide a complete characterization. Moreover, we consider the Colored Cube Isomorphism problem of deciding whether for two colorings of the vertices of \(n^d\) there exists an automorphism of \(n^d\) preserving the colors. We show that this problem is \(\mathsf{GI} \)-complete.On normal subgroups of the braided Thompson groups.https://www.zbmath.org/1456.200512021-04-16T16:22:00+00:00"Zaremsky, Matthew C. B."https://www.zbmath.org/authors/?q=ai:zaremsky.matthew-curtis-burkholderSummary: We inspect the normal subgroup structure of the braided Thompson groups \(V_{\mathrm{br}}\) and \(F_{\mathrm{br}}\). We prove that every proper normal subgroup of \(V_{\mathrm{br}}\) lies in the kernel of the natural quotient \(V_{\mathrm{br}}\twoheadrightarrow V\), and we exhibit some families of interesting such normal subgroups. For \(F_{\mathrm{br}}\), we prove that for any normal subgroup \(N\) of \(F_{\mathrm{br}}\), either \(N\) is contained in the kernel of \(F_{\mathrm{br}}\twoheadrightarrow F\), or else \(N\) contains \([F_{\mathrm{br}},F_{\mathrm{br}}]\). We also compute the Bieri-Neumann-Strebel invariant \(\Sigma^1(F_{\mathrm{br}})\), which is a useful tool for understanding normal subgroups containing the commutator subgroup.Cohomology of group theoretic Dehn fillings. I: Cohen-Lyndon type theorems.https://www.zbmath.org/1456.200532021-04-16T16:22:00+00:00"Sun, Bin"https://www.zbmath.org/authors/?q=ai:sun.binDehn surgery is an operation in 3-dimensional topology which consists in modifying a 3-manifold by cutting off a solid torus and then gluing it back in a different way. A famous result in this direction is Thurston's hyperbolic Dehn filling theorem which asserts the existence of hyperbolic structures on a large class of manifolds obtained by Dehn filling. There is also a group theoretic settings for Dehn filling, which had many applications. For instance, the solution of the virtually Haken, by \textit{I. Agol} [Doc. Math. 18, 1045--1087 (2013; Zbl 1286.57019)] uses Dehn fillings of hyperbolic groups.
In the paper under review, the author studies the cohomology of group theoretic Dehn fillings, in particular free product structure of Dehn filling kernels, which he calls the Cohen-Lyndon property. Using this, he describes the structure of relative relation modules of Dehn fillings.
Reviewer: Athanase Papadopoulos (Strasbourg)The intrinsic hyperplane arrangement in an arbitrary irreducible representation of the symmetric group.https://www.zbmath.org/1456.200082021-04-16T16:22:00+00:00"Tsilevich, N. V."https://www.zbmath.org/authors/?q=ai:tsilevich.natalia-v"Vershik, A. M."https://www.zbmath.org/authors/?q=ai:vershik.anatoli-m"Yuzvinsky, S."https://www.zbmath.org/authors/?q=ai:yuzvinskij.s-aThe main result of the paper says that for every irreducible complex representation \(\pi_\lambda\) of the symmetric group \(S_n\) there exists a canonical ``intrinsic'' hyperplane arrangement \(A_\lambda\) in the space \(V_\lambda\) of this representation. In the case of the natural representation of \(S_n\),
it coincides with the so-called braid arrangement, studied by [\textit{V. I. Arnol'd}, Math. Notes 5, 138--140 (1969; Zbl 0277.55002); translation from Mat. Zametki 5, 227--231 (1969)] in connection
with the cohomology of the group of pure1 braids.
This arrangement has a natural description in terms of invariant subspaces of Young subgroups, and enjoys a number of remarkable properties. An attempt to generalize Arnold's construction to other irreducible representations of symmetric groups has led the authors to quite dissimilar arrangements,
whose complements, in particular, are not \(K(\pi,1)\) spaces.
Reviewer: Marek Golasiński (Olsztyn)Every group is the outer automorphism group of an HNN-extension of a fixed triangle group.https://www.zbmath.org/1456.200342021-04-16T16:22:00+00:00"Logan, Alan D."https://www.zbmath.org/authors/?q=ai:logan.alan-dA remarkable result of \textit{I. Bumagin} and \textit{D. T. Wise} [J. Pure Appl. Algebra 200, No. 1--2, 137--147 (2005; Zbl 1082.20021)] says that every countable group \(Q\) can be realized as the outer automorphism group of a finitely generated group \(G_Q\). These groups \(G_Q\) have, for a given \(Q\), often special additional properties. The main result of the paper is as follows. Fix a triangle group \(T_i=\langle a,b\mid a^i=b^i=(ab)^i=1\rangle\) with \(i\geq 6\). For every countable group, given by a countable presentation \(P\), there exists an automorphism-induced HNN-extension \(T_P\) of \(T_i\) such that Out\((T_P)\cong Q=\pi_1(P)\) and Aut\((T_P)\cong T_P\rtimes Q\).
The HNN-extensions \(T_P\) of \(T_i\) are explicitly constructed. Interesting enough, the single steps lead to nice residual and malnormal properties of the constructed groups.
Reviewer: Gerhard Rosenberger (Hamburg)Sets of prime power order generators of finite groups.https://www.zbmath.org/1456.200112021-04-16T16:22:00+00:00"Stocka, A."https://www.zbmath.org/authors/?q=ai:stocka.agnieszkaA finite group \(G\) is said to have property \(\mathcal B\) if all its minimal generating sets have the same size. In a previous joint paper with \textit{J. Krempa} [J. Algebra 405, 122--134 (2014; Zbl 1319.20029)], the author considered groups satisfying analogous conditions, but only for sets of generators of prime power orders. More precisely, a subset \(X\) of prime power order elements of a finite group \(G\) is called pp-independent if there is no proper subset
\(Y\) of \(X\) such that \(\langle Y,\Phi(G)\rangle=\langle X,\Phi(G)\rangle\)
(where \(\Phi(G)\) is the Frattini
subgroup of \(G\)) and \(G\) has property \(\mathcal{B}_{pp}\) if all pp-independent generating sets of \(G\) have the same size.
Moreover \(G\) has the pp-basis exchange
property if for any pp-independent generating sets \(B_1, B_2\) of \(G\) and
\(x \in B_1\) there exists \(y \in B_2\) such that \((B_1 \setminus \{x\})\cup \{y\}\) is a pp-independent generating set of \(G.\)
The first result in this paper is a description of the finite solvable groups with property
\(\mathcal B_{pp}\): modulo the Frattini subgroup, they are direct products of factors of pairwise coprime orders, each of which is either an elementary abelian \(p\)-group or
a suitable cyclic \(q\)-extensions of an elementary abelian \(p\)-group. The second result describes the finite solvable groups with the pp-basis exchange
property: modulo the Frattini subgroup, they are \(\mathcal B_{pp}\)-groups in which all pp-elements have prime orders.
Reviewer: Andrea Lucchini (Padova)On the image of a word map with constants of a simple algebraic group.https://www.zbmath.org/1456.200572021-04-16T16:22:00+00:00"Gnutov, F. A."https://www.zbmath.org/authors/?q=ai:gnutov.f-a"Gordeev, N. L."https://www.zbmath.org/authors/?q=ai:gordeev.nikolai-lLet \(G\) be an algebraic group, \(\tilde w: G^n \rightarrow {G}\) a word map with constants, let \(T\) be a fixed maximal torus of \(G\), let \(W\) be the Weil group of \(G\), and let \(\pi: G \rightarrow T/W\) be the factor morphism.
The main results that the authors establish is that for the adjoint group of \(G\) of type \(A_r, D_r\), or \(E_r\), the
map \(\pi\circ{\tilde{w}}\) is a constant map only for the words \(vgv^{-1}\), where \(g \in G\) and \(v\) is a word with constants.
Reviewer: Erich W. Ellers (Toronto)No Tits alternative for cellular automata.https://www.zbmath.org/1456.200482021-04-16T16:22:00+00:00"Salo, Ville"https://www.zbmath.org/authors/?q=ai:salo.ville-o|salo.villeSummary: We show that the automorphism group of a one-dimensional full shift (the group of reversible cellular automata) does not satisfy the Tits alternative. That is, we construct a finitely-generated subgroup which is not virtually solvable yet does not contain a free group on two generators. We give constructions both in the two-sided case (spatially acting group \(\mathbb Z)\) and the one-sided case (spatially acting monoid \(\mathbb N\), alphabet size at least eight). Lack of Tits alternative follows for several groups of symbolic (dynamical) origin: automorphism groups of two-sided one-dimensional uncountable sofic shifts, automorphism groups of multidimensional subshifts of finite type with positive entropy and dense minimal points, automorphism groups of full shifts over non-periodic groups, and the mapping class groups of two-sided one-dimensional transitive SFTs. We also show that the classical Tits alternative applies to one-dimensional (multi-track) reversible linear cellular automata over a finite field.Higher dimensional divergence for mapping class groups.https://www.zbmath.org/1456.200412021-04-16T16:22:00+00:00"Behrstock, Jason"https://www.zbmath.org/authors/?q=ai:behrstock.jason-a"Druţu, Cornelia"https://www.zbmath.org/authors/?q=ai:drutu.corneliaSummary: In this paper we investigate the higher dimensional divergence functions of mapping class groups of surfaces. We show that these functions exhibit phase transitions at the quasi-flat rank (as measured by \(3 \cdot \text{ genus }+ \text{ number of punctures }- 3)\).The non-commuting, non-generating graph of a nilpotent group.https://www.zbmath.org/1456.050722021-04-16T16:22:00+00:00"Cameron, Peter J."https://www.zbmath.org/authors/?q=ai:cameron.peter-j"Freedman, Saul D."https://www.zbmath.org/authors/?q=ai:freedman.saul-d"Roney-Dougal, Colva M."https://www.zbmath.org/authors/?q=ai:roney-dougal.colva-mSummary: For a nilpotent group \(G\), let \(\Xi(G)\) be the difference between the complement of the generating graph of \(G\) and the commuting graph of \(G\), with vertices corresponding to central elements of \(G\) removed. That is, \( \Xi(G)\) has vertex set \(G \setminus Z(G)\), with two vertices adjacent if and only if they do not commute and do not generate \(G\). Additionally, let \(\Xi^+(G)\) be the subgraph of \(\Xi(G)\) induced by its non-isolated vertices. We show that if \(\Xi(G)\) has an edge, then \(\Xi^+(G)\) is connected with diameter \(2\) or \(3\), with \(\Xi(G) = \Xi^+(G)\) in the diameter \(3\) case. In the infinite case, our results apply more generally, to any group with every maximal subgroup normal. When \(G\) is finite, we explore the relationship between the structures of \(G\) and \(\Xi(G)\) in more detail.Permanence properties of the second nilpotent product of groups.https://www.zbmath.org/1456.200242021-04-16T16:22:00+00:00"Sasyk, Román"https://www.zbmath.org/authors/?q=ai:sasyk.romanThe following results are given.
1) Amenability, the Haagerup property, the Kazhdan property (T) and exactness are preserved under taking second nilpotent product of groups.
2) For two countable groups \(H\) and \(G\) with the Haagerup property, the restricted second nilpotent wreath product \(Hwr_2G\) has also this property.
3) For two countable groups \(A\) and \(G\), if \(A\) is abelian then \(Awr_2G\) is unitarizable if and only if \(G\) is amenable.
Reviewer: V. A. Roman'kov (Omsk)Coloured Neretin groups.https://www.zbmath.org/1456.220082021-04-16T16:22:00+00:00"Lederle, Waltraud"https://www.zbmath.org/authors/?q=ai:lederle.waltraudLederle establishes and utilises a connection between topological full groups of étale groupoids associated to one-sided shifts after \textit{H. Matui} [Proc. Lond. Math. Soc. (3) 104, No. 1, 27--56 (2012; Zbl 1325.19001); J. Reine Angew. Math. 705, 35--84 (2015; Zbl 1372.22006)] and almost automorphism groups of trees after \textit{Yu. A. Neretin} [Russ. Acad. Sci., Izv., Math. 41, No. 2, 1072--1085 (1992; Zbl 0789.22036); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 56, No. 5, 1072--1085 (1992)] to construct compactly generated, locally compact totally disconnected groups that are virtually simple and do not have lattices.
An étale groupoid is a small category in which all morphisms are isomorphisms and whose object and morphism space are topologised in such a way that all structure maps are continuous, and source and range are local homeomorphisms. The associated topological full group is defined as a subgroup of the homeomorphism group of the object space, which is typically a Cantor space.
An almost automorphism of a locally finite tree \(T\) is an equivalence class of forest isomorphisms that arise from removing finite subtrees from \(T\). It can be viewed as a homeomorphism of the boundary of \(T\), which is also a Cantor space.
Lederle shows, using the newly established connection above and Matui's theory, that the groups of almost automorphisms arising from certain Burger-Mozes universal groups [\textit{M. Burger} and \textit{S. Mozes}, Publ. Math., Inst. Hautes Étud. Sci. 92, 113--150 (2000; Zbl 1007.22012)] have the desired, rare properties. The no-lattice argument follows the approach of [\textit{U. Bader} et al., Bull. Lond. Math. Soc. 44, No. 1, 55--67 (2012; Zbl 1239.22007)].
In particular, Lederle provides a family of candidates of groups without invariant random subgroups.
Reviewer: Stephan Tornier (Newcastle)Cocompact cubulations of mixed 3-manifolds.https://www.zbmath.org/1456.200492021-04-16T16:22:00+00:00"Tidmore, Joseph"https://www.zbmath.org/authors/?q=ai:tidmore.josephSummary: In this paper, we complete the classification of which compact 3-manifolds have a virtually compact special fundamental group by addressing the case of mixed 3-manifolds. A compact aspherical 3-manifold \(M\) is mixed if its JSJ decomposition has at least one JSJ torus and at least one hyperbolic block. We show \(\pi_1M\) is virtually compact special if and only if \(M\) is chargeless, i.e. each interior Seifert fibered block has a trivial Euler number relative to the fibers of adjacent blocks.Relatively extra-large Artin groups.https://www.zbmath.org/1456.200352021-04-16T16:22:00+00:00"Juhász, Arye"https://www.zbmath.org/authors/?q=ai:juhasz.aryeSummary: Let \(n\geq 2\) be an integer and let \(N\) be an \(n \times n\) symmetric matrix with 1's on the main diagonal and natural numbers \(n_{ij}\neq 1\) as off-diagonal entries. (0 is a natural number). Let \(X=\{x_1,\ldots ,x_n\}\) and let \(F\) be the free group on \(X\). For every non-zero off-diagonal entry \(n_{ij}\) of \(N\) define a word \(R_{ij}:=UV^{-1}\) in \(F\), where \(U\) is the initial subword of \((x_ix_j)^{n_{ij}}\) of length \(n_{ij}\) and \(V\) is the initial subword of \((x_jx_i)^{n_{ij}}\) of length \(n_{ij}\), \(1\leq i,j \leq n\). Let \(A\) be the group given by the presentation \(\langle X\mid R_{ij},\,n_{ij}\geq 2 \rangle\). \(A\) is called the \textit{Artin group defined by \(N\), with standard generators \(X\)}. Let \(Y=\{x_1,\ldots ,x_k\}, k<n\) and let \(N_Y\) be the submatrix of \(N\) corresponding to \(Y\). Let \(H=\langle Y \rangle\). We call \(A\) \textit{extra-large relative to \(H\)} if \(N\) subdivides into submatrices \(N_Y,B,C\) and \(D\) of sizes \(k\times k,k\times l,l\times k,l\times l\), respectively \((l+k=n)\) such that every non zero element of \(B\) and \(C\) is at least 4 and every off-diagonal non-zero entry of \(D\) is at least 3. No condition on \(N_Y\). In this work we solve the word problem for such \(A\), show that \(A\) is torsion free and show that \(A\) has property \(K(\pi,1)\), provided that \(H\) has these properties, correspondingly. We also compute the homology and cohomology of \(A\), relying on that of \(H\). The two main tools used are Howie diagrams corresponding to relative presentations of \(A\) with respect to \(H\) and small cancellation theory with mixed small cancellation conditions.Commensurability for certain right-angled Coxeter groups and geometric amalgams of free groups.https://www.zbmath.org/1456.200392021-04-16T16:22:00+00:00"Dani, Pallavi"https://www.zbmath.org/authors/?q=ai:dani.pallavi"Stark, Emily"https://www.zbmath.org/authors/?q=ai:stark.emily"Thomas, Anne"https://www.zbmath.org/authors/?q=ai:thomas.anne|thomas.anne.1Summary: We give explicit necessary and sufficient conditions for the abstract commensurability of certain families of 1-ended, hyperbolic groups, namely right-angled Coxeter groups defined by generalized \(\Theta\)-graphs and cycles of generalized \(\Theta\)-graphs, and geometric amalgams of free groups whose JSJ graphs are trees of diameter \(\leq 4\). We also show that if a geometric amalgamof free groups has JSJ graph a tree, then it is commensurable to a right-angled Coxeter group, and give an example of a geometric amalgam of free groups which is not quasi-isometric (hence not commensurable) to any group which is finitely generated by torsion elements. Our proofs involve a new geometric realization of the right-angled Coxeter groups we consider, such that covers corresponding to torsion-free, finite-index subgroups are surface amalgams.Some virtual limit groups.https://www.zbmath.org/1456.200252021-04-16T16:22:00+00:00"Wise, Daniel T."https://www.zbmath.org/authors/?q=ai:wise.daniel-tSummary: We show that certain graphs of free groups with cyclic edge groups have a finite index subgroup that is fully residually free.Classifying virtually special tubular groups.https://www.zbmath.org/1456.200542021-04-16T16:22:00+00:00"Woodhouse, Daniel J."https://www.zbmath.org/authors/?q=ai:woodhouse.daniel-jSummary: A group is tubular if it acts on a tree with \(\mathbb{Z}^2\) vertex stabilizers and \(\mathbb{Z}\) edge stabilizers. We prove that a tubular group is virtually special if and only if it acts freely on a locally finite \(\mathrm{CAT}(0)\) cube complex. Furthermore, we prove that if a tubular group acts freely on a finite dimensional \(\mathrm{CAT}(0)\) cube complex, then it virtually acts freely on a three dimensional \(\mathrm{CAT}(0)\) cube complex.Quasi-automorphisms of the infinite rooted 2-edge-coloured binary tree.https://www.zbmath.org/1456.200592021-04-16T16:22:00+00:00"Nucinkis, Brita E. A."https://www.zbmath.org/authors/?q=ai:nucinkis.brita-e-a"John-Green, Simon St."https://www.zbmath.org/authors/?q=ai:john-green.simon-stSummary: We study the group \(QV\), the self-maps of the infinite 2-edge coloured binary tree which preserve the edge and colour relations at cofinitely many locations. We introduce related groups \(QF\), \(QT\), \(\widetilde{Q}T\), and \(\widetilde{Q}V\), prove that \(QF\), \(\widetilde{Q}T\), and \(\widetilde{Q}V\) are of type \(\mathrm{F}_\infty\), and calculate finite presentations for them. We calculate the normal subgroup structure of all 5 groups, the Bieri-Neumann-Strebel-Renz invariants of \(QF\), and discuss the relationship of all 5 groups with other generalisations of Thompson's groups.Modules over group rings of groups with restrictions on the system of all proper subgroups.https://www.zbmath.org/1456.200302021-04-16T16:22:00+00:00"Dashkova, Olga"https://www.zbmath.org/authors/?q=ai:dashkova.olga-yuSummary: We consider the class \(\mathfrak{M}\) of \(\mathbf{R}\)-modules where \(\mathbf{R}\) is an associative ring. Let \(A\) be a module over a group ring \(\mathbf{R} G\), \(G\) be a group and let \(\mathfrak{L}(G)\) be the set of all proper subgroups of \(G\). We suppose that if \(H \in \mathfrak{L}(G)\) then \(A/C_A(H)\) belongs to \(\mathfrak{M}\). We investigate an \(\mathbf{R} G\)-module \(A\) such that \(G \neq G'\), \(C_G(A) = 1\). We study the cases: 1) \(\mathfrak{M}\) is the class of all artinian \(\mathbf{R}\)-modules, \(\mathbf{R}\) is either the ring of integers or the ring of \(p\)-adic integers; 2) \(\mathfrak{M}\) is the class of all finite \(\mathbf{R}\)-modules, \(\mathbf{R}\) is an associative ring; 3) \(\mathfrak{M}\) is the class of all finite \(\mathbf{R}\)-modules, \(\mathbf{R}=F\) is a finite field.Conjugacy in relatively extra-large Artin groups.https://www.zbmath.org/1456.200362021-04-16T16:22:00+00:00"Juhász, Arye"https://www.zbmath.org/authors/?q=ai:juhasz.aryeSummary: In this work we consider conjugacy of elements and parabolic subgroups in details, in a new class of Artin groups, introduced in an earlier work, which may contain arbitrary parabolic subgroups. In particular, we find algorithmically minimal representatives of elements in a conjugacy class and also an algorithm to pass from one minimal representative to the others.The local nilpotence theorem for 4-Engel groups revisited.https://www.zbmath.org/1456.200372021-04-16T16:22:00+00:00"Traustason, Gunnar"https://www.zbmath.org/authors/?q=ai:traustason.gunnarSummary: The proof of the local nilpotence theorem for 4-Engel groups was completed by \textit{G. Havas} and \textit{M. R. Vaghan-Lee} [Int. J. Algebra Comput. 15, No. 4, 649--682 (2005; Zbl 1087.20029)]. The complete proof on the other hand is spread over several articles and the aim of this paper is to give a complete coherent linear version. In the process we are also able to make a few simplifications and in particular we are able to merge two of the key steps into one.Finite simple groups of low rank: Hurwitz generation and \((2,3)\)-generation.https://www.zbmath.org/1456.200102021-04-16T16:22:00+00:00"Pellegrini, Marco"https://www.zbmath.org/authors/?q=ai:pellegrini.marco-antonio"Tamburini, Maria"https://www.zbmath.org/authors/?q=ai:tamburini.maria-chiaraSummary: Let us consider the set of non-abelian finite simple groups which admit non-trivial irreducible projective representations of degree \(\leq 7\) over an algebraically closed field \(F\) of characteristic \(p\geq 0\). We survey some recent results which lead to the complete list of the groups in this set which are \((2, 3, 7)\)-generated and of those which are \((2,3)\)-generated.On central endomorphisms of a group.https://www.zbmath.org/1456.200262021-04-16T16:22:00+00:00"Russo, Alessio"https://www.zbmath.org/authors/?q=ai:russo.alessioSummary: Let \(\Gamma\) be a normal subgroup of the full automorphism group \(\mathrm{Aut}(G)\) of a group \(G\), and assume that \(\mathrm{Inn}(G)\leq \Gamma\). An endomorphism \(\sigma\) of \(G\) is said to be \(\Gamma\)-central if \(\sigma\) induces the the identity on the factor group \(G/C_G(\Gamma)\). Clearly, if \(\Gamma =\mathrm{Inn}(G)\), then a \(\Gamma\)-central endomorphism is a central endomorphism. In this article the conditions under which a \(\Gamma\)-central endomorphism of a group is an automorphism are investigated.Bias of group generators in finite and profinite groups: known results and open problems.https://www.zbmath.org/1456.200772021-04-16T16:22:00+00:00"Crestani, Eleonora"https://www.zbmath.org/authors/?q=ai:crestani.eleonora"Lucchini, Andrea"https://www.zbmath.org/authors/?q=ai:lucchini.andreaSummary: We analyze some properties of the distribution \(Q_{G,k}\) of the first component in a \(k\)-tuple chosen uniformly in the set of all the \(k\)-tuples generating a finite group \(G\) (the limiting distribution of the product replacement algorithm). In particular, we concentrate our attention on the study of the variation distance \(\beta_k(G)\) between \(Q_{G,k}\) and the uniform distribution. We review some known results, analyze several examples and propose some intriguing open questions.On double cosets with the trivial intersection property and Kazhdan-Lusztig cells in \(S_n\).https://www.zbmath.org/1456.200012021-04-16T16:22:00+00:00"McDonough, Thomas P."https://www.zbmath.org/authors/?q=ai:mcdonough.thomas-p"Pallikaros, Christos A."https://www.zbmath.org/authors/?q=ai:pallikaros.christos-aSummary: For a composition \(\lambda\) of \(n\) our aim is to obtain reduced forms for all the elements in the \(w_{J(\lambda)}\), the longest element of the standard parabolic subgroup of \(S_n\) corresponding to \(\lambda\). We investigate how far this is possible to achieve by looking at elements of the form \(w_{J(\lambda)}d\), where \(d\) is a prefix of an element of minimum length in a \((W_{J(\lambda)},B)\) double coset with the trivial intersection property, \(B\) being a parabolic subgroup of \(S_n\) whose type is ``dual'' to that of \(W_{J(\lambda)}\).On soluble groups whose subnormal subgroups are inert.https://www.zbmath.org/1456.200182021-04-16T16:22:00+00:00"Dardano, Ulderico"https://www.zbmath.org/authors/?q=ai:dardano.ulderico"Rinauro, Silvana"https://www.zbmath.org/authors/?q=ai:rinauro.silvanaSummary: A subgroup \(H\) of a group \(G\) is called inert if, for each \(g\in G\), the index of \(H\cap H^g\) in \(H\) is finite. We give a classification of soluble-by-finite groups \(G\) in which subnormal subgroups are inert in the cases where \(G\) has no nontrivial torsion normal subgroups or \(G\) is finitely generated.A note on groups with many locally supersoluble subgroups.https://www.zbmath.org/1456.200312021-04-16T16:22:00+00:00"De Giovanni, Francesco"https://www.zbmath.org/authors/?q=ai:de-giovanni.francesco"Trombetti, Marco"https://www.zbmath.org/authors/?q=ai:trombetti.marcoSummary: It is proved here that if \(G\) is a locally graded group satisfying the minimal condition on subgroups which are not locally supersoluble, then \(G\) is either locally supersoluble or a Cernikov group. The same conclusion holds for locally finite groups satisfying the weak minimal condition on non-(locally supersoluble) subgroups. As a consequence, it is shown that any infinite locally graded group whose non-(locally supersoluble) subgroups lie into finitely many conjugacy classes must be locally supersoluble.Groups of infinite rank with a normalizer condition on subgroups.https://www.zbmath.org/1456.200212021-04-16T16:22:00+00:00"De Luca, Anna Valentina"https://www.zbmath.org/authors/?q=ai:de-luca.anna-valentina"Di Grazia, Giovanna"https://www.zbmath.org/authors/?q=ai:di-grazia.giovannaSummary: Groups of infinite rank in which every subgroup is either normal or self-normalizing are characterized in terms of their subgroups of infinite rank.The theorems of Schur and Baer: a survey.https://www.zbmath.org/1456.200292021-04-16T16:22:00+00:00"Dixon, Martyn"https://www.zbmath.org/authors/?q=ai:dixon.martyn-russell"Kurdachenko, Leonid"https://www.zbmath.org/authors/?q=ai:kurdachenko.leonid-a"Pypka, Aleksander"https://www.zbmath.org/authors/?q=ai:pypka.alexsandr-aSummary: This paper gives a short survey of some of the known results generalizing the theorem, credited to I. Schur, that if the central factor group is finite then the derived subgroup is also finite.On Magnus' Freiheitssatz and free polynomial algebras.https://www.zbmath.org/1456.200272021-04-16T16:22:00+00:00"Fine, Benjamin"https://www.zbmath.org/authors/?q=ai:fine.benjamin-l"Kreuzer, Martin"https://www.zbmath.org/authors/?q=ai:kreuzer.martin"Rosenberger, Gerhard"https://www.zbmath.org/authors/?q=ai:rosenberger.gerhardSummary: The Freiheitssatz of Magnus for one-relator groups is one of the cornerstones of combinatorial group theory. In this short note which is mostly expository we discuss the relationship between the Freiheitssatz and corresponding results in free power series rings over fields. These are related to results of Schneerson not readily available in English. This relationship uses a faithful representation of free groups due to Magnus. Using this method in free polynomial algebras provides a proof of the Freiheitssatz for one-relation monoids. We show how the classical Freiheitssatz depends on a condition on certain ideals in power series rings in noncommuting variables over fields. A proof of this result over fields would provide a completely dif erent proof of the classical Freiheitssatz.Homogenous finitary symmetric groups.https://www.zbmath.org/1456.200382021-04-16T16:22:00+00:00"Kegel, Otto H."https://www.zbmath.org/authors/?q=ai:kegel.otto-h"Kuzucuoğlu, Mahmut"https://www.zbmath.org/authors/?q=ai:kuzucuoglu.mahmutSummary: We characterize strictly diagonal type of embeddings of finitary symmetric groups in terms of cardinality and the characteristic. Namely, we prove the following. Let \(\kappa\) be an infinite cardinal. If \(G=\bigcup\limits_{i=1}^\infty G_i\), where \(G_i\cong \mathrm{FSym} (\kappa n_i)\), (\(H=\bigcup\limits_{i=1}^\infty H_i\), where \(H_i\cong \mathrm{Alt} (\kappa n_i)\)), is a group of strictly diagonal type and \(\xi=(p_1, p_2, \dots)\) is an infinite sequence of primes, then \(G\) is isomorphic to the homogenous finitary symmetric group \(\mathrm{FSym} (\kappa)(\xi) (H\) is isomorphic to the homogenous alternating group \(\mathrm{Alt} (\kappa)(\xi))\), where \(n_0=1\), \(n_i=p_1p_2\cdots p_i\).Regular dilation and Nica-covariant representation on right LCM semigroups.https://www.zbmath.org/1456.430042021-04-16T16:22:00+00:00"Li, Boyu"https://www.zbmath.org/authors/?q=ai:li.boyuGeneralizing the celebrated Sz.~Nagy dilation of a single contraction, Brehmer studied regular dilations back in the early sixties. Since then the notion of regular dilation has been investigated by many researchers and has been generalized to product systems, lattice ordered semigroups, and recently to graph products of \(\mathbb{N}\). It was shown by the author of the present paper in [J. Funct. Anal. 273, No. 2, 799--835 (2017; Zbl 06720587)] that for such graph products, the existence of a \(*\)-regular dilation is equivalent to the existence of a minimal isometric Nica-covariant dilation.
In the paper under review, the author extends this result to right LCM semigroups, which are unital left cancellative semigroups \(P\) such that for any \(p, q\in P\), either \(pP\cap qP=\emptyset\) or \(pP\cap qP=rP\) for some \(r\in P\). Such an element \(r\) can be considered as a right least common multiple of \(p\) and \(q\) (hence the name right LCM). The author also shows the equivalence among a \(*\)-regular dilation, minimal isometric Nica-covariant dilation, and a Brehmer-type condition. This result unifies many previous results on regular dilations, including Brehmer's theorem, Frazho-Bunce-Popescu's dilation of noncommutative row contractions, regular dilations on lattice ordered semigroups and graph products of \(\mathbb{N}\). Applications to many important classes of right LCM semigroups are given. In the last section of the paper, the author obtains a characterization of \(*\)-regular representations of graph products of right LCM semigroups and an application to doubly commuting representations of direct sums of right LCM semigroups.
Reviewer: Trieu Le (Toledo)Algebraic laminations for free products and arational trees.https://www.zbmath.org/1456.200202021-04-16T16:22:00+00:00"Guirardel, Vincent"https://www.zbmath.org/authors/?q=ai:guirardel.vincent"Horbez, Camille"https://www.zbmath.org/authors/?q=ai:horbez.camilleIn analogy to curve complexes used to study mapping class groups of surfaces, the free factor graph of a free group \(F_n\) has recently turned to be fruitful in the study of Out(\(F_n\)). It is Gromov hyperbolic, as was proved by \textit{M. Bestvina} and \textit{M. Feighn} [Adv. Math. 256, 104--155 (2014; Zbl 1348.20028)], and the action of an automorphism of \(F_n\) is loxodromic if and only if it is fully irreducible. Its Gromov boundary was described by \textit{M. Bestvina} and \textit{P. Reynolds} [Duke Math. J. 164, No. 11, 2213--2251 (2015; Zbl 1337.20040)] and \textit{U. Hamenstädt} [``The boundary of the free splitting graph and the free factor graph'', Preprint, \url{arXiv:1211.1630}] as the set of equivalence classes of arational trees.
The main goal of the paper under review is to extend the theory of algebraic laminations to the context of free products. A key point for this intended application says that if two trees have a leaf in common in their dual laminations, and if one of the trees is arational and relatively free, then they are equivariantly homeomorphic.
Reviewer: V. A. Roman'kov (Omsk)Products of commutators on a general linear group over a division algebra.https://www.zbmath.org/1456.200552021-04-16T16:22:00+00:00"Egorchenkova, E. A."https://www.zbmath.org/authors/?q=ai:egorchenkova.e-a"Gordeev, N. L."https://www.zbmath.org/authors/?q=ai:gordeev.nikolai-lLet \(D\) be a division ring over its center \(K\) of index \(c > 1\), and let \(\mathrm{ GL}_n(D)\) be the group of invertible
\(n \times n\) matrices over \(D\). Let
\(\mathrm{ SL}_n(D) = \{d \in \mathrm{ GL}_n(D) \,|\, \mathrm{ Nrd} \,d = 1\},\)
where \(\mathrm{ Nrd} \,d\) is the reduced norm of \(d\). Then there exists a form \(\mathcal{G}\) of the group \(\mathrm{ SL}_{cn}\) such that
\(\mathcal{G}(K) = \mathrm{ SL}_n(D).\)
The subgroup \(\mathcal{G}^+(K)\) generated by unipotent elements of \(\mathcal{G}(K)\) is denoted by \(E_n(D).\) This is a group generated by the transvections of \(\mathrm{ GL}_n(D)\). The group \(E_n(D)/Z(E_n(D))\) is simple and
\(E_n(D) \unlhd\mathcal{G}(K) = \mathrm{ SL}_n(D), \,\,\, \mathrm{ SL}_n(D)/E_n(D) \approx \mathrm{ SL}_1(D)/[D^\ast, D^\ast].\)
The authors prove the following theorem:
Let \(w = \prod_{i=1}^{k} [x_i,y_i ]\) and let \(\widetilde{w}: D^{* 2k} \rightarrow D^*\)
be the corresponding word map. Further, let \(\widetilde{w} : \mathrm{ GL}_n(D)^{2k} \rightarrow E_n(D)\)
be a word map on \(\mathrm{ GL}_n(D)\) corresponding to the same word \(w\). Assume that \(\widetilde{w}(D^{* 2k}) =
[D^*, D^*]\).
Then \(\widetilde{w}(\mathrm{ GL}_n(D)^{2k}) \supset E_n(D) \backslash Z(E_n(D)).\)
If in addition \(n > 2\), then
\(\widetilde{w}(E_n(D)^{2k}) \supset E_n(D) \backslash Z(E_n(D)).\)
In particular, if every element of \([D^*,D^*]\) is the commutator of elements of \(D^*\) then each noncentral element of \(E_n(D)\) is a commutator of elements of \(E_n(D).\)
Reviewer: Erich W. Ellers (Toronto)Generators of split extensions of abelian groups by cyclic groups.https://www.zbmath.org/1456.200282021-04-16T16:22:00+00:00"Guyot, Luc"https://www.zbmath.org/authors/?q=ai:guyot.lucSummary: Let \(G\simeq M\rtimes C\) be an \(n\)-generator group which is a split extension of an abelian group \(M\) by a cyclic group \(C\). We study the Nielsen equivalence classes and T-systems of generating \(n\)-tuples of \(G\). The subgroup \(M\) can be turned into a finitely generated faithful module over a suitable quotient \(R\) of the integral group ring of \(C\). When \(C\) is infinite, we show that the Nielsen equivalence classes of the generating \(n\)-tuples of \(G\) correspond bijectively to the orbits of unimodular rows in \(M^{n -1}\) under the action of a subgroup of \(\mathrm{GL}_{n - 1}(R)\). Making no assumption on the cardinality of \(C\), we exhibit a complete invariant of Nielsen equivalence in the case \(M\simeq R\). As an application, we classify Nielsen equivalence classes and T-systems of soluble Baumslag-Solitar groups, split metacyclic groups and lamplighter groups.Spaces of invariant circular orders of groups.https://www.zbmath.org/1456.200402021-04-16T16:22:00+00:00"Baik, Hyungryul"https://www.zbmath.org/authors/?q=ai:baik.hyungryul"Samperton, Eric"https://www.zbmath.org/authors/?q=ai:samperton.ericSummary: Motivated by well known results in low-dimensional topology, we introduce and study a topology on the set \(\mathrm{CO}(G)\) of all left-invariant circular orders on a fixed countable and discrete group \(G\). \(\mathrm{CO}(G)\) contains as a closed subspace \(\mathrm{LO}(G)\), the space of all left-invariant linear orders of \(G\), as first topologized by Sikora. We use the compactness of these spaces to show the sets of non-linearly and non-circularly orderable finitely presented groups are recursively enumerable. We describe the action of \(\mathrm{Aut}(G)\) on \(\mathrm{CO}(G)\) and relate it to results of Koberda regarding the action on \(\mathrm{LO}(G)\). We then study two families of circularly orderable groups: finitely generated abelian groups, and free products of circularly orderable groups. For finitely generated abelian groups \(A\), we use a classification of elements of \(\mathrm{CO}(A)\) to describe the homeomorphism type of the space \(\mathrm{CO}(A)\), and to show that \(\mathrm{Aut}(A)\) acts faithfully on the subspace of circular orders which are not linear. We define and characterize Archimedean circular orders, in analogy with linear Archimedean orders. We describe explicit examples of circular orders on free products of circularly orderable groups, and prove a result about the abundance of orders on free products. Whenever possible, we prove and interpret our results from a dynamical perspective.The action of the mapping class group on the space of geodesic rays of a punctured hyperbolic surface.https://www.zbmath.org/1456.200422021-04-16T16:22:00+00:00"Bowditch, Brian H."https://www.zbmath.org/authors/?q=ai:bowditch.brian-h"Sakuma, Makoto"https://www.zbmath.org/authors/?q=ai:sakuma.makotoSummary: Let \(\Sigma\) be a complete finite-area orientable hyperbolic surface with one cusp, and let \(\mathcal{R}\) be the space of complete geodesic rays in \(\Sigma\) emanating from the puncture. Then there is a natural action of the mapping class group of \(\Sigma\) on \(\mathcal{R}\). We show that this action is ``almost everywhere'' wandering.On locally solvable subgroups in division rings.https://www.zbmath.org/1456.200322021-04-16T16:22:00+00:00"Huỳnh Vi\d{ê}t Khánh"https://www.zbmath.org/authors/?q=ai:huynh-viet-khanh.Let $D$ be division ring with centre $F$. The most striking result of this paper is that if $G$ is a locally soluble subnormal subgroup of $D^*$ such that the $i$-th member $G^{(i)}$ of the derived series of $G$, for some finite $i$, is algebraic over $F$, then $G$ is central in $D$. Various special cases of this are known and are listed in this paper.
The author then widens his study. Suppose $M$ is a non-abelian, locally soluble maximal subgroup of the subnormal subgroup $G$ of $D^*$ such that $M^{(i)}$ is algebraic over $F$, again with $i$ finite. Then $D$ is a cyclic algebra of prime degree. In Theorem 2.7, further information is given concerning the structure of $G$. For example, the FC-centre of $G$ and the Fitting subgroup of $M$ are equal and are described in terms of the cyclic structure of $G$.
Reviewer: B. A. F. Wehrfritz (London)A dense geodesic ray in the \(\mathrm{Out}(F_r)\)-quotient of reduced outer space.https://www.zbmath.org/1456.200332021-04-16T16:22:00+00:00"Algom-Kfir, Yael"https://www.zbmath.org/authors/?q=ai:algom-kfir.yael"Pfaff, Catherine"https://www.zbmath.org/authors/?q=ai:pfaff.catherineSummary: In [Ann. Math. Stud. 97, 417--438 (1981; Zbl 0476.32027)] \textit{H. Masur} proved the existence of a dense geodesic in the moduli space for a surface. We prove an analogue theorem for reduced Outer Space endowed with the Lipschitz metric. We also prove two results possibly of independent interest: we show Brun's unordered algorithm weakly converges and from this prove that the set of Perron-Frobenius eigenvectors of positive integer \(m\times m\) matrices is dense in the positive cone \(\mathbf{R}^m_+\) (these matrices will in fact be the transition matrices of positive automorphisms). We give a proof in the appendix that not every point in the boundary of Outer Space is the limit of a flow line.Weil representations via abstract data and Heisenberg groups: a comparison.https://www.zbmath.org/1456.200022021-04-16T16:22:00+00:00"Cruickshank, J."https://www.zbmath.org/authors/?q=ai:cruickshank.james"Gutiérrez Frez, L."https://www.zbmath.org/authors/?q=ai:gutierrez-frez.luis"Szechtman, F."https://www.zbmath.org/authors/?q=ai:szechtman.fernandoThe paper provides Weil representations of unitary groups with even rank over finite rings via Heisenberg groups. The authors use a constructive approach to obtain the explicit matrix form of the Bruhat elements as well as information on generalized Gauss sums. The result is then shown to be identical to the one following from axiomatic considerations. When the ring is local (not necessarily finite) on the other hand, the index of the subgroup generated by the Bruhat elements is computed. Although the subject of the paper is rather technical, all concepts are explained clearly, results are layed down in great detail and proofs are given in a consistent rigorous manner. The authors also provide several examples at the end as well as a nice selection of references. In view of all this, the article might be interesting not only to specialists in the field, but also to graduate students, due to its pedagogical merits.
Reviewer: Danail Brezov (Sofia)On graph products of multipliers and the Haagerup property for \(C^{\ast}\)-dynamical systems.https://www.zbmath.org/1456.460542021-04-16T16:22:00+00:00"Atkinson, Scott"https://www.zbmath.org/authors/?q=ai:atkinson.scott-eThe following paragraphs are essentially taken from the author's abstract and introduction.
The Haagerup property is an important approximation property for groups and for self-adjoint operator algebras. Since its appearance in Haagerup's seminal article, this property has been the subject of intense study. In 2012, \textit{Z. Dong} and \textit{Z.-J. Ruan} [Integral Equations Oper. Theory 73, No. 3, 431--454 (2012; Zbl 1263.46043)] introduced the Haagerup property for the action of a discrete group \(G\) on a unital \(C^*\)-algebra \(A\).
The author considers the notion of the graph product of actions of discrete groups \(\{G_v\}\) on a \(C^*\)-algebra \(A\) and shows that, under suitable commutativity conditions, the graph product action \(\bigstar_\Gamma \alpha_v:\bigstar_\Gamma G_v\curvearrowright A\) has the Haagerup property if each action \(\alpha_v: G_v\curvearrowright A\) possesses the Haagerup property. This generalizes the known results on graph products of groups with the Haagerup property. To accomplish this, the author introduces the graph product of multipliers associated to the actions and shows that the graph product of positive-definite multipliers is positive definite. These results have impacts on left-transformation groupoids and give an alternative proof of a known result for coarse embeddability. The author also records a cohomological characterization of the Haagerup property for group actions.
Reviewer: Qing Meng (Qufu)