-
A refined combination theorem for hierarchically hyperbolic groups Groups Geom. Dyn. (IF 0.742) Pub Date : 2020-10-22 Federico Berlai; Bruno Robbio
In this work, we are concerned with hierarchically hyperbolic spaces and hierarchically hyperbolic groups. Our main result is a wide generalization of a combination theorem of Behrstock, Hagen, and Sisto. In particular, as a consequence, we show that any finite graph product of hierarchically hyperbolic groups is again a hierarchically hyperbolic group, thereby answering [6, Question D] posed by Behrstock
-
Nonplanar graphs in boundaries of CAT(0) groups Groups Geom. Dyn. (IF 0.742) Pub Date : 2020-10-26 Kevin Schreve; Emily Stark
Croke and Kleiner constructed two homeomorphic locally CAT(0) complexes whose universal covers have visual boundaries that are not homeomorphic. We construct two homeomorphic locally CAT(0) complexes so that the visual boundary of one universal cover contains a nonplanar graph, while the visual boundary of the other does not. In contrast, we prove for any two locally CAT(0) metrics on the Croke–Kleiner
-
Length spectra of flat metrics coming from $q$-differentials Groups Geom. Dyn. (IF 0.742) Pub Date : 2020-10-26 Marissa Loving
When geometric structures on surfaces are determined by the lengths of curves, it is natural to ask: which curves’ lengths do we really need to know? It is a result of Duchin, Leininger, and Rafi that any flat metric induced by a unit-norm quadratic differential is determined by its marked simple length spectrum. We generalize the notion of simple curves to that of $q$-simple curves, for any positive
-
Lamplighters admit weakly aperiodic SFTs Groups Geom. Dyn. (IF 0.742) Pub Date : 2020-10-26 David B. Cohen
Let $A$ be a finite set and $G$ a group. A closed subset $X$ of $A^G$ is called a subshift if the action of $G$ on $A^G$ preserves $X$. If $K$ is a closed subset of $A^G$ such that membership in $K$ is determined by looking at a fixed finite set of coordinates, and $X$ is the intersection of all translates of $K$ under the action of $G$, then $X$ is called a subshift of finite type (SFT). If an SFT
-
Hyperbolic immersions of free groups Groups Geom. Dyn. (IF 0.742) Pub Date : 2020-10-26 Jean Pierre Mutanguha
We prove that the mapping torus of a graph immersion has a word-hyperbolic fundamental group if and only if the corresponding endomorphism does not produce Baumslag–Solitar subgroups. Due to a result by Reynolds, this theorem applies to all injective endomorphisms of $F_2$ and nonsurjective fully irreducible endomorphisms of $F_n$. We also give a framework for extending the theorem to all injective
-
Bestvina complex for group actions with a strict fundamental domain Groups Geom. Dyn. (IF 0.742) Pub Date : 2020-10-27 Nansen Petrosyan; Tomasz Prytuła
We consider a strictly developable simple complex of finite groups $G(\mathcal Q)$. We show that Bestvina's construction for Coxeter groups applies in this more general setting to produce a complex that is equivariantly homotopy equivalent to the standard development. When $G(\mathcal Q)$ is non-positively curved, this implies that the Bestvina complex is a cocompact classifying space for proper actions
-
Arcs on punctured disks intersecting at most hwice with endpoints on the boundary Groups Geom. Dyn. (IF 0.742) Pub Date : 2020-11-03 Assaf Bar-Natan
Let $D_n$ be the $n$-punctured disk. We prove that a family of essential simple arcs starting and ending at the boundary and pairwise intersecting at most twice is of size at most $\binom{n+1}{3}$. On the way, we also show that any nontrivial square complex homeomorphic to a disk whose hyperplanes are simple arcs intersecting at most twice must have a corner or a spur.
-
Non virtually solvable subgroups of mapping class groups have non virtually solvable representations Groups Geom. Dyn. (IF 0.742) Pub Date : 2020-11-12 Asaf Hadari
Let $\Sigma$ be a compact orientable surface of finite type with at least one boundary component. Let $\Gamma \leq \mathrm{Mod}(\Sigma)$ be a non virtually solvable subgroup. We answer a question of Lubotzky by showing that there exists a finite dimensional homological representation $\rho$ of $\mathrm{Mod}(\Sigma)$ such that $\rho(\Gamma)$ is not virtually solvable. We then apply results of Lubotzky
-
Quasi-isometry classification of right-angled Artin groups that split over cyclic subgroups Groups Geom. Dyn. (IF 0.742) Pub Date : 2020-11-12 Alexander Margolis
For a one-ended right-angled Artin group, we give an explicit description of its JSJ tree of cylinders over infinite cyclic subgroups in terms of its defining graph. This is then used to classify certain right-angled Artin groups up to quasi-isometry. In particular, we show that if two right-angled Artin groups are quasi-isometric, then their JSJ trees of cylinders are weakly equivalent. Although the
-
A couple of real hyperbolic disc bundles over surfaces Groups Geom. Dyn. (IF 0.742) Pub Date : 2020-11-24 Sasha Anan'in; Philipy Chiovetto
Applying the techniques developed in [1], we construct new real hyperbolic manifolds whose underlying topology is that of a disc bundle over a closed orientable surface. By the Gromov–Lawson–Thurston conjecture [6], such bundles $M \to S$ should satisfy the inequality $|eM/\chi S|\leq 1$, where $eM$ stands for the Euler number of the bundle and $\chi S$, for the Euler characteristic of the surface
-
On the Hochschild homology of $\ell^1$-rapid decay group algebras Groups Geom. Dyn. (IF 0.742) Pub Date : 2020-12-10 Alexander Engel
We show that for many semi-hyperbolic groups the decomposition into conjugacy classes of the Hochschild homology of the $\ell^1$-rapid decay group algebra is injective.
-
The classification of hyperelliptic threefolds Groups Geom. Dyn. (IF 0.742) Pub Date : 2020-12-10 Fabrizio Catanese; Andreas Demleitner
We complete the classification of hyperelliptic threefolds, describing in an elementary way the hyperelliptic threefolds with group $D_4$. These are algebraic and form an irreducible 2-dimensional family.
-
Entropy and drift for word metrics on relatively hyperbolic groups Groups Geom. Dyn. (IF 0.742) Pub Date : 2020-12-10 Matthieu Dussaule; Ilya Gekhtman
We are interested in the Guivarc’h inequality for admissible random walks on finitely generated relatively hyperbolic groups, endowed with a word metric. We show that for random walks with finite super-exponential moment, if this inequality is an equality, then the Green distance is roughly similar to the word distance, generalizing results of Blachère, Haïssinsky, and Mathieu for hyperbolic groups
-
On groups with $S^2$ Bowditch boundary Groups Geom. Dyn. (IF 0.742) Pub Date : 2020-08-22 Bena Tshishiku; Genevieve Walsh
We prove that a relatively hyperbolic pair $(G,P)$ has Bowditch boundary a 2-sphere if and only if it is a 3-dimensional Poincare duality pair. We prove this by studying the relationship between the Bowditch and Dahmani boundaries of relatively hyperbolic groups.
-
$S$-arithmetic spinor groups with the same finite quotients and distinct $\ell^2$-cohomology Groups Geom. Dyn. (IF 0.742) Pub Date : 2020-10-12 Holger Kammeyer; Roman Sauer
In this note we refine examples by Aka from arithmetic to $S$-arithmetic groups to show that the vanishing of the $i$-th $\ell^2$-Betti number is not a profinite invariant for all $i \geq 2$.
-
Inverted orbits of exclusion processes, diffuse-extensive-amenability, and (non-?)amenability of the interval exchanges Groups Geom. Dyn. (IF 0.742) Pub Date : 2020-10-13 Christophe Garban
The recent breakthrough works [9, 11, 12] which established the amenability for new classes of groups, lead to the following question: is the action $W(\mathbb Z^d) \curvearrowright \mathbb Z^d$ extensively amenable? (Where $W(\mathbb Z^d)$ is the wobbling group of permutations $\sigma\colon \mathbb Z^d \to \mathbb Z^d$ with bounded range). This is equivalent to asking whether the action $(\mathbb
-
Cost vs. integral foliated simplicial volume Groups Geom. Dyn. (IF 0.742) Pub Date : 2020-10-13 Clara Löh
We show that integral foliated simplicial volume of closed manifolds gives an upper bound for the cost of the corresponding fundamental groups.
-
Classification of pro-$p$ PD$^2$ pairs and the pro-$p$ curve complex Groups Geom. Dyn. (IF 0.742) Pub Date : 2020-10-13 Gareth Wilkes
We classify pro-$p$ Poincaré duality pairs in dimension two. We then use this classification to build a pro-$p$ analogue of the curve complex and establish its basic properties. We conclude with some statements concerning separability properties of the mapping class group.
-
Cartan subalgebras in uniform Roe algebras Groups Geom. Dyn. (IF 0.742) Pub Date : 2020-10-21 Stuart White; Rufus Willett
In this paper we study structural and uniqueness questions for Cartan subalgebras of uniform Roe algebras. We characterise when an inclusion $B\subseteq A$ of $C^*$-algebras is isomorphic to the canonical inclusion of $\ell^\infty(X)$ inside a uniform Roe algebra $C^*_u(X)$ associated to a metric space of bounded geometry. We obtain uniqueness results for “Roe Cartans” inside uniform Roe algebras up
-
Words of Engel type are concise in residually finite groups. Part II Groups Geom. Dyn. (IF 0.742) Pub Date : 2020-10-21 Eloisa Detomi; Marta Morigi; Pavel Shumyatsky
This work is a natural follow-up of the article [5]. Given a group-word $w$ and a group $G$, the verbal subgroup $w(G)$ is the one generated by all $w$-values in $G$. The word $w$ is called concise if $w(G)$ is finite whenever the set of $w$-values in $G$ is finite. It is an open question whether every word is concise in residually finite groups. Let $w=w(x_1,\ldots,x_k)$ be a multilinear commutator
-
Lattice deformations in the Heisenberg group Groups Geom. Dyn. (IF 0.742) Pub Date : 2020-10-21 Jayadev S. Athreya; Ioannis Konstantoulas
The space of deformations of the integer Heisenberg group under the action of Aut$(\mathbf H(\mathbb R))$ is a homogeneous space for a non-reductive group. We analyze its structure as a measurable dynamical system and obtain mean and variance estimates for Heisenberg lattice point counting in measurable subsets of $\mathbb R^3$; in particular, we obtain a random Minkowski-type theorem. Unlike the Euclidean
-
On localizations of quasi-simple groups with given countable center Groups Geom. Dyn. (IF 0.742) Pub Date : 2020-10-21 Ramón Flores; José L. Rodríguez
A group homomorphism $i\colon H \to G$ is a localization of $H$, if for every homomorphism $\varphi\colon H\to G$ there exists a unique endomorphism $\psi\colon G\to G$ such that $i \psi=\varphi$ (maps are acting on the right). Göbel and Trlifaj asked in [18, Problem 30.4(4), p. 831] which abelian groups are centers of localizations of simple groups. Approaching this question we show that every countable
-
The polynomial endomorphisms of graph algebras Groups Geom. Dyn. (IF 0.742) Pub Date : 2020-10-21 Rune Johansen; Adam P. W. Sørensen; Wojciech Szymański
We investigate polynomial endomorphisms of graph $C^*$-algebras and Leavitt path algebras. To this end, we define and analyze the coding graph corresponding to each such an endomorphism. We find an if and only if condition for the endomorphism to restrict to an automorphism of the diagonal MASA, which is stated in terms of synchronization of a certain labelling on the coding graph. We show that the
-
Random veering triangulations are not geometric Groups Geom. Dyn. (IF 0.742) Pub Date : 2020-10-22 David Futer; Samuel J. Taylor; William Worden
Every pseudo-Anosov mapping class $\phi$ defines an associated veering triangulation $\tau_\phi$ of a punctured mapping torus. We show that generically, $\tau_\phi$ is not geometric. Here, the word "generic" can be taken either with respect to random walks in mapping class groups or with respect to counting geodesics in moduli space. Tools in the proof include Teichmüller theory, the Ending Lamination
-
A note on homology for Smale spaces Groups Geom. Dyn. (IF 0.742) Pub Date : 2020-08-14 Valerio Proietti
We collect three observations on the homology for Smale spaces defined by Putnam. The definition of such homology groups involves four complexes. It is shown here that a simple convergence theorem for spectral sequences can be used to prove that all complexes yield the same homology. Furthermore, we introduce a simplicial framework by which the various complexes can be understood as suitable "symmetric"
-
The congruence subgroup problem for the free metabelian group on $n\geq4$ generators Groups Geom. Dyn. (IF 0.742) Pub Date : 2020-08-13 David El-Chai Ben-Ezra
The congruence subgroup problem for a finitely generated group $\Gamma$ asks whether the map $\widehat{\mathrm{Aut}(\Gamma)}\to \mathrm{Aut}(\widehat{\Gamma})$ is injective, or more generally, what is its kernel $C(\Gamma)$? Here $\widehat{X}$ denotes the profinite completion of $X$. It is well known that for finitely generated free abelian groups $C(\mathbb{Z}^{n})=\{ 1\}$ for every $n\geq3$, but
-
Finitely $\mathcal{F}$-amenable actions and decomposition complexity of groups Groups Geom. Dyn. (IF 0.742) Pub Date : 2020-08-13 Andrew Nicas; David Rosenthal
In his work on the Farrell–Jones Conjecture, Arthur Bartels introduced the concept of a "finitely $\mathcal F$-amenable" group action, where $\mathcal F$ is a family of subgroups. We show how a finitely $\mathcal F$-amenable action of a countable group $G$ on a compact metric space, where the asymptotic dimensions of the elements of $\mathcal F$ are bounded from above, gives an upper bound for the
-
Representing interpolated free group factors as group factors Groups Geom. Dyn. (IF 0.742) Pub Date : 2020-08-13 Sorin Popa; Dimitri L. Shlyakhtenko
We construct a one parameter family of ICC groups $\{G_t\}_{t > 1}$, with the property that the group factor $L(G_t)$ is isomorphic to the interpolated free group factor $L(\mathbb F_t):=L(\mathbb{F}_2)^{1/\sqrt{t-1}}$, for all $t$. Moreover, the groups $G_t$ have fixed cost $t$, are strongly treeable and freely generate any treeable ergodic equivalence relation of same cost.
-
Normalizer, divergence type, and Patterson measure for discrete groups of the Gromov hyperbolic space Groups Geom. Dyn. (IF 0.742) Pub Date : 2020-05-12 Katsuhiko Matsuzaki; Yasuhiro Yabuki; Johannes Jaerisch
For a non-elementary discrete isometry group $G$ of divergence type acting on a proper geodesic $delta$-hyperbolic space, we prove that its Patterson measure is quasi-invariant under the normalizer of $G$. As applications of this result, we have: (1) under a minor assumption, such a discrete group $G$ admits no proper conjugation, that is, if the conjugate of $G$ is contained in $G$, then it coincides
-
Equicontinuity, orbit closures and invariant compact open sets for group actions on zero-dimensional spaces Groups Geom. Dyn. (IF 0.742) Pub Date : 2020-06-22 Colin D. Reid
Let $X$ be a locally compact zero-dimensional space, let $S$ be an equicontinuous set of homeomorphisms such that $1 \in S = S^{-1}$, and suppose that $\overline{Gx}$ is compact for each $x \in X$, where $G = \langle S \rangle$. We show in this setting that a number of conditions are equivalent: (a) $G$ acts minimally on the closure of each orbit; (b) the orbit closure relation is closed; (c) for every
-
$p$-Adic limits of renormalized logarithmic Euler characteristics Groups Geom. Dyn. (IF 0.742) Pub Date : 2020-06-22 Christopher Deninger
Given a countable residually finite group $\Gamma$, we write $\Gamma_n \to e$ if $(\Gamma_n)$ is a sequence of normal subgroups of finite index such that any infinite intersection of $\Gamma_n$'s contains only the unit element $e$ of $\Gamma$. Given a $\Gamma$-module $M$ we are interested in the multiplicative Euler characteristics \begin{equation} \label{eq:1a} \chi (\Gamma_n , M) = \prod_i |H_i (\Gamma_n
-
On the Dehn functions of Kähler groups Groups Geom. Dyn. (IF 0.742) Pub Date : 2020-06-22 Claudio Llosa Isenrich; Romain Tessera
We address the problem of which functions can arise as Dehn functions of Kähler groups. We explain why there are examples of Kähler groups with linear, quadratic, and exponential Dehn function. We then proceed to show that there is an example of a Kähler group which has Dehn function bounded below by a cubic function and above by $n^6$. As a consequence we obtain that for a compact Kähler manifold
-
On the smallest non-trivial quotients of mapping class groups Groups Geom. Dyn. (IF 0.742) Pub Date : 2020-06-22 Dawid Kielak; Emilio Pierro
We prove that the smallest non-trivial quotient of the mapping class group of a connected orientable surface of genus $g \geq 3$ without punctures is Sp$_{2g}(2)$, thus confirming a conjecture of Zimmermann. In the process, we generalise Korkmaz’s results on $\mathbb C$-linear representations of mapping class groups to projective representations over any field.
-
Universal minimal flow in the theory of topological groupoids Groups Geom. Dyn. (IF 0.742) Pub Date : 2020-06-22 Riccardo Re; Pietro Ursino
We extend the notion of Universal Minimal Flows to groupoid actions of locally trivial groupoids. We also prove that any $G$-bundle with compact fibers has a global section if $G$ is extremely amenable.
-
Linear progress with exponential decay in weakly hyperbolic groups Groups Geom. Dyn. (IF 0.742) Pub Date : 2020-06-24 Matthew H. Sunderland
A random walk $w_n$ on a separable, geodesic hyperbolic metric space $X$ converges to the boundary $\partial X$ with probability one when the step distribution supports two independent loxodromics. In particular, the random walk makes positive linear progress. Progress is known to be linear with exponential decay when (1) the step distribution has exponential tail and (2) the action on $X$ is acylindrical
-
Lamplighter groups, bireversible automata, and rational series over finite rings Groups Geom. Dyn. (IF 0.742) Pub Date : 2020-06-22 Rachel Skipper; Benjamin Steinberg
We realize lamplighter groups $A\wr \mathbb Z$, with $A$ a finite abelian group, as automaton groups via affine transformations of power series rings with coefficients in a finite commutative ring. Our methods can realize $A\wr \mathbb Z$ as a bireversible automaton group if and only if the 2-Sylow subgroup of $A$ has no multiplicity one summands in its expression as a direct sum of cyclic groups of
-
Finiteness of mapping class groups: locally large strongly irreducible Heegaard splittings Groups Geom. Dyn. (IF 0.742) Pub Date : 2020-06-24 Yanqing Zou; Ruifeng Qiu
By Namazi and Johnson’s results, for any distance at least 4 Heegaard splitting, its mapping class group is finite. In contrast, Namazi showed that for a weakly reducible Heegaard splitting, its mapping class group is infinite; Long constructed an irreducible Heegaard splitting where its mapping class group contains a pseudo anosov map. Thus it is interesting to know that for a strongly irreducible
-
Large-scale rank and rigidity of the Weil–Petersson metric Groups Geom. Dyn. (IF 0.742) Pub Date : 2020-06-22 Brian H. Bowditch
We study the large-scale geometry of Weil–Petersson space, that is, Teichmüller space equipped with theWeil–Petersson metric. We show that this admits a natural coarse median structure of a specific rank. Given that this is equal to the maximal dimension of a quasi-isometrically embedded euclidean space,we recover a result of Eskin,Masur and Rafi which gives the coarse rank of the space. We go on
-
Properly convex bending of hyperbolic manifolds Groups Geom. Dyn. (IF 0.742) Pub Date : 2020-06-24 Samuel A. Ballas; Ludovic Marquis
In this paper we show that bending a finite volume hyperbolic $d$-manifold $M$ along a totally geodesic hypersurface $\Sigma$ results in a properly convex projective structure on $M$ with finite volume. We also discuss various geometric properties of bent manifolds and algebraic properties of their fundamental groups. We then use this result to show in each dimension $d\geqslant 3$ there are examples
-
Grigorchuk–Gupta–Sidki groups as a source for Beauville surfaces Groups Geom. Dyn. (IF 0.742) Pub Date : 2020-06-24 Şükran Gül; Jone Uria-Albizuri
If $G$ is a Grigorchuk–Gupta–Sidki group defined over a $p$-adic tree, where $p$ is an odd prime, we study the existence of Beauville surfaces associated to the quotients of $G$ by its level stabilizers $\mathrm {st}_G(n)$. We prove that if $G$ is periodic then the quotients $G/\mathrm {st}_G(n)$ are Beauville groups for every $n\geq 2$ if $p\geq 5$ and $n\geq 3$ if $p = 3$. In this case, we further
-
The word and order problems for self-similar and automata groups Groups Geom. Dyn. (IF 0.742) Pub Date : 2020-06-24 Laurent Bartholdi; Ivan Mitrofanov
We prove that the word problem is undecidable in functionally recursive groups, and that the order problem is undecidable in automata groups, even under the assumption that they are contracting.
-
Weighted cogrowth formula for free groups Groups Geom. Dyn. (IF 0.742) Pub Date : 2020-04-20 Johannes Jaerisch; Katsuhiko Matsuzaki
We investigate the relationship between geometric, analytic and probabilistic indices for quotients of the Cayley graph of the free group ${\rm Cay}(F_n)$ by an arbitrary subgroup $G$ of $F_n$. Our main result, which generalizes Grigorchuk's cogrowth formula to variable edge lengths, provides a formula relating the bottom of the spectrum of weighted Laplacian on $G \backslash {\rm Cay}(F_n)$ to the
-
On self-similar finite $p$-groups Groups Geom. Dyn. (IF 0.742) Pub Date : 2020-03-12 Azam Babai; Khadijeh Fathalikhani; Gustavo A. Fernández-Alcober; Matteo Vannacci
In this paper, we address the following question: when is a finite $p$-group $G$ self-similar, i.e. when can $G$ be faithfully represented as a self-similar group of automorphisms of the $p$-adic tree? We show that, if $G$ is a self-similar finite $p$-group of rank $r$, then its order is bounded by a function of $p$ and $r$. This applies in particular to finite $p$-groups of a given coclass. In the
Contents have been reproduced by permission of the publishers.