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  • 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

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