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Conjugacy and centralizers in groups of piecewise projective homeomorphisms Groups Geom. Dyn. (IF 0.6) Pub Date : 2022-05-13 Francesco Matucci, Altair Santos de Oliveira-Tosti
In 2013, Monod introduced a family of Thompson-like groups which provides natural counterexamples to the von Neumann–Day conjecture.We construct a characterization of conjugacy and an invariant and use them to compute centralizers in one group of this family.
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Hyperbolic groups with almost finitely presented subgroups (with an appendix by Robert Kropholler and Federico Vigolo) Groups Geom. Dyn. (IF 0.6) Pub Date : 2022-05-13 Peter H. Kropholler
In this paper, we create many examples of hyperbolic groups with subgroups satisfying interesting finiteness properties.We give the first examples of subgroups of hyperbolic groups which are of type $FP_2$ but not finitely presented. We give uncountably many groups of type $FP_2$ with similar properties to those subgroups of hyperbolic groups. Along the way we create more subgroups of hyperbolic groups
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Superrigidity, measure equivalence, and weak Pinsker entropy Groups Geom. Dyn. (IF 0.6) Pub Date : 2022-05-13 Lewis Bowen, Robin D. Tucker-Drob
We show that the class $\mathscr{B}$, of discrete groups which satisfy the conclusion of Popa’s cocycle superrigidity theorem for Bernoulli actions, is invariant under measure equivalence. We generalize this to the setting of discrete probability measure preserving (p.m.p.) groupoids, and as a consequence we deduce that any nonamenable lattice in a product of two noncompact, locally compact second
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Signature for piecewise continuous groups Groups Geom. Dyn. (IF 0.6) Pub Date : 2022-04-22 Octave Lacourte
Let $\widehat{\operatorname{PC}^{\bowtie}}$ be the group of bijections from $[0,1[$ to itself which are continuous outside a finite set. Let $\operatorname{PC}^{\bowtie}$ be its quotient by the subgroup of finitely supported permutations. We show that the Kapoudjian class of $\operatorname{PC}^{\bowtie}$ vanishes. That is, the quotient map $\widehat{\operatorname{PC}^{\bowtie}}\rightarrow\operatorname{PC}^{\bowtie}$
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Distal strongly ergodic actions Groups Geom. Dyn. (IF 0.6) Pub Date : 2022-04-13 Eli Glasner, Benjamin Weiss
Let $\eta$ be an arbitrary countable ordinal. Using results of Bourgain, Gamburd, and Sarnak on compact systems with spectral gap, we show the existence of an action of the free group on three generators $F_3$ on a compact metric space $X$, admitting an invariant probability measure $\mu$, such that the resulting dynamical system $(X,\mu,F_3)$ is strongly ergodic and distal of rank $\eta$. In particular
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Hyperbolic quotients of projection complexes Groups Geom. Dyn. (IF 0.6) Pub Date : 2022-04-05 Matt Clay, Johanna Mangahas
This paper is a continuation of our previous work with Margalit where we studied group actions on projection complexes. In that paper, we demonstrated sufficient conditions so that the normal closure of a family of subgroups of vertex stabilizers is a free product of certain conjugates of these subgroups. In this paper, we study both the quotient of the projection complex by this normal subgroup and
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Descriptive chromatic numbers of locally finite and everywhere two-ended graphs Groups Geom. Dyn. (IF 0.6) Pub Date : 2022-03-29 Felix Weilacher
We construct Borel graphs which settle several questions in descriptive graph combinatorics. These include “Can the Baire measurable chromatic number of a locally finite Borel graph exceed the usual chromatic number by more than one?” and “Can marked groups with isomorphic Cayley graphs have Borel chromatic numbers for their shift graphs which differ by more than one?” We also provide a new bound for
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Automaton groups and complete square complexes Groups Geom. Dyn. (IF 0.6) Pub Date : 2022-03-29 Ievgen Bondarenko, Bohdan Kivva
The first example of a non-residually finite group in the classes of finitely presented small-cancelation groups, automatic groups, and $\operatorname{CAT}(0)$ groups was constructed by Wise as the fundamental group of a complete square complex (CSC for short) with twelve squares. At the same time, Janzen and Wise proved that CSCs with at most three, five or seven squares have residually finite fundamental
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Hyperbolic geometry of shapes of convex bodies Groups Geom. Dyn. (IF 0.6) Pub Date : 2022-03-16 Clément Debin, François Fillastre
We use the intrinsic area to define a distance on the space of homothety classes of convex bodies in the $n$-dimensional Euclidean space, which makes it isometric to a convex subset of the infinite dimensional hyperbolic space. The ambient Lorentzian structure is an extension of the intrinsic area form of convex bodies, and Alexandrov–Fenchel inequality is interpreted as the Lorentzian reversed Cauchy–Schwarz
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Cubulation of some triangle-free Artin groups Groups Geom. Dyn. (IF 0.6) Pub Date : 2022-03-14 Thomas Haettel
We prove that some classes of triangle-free Artin groups act properly on locally finite, finite-dimensional CAT(0) cube complexes. In particular, this provides the first examples of Artin groups that are properly cubulated but cannot be cocompactly cubulated, even virtually. The existence of such a proper action has many interesting consequences for the group, notably the Haagerup property, and the
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Topology of leaves for minimal laminations by non-simply-connected hyperbolic surfaces Groups Geom. Dyn. (IF 0.6) Pub Date : 2022-03-02 Sébastien Alvarez, Joaquín Brum
We give the topological obstructions to be a leaf in a minimal lamination by hyperbolic surfaces whose generic leaf is homeomorphic to a Cantor tree. Then, we show that all allowed topological types can be simultaneously embedded in the same lamination. This result, together with results in [arXiv:1906.10029] and [Comment. Math. Helv. 78 (2003), 845–864], completes the panorama of understanding which
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Hyperbolicity of $T$(6) cyclically presented groups Groups Geom. Dyn. (IF 0.6) Pub Date : 2022-02-08 Ihechukwu Chinyere, Gerald Williams
We consider groups defined by cyclic presentations where the defining word has length 3 and the cyclic presentation satisfies the $T$(6) small cancellation condition. We classify when these groups are hyperbolic.When combined with known results, this completely classifies the hyperbolic $T$(6) cyclically presented groups.
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On hereditarily self-similar $p$-adic analytic pro-$p$ groups Groups Geom. Dyn. (IF 0.6) Pub Date : 2022-01-28 Francesco Noseda, Ilir Snopce
A non-trivial finitely generated pro-$p$ group $G$ is said to be strongly hereditarily self-similar of index $p$ if every non-trivial finitely generated closed subgroup of $G$ admits a faithful self-similar action on a $p$-ary tree. We classify the solvable torsion-free $p$-adic analytic pro-$p$ groups of dimension less than $p$ that are strongly hereditarily self-similar of index $p$. Moreover, we
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The Ellis semigroup of bijective substitutions Groups Geom. Dyn. (IF 0.6) Pub Date : 2021-12-20 Johannes Kellendonk, Reem Yassawi
For topological dynamical systems $(X,T,\sigma)$ with abelian group $T$, which admit an equicontinuous factor $\pi:(X,T,\sigma)\to (Y,T,\delta)$, the Ellis semigroup $E(X)$ is an extension of $Y$ by its subsemigroup $E^{\operatorname{fib}}(X)$ of elements which preserve the fibres of $\pi$. We establish methods to compute $E^{\operatorname{fib}}(X)$ and use them to determine the Ellis semigroup of
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Divergence of finitely presented groups Groups Geom. Dyn. (IF 0.6) Pub Date : 2021-12-06 Noel Brady, Hung Cong Tran
We construct families of finitely presented groups exhibiting new divergence behavior; we obtain divergence functions of the form $r^\alpha$ for a dense set of exponents $\alpha \in [2,\infty)$ and $r^n\log(r)$ for integers $n \geq 2$. The same construction also yields examples of finitely presented groups which contain Morse elements that are not contracting.
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Cohomology of hyperfinite Borel actions Groups Geom. Dyn. (IF 0.6) Pub Date : 2021-12-06 Sergey I. Bezuglyi, Shrey Sanadhya
We study cocycles of countable groups $\Gamma$ of Borel automorphisms of a standard Borel space $(X, \mathcal{B})$ taking values in a locally compact second countable group $G$. We prove that for a hyperfinite group $\Gamma$ the subgroup of coboundaries is dense in the group of cocycles. We describe all Borel cocycles of the $2$-odometer and show that any such cocycle is cohomologous to a cocycle with
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On a family of unitary representations of mapping class groups Groups Geom. Dyn. (IF 0.6) Pub Date : 2021-12-06 Biao Ma
For a compact surface $S = S_{g,n}$ with $3g + n \geq 4$, we introduce a family of unitary representations of the mapping class group $\operatorname{Mod}(S)$ based on the space of measured foliations. or this family of representations, we show that none of them has almost invariant vectors. As one application, we obtain an inequality concerning the action of $\operatorname{Mod}(S)$ on the Teichmüller
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Isolated circular orders of PSL(2,$\mathbb{Z}$) Groups Geom. Dyn. (IF 0.6) Pub Date : 2021-12-06 Shigenori Matsumoto
We give a bijection between the isolated circular orders of the group $G=\operatorname{PSL}(2,\mathbb{Z})\approx (\mathbb{Z}/2\mathbb{Z})*(\mathbb{Z}/3\mathbb{Z})$ and the equivalence classes of Markov systems associated to $G$. As applications, we present examples of isolated circular orders of the group $G$.
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Cantor dynamics of renormalizable groups Groups Geom. Dyn. (IF 0.6) Pub Date : 2021-12-06 Steven Hurder, Olga Lukina, Wouter van Limbeek
A group $\Gamma$ is said to be “finitely non-co-Hopfian,” or “renormalizable,” if there exists a self-embedding $\varphi \colon \Gamma \to \Gamma$ whose image is a proper subgroup of finite index. Such a proper self-embedding is called a “renormalization for $\Gamma$.” In this work, we associate a dynamical system to a renormalization $\varphi$ of $\Gamma$. The discriminant invariant ${\mathcal D}_{\varphi}$
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Minimality of the action on the universal circle of uniform foliations Groups Geom. Dyn. (IF 0.6) Pub Date : 2021-12-06 Sérgio R. Fenley, Rafael Potrie
Given a uniform foliation by Gromov hyperbolic leaves on a 3-manifold, we show that the action of the fundamental group on the universal circle is minimal and transitive on pairs of different points. We also prove two other results: we prove that general uniform Reebless foliations are $\mathbb{R}$R-covered and we give a new description of the universal circle of $\mathbb{R}$-covered foliations with
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Elementary subgroups of virtually free groups Groups Geom. Dyn. (IF 0.6) Pub Date : 2021-12-06 Simon André
We give a description of elementary subgroups (in the sense of first-order logic) of finitely generated virtually free groups. In particular, we recover the fact that elementary subgroups of finitely generated free groups are free factors. Moreover, one gives an algorithm that takes as input a finite presentation of a virtually free group $G$ and a finite subset $X$ of $G$, and decides if the subgroup
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Amenability and measure of maximal entropy for semigroups of rational maps Groups Geom. Dyn. (IF 0.6) Pub Date : 2021-10-06 Carlos Cabrera, Peter Makienko
In this article we discuss relations between algebraic and dynamical properties of non-cyclic semigroups of rational maps.
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Locally Roelcke precompact Polish groups Groups Geom. Dyn. (IF 0.6) Pub Date : 2021-10-26 Joseph Zielinski
A Polish group is locally Roelcke precompact if there is a neighborhood of the identity element that is totally bounded in the Roelcke (or lower) group uniformity. These form a subclass of the locally bounded groups, while generalizing the Roelcke precompact and locally compact Polish groups. We characterize these groups in terms of their geometric structure as those locally bounded groups whose coarsely
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Algorithmic aspects of branched coverings III/V. Erasing maps, orbispaces, and the Birman exact sequence Groups Geom. Dyn. (IF 0.6) Pub Date : 2021-11-01 Laurent Bartholdi, Dzmitry Dudko
Let $\tilde{f}\colon(S^2,\widetilde{A})\circlearrowleft$ be a Thurston map and let $M(\tilde{f})$ be its mapping class biset: isotopy classes rel $\widetilde{A}$ of maps obtained by pre- and post-composing $\tilde{f}$ by the mapping class group of $(S^2,\widetilde{A})$. Let $A\subseteq\widetilde{A}$ be an $\tilde{f}$-invariant subset, and let $f\colon(S^2,A)\circlearrowleft$ be the induced map. We
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Patterns in sets of positive density in trees and affine buildings Groups Geom. Dyn. (IF 0.6) Pub Date : 2021-11-10 Michael Björklund, Alexander Fish, James Parkinson
We prove an analogue for homogeneous trees and certain affine buildings of a result of Bourgain on pinned distances in sets of positive density in Euclidean spaces. Furthermore, we construct an example of a non-homogeneous tree with positive Hausdorff dimension, and a subset with positive density thereof, in which not all sufficiently large (even) distances are realised.
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Stability in a group Groups Geom. Dyn. (IF 0.6) Pub Date : 2021-11-10 Gabriel Conant
We develop local stable group theory directly from topological dynamics, and extend the main tools in this subject to the setting of stability “in a model.” Specifically, given a group $G$, we analyze the structure of sets $A \subseteq G$ such that the bipartite relation $xy\in A$ omits infinite half-graphs. Our proofs rely on the characterization of model-theoretic stability via Grothendieck's “double-limit”
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WWPD elements of big mapping class groups Groups Geom. Dyn. (IF 0.6) Pub Date : 2021-07-30 Alexander J. Rasmussen
We study mapping class groups of infinite type surfaces with isolated punctures and their actions on the $loop$ $graphs$ introduced by Bavard and Walker. We classify all of the mapping classes in these actions which are loxodromic with a WWPD action on the corresponding loop graph. The WWPD property is a weakening of Bestvina and Fujiwara’s weak proper discontinuity and is useful for constructing non-trivial
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Maximal subgroups and von Neumann subalgebras with the Haagerup property Groups Geom. Dyn. (IF 0.6) Pub Date : 2021-08-03 Yongle Jiang, Adam Skalski
We initiate a study of maximal subgroups and maximal von Neumann subalgebras which have the Haagerup property. We determine maximal Haagerup subgroups inside $\mathbb{Z}^2 \rtimes \operatorname{SL}_2(\mathbb{Z})$ and obtain several explicit instances where maximal Haagerup subgroups yield maximal Haagerup subalgebras. Our techniques are on one hand based on group-theoretic considerations, and on the
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Convexity of balls in outer space Groups Geom. Dyn. (IF 0.6) Pub Date : 2021-07-30 Yulan Qing, Kasra Rafi
In this paper we study the convexity properties of geodesics and balls in Outer space equipped with the Lipschitz metric. We introduce a class of geodesics called balanced folding paths and show that, for every loop $\alpha$, the length of $\alpha$ along a balanced folding path is not larger than the maximum of its lengths at the endpoints. This implies that out-going balls are weakly convex. We then
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Random walks on the discrete affine group Groups Geom. Dyn. (IF 0.6) Pub Date : 2021-08-03 Jérémie Brieussel, Ryokichi Tanaka, Tianyi Zheng
We introduce the discrete affine group of a regular tree as a finitely generated subgroup of the affine group. We describe the Poisson boundary of random walks on it as a space of configurations. We compute isoperimetric profile and Hilbert compression exponent of the group. We also discuss metric relationship with some lamplighter groups and lamplighter graphs.
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Alternating quotients of right-angled Coxeter groups Groups Geom. Dyn. (IF 0.6) Pub Date : 2021-08-03 Michal Buran
Let $W$ be a right-angled Coxeter group corresponding to a finite non-discrete graph $\mathcal{G}$ with at least $3$ vertices. Our main theorem says that $\mathcal{G}^c$ is connected if and only if for any infinite index convex-cocompact subgroup $H$ of $W$ and any finite subset $\{ \gamma_1, \ldots , \gamma_n \} \subset W \setminus H$ there is a surjection $f$ from $W$ to a finite alternating group
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A complex Euclidean reflection group with a non-positively curved complement complex Groups Geom. Dyn. (IF 0.6) Pub Date : 2021-08-03 Ben Coté, Jon McCammond
The complement of a hyperplane arrangement in $\mathbb{C}^n$ deformation retracts onto an $n$-dimensional cell complex, but the known procedures only apply to complexifications of real arrangements (Salvetti) or the cell complex produced depends on an initial choice of coordinates (Björner–Ziegler). In this article we consider the unique complex Euclidean reflection group acting cocompactly by isometries
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Properly discontinuous actions versus uniform embeddings Groups Geom. Dyn. (IF 0.6) Pub Date : 2021-08-03 Kevin Schreve
Whenever a finitely generated group $G$ acts properly discontinuously by isometries on a metric space $X$, there is an induced uniform embedding (a Lipschitz and uniformly proper map) $\rho\colon G \rightarrow X$ given by mapping $G$ to an orbit. We study when there is a difference between a finitely generated group $G$ acting properly on a contractible $n$-manifold and uniformly embedding into a contractible
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Fluctuations of ergodic averages for amenable group actions Groups Geom. Dyn. (IF 0.6) Pub Date : 2021-08-03 Uri Gabor
We show that for any countable amenable group action, along certain Følner sequences (those that have for any $c>1$ a two-sided $c$-tempered tail), one has a universal estimate for the number of fluctuations in the ergodic averages of $L^{\infty}$ functions. This estimate gives exponential decay in the number of fluctuations. Any two-sided Følner sequence can be thinned out to satisfy the above property
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Almost commuting matrices with respect to the rank metric Groups Geom. Dyn. (IF 0.6) Pub Date : 2021-08-03 Gábor Elek, Łukasz Grabowski
We show that if $A_1,A_2,\ldots, A_n$ are square matrices, each of them is either unitary or self-adjoint, and they almost commute with respect to the rank metric, then one can find commuting matrices $B_1$, $B_2$, $\ldots$, $B_n$ that are close to the matrices $A_i$ in the rank metric.
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Narrow equidistribution and counting of closed geodesics on noncompact manifolds Groups Geom. Dyn. (IF 0.6) Pub Date : 2021-08-04 Barbara Schapira, Samuel Tapie
We prove the equidistribution of (weighted) periodic orbits for the geodesic flow on noncompact negatively curved manifolds toward equilibrium states in the narrow topology, i.e. in the dual of bounded continuous functions. We deduce an exact asymptotic counting for periodic orbits (weighted or not), which was previously known only for geometrically finite manifolds.
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On the existence of free subsemigroups in reversible automata semigroups Groups Geom. Dyn. (IF 0.6) Pub Date : 2021-09-29 Dominik Francoeur, Ivan Mitrofanov
We prove that the semigroup generated by a reversible Mealy automaton contains a free subsemigroup of rank two if and only if it contains an element of infinite order.
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Erratum to “On groups with $S^2$ Bowditch boundary” Groups Geom. Dyn. (IF 0.6) Pub Date : 2021-09-28 Bena Tshishiku,Genevieve Walsh
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Cubulating one-relator products with torsion Groups Geom. Dyn. (IF 0.6) Pub Date : 2021-08-04 Ben Stucky
We generalize results of Lauer and Wise to show that a one-relator product of locally indicable groups whose defining relator has exponent at least 4 admits a proper and cocompact action on a CAT(0) cube complex if the factors do.
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On the entropies of subshifts of finite type on countable amenable groups Groups Geom. Dyn. (IF 0.6) Pub Date : 2021-07-19 Sebastián Barbieri
Let $G,H$ be two countable amenable groups. We introduce the notion of group charts, which gives us a tool to embed an arbitrary $H$-subshift into a $G$-subshift. Using an entropy addition formula derived from this formalism we prove that whenever $H$ is finitely presented and admits a subshift of finite type (SFT) on which $H$ acts freely, then the set of real numbers attained as topological entropies
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Iterated monodromy groups of Chebyshev-like maps on $\mathbb{C}^n$ Groups Geom. Dyn. (IF 0.6) Pub Date : 2021-07-19 Joshua P. Bowman
Every affine Weyl group appears as the iterated monodromy group of a Chebyshev-like polynomial self-map of $\mathbb{C}^n$.
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Quasi-local algebras and asymptotic expanders Groups Geom. Dyn. (IF 0.6) Pub Date : 2021-07-23 Kang Li, Piotr W. Nowak, Ján Špakula, Jiawen Zhang
In this paper, we study the relation between the uniform Roe algebra and the uniform quasi-local algebra associated to a metric space of bounded geometry. In the process, we introduce a weakening of the notion of expanders, called asymptotic expanders. We show that being a sequence of asymptotic expanders is a coarse property under certain connectedness condition, and it implies non-uniformly local
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Abelian subgroups of the fundamental group of a space with no conjugate points Groups Geom. Dyn. (IF 0.6) Pub Date : 2021-08-03 James Dibble
Each Abelian subgroup of the fundamental group of a compact and locally simply connected $d$-dimensional length space with no conjugate points is isomorphic to $\mathbb{Z}^k$ for some $0 \leq k \leq d$. It follows from this and previously known results that each solvable subgroup of the fundamental group is a Bieberbach group. In the Riemannian setting, this may be proved using a novel property of
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Entropy and finiteness of groups with acylindrical splittings Groups Geom. Dyn. (IF 0.6) Pub Date : 2021-07-23 Filippo Cerocchi, Andrea Sambusetti
We prove that there exists a positive, explicit function $F(k, E)$ such that, for any group $G$ admitting a $k$-acylindrical splitting and any generating set $S$ of $G$ with $\operatorname{Ent}$(G,S)
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On spherical unitary representations of groups of spheromorphisms of Bruhat–Tits trees Groups Geom. Dyn. (IF 0.6) Pub Date : 2021-07-23 Yury A. Neretin
Consider an infinite homogeneous tree $\mathcal{T}_n$ of valence $n+1$, its group Aut$(\mathcal{T}_n)$ of automorphisms, and the group Hier$(\mathcal{T}_n)$ of its spheromorphisms (hierarchomorphisms), i.e., the group of homeomorphisms of the boundary of $\mathcal{T}_n$ that locally coincide with transformations defined by automorphisms. We show that the subgroup Aut$(\mathcal{T}_n)$ is spherical in
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Virtually cyclic dimension for 3-manifold groups Groups Geom. Dyn. (IF 0.6) Pub Date : 2021-04-28 Kyle Joecken, Jean-François Lafont, Luis Jorge Sánchez Saldaña
Let $\Gamma$ be the fundamental group of a connected, closed, orientable $3$-manifold. We explicitly compute its virtually cyclic geometric dimension $\underline{\underline{\mathrm{gd}}}(\Gamma)$. Among the tools we use are the prime and JSJ decompositions of $M$, acylindrical splittings of groups, several push-out type constructions, as well as some Bredon cohomology computations.
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Noncommutative joinings II Groups Geom. Dyn. (IF 0.6) Pub Date : 2021-04-07 Jon Bannon, Jan Cameron, Kunal Mukherjee
This paper is a continuation of the authors' previous work on noncommutative joinings, and contains a study of relative independence of W$^*$-dynamical systems. We prove that, given any separable locally compact group $G$, an ergodic W$^{*}$-dynamical $G$-system $\mathfrak{M}$ with compact subsystem $\mathfrak{N}$ is disjoint relative to $\mathfrak{N}$ from its maximal compact subsystem $\mathfrak{M}_{K}$
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Confined subgroups of Thompson's group $F$ and its embeddings into wobbling groups Groups Geom. Dyn. (IF 0.6) Pub Date : 2021-03-31 Maksym Chaudkhari
We obtain a characterisation of confined subgroups of Thompson's group $F$. As a result, we deduce that the orbital graph of a point under an action of $F$ has uniformly subexponential growth if and only if this point is fixed by the commutator subgroup. This allows us to prove non-embeddability of $F$ into wobbling groups of graphs with uniformly subexponential growth.
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Automaticity for graphs of groups Groups Geom. Dyn. (IF 0.6) Pub Date : 2021-03-31 Susan Hermiller, Derek F. Holt, Tim Susse, Sarah Rees
In this article we construct asynchronous and sometimes synchronous automatic structures for amalgamated products and HNN extensions of groups that are strongly asynchronously (or synchronously) coset automatic with respect to the associated automatic subgroups, subject to further geometric conditions. These results are proved in the general context of fundamental groups of graphs of groups. The hypotheses
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Acylindrical hyperbolicity of groups acting on quasi-median graphs and equations in graph products Groups Geom. Dyn. (IF 0.6) Pub Date : 2021-03-25 Motiejus Valiunas
In this paper we study group actions on quasi-median graphs, or "CAT(0) prism complexes", generalising the notion of CAT(0) cube complexes. We consider hyperplanes in a quasi-median graph $X$ and define the contact graph $\mathcal{C}X$ for these hyperplanes. We show that $\mathcal{C}X$ is always quasi-isometric to a tree, generalising a result of Hagen [18], and that under certain conditions a group
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Centering Koebe polyhedra via Möbius transformations Groups Geom. Dyn. (IF 0.6) Pub Date : 2021-03-25 Zsolt Lángi
A variant of the Circle Packing Theorem states that the combinatorial class of any convex polyhedron contains elements, called Koebe polyhedra, midscribed to the unit sphere centered at the origin, and that these representatives are unique up to Möbius transformations of the sphere. Motivated by this result, various papers investigate the problem of centering spherical configurations under Möbius transformations
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Measure equivalence and coarse equivalence for unimodular locally compact groups Groups Geom. Dyn. (IF 0.6) Pub Date : 2021-03-25 Juhani Koivisto, David Kyed, Sven Raum
This article is concerned with measure equivalence and uniform measure equivalence of locally compact, second countable groups. We show that two unimodular, locally compact, second countable groups are measure equivalent if and only if they admit free, ergodic, probability measure preserving actions whose cross section equivalence relations are stably orbit equivalent. Using this we prove that in the
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K- and L-theory of graph products of groups Groups Geom. Dyn. (IF 0.6) Pub Date : 2021-03-25 Daniel Kasprowski, Kevin Li, Wolfgang Lück
We compute the group homology, the algebraic $K$- and $L$-groups, and the topological $K$-groups of right-angled Artin groups, right-angled Coxeter groups, and more generally, graph products.
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CAT(0) cube complexes are determined by their boundary cross ratio Groups Geom. Dyn. (IF 0.6) Pub Date : 2021-03-25 Jonas Beyrer, Elia Fioravanti, Merlin Incerti-Medici
We introduce a $\mathbb{Z}$-valued cross ratio on Roller boundaries of CAT(0) cube complexes. We motivate its relevance by showing that every cross-ratio preserving bijection of Roller boundaries uniquely extends to a cubical isomorphism. Our results are strikingly general and even apply to infinite dimensional, locally infinite cube complexes with trivial automorphism group.
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Acylindrical actions on CAT(0) square complexes Groups Geom. Dyn. (IF 0.6) Pub Date : 2021-03-25 Alexandre Martin
For group actions on hyperbolic CAT(0) square complexes, we show that the acylindricity of the action is equivalent to a weaker form of acylindricity phrased purely in terms of stabilisers of points, which has the advantage of being much more tractable for actions on non-locally compact spaces. For group actions on general CAT(0) square complexes, we show that an analogous characterisation holds for
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Erratum to "Generic free subgroups and statistical hyperbolicity" Groups Geom. Dyn. (IF 0.6) Pub Date : 2021-03-25 Suzhen Han,Wen-Yuan Yang
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CAT(0) cube complexes and inner amenability Groups Geom. Dyn. (IF 0.6) Pub Date : 2021-03-25 Bruno Duchesne, Robin D. Tucker-Drob, Phillip Wesolek
We here consider inner amenability from a geometric and group theoretical perspective. We prove that for every non-elementary action of a group $G$ on a finite dimensional irreducible CAT(0) cube complex, there is a nonempty $G$-invariant closed convex subset such that every conjugation invariant mean on $G$ gives full measure to the stabilizer of each point of this subset. Specializing our result
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Relative entropy and the Pinsker product formula for sofic groups Groups Geom. Dyn. (IF 0.6) Pub Date : 2021-03-25 Ben Hayes
We continue our study of the outer Pinsker factor for probability measure-preserving actions of sofic groups. Using the notion of local and doubly empirical convergence developed by Austin we prove that in many cases the outer Pinsker factor of a product action is the product of the outer Pinsker factors. Our results are parallel to those of Seward for Rokhlin entropy. We use these Pinsker product
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Structure of normally and finitely non-co-Hopfian groups Groups Geom. Dyn. (IF 0.6) Pub Date : 2021-03-25 Wouter van Limbeek
A group $G$ is (finitely) co-Hopfian if it does not contain any proper (finite-index) subgroups isomorphic to itself. We study finitely generated groups $G$ that admit a descending chain of proper normal finite-index subgroups, each of which is isomorphic to $G$. We prove that up to finite index, these are always obtained by pulling back a chain of subgroups from a free abelian quotient. We give two
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Subgroups of word hyperbolic groups in rational dimension 2 Groups Geom. Dyn. (IF 0.6) Pub Date : 2020-12-29 Shivam Arora, Eduardo Martínez-Pedroza
A result of Gersten states that if $G$ is a hyperbolic group with integral cohomological dimension $\mathsf{cd}_{\mathbb{Z}}(G)=2$ then every finitely presented subgroup is hyperbolic. We generalize this result for the rational case $\mathsf{cd}_{\mathbb{Q}}(G)=2$. In particular, the result applies to the class of torsion-free hyperbolic groups $G$ with $\mathsf{cd}_{\mathbb Z}(G)=3$ and $\mathsf{cd}_{\mathbb