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European Mathematical Society Publishing House
2024-03-28 16:05:29
14
https://www.ems-ph.org/meta/jmeta-stream.php?jrn=GGD&vol=11&iss=2&update_since=2024-03-28
Groups, Geometry, and Dynamics
Groups Geom. Dyn.
GGD
1661-7207
1661-7215
Group theory and generalizations
10.4171/GGD
http://www.ems-ph.org/doi/10.4171/GGD
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European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society
11
2017
2
Cubulation of Gromov–Thurston manifolds
Anne
Giralt
Institut Mathématique de Jussieu-Paris Rive Gauche, France
Cube complexes, hyperbolic manifolds, Gromov–Thurston manifolds
In this article we prove that the fundamental group of certain manifolds, introduced by Gromov and Thurston [GT87] and obtained by branched cyclic covering over arithmetic manifolds, acts geometrically on a CAT (0) cube complex. We show in particular that these groups are linear over $\mathbb Z$.
Manifolds and cell complexes
Number theory
Differential geometry
393
414
10.4171/GGD/401
http://www.ems-ph.org/doi/10.4171/GGD/401
Locally compact lacunary hyperbolic groups
Adrien
Le Boudec
Université Paris-Sud 11, Orsay, France
Lacunary hyperbolic groups, asymptotic cones, locally compact groups
We investigate the class of locally compact lacunary hyperbolic groups. We prove that if a locally compact compactly generated group $G$ admits one asymptotic cone that is a real tree and whose natural transitive isometric action is focal, then $G$ must be a focal hyperbolic group. As an application, we characterize connected Lie groups and linear algebraic groups over an ultrametric local eld of characteristic zero having cut-points in one asymptotic cone. We prove several results for locally compact lacunary hyperbolic groups, and extend the characterization of nitely generated lacunary hyperbolic groups to the setting of locally compact groups. We moreover answer a question of Olshanskii, Osin and Sapir about subgroups of lacunary hyperbolic groups.
Group theory and generalizations
Topological groups, Lie groups
415
454
10.4171/GGD/402
http://www.ems-ph.org/doi/10.4171/GGD/402
On the joint behaviour of speed and entropy of random walks on groups
Gideon
Amir
Bar-Ilan University, Ramat Gan, Israel
Random walk, groups, entropy, rate of escape, permutation wreath product, automaton groups
For every $3/4 \le \delta, \beta< 1$ satisfying $\delta\leq \beta < \frac{1+\delta}{2}$ we construct a finitely generated group $\Gamma$ and a (symmetric, finitely supported) random walk $X_n$ on $\Gamma$ so that its expected distance from its starting point satisfies $\mathbf E |X_n|\asymp n^{\beta}$ and its entropy satisfies $H(X_n)\asymp n^\delta$. In fact, the speed and entropy can be set precisely to equal any two nice enough prescribed functions $f,h$ up to a constant factor as long as the functions satisfy the relation $n^{\frac{3}{4}}\leq h(n)\leq f(n)\leq \sqrt{{nh(n)}/{\log (n+1)}}\leq n^\gamma$ for some $\gamma
Combinatorics
Group theory and generalizations
Probability theory and stochastic processes
455
467
10.4171/GGD/403
http://www.ems-ph.org/doi/10.4171/GGD/403
Limit directions for Lorentzian Coxeter systems
Hao
Chen
Freie Universität Berlin, Germany
Jean-Philippe
Labbé
Freie Universität Berlin, Germany
Coxeter groups, Lorentz space, limit set, Coxeter arrangement, infinite root systems, fractal
Every Coxeter group admits a geometric representation as a group generated by refl ections in a real vector space. In the projective representation space of a Coxeter group, limit directions arising from a point are accumulation points of the orbit of this point. In particular, limit directions arising from roots are called limit roots. Recent studies show that limit roots lie on the isotropic cone of the representation space. In this paper, we study limit directions of Coxeter groups arising from any point when the representation space is a Lorentz space. We prove that the limit roots are the only light-like limit directions, and characterize the limit roots using eigenvectors of in finite-order elements. Then we describe the structure of space-like limit directions in terms of the projective Coxeter arrangement. Some non-Lorentzian cases are also discussed.
Group theory and generalizations
469
498
10.4171/GGD/404
http://www.ems-ph.org/doi/10.4171/GGD/404
Full groups of Cuntz–Krieger algebras and Higman–Thompson groups
Kengo
Matsumoto
Joetsu University of Education, Japan
Hiroki
Matui
Chiba University, Japan
Higmann–Thompson group, Thompson group, Cuntz–Krieger algebra, topological Markov shift, full group
In this paper, we will study representations of the continuous full group $\Gamma_A$ of a one-sided topological Markov shift $(X_A,\sigma_A)$ for an irreducible matrix $A$ with entries in $\{0,1\}$ as a generalization of Higman–Thompson groups $V_N, 1 < N \in {\mathbb{N}}$. We will show that the group $\Gamma_A$ can be represented as a group $\Gamma_A^{\operatorname{tab}}$ of matrices, called $A$-adic tables, with entries in admissible words of the shift space $X_A$, and a group $\Gamma_A^{\operatorname{PL}}$ of right continuous piecewise linear functions, called $A$-adic PL functions, on $[0,1]$ with finite singularities.
Group theory and generalizations
Dynamical systems and ergodic theory
Functional analysis
499
531
10.4171/GGD/405
http://www.ems-ph.org/doi/10.4171/GGD/405
On the genericity of pseudo-Anosov braids I: rigid braids
Sandrine
Caruso
Université Rennes 1, France
Braid group, mapping class group, Garside group, pseudo-Anosov, rigid braid
We prove that, in the $l$-ball of the Cayley graph of the braid group with $n \geqslant 3$ strands, the proportion of rigid pseudo-Anosov braids is bounded below independently of $l$ by a positive value.
Group theory and generalizations
533
547
10.4171/GGD/406
http://www.ems-ph.org/doi/10.4171/GGD/406
On the genericity of pseudo-Anosov braids II: conjugations to rigid braids
Sandrine
Caruso
Université Rennes 1, France
Bert
Wiest
Université de Rennes I, France
Braid group, mapping class group, Garside group, pseudo-Anosov, rigid braid
We prove that generic elements of braid groups are pseudo-Anosov, in the following sense: in the Cayley graph of the braid group with $n \geqslant 3$ strands, with respect to Garside's generating set, we prove that the proportion of pseudo-Anosov braids in the ball of radius $l$ tends to $1$ exponentially quickly as $l$ tends to infinity. Moreover, with a similar notion of genericity, we prove that for generic pairs of elements of the braid group, the conjugacy search problem can be solved in quadratic time. The idea behind both results is that generic braids can be conjugated „easily" into a rigid braid.
Group theory and generalizations
549
565
10.4171/GGD/407
http://www.ems-ph.org/doi/10.4171/GGD/407
Minimal models for actions of amenable groups
Bartosz
Frej
Wroclaw University of Science & Technology, Poland
Dawid
Huczek
Wrocław University of Science & Technology, Poland
Topologicalmodel, dynamical system, group action, amenable group, invariant measure, Choquet simplex, Borel isomorphism
We prove that on a metrizable, compact, zero-dimensional space every free action of an amenable group is measurably isomorphic to a minimal $G$-action with the same, i.e. a ffinely homeomorphic, simplex of measures.
Measure and integration
Dynamical systems and ergodic theory
567
583
10.4171/GGD/408
http://www.ems-ph.org/doi/10.4171/GGD/408
Small doubling in ordered groups: generators and structure
Gregory
Freiman
Tel Aviv University, Israel
Marcel
Herzog
Tel Aviv University, Israel
Patrizia
Longobardi
Università di Salerno, Fisciano (Salerno), Italy
Mercede
Maj
Università di Salerno, Fisciano (Salerno), Italy
Alain
Plagne
Ecole polytechnique, Palaiseau, France
Yonutz
Stanchescu
The Open University of Israel, Raanana, and Afeka Academic College, Tel Aviv, Israel
Inverse problems, small doubling, nilpotent groups, ordered groups
We prove several new results on the structure of the subgroup generated by a small doubling subset of an ordered group, abelian or not. We obtain precise results generalizing Freiman’s $3k-3$ and $3k-2$ theorems in the integers and several further generalizations.
Number theory
Group theory and generalizations
585
612
10.4171/GGD/409
http://www.ems-ph.org/doi/10.4171/GGD/409
Embedding mapping class groups into a finite product of trees
David
Hume
UC Louvain, Louvain-La-Neuve, Belgium
Tree-graded space, quasi-tree, embeddings, mapping class group, curve complex
We prove the equivalence between a relative bottleneck property and being quasi-isometric to a tree-graded space. As a consequence, we deduce that the quasi-trees of spaces defined axiomatically by Bestvina-Bromberg-Fujiwara are quasi-isometric to tree-graded spaces. Using this we prove that mapping class groups quasi-isometrically embed into a finite product of simplicial trees. In particular, these groups have finite Assouad–Nagata dimension, direct embeddings exhibiting $\ell^p$ compression exponent $1$ for all $p\geq 1$ and they quasi-isometrically embed into $\ell^1(\N)$. We deduce similar consequences for relatively hyperbolic groups whose parabolic subgroups satisfy such conditions. In obtaining these results we also demonstrate that curve complexes of compact surfaces and coned-off graphs of relatively hyperbolic groups admit quasi-isometric embeddings into finite products of trees.
Group theory and generalizations
613
647
10.4171/GGD/410
http://www.ems-ph.org/doi/10.4171/GGD/410
Chain conditions, elementary amenable groups, and descriptive set theory
Phillip
Wesolek
Université Catholique de Louvain, Louvain-La-Neuve, Belgium
Jay
Williams
California Institute of Technology, Pasadena, USA
Chain conditions, elementary amenable groups, descriptive set theory, space of marked groups
We first consider three well-known chain conditions in the space of marked groups: the minimal condition on centralizers, the maximal condition on subgroups, and the maximal condition on normal subgroups. For each condition, we produce a characterization in terms of well-founded descriptive-set-theoretic trees. Using these characterizations, we demonstrate that the sets given by these conditions are co-analytic and not Borel in the space of marked groups. We then adapt our techniques to show elementary amenable marked groups may be characterized by well-founded descriptive-set-theoretic trees, and therefore, elementary amenability is equivalent to a chain condition. Our characterization again implies the set of elementary amenable groups is co-analytic and non-Borel. As corollary, we obtain a new, non-constructive, proof of the existence of finitely generated amenable groups that are not elementary amenable.
Group theory and generalizations
General topology
649
684
10.4171/GGD/411
http://www.ems-ph.org/doi/10.4171/GGD/411
Non-amenability and visual Gromov hyperbolic spaces
Juhani
Koivisto
University of Helsinki, Finland
Hyperbolic spaces, isoperimetry, amenability
We prove that a uniformly coarsely proper hyperbolic cone over a bounded metric space consisting of a fi nite union of uniformly coarsely connected components each containing at least two points is non-amenable and apply this to visual Gromov hyperbolic spaces.
Differential geometry
Group theory and generalizations
685
704
10.4171/GGD/412
http://www.ems-ph.org/doi/10.4171/GGD/412
Almost algebraic actions of algebraic groups and applications to algebraic representations
Uri
Bader
Technion, Haifa, Israel
Bruno
Duchesne
Université de Lorraine, Vandœuvre-lès-Nancy, France
Jean
Lécureux
Université Paris-Sud 11, France
Complete separable valued fields, probability measures on algebraic varieties, algebraic representations of amenable ergodic actions, Margulis–Zimmer super-rigidity
Let $G$ be an algebraic group over a complete separable valued field $k$. We discuss the dynamics of the $G$-action on spaces of probability measures on algebraic $G$-varieties. We show that the stabilizers of measures are almost algebraic and the orbits are separated by open invariant sets. We discuss various applications, including existence results for algebraic representations of amenable ergodic actions. The latter provides an essential technical step in the recent generalization of Margulis–Zimmer super-rigidity phenomenon [2].
Group theory and generalizations
Field theory and polynomials
Dynamical systems and ergodic theory
705
738
10.4171/GGD/413
http://www.ems-ph.org/doi/10.4171/GGD/413
Anosov structures on Margulis spacetimes
Sourav
Ghosh
University of Paris 11, Orsay, France
Margulis spacetime, stable and unstable leaves, Anosov property
In this paper we describe the stable and unstable leaves for the a ffine fl ow on the space of non-wandering spacelike a ffine lines of a Margulis spacetime and prove contraction properties of the leaves under the fl ow. We also show that monodromies of Margulis spacetimes are “Anosov representations in non semi-simple Lie groups.”
Differential geometry
Dynamical systems and ergodic theory
739
775
10.4171/GGD/414
http://www.ems-ph.org/doi/10.4171/GGD/414