Random groups have fixed points on \(\text{CAT}(0)\) cube complexes
type: Item
instance of: scholarly article
zbMATH Open document ID: 1238.53029
DOI: 10.1090/S0002-9939-2011-11343-1
author: Koji Fujiwara
author: Tetsu Toyoda
published in: Proceedings of the American Mathematical Society
publication date: 2012-04-13T00:00:00Z
full work available at URL: https://arxiv.org/abs/1012.4147
Mathematics Subject Classification ID: 53C23
Mathematics Subject Classification ID: 20F65
Mathematics Subject Classification ID: 20P05
Mathematics Subject Classification ID: 51F99
zbMATH DE Number: 6024109
zbMATH Keywords: random groups
zbMATH Keywords: graph model
zbMATH Keywords: expanders
zbMATH Keywords: isometric action
zbMATH Keywords: cube complex
MaRDI profile type: Publication
arXiv ID: 1012.4147
cites work: Q4720067
cites work: Property \(A\) and \(\text{CAT}(0)\) cube complexes.
cites work: Q3995825
cites work: Addendum to ``Random walk in random groups'' by M. Gromov.
cites work: Fixed-point property of random groups.
cites work: \(N\)-step energy of maps and the fixed-point property of random groups.
cites work: Combinatorial harmonic maps and discrete-group actions on Hadamard spaces
cites work: CAT(0) spaces and expanders.
cites work: Towards a Calculus for Non-Linear Spectral Gaps [Extended Abstract]
cites work: Groups acting on \(\text{CAT}(0)\) cube complexes
cites work: Q4437801
cites work: CAT(0) spaces on which a certain type of singularity is bounded
cites work: Fixed point property for a CAT(0) space which admits a proper cocompact group action
cites work: Generalized harmonic maps and representations of discrete groups
OpenAlex ID: W2962863921
Recommended article: \(N\)-step energy of maps and the fixed-point property of random groups.
Recommended article: Random groups at density 𝑑<3/14 act non-trivially on a CAT(0) cube complex
Recommended article: Poincaré inequalities, embeddings, and wild groups
Recommended article: Balanced walls for random groups
Recommended article: Cubulating random groups in the square model
Recommended article: Fixed-point property for affine actions on a Hilbert space
Recommended article: Fixed-point property of random quotients by plain words.
Recommended article: Addendum to ``Random walk in random groups'' by M. Gromov.
Recommended article: The Furstenberg–Poisson boundary and CAT(0) cube complexes
Recommended article: Random Möbius-Kantor group cobordisms
title: Random groups have fixed points on \(\text{CAT}(0)\) cube complexes
review text: The random groups considered in this paper are built using Gromov's graph model. Specifically, let \(G=(V,E)\) be a finite oriented graph. To each edge one associates a random element of the generating set \(\{s_1,\dots,s_k, s_1^{-1},\dots, s_k^{-1}\}\); these choices are made independently and with uniform probability distribution. Each unoriented cycle in \(G\) gives rise to a relation between the generators in a natural way. Thus, one obtains \(\Gamma(G)\), the \(k\)-generated random group associated to \(G\).NEWLINENEWLINEOf particular interest in this context are the graphs that form a sequence of expanders \(\{G_n\}\). By definition, such graphs have uniformly bounded vertex degree, satisfy a uniform quadratic Poincaré inequality, and the number of their vertices tends to infinity. Under additional conditions on the girth and degree of \(G_n\), the corresponding random groups have Kazhdan's property (T) with high probability, see [\textit{M. Gromov}, Geom. Funct. Anal. 13, No. 1, 73--146 (2003; Zbl 1122.20021)] and [\textit{L. Silberman}, Geom. Funct. Anal. 13, No. 1, 147--177 (2003; Zbl 1124.20027)]. Recall that the property (T) is equivalent to the existence of a common fixed point for every isometric action of \(\Gamma\) on a Hilbert space.NEWLINENEWLINEThe main result of this paper (Theorem 1.5) replaces the Hilbert space by a complete CAT(0) cube complex. The action is not assumed to be simplicial, i.e., it need not preserve individual simplices. Theorem 1.5 states that for any sequence of expanders of vertex degree at least \(2\) and sufficiently large girth, every isometric action of the random group on a complete CAT(0) cube complex has a common fixed point, with high probability.
reviewed by: Leonid V. Kovalev
description: scientific article; zbMATH DE number 6024109