\(\ell^2\) invariants of equivalence relations and groups
type: Item
instance of: scholarly article
title: \(\ell^2\) invariants of equivalence relations and groups
zbMATH Open document ID: 1022.37002
author: Damien Gaboriau
published in: Publications Mathématiques
publication date: 2003-11-02T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/104184
review text: Let \((X,{\mathcal B},\mu)\) be a measure space with a continuous probability measure. A measure preserving action \(\alpha\) of a discrete countable group \(\Gamma\) on \((X,{\mathcal B},\mu)\) forms an equivalence relation \[ {\mathcal R}_\alpha:= \{(x,y)\in X\times X: \alpha(\Gamma)(x)= \alpha(\Gamma) (y)\}. \] In the paper a geometric ergodic study of discrete countable groups, a connection between \(l^2\)-Betti numbers of groups and their measure preserving action and some questions of the classification of standard equivalence relations are studied. The measurable equivalent (ME), orbital equivalent (OE) and stable orbital equivalent (SOE) relations are defined. The \(l^2\)-Betti numbers of the equivalence relation are introduced. They are denoted by \(\beta_n({\mathcal R},\mu)\). The main results of the paper: 1. If \(\Gamma_1\) and \(\Gamma_2\) are two discrete countable groups, and \(\Gamma_1\) is ME to \(\Gamma_2\), then they have proportional \(l^2\)-Betti numbers. For all \(n\in \mathbb{N}: \beta_n(\Gamma_2)= c\beta_n(\Gamma_1)\), \(c\) is constant. In particular, if two groups have free measure preserving actions, which form the same equivalence relation, then they have the same \(l^2\)-Betti numbers. From this some known results and new ones are deduced. Analogous results for SOE relations are received. 2. The geometric dimension of relations \({\mathcal R}\) and the ergodic dimension of a group \(\Gamma\) are introduced. Some results about them are obtained. For example, the SOE relations have the same geometric dimension. 3. Let \(1\to N\to \Gamma\to \Lambda\to 1\) be an exact sequence of infinite groups. If \(\beta_1(N)\) is finite then \(\beta_1(\Gamma)=0\).
reviewed by: Victor Sharapov
Mathematics Subject Classification ID: 37A15
zbMATH DE Number: 1874453
zbMATH Keywords: measure preserving action
zbMATH Keywords: equivalence relation
zbMATH Keywords: \(l^2\)-Betti numbers
zbMATH Keywords: classification
MaRDI profile type: Publication
OpenAlex ID: W2327972468
DOI: 10.1007/S102400200002
description: scientific article; zbMATH DE number 1874453