\(\ell^2\) invariants of equivalence relations and groups
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 \(\beta_1(\Gamma)=0\).
scientific article; zbMATH DE number 1874453