Outer automorphism groups of some ergodic equivalence relations (Q1772475)

Let \((X,{\mathcal B},\mu )\) be a standard probability space and let \(R\) be a countable relation of type \(II_1\) on \((X,{\mathcal B},\mu )\), i.e., measurable, countable, ergodic and measure-preserving equivalence relation \(R\subset X\times X\). Such equivalence relation can be presented as the orbit relation \(R_{X,\Gamma }=\{ (x,y)\in X\times X| \Gamma\cdot x=\Gamma\cdot y \}\) of an ergodic, measure-preserving action of some countable group \(\Gamma\) on the space \((X,{\mathcal B},\mu )\). Given an equivalence relation \(R\) on \((X,\mu )\), the group of relation automorphisms \(\Aut R=\{ T\in \Aut (X,\mu )\mid T\times T(R)=R\}\), the subgroup \(\operatorname{Inn} R\) of inner automorphisms and the outer automorphism group \(\operatorname{Out} R\) are considered. The main results of the paper are: 1. Let \(G\) be a semisimple, connected real Lie group satisfying some additional properties, then \(\operatorname{Out} R_{X,\Gamma }=A^{\ast }(X.\Gamma )\) while \(A^{\ast }(X,\Gamma )\cong \Aut (X,\Gamma )/\Gamma \). 2. There are some generalizations of the preceding result. 3. It is shown in Theorems 1.4--1.9 of the paper that in specific cases it is possible to compute the groups \(\operatorname{Out} R_{X,\Gamma }\) explicitly. The method is based on Zimmer's superrigidity for measurable cocycles, Ratner's theorem and Gromov's measure equivalence construction.