Orbital cohomology and Kahler rigidity

Abstract: In the late $70$'s Feldman and Moore defined the cohomology associated to a countable equivalence relation with coefficients in an Abelian Polish group. When the equivalence relation is the orbital one, that is it is induced by a measure preserving action of a countable group $\Gamma$ on a standard Borel probability space $(X,\mu)$, it still makes sense to consider the Feldmann-Moore $1$-cohomology with $G$-coefficients, where this time $G$ can be any topological group. The latter cohomology, denoted by $H^1(\Gamma \curvearrowright X;G)$, is very misterious and hard to compute, except for some exceptional cases. In this expository paper we are going to focus our attention on the particular case when $\Gamma$ is a finitely generated group and $G$ is a Hermitian Lie group. We are going to give some recent rigidity results in this context and we will see how those results can be used to say something relevant about (some subsets of) the orbital cohomology.