On the bounded cohomology for ergodic nonsingular actions of amenable groups

TL;DR

This paper proves the existence of bounded ergodic G-valued cocycles for ergodic nonsingular actions of amenable groups, extending the understanding of cohomological properties in ergodic theory.

Contribution

It demonstrates the existence of bounded ergodic cocycles valued in certain locally compact groups for amenable group actions, a novel extension in the theory of group cohomology.

Findings

Existence of bounded ergodic G-valued cocycles for amenable groups.

Extension of cohomological results to nonsingular actions.

Application to groups with compact closure of inner automorphisms.

Abstract

Let $\Gamma$ be an amenable countable discrete group. Fix an ergodic free nonsingular action of $\Gamma$ on a nonatomic standard probability space. Let $G$ be a compactly generated locally compact second countable group such that the closure of the group of inner automorphisms of $G$ is compact in the natural topology. It is shown that there exists a {\it bounded} ergodic $G$-valued cocycle of $\Gamma$.