Quasi-invariant means and Zimmer amenability

TL;DR

This paper characterizes when group actions on countable sets admit quasi-invariant means with specific cocycles and introduces a geometric condition called weighted hyperfiniteness that ensures Zimmer amenability of the action on the Stone-c6ch compactification.

Contribution

It provides a combinatorial criterion for quasi-invariant means and introduces weighted hyperfiniteness as a new geometric condition implying Zimmer amenability.

Findings

A necessary and sufficient condition for the existence of a quasi-invariant mean with a given cocycle.

Weighted hyperfiniteness guarantees Zimmer amenability of the action on the Stone-c6ch compactification.

Amenable groups and groups with finite asymptotic dimension are weighted hyperfinite.

Abstract

Let $\Gamma$ be a countable group acting on a countable set $X$ by permutations. We give a necessary and sufficient condition for the action to have a quasi-invariant mean with a given cocycle. This can be viewed as a combinatorial analogue of the condition for the existence of a quasi-invariant measure in the Borel case given by Miller. Then we show a geometric condition that guarantees that the corresponding action on the Stone-\v{C}ech compactification is Zimmer amenable. The geometric condition (weighted hyperfiniteness) resembles Property A. We do not know the exact relation between the two notions, however, we can show that amenable groups and groups of finite asymptotic dimension are weighted hyperfinite.