Amenability of groupoids and asymptotic invariance of convolution powers

TL;DR

This paper extends the concept of amenability from groups to measured groupoids, showing that amenability can be characterized by the asymptotic invariance of convolution powers in this broader context.

Contribution

It generalizes the asymptotic invariance characterization of amenability from groups to measured groupoids, linking group actions and random environments.

Findings

Amenability of measure class preserving group actions is characterized by trivial tail behavior of random walks.

The paper establishes a new equivalence between amenability and asymptotic invariance in the context of groupoids.

It provides a framework connecting amenability, random environments, and convolution powers.

Abstract

The original definition of amenability given by von Neumann in the highly non-constructive terms of means was later recast by Day using approximately invariant probability measures. Moreover, as it was conjectured by Furstenberg and proved by Kaimanovich-Vershik and Rosenblatt, the amenability of a locally compact group is actually equivalent to the existence of a single probability measure on the group with the property that the sequence of its convolution powers is asymptotically invariant. In the present article we extend this characterization of amenability to measured groupoids. It implies, in particular, that the amenability of a measure class preserving group action is equivalent to the existence of a random environment on the group parameterized by the action space, and such that the tail of the random walk in almost every environment is trivial.