Amenable actions preserving a locally finite metric

TL;DR

This paper explores groups acting on sets with locally finite metrics, extending previous work on amenable actions by relaxing transitivity, and characterizes such groups as those embedding into totally disconnected amenable locally compact groups.

Contribution

It introduces a broader class of amenable actions that preserve locally finite metrics without requiring transitivity, and analyzes their properties and examples.

Findings

Groups with such actions embed into totally disconnected amenable locally compact groups.

Includes non-amenable groups with transitive actions preserving locally finite metrics.

Reviews results on amenable actions on locally finite graphs.

Abstract

The class A of countable groups that admit a faithful, transitive, amenable -- in the sense that there is an invariant mean -- action on a set has been widely investigated in the past. In this paper, we no longer require the action to be transitive, but we ask for it to preserve a locally finite metric (and still to be faithful and amenable). The groups having such actions are those that embed into a totally disconnected amenable locally compact group. Then we focus on the subclass A 1 of groups for which the actions are moreover transitive. This class is strictly contained into A and includes non-amenable groups. An important particular case of actions preserving a locally finite metric is given by actions by automorphisms of locally finite connected graphs. We take this opportunity, in our partly expository paper, to review some nice results about amenable actions in this setting.