Quasi-actions and rough Cayley graphs for locally compact groups

TL;DR

This paper introduces the concept of rough Cayley graphs for locally compact groups via quasi-actions, establishing their properties, uniqueness, and applications to growth and amenability, thus extending geometric group theory to a broader class of groups.

Contribution

It defines rough Cayley graphs for locally compact groups, constructs them using quasi-lattices, and relates their properties to group growth and amenability.

Findings

Rough Cayley graphs are unique up to quasi-isometry.

Polynomial, intermediate, and exponential growth are preserved between groups and their rough Cayley graphs.

Amenability of groups corresponds to amenability of their rough Cayley graphs.

Abstract

We define the notion of rough Cayley graph for compactly generated locally compact groups in terms of quasi-actions. We construct such a graph for any compactly generated locally compact group using quasi-lattices and show uniqueness up to quasi-isometry. A class of examples is given by the Cayley graphs of cocompact lattices in compactly generated groups. As an application, we show that a compactly generated group has polynomial growth if and only if its rough Cayley graph has polynomial growth (same for intermediate and exponential growth). Moreover, a unimodular compactly generated group is amenable if and only if its rough Cayley graph is amenable as a metric space.