Amenability of Groupoids Arising from Partial Semigroup Actions and Topological Higher Rank Graphs

TL;DR

This paper establishes conditions under which groupoids from partial semigroup actions and higher rank graphs are amenable, providing a new approach to analyze their properties using groupoid theory.

Contribution

It introduces a general criterion for the amenability of groupoids with a group-valued cocycle, applicable to those from partial semigroup actions and topological higher rank graphs.

Findings

Proves that groupoids with an amenable kernel and certain coverage conditions are amenable.

Shows applicability to groupoids from directed graphs, higher rank graphs, and topological higher rank graphs.

Offers an alternative groupoid approach to studying these structures.

Abstract

We consider the amenability of groupoids $G$ equipped with a group valued cocycle $c:G\to Q$ with amenable kernel $c^{-1}(e)$. We prove a general result which implies, in particular, that $G$ is amenable whenever $Q$ is amenable and if there is countable set $D\subset G$ such that $c(G^{u})D=Q$ for all $u\in G^{(0)}$. We show that our result is applicable to groupoids arising from partial semigroup actions. We explore these actions in detail and show that these groupoids include those arising from directed graphs, higher rank graphs and even topological higher rank graphs. We believe our methods yield a nice alternative groupoid approach to these important constructions.