Representation theory of topological full groups of \'etale groupoids and paradoxicality

TL;DR

This paper unifies various results on the representation theory and paradoxical decompositions of full groups of ample groupoids, providing criteria for amenability and insights into embeddings of Thompson's group V.

Contribution

It offers a unified framework for understanding representations, amenability, and paradoxicality in the context of ample groupoids and their full groups.

Findings

Criterion established for amenability of full groups of ample groupoids.

Comparison of orbit, Koopman, and groupoid-left-regular representations.

Unified treatment of paradoxical decompositions and embeddings of Thompson's group V.

Abstract

We provide a unified treatment of several results concerning full groups of ample groupoids and paradoxical decompositions attached to them. This includes a criterion for the full group of an ample groupoid being amenable as well as comparison of its orbit, Koopman and groupoid-left-regular representations. Besides that, we unify several recent results about paradoxicality in semigroups and groupoids, relating embeddings of Thompson's group V into full groups of ample \'etale groupoids.