Topological Full Groups of Ample Groupoids with Applications to Graph Algebras

TL;DR

This paper extends the concept of topological full groups to locally compact ample groupoids, providing new invariance results, applications to graph algebras, and an embedding theorem analogous to Kirchberg's for $C^*$-algebras.

Contribution

It generalizes the topological full group definition to locally compact groupoids and establishes new isomorphism invariants, with applications to graph algebras and embedding theorems.

Findings

Topological full groups are complete invariants for certain groupoids.

Sharper results obtained for graph groupoids.

An embedding theorem for ample groupoids similar to Kirchberg's theorem.

Abstract

We study the topological full group of ample groupoids over locally compact spaces. We extend Matui's definition of the topological full group from the compact, to the locally compact case. We provide two general classes of groupoids for which the topological full group, as an abstract group, is a complete isomorphism invariant. Hereby extending Matui's Isomorphism Theorem. As an application, we study graph groupoids and their topological full groups, and obtain sharper results for this class. The machinery developed in this process is used to prove an embedding theorem for ample groupoids, akin to Kirchberg's Embedding Theorem for $C^*$-algebras. Consequences for graph $C^*$-algebras and Leavitt path algebras are also spelled out. In particular, we improve on a recent embedding theorem of Brownlowe and S{\o}rensen for Leavitt path algebras.