A Groupoid Picture of Elek Algebras

TL;DR

This paper reinterprets Elek's construction of C*-algebras from uniformly recurrent subgroups using groupoid theory, simplifying proofs and characterizing nuclearity while linking to group dynamics.

Contribution

It provides a groupoid framework for Elek algebras, clarifying their structure and properties, and relates them to the dynamics of group actions.

Findings

Simplified proofs of Elek's results

Full characterization of nuclearity of the algebras

Connection established between groupoid models and group dynamics

Abstract

We describe a construction by G\'abor Elek, associating C*-algebras with uniformly recurrent subgroups, in the language of groupoid C*-algebras. This allows us to simplify several proofs in the original paper and fully characterise their nuclearity. We furthermore relate our groupoids to the dynamics of the group acting on its uniformly recurrent subgroup.