A groupoid picture of the Elek algebras

Abstract

We reformulate a construction by Gábor Elek, which associates $C^{\ast} $-algebras with uniformly recurrent subgroups, in the language of groupoid $C^{\ast} $-algebras. This allows us to simplify several proofs from the original paper and add the converse direction to Elek's characterisation of nuclearity, showing that his sufficient condition is in fact necessary. We furthermore relate our groupoids to the dynamics of the group acting on its uniformly recurrent subgroup.