Comparison and Simplicity of Commutator Subgroups of Full Groups

TL;DR

This paper proves that for certain minimal, locally compact Hausdorff étale groupoids with comparison, the commutator subgroup of their full group is simple, extending previous results in the field.

Contribution

It generalizes the simplicity of commutator subgroups to a broader class of groupoids, building on and extending prior work on Cantor minimal systems and topological full groups.

Findings

Commutator subgroup of full group is simple under given conditions

Generalizes previous results to more general groupoids

Connects properties of groupoids with algebraic simplicity

Abstract

We show that for a minimal, second countable, locally compact Hausdorff \'etale groupoid whose unit space is homeomorphic to the Cantor set, if the groupoid has comparison then the commutator subgroup of its full group is simple. This generalizes a result of Bezuglyi and Medynets for Cantor minimal systems and complements Matui's results for topological full groups.