Given an ample Hausdorff groupoid, there is a natural representation of its topological full group into its complex Steinberg algebra. In this talk, we will characterize when this representation is injective/surjective Restricting to discrete groupoids, we will see some unexpected properties about its extension into the groupoid C*-algebra.

This is joint work with Becky Armstrong, Lisa Clark, Mahya Ghandehari, and Eun Ji Kang.