Topological full groups are a useful groupoid invariant that have been used to solve important open problems in group theory. Steinberg algebras are a purely algebraic analogue of groupoid C*-algebras that generalise both Leavitt path algebras and Kumjian–Pask algebras. The Steinberg algebra of an ample Hausdorff groupoid is a quotient of the algebra generated by the inverse semigroup of compact open bisections of the groupoid. Since the topological full group of an ample Hausdorff groupoid sits inside this inverse semigroup, it is natural to ask what the relationship is between the algebra of the topological full group and the Steinberg algebra of the groupoid. In this talk I will present recent results answering this question. (This is joint work with Lisa Orloff Clark, Mahya Ghandehari, Eun Ji Kang, and Dilian Yang.)
