Topological full groups and stable rank one

Abstract

We establish stable rank one for the reduced group C$^*$-algebras of the C$^*$-simple topological full groups and dynamical alternating groups constructed by Kerr and Tucker-Drob. The proof relies on both type II$_1$ and type III phenomena, in the first case via the use of Følner towers and in the second via Ozawa's recent results on selflessness as applied to direct products of free products.