Topological full groups and stable rank one

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 F{\o}lner towers and in the second via Ozawa's recent results on selflessness as applied to direct products of free products.