$C$*-simple groups without free subgroups

Abstract

We construct first examples of non-trivial groups without non-cyclic free subgroups whose reduced $C$*-algebra is simple and has unique trace. This answers a question of de la Harpe. Both torsion and torsion free examples are provided. In particular, we show that the reduced $C$*-algebra of the free Burnside group $B(m,n)$ of rank $m$ ≥ 2 and any sufficiently large odd exponent $n$ is simple and has unique trace.