Finite factor representations of Higman–Thompson groups

Abstract

We prove that the only finite factor representations of the Higman–Thompson groups $\{F_{n,r}\}$ and $\{G_{n,r}\}$ are the regular representations and scalar representations arising from group abelianizations. As a corollary, we obtain that any measure-preserving ergodic action of the commutator subgroup of a Higman–Thompson group must be essentially free. Finite factor representations of other classes of groups are also discussed.