Finite Factor Representations of Higman-Thompson groups

TL;DR

This paper classifies all finite factor representations of Higman-Thompson groups, showing they are either regular or scalar from abelianizations, and explores implications for ergodic actions.

Contribution

It provides a complete classification of finite factor representations of Higman-Thompson groups and links these to measure-preserving ergodic actions.

Findings

Only regular and scalar representations exist for these groups.

Any ergodic measure-preserving action of a simple Higman-Thompson group is essentially free.

Discussion of finite factor representations in other group classes.

Abstract

We prove that the only finite factor-representations of the Higman-Thompson groups $\{F_{n,r}\}$, $ \{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 a simple Higman-Thompson group must be essentially free. Finite factor representations of other classes of groups are also discussed.