Boundary actions of outer automorphism groups of Thompson-like groups

TL;DR

This paper demonstrates that outer automorphism groups of Thompson-like groups act freely on the Hilbert cube boundary, establishing their C*-simplicity and extending to Higman--Thompson groups.

Contribution

It proves the existence of topologically free boundary actions for outer automorphism groups of certain groupoids, showing their C*-simplicity, including for Higman--Thompson groups.

Findings

Outer automorphism groups act freely on the Hilbert cube boundary.

These groups are C*-simple.

Results apply to all Higman--Thompson groups.

Abstract

For every Cuntz--Krieger groupoid, we show that there is a topologically free boundary action of the outer automorphism group of its topological full group on the Hilbert cube. In particular, these outer automorphism groups, including the outer automorphism groups of all Higman--Thompson groups, are C*-simple.