Non-inner amenability of the Higman-Thompson groups

TL;DR

This paper proves that all Higman-Thompson groups $T_n$ and $V_n$ are non-inner amenable for all $n\,\geq 2$, extending previous results from the specific case of Thompson's groups.

Contribution

It introduces new methods to establish non-inner amenability for all Higman-Thompson groups, generalizing prior work limited to the case $n=2$.

Findings

All Higman-Thompson groups $T_n$ and $V_n$ are non-inner amenable for $n\geq 2$.

Provides an alternative proof for the $n=2$ case using different tools.

Extends the class of groups known to be non-inner amenable.

Abstract

We prove that the Higman-Thompson groups $T_n$ and $V_n$ are non-inner amenable for all $n\ge 2$. This extends Haagerup and Olesen's result that Thompson's groups $T=T_2$ and $V=V_2$ are non-inner amenable. Their proof relied on machinery only available in the $n=2$ case, namely Thurston's piecewise-projective model for Thompson's group $T$, so our approach necessarily utilizes different tools. This also provides an alternate proof of Haagerup-Olesen's result when $n=2$.