Semi-simple actions of the Higman-Thompson groups $T_n$ on finite-dimensional CAT(0) spaces

TL;DR

This paper proves that Higman-Thompson groups $T_n$ have fixed points in all semi-simple isometric actions on finite-dimensional CAT(0) spaces, extending known results for Thompson's group $T$.

Contribution

It generalizes the fixed point property from Thompson's group $T$ to the broader class of Higman-Thompson groups $T_n$ for finite-dimensional CAT(0) spaces.

Findings

Every semi-simple action of $T_n$ on finite-dimensional CAT(0) spaces has a global fixed point.

Uses ring group representations of $T_n$ as homeomorphisms of $S^1$.

Extends fixed point results to a wider class of groups.

Abstract

In this paper, we study isometric actions on finite-dimensional CAT(0) spaces for the Higman-Thompson groups $T_n$, which are generalizations of Thompson's group $T$. It is known that every semi-simple action of $T$ on a complete CAT(0) space of finite covering dimension has a global fixed point. After this result, we show that every semi-simple action of $T_n$ on a complete CAT(0) space of finite covering dimension has a global fixed point. In the proof, we regard $T_n$ as ring groups of homeomorphisms of $S^1$ introduced by Kim, Koberda and Lodha, and use general facts on these groups.