On groups whose actions on finite-dimensional CAT(0) spaces have global fixed points

TL;DR

This paper establishes a criterion for fixed points of group elements acting semi-simply on finite-dimensional CAT(0) spaces and applies it to show that certain Thompson groups have fixed points under such actions.

Contribution

It introduces a new criterion for fixed points in semi-simple actions on finite-dimensional CAT(0) spaces and applies it to Thompson's groups.

Findings

Thompson's group T has a global fixed point in finite-dimensional CAT(0) spaces.

Certain generalizations of Thompson's group V also have fixed points.

The fixed point criterion applies to semi-simple actions on complete CAT(0) spaces.

Abstract

We give a criterion for group elements to have fixed points with respect to a semi-simple action on a complete CAT(0) space of finite topological dimension. As an application, we show that Thompson's group T and various generalizations of Thompson's group V have global fixed points when they act semi-simply on finite-dimensional complete CAT(0) spaces.