Hyperbolic and cubical rigidities of Thompson's group V

TL;DR

This paper establishes criteria for hyperbolic rigidity in groups, applies it to Thompson's group V, and shows it has fixed-point properties on finite-dimensional CAT(0) cube complexes, providing new insights into its geometric actions.

Contribution

It introduces a general criterion for hyperbolic elementary groups and applies it to prove Thompson's group V is hyperbolically elementary with fixed-point properties.

Findings

Thompson's group V is hyperbolically elementary.

V satisfies Property (FW_∞), fixing points on finite-dimensional CAT(0) cube complexes.

First example of a finitely presented group acting properly on an infinite-dimensional CAT(0) cube complex with fixed points on all finite-dimensional actions.

Abstract

In this article, we state and prove a general criterion allowing us to show that some groups are hyperbolically elementary, meaning that every isometric action of one of these groups on a Gromov-hyperbolic space either fixes a point at infinity or has bounded orbits. Also, we show how such a hyperbolic rigidity leads to fixed-point properties on finite-dimensional CAT(0) cube complexes. As an application, we prove that Thompson's group $V$ is hyperbolically elementary, and we deduce that it satisfies Property $(FW_{\infty})$, ie., every isometric action of $V$ on a finite-dimensional CAT(0) cube complex fixes a point. It provides the first example of a (finitely presented) group acting properly on an infinite-dimensional CAT(0) cube complex such that all its actions on finite-dimensional CAT(0) cube complexes have global fixed points.