Hyperbolic and cubical rigidities of Thompson's group V

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 $(F{W}_{\infty })$, 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.

24 pages, 5 figures

References in corpus (2)

• Poc Sets, Median Algebras and Group Actions
• Commensurating actions for groups of piecewise continuous transformations

Cited by in corpus (1)

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