Non-inner amenability of the Thompson groups T and V

TL;DR

This paper proves that the Thompson groups T and V are not inner amenable, explores implications for their von Neumann algebras and the amenability of F, and provides new characterizations of F's amenability.

Contribution

It establishes non-inner amenability of T and V, links the simplicity of T's C*-algebra to F's non-amenability, and offers new characterizations of F's amenability.

Findings

T and V are not inner amenable.

If T's reduced C*-algebra is simple, then F is non-amenable.

New equivalent characterizations of F's amenability.

Abstract

In this paper we prove that the Thompson groups $T$ and $V$ are not inner amenable. In particular, their group von Neumann algebras do not have property $\Gamma$. Moreover, we prove that if the reduced group $C^\ast$-algebra of $T$ is simple, then the Thompson group $F$ is non-amenable. Furthermore, we give a few new equivalent characterizations of amenability of $F$.