Relatively amenable actions of Thompson's groups

TL;DR

This paper explores the concept of relative amenability in topological actions, demonstrating that Thompson's group T acting on the circle is relatively amenable with respect to F, and linking their exactness properties.

Contribution

It introduces the notion of relative amenability for topological actions and establishes the relative amenability of T's action on the circle with respect to F, connecting their exactness.

Findings

T's action on S^1 is relatively amenable with respect to F

F is exact if and only if T is exact

The groupoid of germs of T's action on S^1 is Borel amenable

Abstract

We investigate the notion of relatively amenable topological action and show that the action of Thompson's group $T$ on $S^1$ is relatively amenable with respect to Thompson's group $F$. We use this to conclude that $F$ is exact if and only if $T$ is exact. Moreover, we prove that the groupoid of germs of the action of $T$ on $S^1$ is Borel amenable.