The Action of the Thompson Group $F$ on Infinite Trees

Abstract

We construct an action of the Thompson group $F$ on a compact space built from pairs of infinite, binary rooted trees. The action arises as an $F$-equivariant compactification of the action of $F$ by translations on one of its homogeneous spaces, $F/H_2$, corresponding to a certain subgroup $H_2$ of $F$. The representation of $F$ on the Hilbert space $\ell^2(F/H_2)$ is faithful on the complex group algebra $\mathbb{C}[F]$.