Thompson's group $F$ and uniformly finite homology

Abstract

We use the uniformly finite homology developed by Block and Weinberger to study the geometry of the Cayley graph of Thompson’s group $F$. In particular, a certain class of subgraph of $F$ is shown to be nonamenable (in the Følner sense). This shows that if the Cayley graph of $F$ is amenable, these subsets, which include every finitely generated submonoid of the positive monoid of $F$, must necessarily have measure zero.