We use the uniformly finite homology developed by Block and Weinberger to study the geometry of the Cayley graph of Thompson’s group . In particular, a certain class of subgraph of is shown to be nonamenable (in the Følner sense). This shows that if the Cayley graph of is amenable, these subsets, which include every finitely generated submonoid of the positive monoid of , must necessarily have measure zero.
"Thompson's group $F$ and uniformly finite homology." Algebr. Geom. Topol. 9 (4) 2349 - 2360, 2009. https://doi.org/10.2140/agt.2009.9.2349