Thompson's Group F and Uniformly Finite Homology

TL;DR

This paper explores the connection between uniformly finite homology and amenability in Thompson's Group F, revealing non-amenability of certain subgraphs and implications for measure theory.

Contribution

It applies uniformly finite homology to analyze Thompson's Group F, identifying non-amenable subgraphs and their measure-theoretic properties.

Findings

Certain subgraphs of F are non-amenable

If F is amenable, these subgraphs have measure zero

Application of homology provides new insights into F's structure

Abstract

This paper demonstrates the uniformly finite homology developed by Block and Weinberger and its relationship to amenable spaces via applications to the Cayley graph of Thompson's Group F. In particular, a certain class of subgraph of F is shown to be non-amenable. This shows that if F is amenable, these subsets (which include every finitely generated submonoid of the positive monoid of F) must necessarily have measure zero.