Richard Thompson group F is not amenable

TL;DR

This paper proves that Richard Thompson's group F, known for its complex structure, is not amenable, resolving a long-standing open problem in group theory.

Contribution

It provides a rigorous proof that Thompson's group F is non-amenable, settling a major open question in the field.

Findings

Group F is proven to be non-amenable.

The proof advances understanding of the properties of Thompson's groups.

This result impacts the study of group actions and geometric group theory.

Abstract

Richard Thompson's group F is the group of piecewise linear homeomorphisms of the unit interval with a finite number of break points, all at dyadic rational numbers (their denominators are powers of 2) and with slopes which are powers of 2. A discrete group G is amenable if there exists a finitely-additive probability measure on G which is invariant under left translations and is defined on all subsets of G. The amenability question for F is a well known open problem. In this paper we prove that group F is not amenable.