The classification theorem for groups of homeomorphisms of the line. Nonamenability of Thompson's group $F$

TL;DR

This paper establishes a criterion based on combinatorial properties for the existence of a projectively invariant measure, enabling classification of groups of line homeomorphisms and demonstrating the nonamenability of Thompson's group F.

Contribution

It introduces a new criterion linking group amenability to combinatorial properties, aiding classification of line homeomorphism groups and proving Thompson's group F is nonamenable.

Findings

Criterion for projectively invariant measure based on combinatorial properties

Classification scheme for groups of homeomorphisms of the line

Proof of nonamenability of Thompson's group F

Abstract

This paper allows one to obtain a criterion for the existence of a projectively invariant measure formulated in terms of combinatorial properties of a group (amenability of some canonical quotient group). Such necessary and sufficient condition is a basis of the classification scheme for groups of homeomorphisms of the line. In particular, a nonamenability of Thompson's group $F$ follows from the obtained criterion.