About amenability of subgroups of the group of diffeomorphisms of the interval [0,1]

TL;DR

This paper investigates the amenability of certain subgroups within the group of diffeomorphisms of the interval, providing new insights and solutions, including for Thompson's group F.

Contribution

It introduces an averaging functional on continuous functions over the diffeomorphism group and proves amenability for several discrete subgroups, notably solving the amenability problem for Thompson's group F.

Findings

Averaging linear functional on the diffeomorphism group is constructed.

Amenability of specific discrete subgroups of $ ext{Diff}^3([0,1])$ is established.

Thompson's group F is shown to be amenable.

Abstract

Averaging linear functional on the space continuous functions of the group of diffeomorphisms of interval is found. Amenability of several discrete subgroups of the group of diffeomorphisms $\Diff^3([0,1])$ of interval is prove. In particular, a solution of the problem of amenability of the Thompson's group $F$ is given.