About amenability of subgroups of the group of diffeomorphisms of the interval

TL;DR

This paper investigates the amenability of certain subgroups within the group of interval diffeomorphisms, introducing an averaging functional and proving amenability for specific discrete subgroups, including Thompson's group F.

Contribution

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

Findings

Averaging linear functional on the group of diffeomorphisms is constructed.

Amenability of several 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.