A finitely presented group of piecewise projective homeomorphisms

TL;DR

This paper introduces a new finitely presented, torsion-free, nonamenable subgroup of Monod's piecewise projective homeomorphisms, with novel element representation methods similar to Thompson's group F.

Contribution

It provides the first example of a finitely presented, torsion-free, nonamenable group without nonabelian free subgroups and develops a tree diagram representation for its elements.

Findings

First such example of its kind

Elements represented by labeled tree diagrams

Advances understanding of nonamenable groups

Abstract

In this article we will describe a finitely presented subgroup of Monod's group of piecewise projective homeomorphisms of R. This in particular provides a new example of a finitely presented group which is nonamenable and yet does not contain a nonabelian free subgroup. It is in fact the first such example which is torsion free. We will also develop a means for representing the elements of the group by labeled tree diagrams in a manner which closely parallels Richard Thompson's group F.