A nonamenable finitely presented group of piecewise projective homeomorphisms

In this article we will describe a finitely presented subgroup of the group of piecewise projective homeomorphisms of the real projective line. Th is 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 fi rst 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 $F$.