  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.
