Commutators in groups of piecewise projective homeomorphisms

In [7] Monod introduced examples of groups of piecewise projective homeomorphisms which are not amenable and which do not contain free subgroups, and in [6] Lodha and Moore introduced examples of finitely presented groups with the same property. In this article we examine the normal subgroup structure of these groups. Two important cases of our results are the groups H and G0. We show that the group H of piecewise projective homeomorphisms of R has the property that H″ is simple and that every proper quotient of H is metabelian. We establish simplicity of the commutator subgroup of the group G0, which admits a presentation with 3 generators and 9 relations. Further, we show that every proper quotient of G0 is abelian. It follows that the normal subgroups of these groups are in bijective correspondence with those of the abelian (or metabelian) quotient.