Commutators in groups of piecewise projective homeomorphisms

TL;DR

This paper investigates the structure of commutator subgroups in certain non-amenable, finitely presented groups of piecewise projective homeomorphisms, revealing their simplicity and the nature of their quotients.

Contribution

It proves the simplicity of the commutator subgroup of specific groups and characterizes their proper quotients as abelian or metabelian, advancing understanding of their normal subgroup structure.

Findings

H's second derived subgroup is simple.

Every proper quotient of H is metabelian.

G_0's commutator subgroup is simple and all proper quotients are abelian.

Abstract

In 2012 Monod introduced examples of groups of piecewise projective homeomorphisms which are not amenable and which do not contain free subgroups, and later 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 $G_0$. We show that the group $H$ of piecewise projective homeomorphisms of $\mathbb{R}$ 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 $G_0$, which admits a presentation with $3$ generators and $9$ relations. Further we show that every proper quotient of $G_0$ is abelian. It follows that the normal subgroups of these groups are in bijective correspondence with those of the abelian (or metabelian) quotient.