Commutators in groups of piecewise projective homeomorphisms

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

Marla Reeves and Dennis Lawrence in a Joint Junior Voice Recital

This is the program for the joint junior voice recital of soprano Marla Reeves and baritone Dennis Lawrence. Pianist Janine Reeves accompanied Marla Reeves; pianist Retha Kilmer accompanied Lawrence. The recital took place on February 8, 1985, in the Mabee Fine Arts Center Recital Hall

Marla Reeves and Dennis Lawrence in a Joint Sophomore Voice Recital

This is the program for the joint sophomore voice recital of soprano Marla Reeves and baritone Dennis Lawrence. Pianist Janine Reeves and flutist Ralph Rauch accompanied Reeves; pianist Janet Tullos accompanied Lawrence. The recital took place on March 30, 1984, in the Mabee Fine Arts Center Recital Hall

A combination theorem for strong relative hyperbolicity

We prove a combination theorem for trees of (strongly) relatively hyperbolic spaces and finite graphs of (strongly) relatively hyperbolic groups. This gives a geometric extension of Bestvina and Feighn's Combination Theorem for hyperbolic groups and answers a question of Swarup. We also prove a converse to the main Combination Theorem