A complex hyperbolic Riley slice

Abstract

We study subgroups of $PU\left(2,1\right)$ generated by two noncommuting unipotent maps $A$ and $B$ whose product $AB$ is also unipotent. We call $\mathcal{U}$ the set of conjugacy classes of such groups. We provide a set of coordinates on $\mathcal{U}$ that make it homeomorphic to ${ℝ}^2$. By considering the action on complex hyperbolic space $H_{ℂ}^2$ of groups in $\mathcal{U}$, we describe a two-dimensional disc $\mathcal{Z}$ in $\mathcal{U}$ that parametrises a family of discrete groups. As a corollary, we give a proof of a conjecture of Schwartz for $\left(3,3,\infty \right)$–triangle groups. We also consider a particular group on the boundary of the disc $\mathcal{Z}$ where the commutator $\left[A,B\right]$ is also unipotent. We show that the boundary of the quotient orbifold associated to the latter group gives a spherical CR uniformisation of the Whitehead link complement.