A complex hyperbolic Riley slice

We study subgroups of $\mathrm{P}\mathrm{U}(2,1)$ generated by two non-commuting 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 ${\mathbb{R}}^2$ . By considering the action on complex hyperbolic space ${\mathbf{H}}_{\mathbb{C}}^2$ 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 $(3,3,\mathrm{\infty })$-triangle groups. We also consider a particular group on the boundary of the disc $\mathcal{Z}$ where the commutator $[A,B]$ 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.