A complex hyperbolic Riley slice

TL;DR

This paper explores a specific family of complex hyperbolic groups generated by unipotent maps, providing a parameter space, describing discrete subgroups, and proving a conjecture related to triangle groups and link complements.

Contribution

It introduces a coordinate system for a class of complex hyperbolic groups, describes a 2D parameter space for discrete groups, and proves a conjecture on triangle groups and link complement uniformization.

Findings

Parameter space for groups is homeomorphic to ^2.

Identified a 2D disc parametrizing discrete groups.

Proved Schwartz's conjecture for (3,3,) triangle groups.

Abstract

We study subgroups of ${\rm PU}(2,1)$ 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 $\mathbf{H}^2_{\mathbb{C}}$ 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 $(3,3,\infty)$-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.