A complex hyperbolic Riley slice.

Abstract

We study subgroups of PU(2,1) generated by two non-commuting unipotent maps A and B whose product AB is also unipotent. We call U the set of conjugacy classes of such groups. We provide a set of coordinates on U that make it homeomorphic to R2 . By considering the action on complex hyperbolic space H2C of groups in U, we describe a two dimensional disc Z in U that parametrises a family of discrete groups. As a corollary, we give a proof of a conjecture of Schwartz for (3,3,∞)-triangle groups. We also consider a particular group on the boundary of the disc 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.