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A complex hyperbolic Riley slice

John R Parker and Pierre Will

Geometry & Topology 21 (2017) 3391–3451

We study subgroups of PU(2,1) generated by two noncommuting 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 2. By considering the action on complex hyperbolic space H2 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.

discrete subgroups of Lie groups, complex hyperbolic geometry, spherical CR structures, complex hyperbolic quasi-Fuchsian groups
Mathematical Subject Classification 2010
Primary: 20H10, 22E40, 51M10
Secondary: 57M50
Received: 2 October 2015
Revised: 17 May 2016
Accepted: 28 June 2016
Published: 31 August 2017
Proposed: Walter Neumann
Seconded: Danny Calegari, Jean-Pierre Otal
John R Parker
Department of Mathematical Sciences
Durham University
United Kingdom
Pierre Will
Université Grenoble Alpes
Institut Fourier