#### Volume 21, issue 6 (2017)

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A complex hyperbolic Riley slice

### John R Parker and Pierre Will

Geometry & Topology 21 (2017) 3391–3451
##### 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 $\mathsc{U}$ the set of conjugacy classes of such groups. We provide a set of coordinates on $\mathsc{U}$ that make it homeomorphic to ${ℝ}^{2}$. By considering the action on complex hyperbolic space ${H}_{ℂ}^{2}$ of groups in $\mathsc{U}$, we describe a two-dimensional disc $\mathsc{Z}$ in $\mathsc{U}$ 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 $\mathsc{Z}$ 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.

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##### Keywords
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
##### Publication
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
##### Authors
 John R Parker Department of Mathematical Sciences Durham University Durham United Kingdom http://maths.dur.ac.uk/~dma0jrp/ Pierre Will Université Grenoble Alpes Institut Fourier Saint-Martin-d’Hères France https://www-fourier.ujf-grenoble.fr/~will/