A complex hyperbolic Riley slice

Parker, John; Will, Pierre

doi:10.2140/gt.2017.21.3391

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A complex hyperbolic Riley slice

John R Parker and Pierre Will

Geometry & Topology 21 (2017) 3391–3451

DOI: 10.2140/gt.2017.21.3391

Abstract

We study subgroups of $PU (2, 1)$ generated by two noncommuting unipotent maps $A$ and $B$ whose product $A B$ 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 $ℝ^{2}$ . By considering the action on complex hyperbolic space $H_{ℂ}^{2}$ 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 $(3, 3, \infty)$ –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.

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

References

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/

Volume 21, issue 6 (2017)