We use cookies to ensure that we give you the best experience on our website. By continuing to browse this repository, you give consent for essential cookies to be used. You can read more about our Privacy and Cookie Policy.

Durham Research Online
You are in:

A complex hyperbolic Riley slice.

Parker, John R. and Will, Pierre (2017) 'A complex hyperbolic Riley slice.', Geometry and topology., 21 (6). pp. 3391-3451.


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.

Item Type:Article
Full text:Publisher-imposed embargo
(AM) Accepted Manuscript
File format - PDF (Copyright agreement prohibits open access to the full-text)
Full text:(VoR) Version of Record
Download PDF
Publisher Web site:
Publisher statement:First published in Geometry & Topology, 21(6), 2017, published by Mathematical Sciences Publishers. © 2017 Mathematical Sciences Publishers. All rights reserved.
Date accepted:28 June 2016
Date deposited:05 July 2016
Date of first online publication:31 August 2017
Date first made open access:06 September 2017

Save or Share this output

Look up in GoogleScholar