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A complex hyperbolic Riley slice

Parker, John R.; Will, Pierre

A complex hyperbolic Riley slice Thumbnail


Pierre Will


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.


Parker, J. R., & Will, P. (2017). A complex hyperbolic Riley slice. Geometry & Topology, 21(6), 3391-3451.

Journal Article Type Article
Acceptance Date Jun 28, 2016
Online Publication Date Aug 31, 2017
Publication Date Aug 31, 2017
Deposit Date Jul 5, 2016
Publicly Available Date Sep 6, 2017
Journal Geometry and Topology
Print ISSN 1465-3060
Electronic ISSN 1364-0380
Publisher Mathematical Sciences Publishers (MSP)
Peer Reviewed Peer Reviewed
Volume 21
Issue 6
Pages 3391-3451
Related Public URLs


Published Journal Article (965 Kb)

Copyright Statement
First published in Geometry & Topology, 21(6), 2017, published by Mathematical Sciences Publishers. © 2017 Mathematical Sciences Publishers. All rights reserved.

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