A complex hyperbolic Riley slice

Parker, John R.; Will, Pierre

doi:10.2140/gt.2017.21.3391

A complex hyperbolic Riley slice

Parker, John R.; Will, Pierre

Authors

Professor John Parker j.r.parker@durham.ac.uk
Professor

Pierre Will

Abstract

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.

Citation

Parker, J. R., & Will, P. (2017). A complex hyperbolic Riley slice. Geometry & Topology, 21(6), 3391-3451. https://doi.org/10.2140/gt.2017.21.3391

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
DOI	https://doi.org/10.2140/gt.2017.21.3391
Related Public URLs	http://arxiv.org/pdf/1510.01505v2.pdf

Files

Published Journal Article (965 Kb)
PDF

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