A complex hyperbolic Riley slice

Parker, John R.; Will, Pierre

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
• Public URL: https://durham-repository.worktribe.com/output/1379939
• Related Public URLs: http://arxiv.org/pdf/1510.01505v2.pdf