Ubiquitous quasi-Fuchsian surfaces in cusped hyperbolic 3-manifolds
Abstract
This paper proves that every finite volume hyperbolic 3-manifold M contains a ubiquitous collection of closed, immersed, quasi-Fuchsian surfaces. These surfaces are ubiquitous in the sense that their preimages in the universal cover separate any pair of disjoint, non-asymptotic geodesic planes. The proof relies in a crucial way on the corresponding theorem of Kahn and Markovic for closed 3-manifolds. As a corollary of this result and a companion statement about surfaces with cusps, we recover Wise's theorem that the fundamental group of M acts freely and cocompactly on a CAT(0) cube complex.
Keywords
Cite
@article{arxiv.1705.02890,
title = {Ubiquitous quasi-Fuchsian surfaces in cusped hyperbolic 3-manifolds},
author = {Daryl Cooper and David Futer},
journal= {arXiv preprint arXiv:1705.02890},
year = {2019}
}
Comments
34 pages, 2 figures. v2 contains added references and a strengthened statement of Corollary 1.3. v3 contains minor corrections and revisions, including a discussion of virtual specialness. This version will appear in Geometry & Topology