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.

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

References in corpus (4)

• Effective bilipschitz bounds on drilling and filling
• Nearly Fuchsian surface subgroups of finite covolume Kleinian groups
• Finite-volume Hyperbolic 3-Manifolds contain immersed Quasi-Fuchsian surfaces
• Quasi-Fuchsian Surfaces In Hyperbolic Link Complements

Cited by in corpus (11)

• Hyperbolic groups acting improperly
• Nearly Fuchsian surface subgroups of finite covolume Kleinian groups
• Relative cubulations and groups with a 2-sphere boundary
• Constructing thin subgroups of SL(n+1,R) via bending
• Torsion in the homology of finite covers of 3-manifolds
• Quasi-isometric rigidity of three manifold groups
• Relatively geometric actions of Kähler groups on CAT(0) cube complexes
• Subgroups of Genus-2 Quasi-Fuchsian groups and Cocompact Kleinian Groups
• Thin subgroups isomorphic to Gromov--Piatetski-Shapiro lattices
• Prescribed virtual homological torsion of 3-manifolds
• On the profinite distinguishability of hyperbolic Dehn fillings of finite-volume 3-manifolds