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Ubiquitous quasi-Fuchsian surfaces in cusped hyperbolic $3$–manifolds

Daryl Cooper and David Futer

Geometry & Topology 23 (2019) 241–298
Abstract

We prove 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, nonasymptotic 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.

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Keywords
hyperbolic 3-manifold, immersed surface, quasifuchsian, cubulation
Mathematical Subject Classification 2010
Primary: 20F65, 20H10, 30F40, 57M50
References
Publication
Received: 17 May 2017
Revised: 30 April 2018
Accepted: 11 July 2018
Published: 5 March 2019
Proposed: Ian Agol
Seconded: Benson Farb, Bruce Kleiner
Authors
Daryl Cooper
Department of Mathematics
University of California at Santa Barbara
Santa Barbara, CA
United States
http://web.math.ucsb.edu/~cooper/
David Futer
Department of Mathematics
Temple University
Philadelphia, PA
United States
https://math.temple.edu/~dfuter