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

Daryl Cooper and David Futer

Geometry & Topology 23 (2019) 241–298

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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hyperbolic 3-manifold, immersed surface, quasifuchsian, cubulation
Mathematical Subject Classification 2010
Primary: 20F65, 20H10, 30F40, 57M50
Received: 17 May 2017
Revised: 30 April 2018
Accepted: 11 July 2018
Published: 5 March 2019
Proposed: Ian Agol
Seconded: Benson Farb, Bruce Kleiner
Daryl Cooper
Department of Mathematics
University of California at Santa Barbara
Santa Barbara, CA
United States
David Futer
Department of Mathematics
Temple University
Philadelphia, PA
United States