Ubiquitous quasi-Fuchsian surfaces in cusped hyperbolic 3–manifolds

Cooper, Daryl; Futer, David

doi:10.2140/gt.2019.23.241

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

Daryl Cooper and David Futer

Geometry & Topology 23 (2019) 241–298

DOI: 10.2140/gt.2019.23.241

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.

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

Volume 23, issue 1 (2019)