Open Access
2019 Ubiquitous quasi-Fuchsian surfaces in cusped hyperbolic $3$–manifolds
Daryl Cooper, David Futer
Geom. Topol. 23(1): 241-298 (2019). 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.

Citation

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Daryl Cooper. David Futer. "Ubiquitous quasi-Fuchsian surfaces in cusped hyperbolic $3$–manifolds." Geom. Topol. 23 (1) 241 - 298, 2019. https://doi.org/10.2140/gt.2019.23.241

Information

Received: 17 May 2017; Revised: 30 April 2018; Accepted: 11 July 2018; Published: 2019
First available in Project Euclid: 12 March 2019

zbMATH: 07034546
MathSciNet: MR3921320
Digital Object Identifier: 10.2140/gt.2019.23.241

Subjects:
Primary: 20F65 , 20H10 , 30F40 , 57M50

Keywords: cubulation , hyperbolic 3-manifold , immersed surface , quasifuchsian

Rights: Copyright © 2019 Mathematical Sciences Publishers

Vol.23 • No. 1 • 2019
MSP
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