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Ubiquitous quasi-Fuchsian
surfaces in cusped hyperbolic $3$–manifolds
Daryl Cooper and David Futer |
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Geometry & Topology 23 (2019) 241–298
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Abstract |
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We prove that every finite-volume hyperbolic –manifold 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 –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 acts freely and cocompactly on a cube complex. |
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Keywords
hyperbolic 3-manifold, immersed surface, quasifuchsian,
cubulation
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Mathematical Subject Classification 2010
Primary: 20F65, 20H10, 30F40, 57M50
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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
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Authors |
| Daryl Cooper | |
| Department of Mathematics University of California at Santa Barbara Santa Barbara, CA United States |
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| http://web.math.ucsb.edu/~cooper/ | |
| David Futer | |
| Department of Mathematics Temple University Philadelphia, PA United States |
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| https://math.temple.edu/~dfuter | |
