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Ubiquitous quasi-Fuchsian surfaces in cusped hyperbolic 3-manifolds
Cooper, Daryl Futer, David
Cooper, Daryl
Futer, David
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Pre-print
Date
2019
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DOI
10.2140/gt.2019.23.241
Abstract
This paper proves 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, non-asymptotic 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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Mathematical Sciences Publishers
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GEOMETRY & TOPOLOGY
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