Ubiquitous quasi-Fuchsian surfaces in cusped hyperbolic 3-manifolds

Daryl Cooper; David Futer

doi:10.2140/gt.2019.23.241

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arXiv · 1705.02890

Ubiquitous quasi-Fuchsian surfaces in cusped hyperbolic 3-manifolds

Daryl Cooper·David Futer

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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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BibTeX RIS

Daryl Cooper, David Futer. 2017-05-08. Ubiquitous quasi-Fuchsian surfaces in cusped hyperbolic 3-manifolds. https://doi.org/10.2140/gt.2019.23.241

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