Geometry & Topology

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

Daryl Cooper and David Futer

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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.

Article information

Geom. Topol., Volume 23, Number 1 (2019), 241-298.

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

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Zentralblatt MATH identifier

Primary: 20F65: Geometric group theory [See also 05C25, 20E08, 57Mxx] 20H10: Fuchsian groups and their generalizations [See also 11F06, 22E40, 30F35, 32Nxx] 30F40: Kleinian groups [See also 20H10] 57M50: Geometric structures on low-dimensional manifolds

hyperbolic 3-manifold immersed surface quasifuchsian cubulation


Cooper, Daryl; Futer, David. Ubiquitous quasi-Fuchsian surfaces in cusped hyperbolic $3$–manifolds. Geom. Topol. 23 (2019), no. 1, 241--298. doi:10.2140/gt.2019.23.241.

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