## Geometry & Topology

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

#### 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.

#### Article information

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

Dates
Revised: 30 April 2018
Accepted: 11 July 2018
First available in Project Euclid: 12 March 2019

https://projecteuclid.org/euclid.gt/1552356082

Digital Object Identifier
doi:10.2140/gt.2019.23.241

Mathematical Reviews number (MathSciNet)
MR3921320

Zentralblatt MATH identifier
07034546

#### Citation

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. https://projecteuclid.org/euclid.gt/1552356082

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