#### Volume 23, issue 1 (2019)

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

### Daryl Cooper and David Futer

Geometry & Topology 23 (2019) 241–298
##### 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 $CAT\left(0\right)$ cube complex.

##### Keywords
hyperbolic 3-manifold, immersed surface, quasifuchsian, cubulation
##### Mathematical Subject Classification 2010
Primary: 20F65, 20H10, 30F40, 57M50