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 $CAT\left(0\right)$ cube complex.