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

## TL;DR

This paper demonstrates that every finite volume hyperbolic 3-manifold contains a widespread collection of quasi-Fuchsian surfaces, which play a key role in understanding the manifold's geometric and group-theoretic properties.

## Contribution

It extends the existence of ubiquitous quasi-Fuchsian surfaces to all cusped hyperbolic 3-manifolds, building on Kahn and Markovic's work for closed manifolds.

## Key findings

- Existence of ubiquitous quasi-Fuchsian surfaces in all cusped hyperbolic 3-manifolds
- Surfaces' preimages separate disjoint geodesic planes in the universal cover
- Reproof of Wise's theorem on fundamental group actions on CAT(0) cube complexes

## 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.