Quasi-Fuchsian Surfaces In Hyperbolic Link Complements

TL;DR

This paper proves that all hyperbolic link complements contain closed quasi-Fuchsian surfaces, and demonstrates that specific Dehn fillings produce manifolds with closed immersed incompressible surfaces, revealing new geometric structures in these manifolds.

Contribution

It establishes the existence of closed quasi-Fuchsian surfaces in all hyperbolic link complements and links this to the behavior of Dehn fillings.

Findings

Every hyperbolic link complement contains closed quasi-Fuchsian surfaces.

Certain Dehn fillings on hyperbolic link complements yield manifolds with closed immersed incompressible surfaces.

The result connects geometric structures with Dehn filling outcomes.

Abstract

We show that every hyperbolic link complement contains closed quasi-Fuchsian surfaces. As a consequence, we obtain the result that on a hyperbolic link complement, if we remove from each cusp of the manifold a certain finite set of slopes, then all remaining Dehn fillings on the link complement yield manifolds with closed immersed incompressible surfaces.