Noncompact Fuchsian and quasi-Fuchsian surfaces in hyperbolic 3–manifolds

Abstract

Given a noncompact quasi-Fuchsian surface in a finite volume hyperbolic 3–manifold, we introduce a new invariant called the cusp thickness, that measures how far the surface is from being totally geodesic. We relate this new invariant to the width of a surface, which allows us to extend and generalize results known for totally geodesic surfaces. We also show that checkerboard surfaces provide examples of such surfaces in alternating knot complements and give examples of how the bounds apply to particular classes of knots. We then utilize the results to generate closed immersed essential surfaces.