Quasi-convexity and shrinkwrapping

Abstract

We extend a result of Minsky to show that, for a map of a surface to a hyperbolic $3$–manifold, which is $2$–incompressible rel a geodesic link with a definite tube radius, the set of noncontractible simple loops with bounded length representatives is quasi-convex in the complex of curves of the surface. We also show how wide product regions can be used to find a geodesic link with a definite tube radius with respect to which a map is $2$–incompressible.