We extend a result of Minsky to show that, for a map of a surface to a hyperbolic –manifold, which is –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 –incompressible.
"Quasi-convexity and shrinkwrapping." Algebr. Geom. Topol. 9 (4) 2443 - 2478, 2009. https://doi.org/10.2140/agt.2009.9.2443