Covers of surfaces, Kleinian groups and the curve complex

We show that curve complex distance is coarsely equal to electric distance in hyperbolic manifolds associated to Kleinian surface groups, up to errors that are polynomial in the complexity of the underlying surface. We then use this to control the quasi-isometry constants of maps between curve complexes induced by finite covers of surfaces. This makes effective previously known results, in the sense that the error terms are explicitly determined, and allows us to give several applications. In particular, we effectively relate the electric circumference of a fibered manifold to the curve complex translation length of its monodromy, and we give quantitative bounds on virtual specialness for cube complexes dual to curves on surfaces.