Symmetry, isotopy, and irregular covers

Abstract

We say that a covering space of the surface S over X has the Birman--Hilden property if the subgroup of the mapping class group of X consisting of mapping classes that have representatives that lift to S embeds in the mapping class group of S modulo the group of deck transformations. We identify one necessary condition and one sufficient condition for when a covering space has this property. We give new explicit examples of irregular branched covering spaces that do not satisfy the necessary condition as well as explicit covering spaces that satisfy the sufficient condition. Our criteria are conditions on simple closed curves, and our proofs use the combinatorial topology of curves on surfaces.