  We say that a cover of surfaces S -> 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 cover has this property. We give new
explicit examples of irregular branched covers that do not satisfy the
necessary condition as well as explicit covers that satisfy the sufficient
condition. Our criteria are conditions on simple closed curves, and our proofs
use the combinatorial topology of curves on surfaces.
