  We prove uniform Ancona-Gou\"ezel-Lalley inequalities for an extension by a
hyperbolic group $G$ of a Markov map which allows to deduce that the visual
boundary of the group and the Martin boundary are H\"older equivalent. As
application, we identify the set of minimal conformal measures of a regular
cover of a convex-cocompact CAT(-1)-manifold with the visual boundary of the
covering group, provided that this group is hyperbolic.
