  In this paper, we study the conformal measures of a normal subgroup of a
cocompact Fuchsian group. In particular, we relate the extremal conformal
measures to the eigenmeasures of a suitable Ruelle operator. Using Ancona's
theorem, adapted to the Ruelle operator setting, we show that if the group of
deck transformations $G$ is hyperbolic then the extremal conformal measures and
the hyperbolic boundary of $G$ coincide. We then interpret these results in
terms of the asymptotic behavior of cutting sequences of geodesics on a regular
cover of a compact hyperbolic surface.
