The conformal measures of a normal subgroup of a cocompact Fuchsian group

Abstract

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.