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

TL;DR

This paper investigates conformal measures associated with normal subgroups of cocompact Fuchsian groups, linking extremal measures to Ruelle operator eigenmeasures and exploring their relation to hyperbolic geometry and geodesic behavior.

Contribution

It establishes a connection between extremal conformal measures and Ruelle operator eigenmeasures for hyperbolic groups, extending Ancona's theorem to this setting.

Findings

Extremal conformal measures coincide with the hyperbolic boundary for hyperbolic groups.

The results relate conformal measures to the asymptotic behavior of geodesic cutting sequences.

The study provides a new perspective on the boundary theory of hyperbolic groups.

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.