Abstract
AbstractIn 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 transformationsGis hyperbolic then the extremal conformal measures and the hyperbolic boundary ofGcoincide. 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.
Highlights
Let D = {z ∈ C : |z| < 1} be the open hyperbolic unit disc and let ∂D = {z ∈ C : |z| = 1}
In this work we study the conformal measures of a normal subgroup of a cocompact Fuchsian group, namely under the assumption that there exists a cocompact Fuchsian group Ŵ0 with Ŵ ⊳ Ŵ0
We show that if the potential is uniformly irreducible with respect to a hyperbolic graph for all λ > ρ(φ), the minimal Martin boundary Mm(λ) and the hyperbolic boundary coincide
Summary
Using an extended version of Ancona’s theorem to the Ruelle operator setting (see §2.4), for every δ > δŴ we relate the (Ŵ, δ)-conformal measures, via a suitable Martin boundary, to the hyperbolic boundary of G, denoted by ∂G. By Proposition 2.13, we can narrow our discussion to non-atomic measures and we may ignore the values of fY on ∂Ia. To prove Theorem 1.1, we map, in several steps, the Radon eigenmeasures of LXφ ,δ for eigenvalue 1 to the Radon measures on Y which satisfy Ŵ0-regularity condition; see equation (8) in the following lemma. According to the explicit formula for φ (see [2]) there exists an absolute constant N (that depends only on the group Ŵ0) such that |γn| ≤ N for all n, where | · | denotes the word length with respect to the generating set {ea}.
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