J un 2 00 3 Holography and the Geometry of Certain

Applying the idea of AdS/CFT correspondence, Krasnov [Kra00] studied a class of convex cocompact hyperbolic 3-manifolds. In physics literature they are known as Euclidean BTZ black holes. Mathematically they can be described as H/Γ, where Γ ⊂ PSL(2,C) is a Schottky group. His main result, roughly speaking, identifies the renormalized volume of such a manifold with the action for the Liouville theory on the conformal infinity. See Takhtajan and Teo [TT] for a rigorous proof and related topics. This is a nice result establishing another holography correspondence. But the Liouville theory is not yet fully established and the action which was proposed by Takhtajan and Zograf [ZT87] is quite complicated, so it is desirable to clarify the meaning of the renormalized volume in a more geometric and transparent way. This question was first raised by Manin and Marcolli [MM01] and they speculated that the renormalized volume could be calculated through the volume of the convex core of the bulk space based on an explicit example and a recent result by Brock [Bro] in a different but related situation. In this paper we try to compute the renormalized volume in terms of geometric data. As the first step, we compute the renormalized volume using a different normalization which geometrically is very natural as it uses the distance function to the convex core. The result is very simple and geometric. We first describe the result in the Fuchsian case. Let Γ ⊂ PSL(2,R) be a Fuchsian Schottky group with 2g