Holography and the Geometry of Certain Convex Cocompact Hyperbolic 3-Manifolds

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. 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 compute the renormalized volume in terms of geometric data. We first describe the result in the Fuchsian case for simplicity. Let Γ ⊂ PSL(2,R) be a Fuchsian Schottky group with 2g generators. Let Ω(Γ) ⊂ S be its ordinary set. In physics, X = H/Γ is known as the Euclidean version of a non-rotating BTZ black hole. Mathematically X is a convex cocompact hyperbolic 3-manifold with the conformal infinity Σ = Ω(Γ)/Γ which is a compact Riemann surface of genus g. X has