Upper bounds on Renormalized Volume for Schottky groups

Abstract

In this article we show that for any given Riemann surface $\Sigma$ of genus $g$, we can bound (from above) the renormalized volume of a (hyperbolic) Schottky group with boundary at infinity conformal to $\Sigma$ in terms of the genus and the combined extremal lengths on $\Sigma$ of $(g-1)$ disjoint, non-homotopic, simple closed compressible curves. This result is used to partially answer a question posed by Maldacena about comparing renormalized volumes of Schottky and Fuchsian manifolds with the same conformal boundary.

17 pages, 5 figures. Typos and constants in the main theorem have been corrected. Proof of the main result was restructured. To appear in Mathematische Annalen

References in corpus (2)

• On the renormalized volume of hyperbolic 3-manifolds
• Additive continuity of the renormalized volume under geometric limits

Cited by in corpus (2)

• Comparing hyperbolic and extremal lengths for shortest curves
• The infimum of the dual volume of convex co-compact hyperbolic $3$-manifolds