Seminar

The notion of "renormalized volume" gives a way to assign a finite volume to an infinite volume hyperbolic 3-manifold. Krasnov and Schlenker showed (following a similar result for Schottky space by Takhtjan-Teo) that the renormalized volume on a Bers slice gives a potential for the Weil-Petersson metric on Teichmüller space - Schlenker, in turn, used this idea to give an effective version of bounds due to Brock on the volume of the convex core of quasifuchsian manifolds in terms the Weil-Petersson distance between the two components of the conformal boundary. Kojima-McShane connected Schlenker's idea to volumes of hyperbolic 3-manifolds that fiber of the circle and the normalized entropy on pseudo-Anosov mapping class (Teichmüller translation length). We discuss a proof of improved bounds (originally due to Brock) in terms of the Weil-Petersson translation length to volumes of mapping tori as well as new implications for the length spectrum and systolic geometry of moduli space with the Weil-Petersson metric.  We will give an elementary overview and introduction to the above, with a particular emphasis on new results on the geometry of the Weil-Petersson metric.