Weil-Petersson translation distance and volumes of mapping tori
Jeffrey F. Brock

TL;DR
This paper establishes a quantitative relationship between the Weil-Petersson translation distance of monodromy maps and the volume of the corresponding hyperbolic 3-manifolds, providing bounds that depend only on the surface topology.
Contribution
It proves that the volume of hyperbolic mapping tori is linearly bounded by the Weil-Petersson translation distance, with constants depending solely on the surface topology.
Findings
Volume bounds proportional to Weil-Petersson translation distance
Constants depend only on surface topology
Provides a quantitative link between geometry and topology
Abstract
Given a closed hyperbolic 3-manifold T_\psi that fibers over the circle with monodromy \psi : S -> S, the monodromy determines an isometry of Teichmuller space with its Weil-Petersson metric whose translation distance ||\psi||_WP is positive. We show there is a constant K >= 1 depending only on the topology of S so that the volume of T_\psi satisfies ||\psi||_WP/K <= vol(T_\psi) <= K ||\psi||_WP.
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Taxonomy
TopicsGeometric and Algebraic Topology · Geometry and complex manifolds · Mathematical Dynamics and Fractals
