Weil-Petersson translation distance and volumes of mapping tori

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 $\psi$ 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.