Weil-Petersson translation distance and volumes of mapping tori

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.