Inflexibility, Weil-Petersson distance, and volumes of fibered 3-manifolds

TL;DR

This paper establishes explicit bounds relating hyperbolic volume, Weil-Petersson translation distance, and Teichmüller translation distance for fibered 3-manifolds, providing new quantitative insights into the geometry of moduli space.

Contribution

It introduces a geometric inflexibility estimate to connect hyperbolic volume with Weil-Petersson distance, leading to the first explicit bounds on Weil-Petersson systoles and moduli space diameter.

Findings

Derived explicit lower bounds for Weil-Petersson translation distance in terms of hyperbolic volume.

Provided the first explicit estimates on Weil-Petersson systoles of moduli space.

Recovered and extended previous estimates on Teichmüller translation distance.

Abstract

A recent preprint of S. Kojima and G. McShane [KM] observes a beautiful explicit connection between Teichm\"uller translation distance and hyperbolic volume. It relies on a key estimate which we supply here: using geometric inflexibility of hyperbolic 3-manifolds, we show that for $S$ a closed surface, and $\psi \in \text{Mod}(S)$ pseudo-Anosov, the double iteration $Q(\psi^{-n}(X),\psi^n(X))$ has convex core volume differing from $2n \text{vol}(M_\psi)$ by a uniform additive constant, where $M_\psi$ is the hyperbolic mapping torus for $\psi$. We combine this estimate with work of Schlenker, and a branched covering argument to obtain an explicit lower bound on Weil-Petersson translation distance of a pseudo-Anosov $\psi \in \text{Mod}(S)$ for general compact $S$ of genus $g$ with $n$ boundary components: we have $$ \text{vol}(M_\psi) \le 3 \sqrt{\pi/2(2g - 2 +n)} \, \| \psi \|_{WP}.$$ This gives the first explicit estimates on the Weil-Petersson systoles of moduli space, of the minimal distance between nodal surfaces in the completion of Teichm\"uller space, and explicit lower bounds to the Weil-Petersson diameter of the moduli space via [CP]. In the process, we recover the estimates of [KM] on Teichm\"uller translation distance via a Cauchy-Schwarz estimate (see [Lin]).