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

Abstract

A recent preprint of S. Kojima and G. McShane [KM] observes a beautiful explicit connection between Teichmüller 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 $ψ\in \text{Mod}(S)$ pseudo-Anosov, the double iteration $Q(ψ^{-n}(X),ψ^n(X))$ has convex core volume differing from $2n \text{vol}(M_ψ)$ by a uniform additive constant, where $M_ψ$ is the hyperbolic mapping torus for $ψ$. 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 $ψ\in \text{Mod}(S)$ for general compact $S$ of genus $g$ with $n$ boundary components: we have $$ \text{vol}(M_ψ) \le 3 \sqrt{π/2(2g - 2 +n)} \, \| ψ\|_{WP}.$$ 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 Teichmüller 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 Teichmüller translation distance via a Cauchy-Schwarz estimate (see [Lin]).