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

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)$byauniformadditiveconstant,where$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)$forgeneralcompact$S$ofgenus$g$with$n$ boundary components: we have $$\text{vol}(M_{\psi })\le 3\sqrt{\pi \mathrm{/}2(2g-2+n)}\mathrm{\parallel }\psi {\mathrm{\parallel }}_{WP}\mathrm{.}$$ 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]).