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

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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∈ Mod(S) pseudo-Anosov, the double iteration Q(ψ⁻ⁿ(X),ψⁿ(X)) has convex core volume differing from 2n vol(Mψ) by a uniform additive constant, where Mψ 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∈ Mod(S) for general compact S of genus g with n boundary components: we have vol(Mψ) ≤ 3 √π/2(2g - 2 +n) ψWP. 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]).