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

A recent preprint of S. Kojima and G. McShane [27] 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 $\psi \in \mathrm{Mod}(S)$ pseudo-Anosov, the double iteration $Q(\psi^{-n} (X), \psi^n (X))$ has convex core volume differing from $2n \, \mathrm{vol}(M_{\psi})$ 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 $\psi \in \mathrm{Mod}(S)$ for general compact $S$ of genus $g$ with $n$ boundary components: we have$$\mathrm{vol}(M_{\psi}) \leq \frac{3}{2} \sqrt{\mathrm{area}(S)} \, {\lVert \psi \rVert}_{\mathrm{WP}}$$where $\mathrm{area}(S) = 2 \pi (2g - 2 + n)$ is the usual Poincaré area of any complete finite area hyperbolic structure on $\mathrm{int}(S)$. This gives the first explicit estimates on the lengths of 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 [20]. In the process, we recover the estimates of [27] on Teichmüller translation distance via a Cauchy–Schwarz estimate (see [29]).

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Accepted 19 July 2015

Published 8 July 2016