Inflexibility, Weil-Petersson distance, and volumes of fibered 3-manifolds
Inflexibility, Weil-Petersson distance, and volumes of fibered 3-manifolds
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不灵活性、Weil-Petersson 距离和纤维 3 流形的体积
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发表时间：
2014
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通讯作者：
K. Bromberg
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作者：
Jeffrey F. Brock;K. Bromberg
S. Kojima和G. McShane [KM]观察到Teichm\“uller平移距离和双曲体积之间的一个美丽的显式连接。它依赖于我们在这里提供的一个关键估计：利用双曲三维流形的几何可解性，我们证明了对于$S$一个闭曲面，$\psi \in \text{Mod}（S）$ pseudo-Anosov，双重迭代$Q（\psi^{-n}（X），\psi^n（X））$的凸核体积与2n\text{vol}（M_\psi）$相差一个均匀的加性常数，其中$M_\psi$是$\psi$的双曲映射环面。我们结合联合收割机这个估计与工作的Schlenker，和分支覆盖的论点，以获得一个明确的下界Weil-Petersson平移距离的伪Anosov $\psi \in \text{Mod}（S）$一般紧$S$的亏格$g$与$n$边界组件：我们有$$ \text{vol}（M_\psi）\le 3 \sqrt{\pi/2（2g - 2 +n）} \，\| \psi \|_{WP}.$$本文通过[CP]给出了模空间的Weil-Petersson系统的第一个显式估计，Teichm\“uller空间的完备化中节点曲面间的最小距离，以及模空间的Weil-Petersson直径的显式下界.在这个过程中，我们通过Cauchy-Schwarz估计恢复了[KM]对Teichm\“uller平移距离的估计（见[Lin]）。
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]).