Let $S$$S$ be a surface of negative Euler characteristic and consider a finite filling collection $\mathrm{\Gamma }$$\mathrm{\Gamma }$ of closed curves on $S$$S$ in minimal position. An observation of Foulon and Hasselblatt shows that $PT(S)\setminus \widehat{\mathrm{\Gamma }}$$PT(S)\setminus \widehat{\mathrm{\Gamma }}$ is a finite-volume hyperbolic 3-manifold, where $PT(S)$$PT(S)$ is the projectivized tangent bundle and $\widehat{\mathrm{\Gamma }}$$\widehat{\mathrm{\Gamma }}$ is the set of tangent lines to $\mathrm{\Gamma }$$\mathrm{\Gamma }$. In particular, $vol(PT(S)\setminus \widehat{\mathrm{\Gamma }})$$vol(PT(S)\setminus \widehat{\mathrm{\Gamma }})$ is a mapping class group invariant of the collection $\mathrm{\Gamma }$$\mathrm{\Gamma }$. When $\mathrm{\Gamma }$$\mathrm{\Gamma }$ is a filling pair of simple closed curves, we show that this volume is coarsely comparable to Weil–Petersson distance between strata in Teichmüller space. Our main tool is the study of stratified hyperbolic links $\overline{\mathrm{\Gamma }}$$\overline{\mathrm{\Gamma }}$ in a Seifert-fibered space $N$$N$ over $S$$S$. For such links, the volume of $N\setminus \overline{\mathrm{\Gamma }}$$N\setminus \overline{\mathrm{\Gamma }}$ is coarsely comparable to expressions involving distances in the pants graph.

中文翻译：

---

关于多曲线的体积和填充集合

让$\mathrm{小号}$$S$是一个负欧拉特征的表面，并考虑一个有限的填充集合$\mathrm{\Gamma }$$\mathrm{\Gamma }$上的闭合曲线$\mathrm{小号}$$S$在最小的位置。对 Foulon 和 Hasselblatt 的观察表明，$磷吨(\mathrm{小号})\setminus \widehat{\mathrm{\Gamma }}$$PT(S)\setminus \widehat{\mathrm{\Gamma }}$是有限体积双曲 3 流形，其中$磷吨(\mathrm{小号})$$PT(S)$是投影的切丛和$\widehat{\mathrm{\Gamma }}$$\widehat{\mathrm{\Gamma }}$是切线的集合$\mathrm{\Gamma }$$\mathrm{\Gamma }$. 尤其是，$v○l(磷吨(\mathrm{小号})\setminus \widehat{\mathrm{\Gamma }})$$vol(PT(S)\setminus \widehat{\mathrm{\Gamma }})$是集合的映射类组不变量$\mathrm{\Gamma }$$\mathrm{\Gamma }$. 什么时候$\mathrm{\Gamma }$$\mathrm{\Gamma }$是一对简单闭合曲线的填充，我们表明该体积与 Teichmüller 空间中地层之间的 Weil-Petersson 距离大致相当。我们的主要工具是研究分层双曲链接$\overline{\mathrm{\Gamma }}$$\overline{\mathrm{\Gamma }}$在塞弗特纤维空间$ñ$$N$超过$\mathrm{小号}$$S$. 对于此类链接，$ñ\setminus \overline{\mathrm{\Gamma }}$$N\setminus \overline{\mathrm{\Gamma }}$粗略地与裤子图中涉及距离的表达式进行比较。