标题： 映射类群的约化理论及其在模空间中的应用 显示英文标题

标题： Reduction theory for mapping class groups and applications to moduli spaces

摘要： 设$S=S_{g,p}$为一个亏格为$g$的紧致可定向曲面，带有$p$个洞，并且满足$d(S):=3g-3+p>0$。映射类群$\textup{Mod}_S$在$S$上的标记双曲结构的Teichmüller空间$\mathcal T(S)$上作用真正不连续。 结果商$\mathcal M(S)$是双曲曲面的等距类的模空间。 我们提供了有限指数子群$\textup{Mod}_S$的精确约化理论的一种版本，即精确基本区域的描述。 作为应用，我们证明了赋予Teichmüller度量的模空间$\mathcal M(S)$的渐近锥与欧几里得锥在有限单纯（或带轨道结构的）复形$ \textup{Mod}_S\backslash\mathcal C(S)$上是双利普希茨等价的，其中$\mathcal C(S)$是$S$的曲线复形，而$S$是其对象。 我们还证明，如果 $d(S)\geq 2$，那么 $\mathcal M(S)$ 不 \emph{不} 在 Teichmüller 度量的双 Lipschitz 类中承认有限体积的 (均匀有界) 正标量曲率的黎曼度量。 这两个应用证实了 Farb 的猜想。 显示英文摘要

摘要： Let $S=S_{g,p}$ be a compact, orientable surface of genus $g$ with $p$ punctures and such that $d(S):=3g-3+p>0$. The mapping class group $\textup{Mod}_S$ acts properly discontinuously on the Teichm\"uller space $\mathcal T(S)$ of marked hyperbolic structures on $S$. The resulting quotient $\mathcal M(S)$ is the moduli space of isometry classes of hyperbolic surfaces. We provide a version of precise reduction theory for finite index subgroups of $\textup{Mod}_S$, i.e., a description of exact fundamental domains. As an application we show that the asymptotic cone of the moduli space $\mathcal M(S)$ endowed with the Teichm\"uller metric is bi-Lipschitz equivalent to the Euclidean cone over the finite simplicial (orbi-) complex $ \textup{Mod}_S\backslash\mathcal C(S)$, where $\mathcal C(S)$ of $S$ is the complex of curves of $S$. We also show that if $d(S)\geq 2$, then $\mathcal M(S)$ does \emph{not} admit a finite volume Riemannian metric of (uniformly bounded) positive scalar curvature in the bi-Lipschitz class of the Teichm\"uller metric. These two applications confirm conjectures of Farb.