Reduction theory for mapping class groups and applications to moduli spaces

Abstract

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.