Reduction theory for mapping class groups and applications to moduli spaces
| dc.creator | Leuzinger, Enrico | |
| dc.date | 2008-01-10 | |
| dc.date.accessioned | 2026-07-07T09:49:14Z | |
| dc.date.available | 2026-07-07T09:49:14Z | |
| dc.description | 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 Teichmüller 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 Teichmüller 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 Teichmüller metric. These two applications confirm conjectures of Farb. | |
| dc.identifier | https://arxiv.org/abs/0801.1589 | |
| dc.identifier | http://arxiv.org/abs/0801.1589 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/164513 | |
| dc.subject | Geometric Topology | |
| dc.subject | Group Theory | |
| dc.subject | 32G15; 30F60; 20H10; 20F67; 51K10; 53C; 58B | |
| dc.title | Reduction theory for mapping class groups and applications to moduli spaces | |
| dc.type | text |