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Reduction theory for mapping class groups and applications to moduli spaces

Reduction theory for mapping class groups and applications to moduli spaces,Enrico Leuzinger

Reduction theory for mapping class groups and applications to moduli spaces   (Citations: 2)
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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.
Published in 2008.
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    • ...It is crucial for our aproach to obtain information about the image of the Jacobian map J : Mg −→ Ag when restricted to certain “thin parts” of moduli space Ag, i.e., subsets of Mg consisting of Riemann surfaces (endowed with a hyperbolic metric) which contain at least one closed geodesic of length less than some fixed small number (see [24] for a precise description of these sets)...

    Lizhen Jiet al. The Asymptotic Schottky Problem

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