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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
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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Citation Context (1)

• ...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)...

References (21)

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Citations (2)

The Asymptotic Schottky Problem

Journal: Geometric and Functional Analysis - GEOM FUNCT ANAL , vol. 19, no. 6, pp. 1693-1712, 2010

Almost-isometry between Teichm\"{u}ller metric and length-spectra metric on moduli space

Published in 2010.