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Keywords
(9)
Asymptotic Cone
Complex of Curves
Finite Volume
Group Theory
Indexation
Mapping Class Group
Moduli Space
Positive Scalar Curvature
riemannian metric
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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
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Reduction theory for mapping class groups and applications to moduli spaces
(
Citations: 2
)
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Enrico Leuzinger
Let $S=S_{g,p}$ be a compact, orientable surface of genus $g$ with $p$ punctures and such that $d(S):=3g3+p>0$. 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 biLipschitz 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 biLipschitz 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)...
Lizhen Ji
,
et al.
The Asymptotic Schottky Problem
References
(21)
Dimension and rank for mapping class groups
(
Citations: 13
)
Jason A. Behrstock
,
Yair N. Minsky
Journal:
Annals of Mathematics  ANN MATH
, vol. 167, no. 3, pp. 10551077, 2008
Arithmetic manifolds of positive scalar curvature
(
Citations: 10
)
Jonathan Block
,
Shmuel Weinberger
Journal:
Journal of Differential Geometry  J DIFFEREN GEOM
, vol. 52, no. 1999, pp. 375406, 1999
Compacti cation of symmetric and locally symmetric spaces
(
Citations: 23
)
A. Borel
,
L. Ji
Published in 2005.
Metric Spaces of Nonpositive Curvature
(
Citations: 465
)
Martin R. Bridson
,
Andr'e Haefliger
Published in 2000.
Coarse Obstructions to Positive Scalar Curvature in Noncompact Arithmetic Manifolds
(
Citations: 4
)
Stanley S. Chang
Journal:
Journal of Differential Geometry  J DIFFEREN GEOM
, vol. 57, no. 2001, pp. 121, 2001
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Citations
(2)
The Asymptotic Schottky Problem
Lizhen Ji
,
Enrico Leuzinger
Journal:
Geometric and Functional Analysis  GEOM FUNCT ANAL
, vol. 19, no. 6, pp. 16931712, 2010
Almostisometry between Teichm\"{u}ller metric and lengthspectra metric on moduli space
Lixin Liu
,
Weixu Su
Published in 2010.