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Fundamental groups of moduli and the Grothendieck-Teichmüller group


Authors: David Harbater and Leila Schneps
Journal: Trans. Amer. Math. Soc. 352 (2000), 3117-3148
MSC (1991): Primary 11R32, 14E20, 14H10; Secondary 20F29, 20F34, 32G15
Published electronically: March 31, 2000
MathSciNet review: 1615979
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Abstract: Let ${\mathcal{M}}_{0,n}$ denote the moduli space of Riemann spheres with $n$ ordered marked points. In this article we define the group $\operatorname{Out}^{\sharp }_{n}$ of quasi-special symmetric outer automorphisms of the algebraic fundamental group $\widehat \pi _{1}({\mathcal{M}}_{0,n})$ for all $n\ge 4$ to be the group of outer automorphisms respecting the conjugacy classes of the inertia subgroups of $\widehat \pi _{1}({\mathcal{M}}_{0,n})$ and commuting with the group of outer automorphisms of $\widehat \pi _{1}({\mathcal{M}}_{0,n})$ obtained by permuting the marked points. Our main result states that $\operatorname{Out}^{\sharp }_{n}$ is isomorphic to the Grothendieck-Teichmüller group $\widehat {\operatorname{GT}}$for all $n\ge 5$.


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Additional Information

David Harbater
Affiliation: Department of Mathematics, University of Pennsylvania, Philadelphia, Pennsylvania 19104-6395
Email: harbater@math.upenn.edu

Leila Schneps
Affiliation: Faculté des Sciences, Université de Franche-Comté, 25030 Besançon Cedex, France
Email: Leila.Schneps@ens.fr

DOI: https://doi.org/10.1090/S0002-9947-00-02347-3
Keywords: Moduli space, Galois actions, fundamental groups, Grothendieck-Teichm\"{u}ller group, mapping class group
Received by editor(s): July 21, 1997
Received by editor(s) in revised form: February 12, 1998
Published electronically: March 31, 2000
Additional Notes: Supported in part by NSF Grant DMS94-00836.
Article copyright: © Copyright 2000 American Mathematical Society