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Harbater, David; Schneps, Leila

Fundamental groups of moduli and the Grothendieck-Teichmüller group. (English) Zbl 0956.14013

Trans. Am. Math. Soc. 352, No. 7, 3117-3148 (2000).
Author’s summary: Let \({\mathcal M}_{0,n}\) denote the moduli space of Riemann spheres with \(n\) ordered marked points. In this article we define the group \(\text{Out}_n^\#\) of quasi-special symmetric outer automorphisms of the algebraic fundamental group \(\widehat\pi_1({\mathcal M}_{0,n})\) for all \(n\geq 4\) 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 \(\text{Out}_n^\#\) is isomorphic to the Grothendieck-Teichmüller group \(\widehat {GT}\) for all \(n\geq 5\).
Reviewer: Hiroaki Nakamura (Tokyo)

Cited in 1 Review
Cited in 10 Documents

MSC:

14F35 Homotopy theory and fundamental groups in algebraic geometry
32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
14H10 Families, moduli of curves (algebraic)
20F34 Fundamental groups and their automorphisms (group-theoretic aspects)

Keywords:

braid group; moduli space of Riemann spheres; outer automorphisms; algebraic fundamental group; Grothendieck-Teichmüller group
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References:

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