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Galois rigidity of pure sphere braid groups and profinite calculus. (English) Zbl 0901.14012
Summary: Let \({\mathfrak C}\) be a class of finite groups closed under the formation of subgroups, quotients, and group extensions. For an algebraic variety \(X\) over a number field \(k\), let \(\pi_1^{\mathfrak C} (X)\) denote the (\(\mathbb{C}\)-modified) profinite fundamental group of \(X\) having the absolute Galois group \(\text{Gal} (k/k)\) as a quotient with kernel \(\pi_1^{\mathfrak C}(X_{\overline{k}})\), the maximal pro-\({\mathfrak C}\) quotient of the geometric fundamental group of \(X\). The purpose of this paper is to show certain rigidity properties of \(\pi_1^{\mathfrak C}(X)\) for \(X\) of hyperbolic type by the study of the outer automorphism group \(\text{Out } \pi_1^{\mathfrak C}(X)\) of \(\pi_1^{\mathfrak C}(X)\). In particular, we show finiteness of \(\text{Out } \pi_1^{\mathfrak C}(X)\) when \(X\) is a certain typical hyperbolic variety and \({\mathfrak k}\) is the class of finite \(l\)-groups (\(l\): odd prime).
Indeed, we have a criterion of Gottlieb type for center-triviality of \(\pi_1^{\mathfrak C}(X_{\overline{k}})\) under certain good hyperbolicity condition on \(X\). Then our question on finiteness of \(\text{Out } \pi_1^{\mathfrak C}(X)\) for such \(X\) is reduced to the study of the exterior Galois representation \(\varphi_X^{\mathfrak C}: \text{Gal} (\overline{k}/k)\to \text{Out } \pi_1^{\mathfrak C}(X_{\overline{k}})\), especially to the estimation of the centralizer of the Galois image of \(\varphi_X^{\mathfrak C}\). In §2, we study the case where \(X\) is an algebraic curve of hyperbolic type, and give fundamental tools and basic results. We devote §3, §4 and the appendix to detailed studies of the special case \(X= M_{0,n}\), the moduli space of the \(n\)-point punctured projective lines \((n\geq 3)\), which are closely related with topological work of N. V. Ivanov, arithmetic work of P. Deligne and {Y. Ihara}, and categorical work of V. G. Drinfeld. Section 4 deals with a Lie variant suggested by P. Deligne.

14F35 Homotopy theory and fundamental groups in algebraic geometry
20F36 Braid groups; Artin groups
12F12 Inverse Galois theory
20F34 Fundamental groups and their automorphisms (group-theoretic aspects)
14E20 Coverings in algebraic geometry
11R32 Galois theory
20E18 Limits, profinite groups