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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.

##### MSC:
 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