Galois rigidity of pure sphere braid groups and profinite calculus
Vol. 1 (1994), No. 1, Page 71--136.
Nakamura, Hiroaki
Galois rigidity of pure sphere braid groups and profinite calculus
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Abstract:
Let 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 Ï^{\frak C}_1(X) denote the (\frak C-modified) profinite fundamental group of X having the absolute Galois group Gal(\bar k/k) as a quotient with kernel Ï^{\frak C}_1(X_{\bar k}) the maximal pro-\frak C quotient of the geometric fundamental group of X. The purpose of this paper is to show certain rigidity properties of Ï^{\frak C}_1(X) for X of hyperbolic type through the study of outer automorphism group OutÏ^{\frak C}_1(X) of Ï^{\frak C}_1(X). In particular, we show finiteness of OutÏ^{\frak C}_1(X) when X is a certain typical hyperbolic variety and \frak C is the class of finite l-groups (l: odd prime). Indeed, we have a criterion of Gottlieb type for center-triviality of Ï^{\frak C}_1(X_{\bar k}) under certain good hyperbolicity condition on X. Then our question on finiteness of OutÏ^{\frak C}_1(X) for such X is reduced to the study of the exterior Galois representation \varphi^{\frak C}_X:Gal(\bar k/k)\to OutÏ^{\frak C}_1(X_{\bar k}), especially to the estimation of the centralizer of the Galois image of \varphi^{\frak C}_X (\S 1.6). In \S 2, we study the case where X is an algebraic curve of hyperbolic type, and give fundamental tools and basic results. We devote \S 3, \S 4 and Appendix to detailed studies of the special case X=M_{0, n}, the moduli space of the n-point punctured projective lines (n\ge 3), which are closely related with topological work of N. V. Ivanov, arithmetic work of P. Delinge, Y. Ihara, and categorical work of V. G. Drinfeld. Section 4 deal with a Lie variant suggested by P. Deligne.
Mathematics Subject Classification (1991): Primary 14E20; Secondary 14F35, 20F34, 20F36
Mathematical Reviews Number: MR1298541
Received: 1992-03-10
