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タイトル: Galois rigidity of pure sphere braid groups and profinite calculus
著者: Nakamura, Hiroaki
発行日: 1994年
出版者: Graduate School of Mathematical Sciences, The University of Tokyo
掲載誌情報: Journal of Mathematical Sciences, The University of Tokyo. Vol. 1 (1994), No. 1, Page 71-136
抄録: Let $\frak C$ 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 $π^{\frak C}_1(X)$ denote the ($\frak C$-modified) profinite fundamental group of $X$ having the absolute Galois group $Gal(\bar k/k)$ as a quotient with kernel $π^{\frak C}_1(X_{\bar k})$ 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 $π^{\frak C}_1(X)$ for $X$ of hyperbolic type through the study of outer automorphism group $Outπ^{\frak C}_1(X)$ of $π^{\frak C}_1(X)$. In particular, we show finiteness of $Outπ^{\frak C}_1(X)$ 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 $π^{\frak C}_1(X_{\bar k})$ under certain good hyperbolicity condition on $X$. Then our question on finiteness of $Outπ^{\frak C}_1(X)$ for such $X$ is reduced to the study of the exterior Galois representation $\varphi^{\frak C}_X:Gal(\bar k/k)\to Outπ^{\frak C}_1(X_{\bar k})$, 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.
URI: http://hdl.handle.net/2261/1590
ISSN: 13405705
出現カテゴリ:Journal of Mathematical Sciences, the University of Tokyo
Journal of Mathematical Sciences, the University of Tokyo

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