## Galois rigidity of pure sphere braid groups and profinite calculus

J. Math. Sci. Univ. Tokyo
Vol. 1 (1994), No. 1, Page 71--136.

Nakamura, Hiroaki
Galois rigidity of pure sphere braid groups and profinite calculus
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 $Ï^{\frak C}_1(X)$ 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.