Relative pro-l completions of mapping class groups

JOURNAL OF ALGEBRA

卷:321

Article; Proceedings Paper

Hain, Richard¹  Matsumoto, Makoto² 

[1] Duke Univ, Dept Math, Durham, NC 27708 USA

[2] Hiroshima Univ, Grad Sch Sci, Dept Math, Hiroshima 7398526, Japan

关键词: Mapping class group;    Moduli space of curves;    Pro-l completion;    Torelli group;    Johnson homomorphism;    Unramified Galois representation;   

DOI  :  10.1016/j.jalgebra.2009.02.014

来源: Elsevier

【 摘 要 】

Fix a prime number l. In this paper we develop the theory of relative pro-l completion of discrete and profinite groups-a natural generalization of the classical notion of pro-l completion-and show that the pro-l completion of the Torelli group does not inject into the relative pro-l completion of the corresponding mapping class group when the genus is at least 2. (See Theorem 1 below.) As an application, we prove that when g >= 2, the action of the pro-l completion of the Torelli group T-g,T- 1 on the pro-l fundamental group of a pointed genus g surface is not faithful. The choice of a first-order deformation of it maximally degenerate stable curve of genus g determines an action of the absolute Galois group G(Q) on the relative pro-l completion of the corresponding mapping class group. We prove that for all g all such representations are unramified at all primes not equal l when the first order deformation is Suitably chosen. This proof was communicated to us by Mochizuki and Tamagawa. (C) 2009 Elsevier Inc. All rights reserved.

【 授权许可】

Free