Relative Pro-$\ell$ Completions of Mapping Class Groups

Details Relative Pro-$\ell$ Completions of Mapping Class Groups Relative Pro-$\ell$ Completions of Mapping Class Groups

2008-02-06

arxiv.org/abs/0802.0806v4

Mathematics

Number Theory

A few minor changes. Will appear in Lehrer volume of J. Algebra

Scientific paper

Fix a prime number ell. In this paper we develop the theory of relative pro-ell completion of discrete and profinite groups -- a natural generalization of the classical notion of pro-ell completion -- and show that the pro-ell completion of the Torelli group does not inject into the relative pro-ell completion of the corresponding mapping class group when the genus is at least 3. As an application, we prove that when g > 2, the action of the pro-ell completion of the Torelli group T_{g,1} on the pro-ell fundamental group of a pointed genus g surface is not faithful. The choice of a first-order deformation of a maximally degenerate stable curve of genus g determines an action of the absolute Galois group G_Q on the relative pro-ell completion of the corresponding mapping class group. We prove that for all g all such representations are unramified at all primes \neq ell when the first order deformation is suitably chosen. This proof was communicated to us by Mochizuki and Tamagawa.

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