Relative Pro- Completions of Mapping Class Groups

Abstract

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 Tg,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 GQ 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 ≠ ell when the first order deformation is suitably chosen. This proof was communicated to us by Mochizuki and Tamagawa.