Relative pro-ℓ completions of mapping class groups

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 Tg,1 on the pro-ℓ 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-ℓ completion of the corresponding mapping class group. We prove that for all g all such representations are unramified at all primes ≠ℓ when the first order deformation is suitably chosen. This proof was communicated to us by Mochizuki and Tamagawa.