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 6= l when the first order deformation is suitably chosen. This proof was communicated to us by Mochizuki and Tamagawa.