Profinite Teichmuller theory

For 2g - 2 + n > 0, let Gamma(g,n) be the Teichmuller group of a compact Riemann surface of genus g with n points removed S-g,S-n, i.e., the group of homotopy classes of diffeomorphisms of S-g,S-n which preserve the orientation of S-g,S-n and a given order of its punctures. There is a natural faithful representation Gamma(g,n) --> Out(pi(1) (S-g,S-n)). For any given finite index subgroup Gamma(lambda) of Gamma(g,n) the congruence subgroup problem asks whether there exists a finite index characteristic subgroup K of pi(1) (S-g,S-n) such that the kernel of the induced representation Gamma(g,n) --> Out(pi(1) (S-g,S-n)/K) is contained in Gamma(lambda). The main result of the paper is an affirmative answer to this question. (C) 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.