• Teichmüller theory;
  • moduli of curves;
  • profinite groups;
  • fundamental groups;
  • Riemann surfaces;
  • Galois theory


For 2g – 2 + n > 0, let Γg, n be the Teichmüller group of a compact Riemann surface of genus g with n points removed Sg, n , i.e., the group of homotopy classes of diffeomorphisms of Sg, n which preserve the orientation of Sg, n and a given order of its punctures. There is a natural faithful representation Γg, n → Out(π1(Sg, n )). For any given finite index subgroup Γλ of Γg, n, the congruence subgroup problem asks whether there exists a finite index characteristic subgroup K of π1(Sg, n ) such that the kernel of the induced representation Γg, n → Out(π1(Sg, n )/K ) is contained in Γλ. The main result of the paper is an affirmative answer to this question. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)