Profinite surface groups and the congruence kernel of arithmetic lattices in SL2 (R)

LetX be a proper, nonsingular, connected algebraic curve of genusg over the fieldC of complex numbers. The algebraic fundamental group Γ = π1 (X) in the sense of [SGA-1] (1971) is the profinite completion of the fundamental group π1top (X) of a compact oriented 2-manifold. We prove that every projective normal (respectively, characteristic, accessible) subgroup of Γ is isomorphic to a normal (respectively, characteristic, accessible) subgroup of a free profinite group. We use this description to give a complete solution of the congruence subgroup problem for arithmetic lattices in SL2 (R).