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

Let X be a proper, nonsingular, connected algebraic curve of genus g over the field C of complex numbers. The algebraic fundamental group Gamma = pi(1)(X) in the sense of [SGA-1] (1971) is the profinite completion of the fundamental group pi(top)(1)(X) of a compact oriented 2-manifold. We prove that every projective normal (respectively, characteristic, accessible) subgroup of Gamma 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).