Profinite surface groups and the congruence kernel of arithmetic lattices in \(\text{SL}_2(\mathbb{R})\).
Let \(X\) be a proper, nonsingular, connected algebraic curve of genus \(g\) over the field of complex numbers. The algebraic fundamental group \(\Gamma=\pi(X)\) is the profinite completion of the fundamental group of a compact oriented 2-manifold. The aim of the paper is to analyze the subgroup structure of \(\Gamma\), proving in particular: every projective (resp. characteristic, accessible) subgroup of \(\Gamma\) is isomorphic to a normal (resp. characteristic, accessible) subgroup of a free profinite group of countable rank. The author uses this description to give a solution of the congruence subgroup problem for arithmetic lattices in \(\text{SL}_2(\mathbb{R})\): namely he proves that the congruence kernel of any such arithmetic group is isomorphic to a free profinite group of countable rank.
scientific article; zbMATH DE number 2174705