Profinite mapping class groups

Abstract

It is proved that the profinite completion of the mapping class group Mod (g,n) of a surface of genus g with n boundary components is isomorphic to such of the arithmetic group GL(6g-6+2n, Z). We establish a relation between the normal subgroups of Mod (g,n) and the absolute Galois group G(K) of a number field K. Using the Tits alternative, we prove the Shafarevich Conjecture saying that the group G(Q^ab) of the maximal abelian extension of the field of rationals is isomorphic to a free profinite group.