Profinite mapping class groups

TL;DR

This paper proves the isomorphism between the profinite completion of surface mapping class groups and certain arithmetic groups, relates their normal subgroups to Galois groups, and confirms the Shafarevich Conjecture for G(Q^ab).

Contribution

It establishes a deep connection between geometric mapping class groups, arithmetic groups, and Galois groups, and proves the Shafarevich Conjecture using the Tits alternative.

Findings

Profinite completion of Mod(g,n) is isomorphic to GL(6g-6+2n, Z)

Normal subgroups of Mod(g,n) relate to Galois groups G(K)

G(Q^ab) is isomorphic to a free profinite group

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.