Profinite mapping class groups
Number Theory
2020-04-10 v3 Geometric Topology
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.
Cite
@article{arxiv.1812.05390,
title = {Profinite mapping class groups},
author = {Igor Nikolaev},
journal= {arXiv preprint arXiv:1812.05390},
year = {2020}
}
Comments
10 pages; remark 4.5 is added