Congruence topologies on the mapping class group

JOURNAL OF ALGEBRA

卷:546

Article

Boggi, Marco¹ 

[1] Univ Fed Minas Gerais, Dept Matemat, Av Antonio Carlos,6627 Caixa Postal 702, BR-31270901 Belo Horizonte, MG, Brazil

关键词: Mapping class groups;    Profinite groups;    Profinite topologies and completions;   

DOI  :  10.1016/j.jalgebra.2019.11.004

来源: Elsevier

【 摘 要 】

Let Gamma(S) be the pure mapping class group of a connected orientable surface S of negative Euler characteristic. For l a class of finite groups, let (pi) over cap (1) (S)(l) be the pro-l completion of the fundamental group of S. The l-congruence completion Gamma(S)(l) of Gamma(S) is the profinite completion induced by the embedding Gamma(S) -> Out(pi(1)(S)(l)). In this paper, we begin a systematic study of such completions for different l. We show that the combinatorial structure of the profinite groups Gamma(S)(l) closely resemble that of Gamma(S). A fundamental question is how l-congruence completions compare with pro-l completions. Even though, in general (e.g. for l the class of finite solvable groups), Gamma(S)(l) is not even virtually a pro-l group, we show that, for Z/2 is an element of l, g(S) <= 2 and S open, there is a natural epimorphism from the l-congruence completion Gamma(S)(2)(l) of the abelian level of order 2 to its pro-l completion Gamma(S)(2)(l). In particular, this is an isomorphism for the class of finite groups and for the class of 2-groups. Moreover, in these two cases, the result also holds for a closed surface. (C) 2019 Elsevier Inc. All rights reserved.

【 授权许可】

Free