Congruence topologies on the mapping class group

TL;DR

This paper systematically studies the ${ m ext{ extbf{C}}}$-congruence completions of the pure mapping class group of surfaces, comparing them with pro-${ m ext{ extbf{C}}}$ completions, revealing structural similarities and specific isomorphisms in certain cases.

Contribution

It introduces a systematic framework for ${ m ext{ extbf{C}}}$-congruence completions of mapping class groups and compares these with pro-${ m extbf{C}}$ completions, establishing key isomorphisms.

Findings

Pro-${ m extbf{C}}$} structure closely resembles the original group.

Existence of a natural epimorphism between ${ m extbf{C}}$-congruence and pro-${ m extbf{C}}$} completions under certain conditions.

Isomorphism between completions for finite groups and 2-groups cases.

Abstract

Let $\Gamma(S)$ be the pure mapping class group of a connected orientable surface $S$ of negative Euler characteristic. For ${\mathscr C}$ a class of finite groups, let $\hat{\pi}_1(S)^{\mathscr C}$ be the pro-${\mathscr C}$ completion of the fundamental group of $S$. The \emph{${\mathscr C}$-congruence completion $\check{\Gamma}(S)^{\mathscr C}$ of $\Gamma(S)$} is the profinite completion induced by the embedding $\Gamma(S)\hookrightarrow{\operatorname{Out}}(\hat{\pi}_1(S)^{\mathscr C})$. In this paper, we begin a systematic study of such completions for different ${\mathscr C}$. We show that the combinatorial structure of the profinite groups $\check{\Gamma}(S)^{\mathscr C}$ closely resemble that of $\Gamma(S)$. A fundamental question is how ${\mathscr C}$-congruence completions compare with pro-${\mathscr C}$ completions. Even though, in general (e.g.\ for ${\mathscr C}$ the class of finite solvable groups), $\check{\Gamma}(S)^{\mathscr C}$ is not even virtually a pro-${\mathscr C}$ group, we show that, for ${\mathbb Z}/2\in{\mathscr C}$, $g(S)\leq 2$ and $S$ open, there is a natural epimorphism from the ${\mathscr C}$-congruence completion $\check{\Gamma}(S)(2)^{\mathscr C}$ of the abelian level of order $2$ to its pro-${\mathscr C}$ completion $\widehat{\Gamma}(S)(2)^{\mathscr C}$. 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.