A generalized congruence subgroup property for the hyperelliptic modular group

Marco Boggi

Published 2008-03-27, updated 2016-09-29Version 5

Let $S_{g,n}$, for $2g-2+n>0$, be obtained from a closed Riemann surface of genus $g$ removing $n$ points. The mapping class group $\Gamma_{g,n}$ is the group of isotopy classes of diffeomorphisms of the surface $S_{g,n}$ which preserve the orientation and a given order of the punctures. For a fixed hyperelliptic involution $\iota\in\Gamma_g$, the hyperelliptic mapping class group $\Upsilon_g$ is the centralizer of $\iota$ in $\Gamma_g$. For $n> 0$, let $p\colon\Gamma_{g,n}\to\Gamma_g$ be the natural epimorphism. The hyperelliptic mapping class group $\Upsilon_{g,n}$ is the inverse image $p^{-1}(\Upsilon_g)\leq\Gamma_{g,n}$. In genus one, we just let $\Upsilon_{1,n}:=\Gamma_{1,n}$. As a subgroup of $\Gamma_{g,n}$, the hyperelliptic modular group admits a natural faithful representation: $\Upsilon_{g,n}\to\operatorname{Out}(\pi_1(S_{g,n}))$. For $\mathcal S$ a class of finite groups, let $\hat{\pi}_1(S_{g,n})^{\mathcal S}$ be the pro-$\mathcal S$ completion of the fundamental group $\pi_1(S_{g,n})$. There is a natural monomorphism $\Upsilon_{g,n}\to\operatorname{Out}(\hat{\pi}_1(S_{g,n})^{\mathcal S})$ and we let $\check{\Upsilon}_{g,n}^{\mathcal S}$ be the closure of $\Upsilon_{g,n}$ inside the profinite group $\operatorname{Out}(\hat{\pi}_1(S_{g,n})^{\mathcal S})$. The main result of the paper is that, if ${\mathbb Z}/2\in{\mathcal S}$, the profinite group $\check{\Upsilon}_{g,n}^{\mathcal S}$ is, virtually, the pro-$\mathcal S$ completion of the hyperelliptic mapping class group $\Upsilon_{g,n}$.