The congruence subgroup property for the hyperelliptic modular group: the open surface case

Let ${\mathcal{M}}_{g,n}$ and ${\mathcal{H}}_{g,n}$, for $2g-2+n>0$, be, respectively, the moduli stack of $n$-pointed, genus $g$ smooth curves and its closed substack consisting of hyperelliptic curves. Their topological fundamental groups can be identified, respectively, with ${\mathrm{\Gamma }}_{g,n}$ and $H_{g,n}$, the so called Teichm{"u}ller modular group and hyperelliptic modular group. A choice of base point on ${\mathcal{H}}_{g,n}$ defines a monomorphism $H_{g,n}\hookrightarrow {\mathrm{\Gamma }}_{g,n}$. Let $S_{g,n}$ be a compact Riemann surface of genus $g$ with $n$ points removed. The Teichm"uller group ${\mathrm{\Gamma }}_{g,n}$ 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. As a subgroup of ${\mathrm{\Gamma }}_{g,n}$, the hyperelliptic modular group then admits a natural faithful representation $H_{g,n}\hookrightarrow \mathrm{Out}({\pi }_1(S_{g,n}))$. The congruence subgroup problem for $H_{g,n}$ asks whether, for any given finite index subgroup $H^{\lambda }$ of $H_{g,n}$, there exists a finite index characteristic subgroup $K$ of ${\pi }_1(S_{g,n})$ such that the kernel of the induced representation $H_{g,n}\to \mathrm{Out}({\pi }_1(S_{g,n})\mathrm{/}K)$ is contained in $H^{\lambda }$. The main result of the paper is an affirmative answer to this question for $n\ge 1$.