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

Let ${\cal M}_{g,n}$ and ${\cal H}_{g,n}$, 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 $\Gamma_{g,n}$ and $H_{g,n}$, the so called Teichm{\"u}ller modular group and hyperelliptic modular group. A choice of base point on ${\cal H}_{g,n}$ defines a monomorphism $H_{g,n}\hookrightarrow\Gamma_{g,n}$. Let $S_{g,n}$ be a compact Riemann surface of genus $g$ with $n$ points removed. The Teichm\"uller 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. As a subgroup of $\Gamma_{g,n}$, the hyperelliptic modular group then admits a natural faithful representation $H_{g,n}\hookrightarrow\operatorname{Out}(\pi_1(S_{g,n}))$. 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 $H_{g,n}\to\operatorname{Out}(\pi_1(S_{g,n})/K)$ is contained in $H^\lambda$. The main result of the paper is an affirmative answer to this question for $n\geq 1$.