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

Abstract

Let $\cM_{g,n}$ and $\cH_{g,n}$, 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 $\GG_{g,n}$ and $H_{g,n}$, the so called Teichmüller modular group and hyperelliptic modular group. A choice of base point on $\cH_{g,n}$ defines a monomorphism $H_{g,n}\hookra\GG_{g,n}$.

Let $S_{g,n}$ be a compact Riemann surface of genus $g$ with $n$ points removed. The Teichmüller group $\GG_{g,n}$ 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 $\GG_{g,n}$, the hyperelliptic modular group then admits a natural faithful representation $H_{g,n}\hookra\out(\pi_1(S_{g,n}))$.

The congruence subgroup problem for $H_{g,n}$ asks whether, for any given finite index subgroup $H^\ld$ 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 $H_{g,n}\ra\out(\pi_1(S_{g,n})/K)$ is contained in $H^\ld$. The main result of the paper is an affirmative answer to this question for $n\geq 1$.