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 $n\geq 1$.