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

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.