The congruence subgroup property for the hyperelliptic modular group

Let Mg,n and Hg,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 Γg,n and Hg,n, the so called Teichmüller modular group and hyperelliptic modular group. A choice of base point on Hg,n defines a monomorphism Hg,n →֒ Γg,n. Let Sg,n be a compact Riemann surface of genus g with n points removed. The Teichmüller group Γg,n is the group of isotopy classes of diffeomorphisms of the surface Sg,n which preserve the orientation and a given order of the punctures. As a subgroup of Γg,n, the hyperelliptic modular group then admits a natural faithful representation Hg,n →֒ Out(π1(Sg,n)). The congruence subgroup problem for Hg,n asks whether, for any given finite index subgroup Hλ of Hg,n, there exists a finite index characteristic subgroup K of π1(Sg,n) such that the kernel of the induced representation Hg,n → Out(π1(Sg,n)/K) is contained in Hλ. The main result of the paper is an affermative answer to this question. Mathematics Subject Classifications (2000): 14H10, 14H15, 14F35, 11R34.