For 2g − 2 + n > 0, the Teichmüller modular group Γ g

For 2g – 2 + n > 0, let Γg, n be the Teichmüller group of a compact Riemann surface of genus g with n points removed Sg, n , i.e., the group of homotopy classes of diffeomorphisms of Sg, n which preserve the orientation of Sg, n and a given order of its punctures. There is a natural faithful representation Γg, n → Out(π 1(Sg, n )). For any given finite index subgroup Γλ of Γg, n , the congruence subgroup problem asks whether there exists a finite index characteristic subgroup K of π 1(Sg, n ) such that the kernel of the induced representation Γg, n → Out(π 1(Sg, n )/K ) is contained in Γλ . The main result of the paper is an affirmative answer to this question. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Let M g,n and 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 Γ g,n and H g,n , the so called Teichmüller modular group and hyperelliptic modular group. A choice of base point on H g,n defines a monomorphism H g,n ֒→ Γ g,n .Let S g,n be a compact Riemann surface of genus g with n points removed. The Teichmüller group Γ 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 Γ g,n , the hyperelliptic modular group then admits a natural faithful representation H g,n ֒→ Out(π 1 (S g,n )).The congruence subgroup problem for H g,n asks whether, for any given finite index subgroup H λ of H g,n , there exists a finite index characteristic subgroup K of π 1 (S g,n ) such that the kernel of the induced representation H g,n → Out(π 1 (S g,n )/K) is contained in H λ . The main result of the paper is an affirmative answer to this question for n ≥ 1.

Let M g, [n] , for 2g − 2 + n > 0, be the stack of genus g, stable algebraic curves over C, endowed with n unordered marked points.In [15], Looijenga introduced the notion of Prym level structures in order to construct smooth projective Galois coverings of the stack M g .In §2 of this paper, we introduce the notion of Looijenga level structure which generalizes Looijenga construction and provides a tower of Galois coverings of M g, [n] equivalent to the tower of all geometric level structures over M g, [n] .In §3, Looijenga level structures are interpreted geometrically in terms of moduli of curves with symmetry. A byproduct of this characterization is a simple criterion for their smoothness. As a consequence of this criterion, it is shown that Looijenga level structures are smooth under mild hypotheses.The second part of the paper, from §4, deals with the problem of describing the D-M boundary of Looijenga level structures. In §6, a description is given of the nerve of the D-M boundary of abelian level structures. In §7, it is shown how this construction can be used to "approximate" the nerve of Looijenga level structures. These results are then applied to elaborate a new approach to the congruence subgroup problem for the Teichmüller modular group along the lines of [6].

Let Γ(S) be the pure mapping class group of a connected orientable surface S of negative Euler characteristic. For C a class of finite groups, letπ 1 (S) C be the pro-C completion of the fundamental group of S. The C -congruence completionΓ(S) C of Γ(S) is the profinite completion induced by the embedding Γ(S) → Out(π 1 (S) C ). In this paper, we begin a systematic study of such completions for different C . We show that the combinatorial structure of the profinite groupsΓ(S) C closely resemble that of Γ(S). A fundamental question is how C -congruence completions compare with pro-C completions. Even though, in general (e.g. for C the class of finite solvable groups),Γ(S) C is not even virtually a pro-C group, we show that, for Z/2 ∈ C , g(S) ≤ 2 and S open, there is a natural epimorphism from the C -congruence com-pletionΓ(S)(2) C of the abelian level of order 2 to its pro-C completion Γ(S)(2) C . In particular, this is an isomorphism for the class of finite groups and for the class of 2-groups. Moreover, in these two cases, the result also holds for a closed surface.