Profinite Teichmüller theory

Abstract

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)

Congruence topologies on the mapping class group

“…Given a simplex σ ∈ C(S), let us denote also by σ its image in L( Π C ) and byΓ(S) C σ the correspondingΓ(S) C -stabilizer. From Proposition 6.5 in [3] and Theorem 3.5, it follows thatΓ(S) C σ is the closure of the stabilizer Γ(S) σ (for the action of Γ(S) on C(S)) inΓ(S) C . Since every σ ∈ L( Π C ) is in theΓ(S) C -orbit of some simplex in the image of C(S), this gives a complete description of stabilizers for the action ofΓ(S) C on the pro-C curve complex L( Π C ).…”

“…Indeed, there is a natural Z/2-gerbe H g → M 0,[2g+2] , for g ≥ 2, defined assigning, to a genus g hyperelliptic curve C, the genus zero quotient curve C/ι marked by the branch locus of the finite morphism C → C/ι, where ι, as usual, denotes the hyperelliptic involution. Similarly, in the genus 1 case, there is a Z/2-gerbe M 1,1 → M 0,1 [3] , where, by the notation "1 [3]", we mean that one label is distinguished from the others (which instead are unordered). For 2g − 2 + n > 0, there is also a natural representable morphism H g,n+1 → H g,n , forgetting the (n + 1)-th marked point, which is isomorphic to the universal n-punctured curve over H g,n .…”

“…where, for all n ≥ 0, the groups π alg 1 (H g,n ) and π alg 1 (M 0,[2g+2] ) identify, respectively, with the profinite completions Υ g,n of Υ g,n and Γ 0,[2g+2] of Γ 0,[2g+2] . For g = 1, we just need to replace the short exact sequences which appear on the left in (10) and (11) above with 1 → Z/2 → Γ 1,1 → Γ 0,1 [3] → 1 and 1 → Z/2 → π alg 1 (M 1,1 ) → π alg 1 (M 0,1 [3] ) → 1, respectively.…”

Fundamental groups of moduli stacks of stable curves of compact type

“…For a more complete treatment of level structures and Teichmüller theory, we refer the reader, for instance, to Section 1 of Boggi [4]. Let S M g;n , for 2g 2 C n > 0, be the stack of n-pointed, genus g , stable algebraic curves over C .…”

“…Let M g;n be the real oriented blow-up of M g;n along the divisor M g;n X M g;n (more details on this construction can be found in Section 3 of Boggi [4]). There is a natural embedding M g;n ,!…”