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

Abstract

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.

Congruence topologies on the mapping class group

“…In Section 3 of [4], we saw that this decomposition induces one: Υ g,2g+2 (2) ∼ = Π g (2g + 2) Υ g (2).…”

“…The proof of Theorem 1.4 is based on the interpretation of hyperelliptic mapping class groups as fundamental groups of moduli stacks of complex hyperelliptic curves. Let us recall a few facts from Section 3 of [4]. The moduli stack of n-pointed, genus g smooth hyperelliptic curves H g,n can be described in terms of moduli of pointed genus 0 curves.…”

“…There is a standard procedure to simplify the structure of an algebraic stack X by erasing a generic group of automorphisms G. The algebraic stack thus obtained is usually denoted by X G. The natural map H g → M 0,[2g+2] yields an isomorphism H g ι ∼ = M 0,[2g+2] . In Proposition 3.3 in [4], it was observed that the pull-back over H g of the Galoisétale covering M 0,2g+2 → M 0,[2g+2] is the abelian level structure H (2) g . More precisely: Proposition 6.1.…”

“…By Proposition 6.1, since ι / ∈ Υ(4), there is a natural representableétale Galois morphism H (4) → M 0,2g+2 . Let C (4) 0 → H (4) and R (4) → H (4) be, respectively, the pull-backs of the curve C 0 → H g and of the universal (2g + 2)-punctured, genus 0 curve R → M 2g+2 over H (4) . There is then a commutative diagram:…”

“…where ψ is theétale, degree 2 map which, fibrewise, is the natural map to the quotient by the hyperelliptic involution. Pick a base point of H (4) so that the fiber over it of the curve C (4) 0 → H (4) identifies with C 0 and let R be the fiber over the chosen base point of the curve R (4) → H (4) . From the above diagram, we then get the algebraic monodromy representations: Υ(4) → Out(π 1 (C 0 ) C ), which is just the restriction ofˆ C 0 to Υ(4), has its same kernel and we denote in the same way, andˆ C R : Υ(4) → Out(π 1 (R) C ).…”