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

TL;DR

This paper proves that the hyperelliptic modular group associated with open surfaces satisfies the congruence subgroup property, confirming a long-standing conjecture in the context of moduli spaces of hyperelliptic curves.

Contribution

It establishes the congruence subgroup property for the hyperelliptic modular group in the open surface case, extending known results to this specific class of groups.

Findings

Affirmative answer to the congruence subgroup problem for hyperelliptic modular groups with punctures.

Provides a faithful representation of the hyperelliptic modular group into the outer automorphism group of the fundamental group.

Extends the understanding of the structure of hyperelliptic modular groups in relation to the congruence subgroup property.

Abstract

Let ${\cal M}_{g,n}$ and ${\cal H}_{g,n}$, 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 $\Gamma_{g,n}$ and $H_{g,n}$, the so called Teichm{\"u}ller modular group and hyperelliptic modular group. A choice of base point on ${\cal H}_{g,n}$ defines a monomorphism $H_{g,n}\hookrightarrow\Gamma_{g,n}$. Let $S_{g,n}$ be a compact Riemann surface of genus $g$ with $n$ points removed. The Teichm\"uller group $\Gamma_{g,n}$ 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 $\Gamma_{g,n}$, the hyperelliptic modular group then admits a natural faithful representation $H_{g,n}\hookrightarrow\operatorname{Out}(\pi_1(S_{g,n}))$. The congruence subgroup problem for $H_{g,n}$ asks whether, for any given finite index subgroup $H^\lambda$ 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 $H_{g,n}\to\operatorname{Out}(\pi_1(S_{g,n})/K)$ is contained in $H^\lambda$. The main result of the paper is an affirmative answer to this question for $n\geq 1$.