## Hiroshima Mathematical Journal

- Hiroshima Math. J.
- Volume 39, Number 3 (2009), 351-362.

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

#### Abstract

Let $\cM_{g,n}$ and $\cH_{g,n}$, 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 $\GG_{g,n}$ and $H_{g,n}$, the so called
*Teichmüller modular group* and *hyperelliptic modular
group*. A choice of base point on $\cH_{g,n}$ defines a monomorphism
$H_{g,n}\hookra\GG_{g,n}$.

Let $S_{g,n}$ be a compact Riemann surface of genus $g$ with $n$ points removed. The Teichmüller group $\GG_{g,n}$ 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 $\GG_{g,n}$, the hyperelliptic modular group then admits a natural faithful representation $H_{g,n}\hookra\out(\pi_1(S_{g,n}))$.

The *congruence subgroup problem for* $H_{g,n}$ asks whether, for any given
finite index subgroup $H^\ld$ 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 $H_{g,n}\ra\out(\pi_1(S_{g,n})/K)$ is contained in
$H^\ld$. The main result of the paper is an affirmative answer to this question
for $n\geq 1$.

#### Article information

**Source**

Hiroshima Math. J., Volume 39, Number 3 (2009), 351-362.

**Dates**

First available in Project Euclid: 6 November 2009

**Permanent link to this document**

https://projecteuclid.org/euclid.hmj/1257544213

**Digital Object Identifier**

doi:10.32917/hmj/1257544213

**Mathematical Reviews number (MathSciNet)**

MR2569009

**Zentralblatt MATH identifier**

1209.14023

**Subjects**

Primary: 14H10: Families, moduli (algebraic) 14H15: Families, moduli (analytic) [See also 30F10, 32G15] 14F35: Homotopy theory; fundamental groups [See also 14H30] 11R34: Galois cohomology [See also 12Gxx, 19A31]

**Keywords**

congruence subgroups Teichmüller theory moduli of curves profinite groups

#### Citation

Boggi, Marco. The congruence subgroup property for the hyperelliptic modular group: the open surface case. Hiroshima Math. J. 39 (2009), no. 3, 351--362. doi:10.32917/hmj/1257544213. https://projecteuclid.org/euclid.hmj/1257544213