The congruence subgroup property for the hyperelliptic modular group: the open surface case
Boggi, Marco
Hiroshima Math. J., Tome 39 (2009) no. 1, p. 351-362 / Harvested from Project Euclid
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 $n\geq 1$.
Publié le : 2009-11-15
Classification:  congruence subgroups,  Teichmüller theory,  moduli of curves,  profinite groups,  14H10,  14H15,  14F35,  11R34
@article{1257544213,
     author = {Boggi, Marco},
     title = {The congruence subgroup property for the hyperelliptic modular group: the open
				surface case},
     journal = {Hiroshima Math. J.},
     volume = {39},
     number = {1},
     year = {2009},
     pages = { 351-362},
     language = {en},
     url = {http://dml.mathdoc.fr/item/1257544213}
}
Boggi, Marco. The congruence subgroup property for the hyperelliptic modular group: the open
				surface case. Hiroshima Math. J., Tome 39 (2009) no. 1, pp.  351-362. http://gdmltest.u-ga.fr/item/1257544213/