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Register a new userThe congruence subgroup property for the hyperelliptic modular group: the open surface case
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Marco Boggi
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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 $H_{g,n}hookrightarrowGamma_{g,n}$. Let $S_{g,n}$ be a compact Riemann surface of genus $g$ with $n$ points removed. The Teichmuller 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 $H_{g,n}hookrightarrowoperatorname{Out}(pi_1(S_{g,n}))$. 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 $H_{g,n}tooperatorname{Out}(pi_1(S_{g,n})/K)$ is contained in $H^lambda$. The main result of the paper is an affirmative answer to this question for $ngeq 1$.
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