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November 2009 The congruence subgroup property for the hyperelliptic modular group: the open surface case
Marco Boggi
Hiroshima Math. J. 39(3): 351-362 (November 2009). DOI: 10.32917/hmj/1257544213

Abstract

Let \cMg,n and \cHg,n, for 2g2+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 \GGg,n and Hg,n, the so called Teichmüller modular group and hyperelliptic modular group. A choice of base point on \cHg,n defines a monomorphism Hg,n\hookra\GGg,n.

Let Sg,n be a compact Riemann surface of genus g with n points removed. The Teichmüller group \GGg,n is the group of isotopy classes of diffeomorphisms of the surface Sg,n which preserve the orientation and a given order of the punctures. As a subgroup of \GGg,n, the hyperelliptic modular group then admits a natural faithful representation Hg,n\hookra\out(π1(Sg,n)).

The congruence subgroup problem for Hg,n asks whether, for any given finite index subgroup H\ld of Hg,n, there exists a finite index characteristic subgroup K of π1(Sg,n) such that the kernel of the induced representation Hg,n\ra\out(π1(Sg,n)/K) is contained in H\ld. The main result of the paper is an affirmative answer to this question for n1.

Citation

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Marco Boggi. "The congruence subgroup property for the hyperelliptic modular group: the open surface case." Hiroshima Math. J. 39 (3) 351 - 362, November 2009. https://doi.org/10.32917/hmj/1257544213

Information

Published: November 2009
First available in Project Euclid: 6 November 2009

zbMATH: 1209.14023
MathSciNet: MR2569009
Digital Object Identifier: 10.32917/hmj/1257544213

Subjects:
Primary: 11R34 , 14F35 , 14H10 , 14H15

Keywords: congruence subgroups , moduli of curves , profinite groups , Teichmüller theory

Rights: Copyright © 2009 Hiroshima University, Mathematics Program

Vol.39 • No. 3 • November 2009
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