Abstract
Let and , for , be, respectively, the moduli stack of -pointed, genus smooth curves and its closed substack consisting of hyperelliptic curves. Their topological fundamental groups can be identified, respectively, with and , the so called Teichmüller modular group and hyperelliptic modular group. A choice of base point on defines a monomorphism .
Let be a compact Riemann surface of genus with points removed. The Teichmüller group is the group of isotopy classes of diffeomorphisms of the surface which preserve the orientation and a given order of the punctures. As a subgroup of , the hyperelliptic modular group then admits a natural faithful representation .
The congruence subgroup problem for asks whether, for any given finite index subgroup of , there exists a finite index characteristic subgroup of such that the kernel of the induced representation is contained in . The main result of the paper is an affirmative answer to this question for .
Citation
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
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