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Notes on hyperelliptic mapping class groups

Part of: Arithmetic problems. Diophantine geometry (Co)homology theory Curves

Published online by Cambridge University Press:  31 October 2023

Marco Boggi Open the ORCID record for Marco Boggi [Opens in a new window]
Marco Boggi*
Affiliation:
Instituto de Matemática e Estatística, Universidade Federal Fluminense, Niterói, RJ, 24210-200, Brazil
*
Email: marco.boggi@gmail.com
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Abstract

Hyperelliptic mapping class groups are defined either as the centralizers of hyperelliptic involutions inside mapping class groups of oriented surfaces of finite type or as the inverse images of these centralizers by the natural epimorphisms between mapping class groups of surfaces with marked points. We study these groups in a systematic way. An application of this theory is a counterexample to the genus $2$ case of a conjecture by Putman and Wieland on virtual linear representations of mapping class groups. In the last section, we study profinite completions of hyperelliptic mapping class groups: we extend the congruence subgroup property to the general class of hyperelliptic mapping class groups introduced above and then determine the centralizers of multitwists and of open subgroups in their profinite completions.

Keywords

Hyperelliptic mapping class groupstopological surfacesmoduli of curvesvirtual linear representations of mapping class groupsprofinite hyperelliptic mapping class groups

MSC classification

Primary: 14H10: Families, moduli (algebraic)
Secondary: 14F45: Topological properties 14G32: Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory)

Information

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 66 , Issue 1 , January 2024 , pp. 126 - 161
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Glasgow Mathematical Journal Trust

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