On the mapping class group action on the homology of surface covers

Let \(\phi \in {\rm Mod}(\Sigma)\) be an arbitrary element of the mapping class group of a closed orientable surface \(\Sigma\) of genus at least \(2\). For any characteristic cover \(\widetilde{\Sigma} \to \Sigma\) one can consider the linear subspace \({\rm H}_1^{f.o.}(\widetilde{\Sigma}, \mathbb{...

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Bibliographic Details
Published in:arXiv.org 2024-06
Main Author: Spiridonov, Igor
Format: Article
Language:English
Subjects:
Homology
Mapping
Online Access:Get full text
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Summary:Let \(\phi \in {\rm Mod}(\Sigma)\) be an arbitrary element of the mapping class group of a closed orientable surface \(\Sigma\) of genus at least \(2\). For any characteristic cover \(\widetilde{\Sigma} \to \Sigma\) one can consider the linear subspace \({\rm H}_1^{f.o.}(\widetilde{\Sigma}, \mathbb{Q})^\phi \subseteq {\rm H}_1(\widetilde{\Sigma}, \mathbb{Q})\) consisting of all homology classes with finite \(\phi\)-orbit. We prove that \(\dim {\rm H}_1^{f.o.}(\widetilde{\Sigma}, \mathbb{Q})^\phi\) can be arbitrary large for any fixed \(\phi \in {\rm Mod}(\Sigma)\).
ISSN:2331-8422