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

Abstract: 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 $\phi \in {\rm Mod}(\Sigma)$.