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 $\phi \in { \rm Mod}(\Sigma)$.
Categories: math.GT