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

Abstract

Let $\varphi \in \mathrm{Mod}(\Sigma )$ be an arbitrary element of the mapping class group of a closed orientable surface $\Sigma$ of genus at least $2$. For any characteristic cover $\tilde{\Sigma }\to \Sigma$ one can consider the linear subspace ${\mathrm{H}}_1^{f.o.}(\tilde{\Sigma },\mathbb{Q}{)}^{\varphi }\subseteq {\mathrm{H}}_1(\tilde{\Sigma },\mathbb{Q})$ consisting of all homology classes with finite $\varphi$-orbit. We prove that $\dim {\mathrm{H}}_1^{f.o.}(\tilde{\Sigma },\mathbb{Q}{)}^{\varphi }$ can be arbitrary large for any fixed $\varphi \in \mathrm{Mod}(\Sigma )$.

11 pages, 2 figures, minor corrections