Simple closed geodesics on hyperelliptic translation surfaces and classification theorem for translation surfaces in $\mathcal{H}^{\mathrm{hyp}}(4)$

Abstract

In this paper, we give the maximum of the numbers $n$ such that we can take $n$ simple closed geodesics without singularities that are disjoint to each other for translation surfaces in the hyperelliptic components $\mathcal{H}^{\rm {hyp}}(2g-2)$ and $\mathcal{H}^{\rm{hyp}}(g-1, g-1)$. The maximum is different from that of the case of hyperbolic surfaces. We also give a classification theorem for translation surfaces in $\mathcal{H}^{\mathrm{hyp}}(4)$ with respect to their Euclidean structures.