Simple closed geodesics on hyperelliptic translation surfaces and a classification theorem for translation surfaces in $\mathcal{H}$^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 ^hyp(2g−2) and ^hyp(g−1, g−1). The maximum is different from the maximum number of pairwise disjoint simple closed geodesics on a hyperbolic surface. We also give a classification theorem for translation surfaces in ^hyp(4) with respect to their Euclidean structures.