Short homology bases for translation surfaces

Abstract

Let $S$ be a closed translation surface of genus $g\ge 2$ with $\mathrm{area}(S)=4\pi g$. We show that for any $\lambda \in (0,1)$ there exist $\lfloor \lambda \cdot g\rfloor$ homologically independent simple closed curves of length at most $C(\lambda )\cdot \log (g)$, where $C(\lambda )$ is a constant that depends only on $\lambda$. This result is obtained from a mixed graph construction based on a Voronoi graph of the surface with the cone points as seeds and its dual graph. We also give a complementary result using only the Voronoi graph that produces $2g$ short loops that form a homology basis. This construction, however, depends on the length of a shortest saddle connection of the surface $S$.

14 page, 4 figures, minor revisions