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arXiv · 2609.30561

Short homology bases for translation surfaces

Peter Buser·Achintya Dey·Eran Makover·Bjoern Muetzel
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Abstract

Let $S$ be a closed translation surface of genus $g\ge 2$ with $\mathop{area}(S)= 4πg$. We show that for any $λ\in (0, 1)$ there exist $\lfloor λ\cdot g\rfloor$ homologically independent simple closed curves of length at most $ C(λ) \cdot \log(g)$, where $C(λ)$ is a constant that depends only on $λ$. 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 $S$.

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Peter Buser, Achintya Dey, Eran Makover, Bjoern Muetzel. 2026-10-04. Short homology bases for translation surfaces. https://arxiv.org/abs/2609.30561

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