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PreprintCC BY 4.0arXiv

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≥ 2 with mathoparea(S)= 4πg. We show that for any λ∈ (0, 1) there exist ⌊ λ· g⌋ 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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Cite

BibTeXRIS
BibTeX
@misc{buser2026short,
  title = {{Short homology bases for translation surfaces}},
  author = {Peter Buser and Achintya Dey and Eran Makover and Bjoern Muetzel},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.30561},
  eprint = {2609.30561},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.30561},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Short homology bases for translation surfaces
AU  - Peter Buser
AU  - Achintya Dey
AU  - Eran Makover
AU  - Bjoern Muetzel
PY  - 2026
DA  - 2026/10/04
PB  - arXiv
DO  - 10.48550/arxiv.2609.30561
UR  - https://arxiv.org/abs/2609.30561
AB  - Let S be a closed translation surface of genus g≥ 2 with mathoparea(S)= 4πg. We show that for any λ∈ (0, 1) there exist ⌊ λ· g⌋ 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.
ER  -