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Hyperbolic Punctured Spheres Without Arithmetic Systole Maximizers

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Abstract

We find bounds for the length of the systole—the shortest essential, non-peripheral closed curve—for arithmetic punctured spheres with n cusps, for \(n=4\) through \(n=12\), some of which were previously known due to Schmutz. This is shown using a correspondence between such surfaces and planar triangulations. We show that for \(n=7,10,11\), arithmetic surfaces do not achieve the maximal systole length.

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Acknowledgements

We thank Rick Anderson for helping initiate this project, Alan Reid for suggesting a shorter proof of Lemma 2.1 than was given in an earlier draft, and Brendan McKay for graph theory help with the 8-cusped case via comments to a MathOverflow question [9]. We also thank the anonymous referees for their careful reading of the manuscript and helpful suggestions for improving the paper.

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  1. Department of Mathematics & Computer Science, Eastern Illinois University, 600 Lincoln Avenue, Charleston, IL, 61920, USA

    Grant S. Lakeland & Clayton Young

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  1. Grant S. Lakeland
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Correspondence to Grant S. Lakeland.

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Lakeland, G.S., Young, C. Hyperbolic Punctured Spheres Without Arithmetic Systole Maximizers. La Matematica 2, 934–961 (2023). https://doi.org/10.1007/s44007-023-00066-x

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  • DOI: https://doi.org/10.1007/s44007-023-00066-x

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