Hyperbolic punctured spheres without arithmetic systole maximizers

Grant S. Lakeland; Clayton Young

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arXiv · 2209.01748

Hyperbolic punctured spheres without arithmetic systole maximizers

Grant S. Lakeland·Clayton Young

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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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Grant S. Lakeland, Clayton Young. 2022-09-05. Hyperbolic punctured spheres without arithmetic systole maximizers. https://arxiv.org/abs/2209.01748

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