Non-simple systoles on random hyperbolic surfaces for large genus

Yuxin He; Yang Shen; Yunhui Wu; Yuhao Xue

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arXiv · 2308.16447

Non-simple systoles on random hyperbolic surfaces for large genus

Yuxin He·Yang Shen·Yunhui Wu·Yuhao Xue

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Abstract

In this paper, we investigate the asymptotic behavior of the non-simple systole, which is the length of a shortest non-simple closed geodesic, on a random closed hyperbolic surface on the moduli space $\mathcal{M}_g$ of Riemann surfaces of genus $g$ endowed with the Weil-Petersson measure. We show that as the genus $g$ goes to infinity, the non-simple systole of a generic hyperbolic surface in $\mathcal{M}_g$ behaves exactly like $\log g$.

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Yuxin He, Yang Shen, Yunhui Wu, Yuhao Xue. 2023-08-31. Non-simple systoles on random hyperbolic surfaces for large genus. https://arxiv.org/abs/2308.16447

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