Low-lying geodesics on the modular surface and necklaces

Ara Basmajian; Mingkun Liu

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arXiv · 2311.16041

Low-lying geodesics on the modular surface and necklaces

Ara Basmajian·Mingkun Liu

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Abstract

The m-thick part of the modular surface X is the smallest compact subsurface of X with horocycle boundary containing all the closed geodesics which wind around the cusp at most m times. The m-thick parts form a compact exhaustion of X. We are interested in the geodesics that lie in the m-thick part (so called m low-lying geodesics). We produce a complete asymptotic expansion for the number of m low-lying geodesics of length equal to 2n in the modular surface. In particular, we obtain the asymptotic growth rate of the m low-lying geodesics in terms of their word length using the natural generators of the modular group. After establishing a correspondence between this counting problem and the problem of counting necklaces with n beads, we perform a careful singularity analysis on the associated generating function of the sequence.

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Ara Basmajian, Mingkun Liu. 2023-11-27. Low-lying geodesics on the modular surface and necklaces. https://arxiv.org/abs/2311.16041

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