Low-lying geodesics on the modular surface and necklaces
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.
Cite
@article{arxiv.2311.16041,
title = {Low-lying geodesics on the modular surface and necklaces},
author = {Ara Basmajian and Mingkun Liu},
journal= {arXiv preprint arXiv:2311.16041},
year = {2023}
}