Geodesic
Counting geodesics of given commutator length
Forum of Mathematics, Sigma, Tome 11 (2023)

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Let $\Sigma $ be a closed hyperbolic surface. We study, for fixed g, the asymptotics of the number of those periodic geodesics in $\Sigma $ having at most length L and which can be written as the product of g commutators. The basic idea is to reduce these results to being able to count critical realizations of trivalent graphs in $\Sigma $. In the appendix, we use the same strategy to give a proof of Huber’s geometric prime number theorem. 
@article{10_1017_fms_2023_114,
     author = {Viveka Erlandsson and Juan Souto},
     title = {Counting geodesics of given commutator length},
     journal = {Forum of Mathematics, Sigma},
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Viveka Erlandsson; Juan Souto. Counting geodesics of given commutator length. Forum of Mathematics, Sigma, Tome 11 (2023). doi: 10.1017/fms.2023.114

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