Counting geodesics of given commutator length

Viveka  Erlandsson; Juan  Souto

doi:10.1017/fms.2023.114

Counting geodesics of given commutator length

Viveka Erlandsson, Juan Souto

School of Mathematics

Research output: Contribution to journal › Article (Academic Journal) › peer-review

Abstract

Let Σ be a closed hyperbolic surface. We study, for fixed g, the asymptotics of the number of those periodic geodesics in Σ 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 Σ. In the appendix, we use the same strategy to give a proof of Huber’s geometric prime number theorem.

Original language	English
Article number	e114
Journal	Forum of Mathematics, Sigma
Volume	11
DOIs	https://doi.org/10.1017/fms.2023.114
Publication status	Published - 15 Dec 2023

Bibliographical note

Funding Information:
The first author gratefully acknowledges support from EPSRC grant EP/T015926/1 and UiT Aurora Center for Mathematical Structures in Computations (MASCOT).

Publisher Copyright:
© The Author(s), 2023. Published by Cambridge University Press.

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10.1017/fms.2023.114Licence: CC BY

https://arxiv.org/abs/2304.10274Licence: CC BY

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title = "Counting geodesics of given commutator length",

abstract = "Let Σ be a closed hyperbolic surface. We study, for fixed g, the asymptotics of the number of those periodic geodesics in Σ 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 Σ. In the appendix, we use the same strategy to give a proof of Huber{\textquoteright}s geometric prime number theorem.",

author = "Viveka Erlandsson and Juan Souto",

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AU - Souto, Juan

N1 - Funding Information: The first author gratefully acknowledges support from EPSRC grant EP/T015926/1 and UiT Aurora Center for Mathematical Structures in Computations (MASCOT). Publisher Copyright: © The Author(s), 2023. Published by Cambridge University Press.

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N2 - Let Σ be a closed hyperbolic surface. We study, for fixed g, the asymptotics of the number of those periodic geodesics in Σ 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 Σ. In the appendix, we use the same strategy to give a proof of Huber’s geometric prime number theorem.

AB - Let Σ be a closed hyperbolic surface. We study, for fixed g, the asymptotics of the number of those periodic geodesics in Σ 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 Σ. In the appendix, we use the same strategy to give a proof of Huber’s geometric prime number theorem.

U2 - 10.1017/fms.2023.114

DO - 10.1017/fms.2023.114

M3 - Article (Academic Journal)

SN - 2050-5094

VL - 11

JO - Forum of Mathematics, Sigma

JF - Forum of Mathematics, Sigma

M1 - e114

ER -

Counting geodesics of given commutator length

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