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Density of length ratios on Teichmüller space
Abstract
We study length ratios of simple closed geodesics on finite-type hyperbolic surfaces of positive genus. We allow cusps and fix the geodesic boundary lengths. Let d be the dimension of the corresponding Teichmüller space 𝒯L(S). We prove that for every positive vector (r0,…,rd), the set of surfaces carrying d+1 distinct essential interior simple closed geodesics with lengths proportional to this vector is dense in 𝒯L(S). In particular, one-holed tori with three distinct simple closed geodesics of equal length are dense. The proof uses high powers of Dehn twists, asymptotics of length differentials, and the Brouwer fixed-point theorem.
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BibTeX
@misc{doan2026density,
title = {{Density of length ratios on Teichmüller space}},
author = {Nhat Minh Doan and Hanh Vo},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.24700},
eprint = {2610.24700},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.24700},
}
RIS
TY - UNPB
TI - Density of length ratios on Teichmüller space
AU - Nhat Minh Doan
AU - Hanh Vo
PY - 2026
DA - 2026/10/17
PB - arXiv
DO - 10.48550/arxiv.2610.24700
UR - https://arxiv.org/abs/2610.24700
AB - We study length ratios of simple closed geodesics on finite-type hyperbolic surfaces of positive genus. We allow cusps and fix the geodesic boundary lengths. Let d be the dimension of the corresponding Teichmüller space 𝒯_L(S). We prove that for every positive vector (r_0,…,r_d), the set of surfaces carrying d+1 distinct essential interior simple closed geodesics with lengths proportional to this vector is dense in 𝒯_L(S). In particular, one-holed tori with three distinct simple closed geodesics of equal length are dense. The proof uses high powers of Dehn twists, asymptotics of length differentials, and the Brouwer fixed-point theorem.
ER -