Lifting curves simply

We provide linear lower bounds for \(f_\rho(L)\), the smallest integer so that every curve on a fixed hyperbolic surface \((S,\rho)\) of length at most \(L\) lifts to a simple curve on a cover of degree at most \(f_\rho(L)\). This bound is independent of hyperbolic structure \(\rho\), and improves o...

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Bibliographic Details
Published in:arXiv.org 2015-01
Main Author: Gaster, Jonah
Format: Article
Language:English
Subjects:
Lower bounds
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Summary:We provide linear lower bounds for \(f_\rho(L)\), the smallest integer so that every curve on a fixed hyperbolic surface \((S,\rho)\) of length at most \(L\) lifts to a simple curve on a cover of degree at most \(f_\rho(L)\). This bound is independent of hyperbolic structure \(\rho\), and improves on a recent bound of Gupta-Kapovich. When \((S,\rho)\) is without punctures, using work of Patel we conclude asymptotically linear growth of \(f_\rho\). When \((S,\rho)\) has a puncture, we obtain exponential lower bounds for \(f_\rho\).
ISSN:2331-8422