The stable neighborhood theorem and lengths of closed geodesics

Basmajian, Ara

doi:10.1090/S0002-9939-1993-1152271-0

The stable neighborhood theorem and lengths of closed geodesics
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by Ara Basmajian

Proc. Amer. Math. Soc. 119 (1993), 217-224

DOI: https://doi.org/10.1090/S0002-9939-1993-1152271-0

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Abstract:

We derive a generalized collar lemma, called the stable neighborhood theorem, for nonsimple closed geodesics. As an application, we show that there is a lower bound for the length of a closed geodesic having crossing number $k$ on a hyperbolic surface. This lower bound only depends on $k$ and tends to infinity as $k$ goes to infinity. Also, we show that the shortest nonsimple closed geodesic on a closed hyperbolic surface has (geometric) crossing number bounded above by a constant which only depends on the genus.

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