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The stable neighborhood theorem and lengths of closed geodesics

Author: Ara Basmajian
Journal: Proc. Amer. Math. Soc. 119 (1993), 217-224
MSC: Primary 30F35; Secondary 53C22
MathSciNet review: 1152271
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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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Keywords: Closed geodesics, collar, hyperbolic surface, self-intersection number
Article copyright: © Copyright 1993 American Mathematical Society

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