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

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

MathSciNet review:
1152271

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Abstract | References | Similar Articles | Additional Information

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 on a hyperbolic surface. This lower bound only depends on and tends to infinity as 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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Additional Information

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

Keywords:
Closed geodesics,
collar,
hyperbolic surface,
self-intersection number

Article copyright:
© Copyright 1993
American Mathematical Society