## 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.## References

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## Bibliographic Information

- © Copyright 1993 American Mathematical Society
- 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