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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
Full-text PDF Free Access
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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  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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- [A]
- William Abikoff, The real analytic theory of Teichmüller space, Lecture Notes in Math., vol. 820, Springer-Verlag, New York, 1980. MR 590044 (82a:32028)
- [Ba]
- Ara Basmajian, Constructing pairs of pants, Ann. Acad. Sci. Fenn. Ser. AI Math. 15 (1990), 65-74. MR 1050782 (91g:57041)
- [Bea]
- Alan Beardon, The geometry of discrete groups, Springer-Verlag, New York, 1983. MR 698777 (85d:22026)
- [Ber]
- Lipman Bers, An inequality for Riemann surfaces, Differential Geometry and Complex Analysis, H. E. Rauch Memorial volume (Isaac Chavel and Herschel M. Farkas, eds.), Springer-Verlag, New York, 1985, pp. 87-93. MR 780038 (86h:30076)
- [Bu]
- Peter Buser, The collar theorem and examples, Manuscripta Math. 25 (1978), 349-357. MR 509590 (80h:53046)
- [GM]
- Jane Gilman and Bernard Maskit, An algorithm for
 -generator Fuchsian groups, Michigan Math. J. 38 (1991), 13-32. MR 1091506 (92f:30062)
- [Ha]
- Naomi Halpern, A proof of the collar lemma, Bull. London Math. Soc. 13 (1981), 141-144. MR 608099 (82e:30064)
- [He]
- John Hempel, Traces, lengths, and simplicity of loops on surfaces, Topology Appl. 18 (1984), 153-161. MR 769288 (86c:32023)
- [K]
- Linda Keen, Collars on Riemann surfaces, Discontinuous Groups and Riemann Surfaces, Ann. of Math. Stud., no. 79, Princeton Univ. Press, Princeton, NJ, 1974, pp. 263-268. MR 0379833 (52:738)
- [M]
- Bernard Maskit, Comparison of hyperbolic and extremal lengths, Ann. Acad. Sci. Fenn. Ser. AI Math. 10 (1985), 381-386. MR 802500 (87c:30062)
- [Ma]
- Peter Matelski, A compactness theorem for Fuchsian groups of the second kind, Duke Math. J. 43 (1976), 829-840. MR 0432921 (55:5900)
- [N]
- Toshihiro Nakanishi, The lengths of the closed geodesics on a Riemann surface with self-intersection, Tohoku Math. J. (2) 41 (1989), 527-541. MR 1025320 (91b:30132)
- [R]
- Burton Randol, Cylinders in Riemann surfaces, Comment. Math. Helv. 54 (1979), 1-5. MR 522028 (80j:30065)
- [Y1]
- Akira Yamada, On Marden's universal constant of Fuchsian groups, Kodai Math. J. 4 (1981), 266-277. MR 630246 (83k:30047)
- [Y2]
- -, On Marden's universal constant of Fuchsian group. II, J. Analyse Math. 41 (1982), 234-248. MR 687954 (84h:30073)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S0002-9939-1993-1152271-0
PII:
S 0002-9939(1993)1152271-0
Keywords:
Closed geodesics,
collar,
hyperbolic surface,
self-intersection number
Article copyright:
© Copyright 1993 American Mathematical Society
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