The stable neighborhood theorem and lengths of closed geodesics
Author:
Ara Basmajian
Journal:
Proc. Amer. Math. Soc. 119 (1993), 217224
MSC:
Primary 30F35; Secondary 53C22
MathSciNet review:
1152271
Fulltext PDF Free Access
Abstract 
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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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 William Abikoff, The real analytic theory of Teichmüller space, Lecture Notes in Math., vol. 820, SpringerVerlag, New York, 1980. MR 590044 (82a:32028)
 [Ba]
 Ara Basmajian, Constructing pairs of pants, Ann. Acad. Sci. Fenn. Ser. AI Math. 15 (1990), 6574. MR 1050782 (91g:57041)
 [Bea]
 Alan Beardon, The geometry of discrete groups, SpringerVerlag, 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.), SpringerVerlag, New York, 1985, pp. 8793. MR 780038 (86h:30076)
 [Bu]
 Peter Buser, The collar theorem and examples, Manuscripta Math. 25 (1978), 349357. MR 509590 (80h:53046)
 [GM]
 Jane Gilman and Bernard Maskit, An algorithm for generator Fuchsian groups, Michigan Math. J. 38 (1991), 1332. MR 1091506 (92f:30062)
 [Ha]
 Naomi Halpern, A proof of the collar lemma, Bull. London Math. Soc. 13 (1981), 141144. MR 608099 (82e:30064)
 [He]
 John Hempel, Traces, lengths, and simplicity of loops on surfaces, Topology Appl. 18 (1984), 153161. 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. 263268. MR 0379833 (52:738)
 [M]
 Bernard Maskit, Comparison of hyperbolic and extremal lengths, Ann. Acad. Sci. Fenn. Ser. AI Math. 10 (1985), 381386. MR 802500 (87c:30062)
 [Ma]
 Peter Matelski, A compactness theorem for Fuchsian groups of the second kind, Duke Math. J. 43 (1976), 829840. MR 0432921 (55:5900)
 [N]
 Toshihiro Nakanishi, The lengths of the closed geodesics on a Riemann surface with selfintersection, Tohoku Math. J. (2) 41 (1989), 527541. MR 1025320 (91b:30132)
 [R]
 Burton Randol, Cylinders in Riemann surfaces, Comment. Math. Helv. 54 (1979), 15. MR 522028 (80j:30065)
 [Y1]
 Akira Yamada, On Marden's universal constant of Fuchsian groups, Kodai Math. J. 4 (1981), 266277. MR 630246 (83k:30047)
 [Y2]
 , On Marden's universal constant of Fuchsian group. II, J. Analyse Math. 41 (1982), 234248. MR 687954 (84h:30073)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939199311522710
PII:
S 00029939(1993)11522710
Keywords:
Closed geodesics,
collar,
hyperbolic surface,
selfintersection number
Article copyright:
© Copyright 1993
American Mathematical Society
