Generalizing the hyperbolic collar lemma
Author:
Ara Basmajian
Journal:
Bull. Amer. Math. Soc. 27 (1992), 154158
MSC (2000):
Primary 30F40; Secondary 53C22, 57M50
MathSciNet review:
1145576
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Abstract: We discuss two generalizations of the collar lemma. The first is the stable neighborhood theorem which says that a (not necessarily simple) closed geodesic in a hyperbolic surface has a "stable neighborhood" whose width only depends on the length of the geodesic. 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 tends to infinity with k. Our second generalization is to totally geodesic hypersurfaces of hyperbolic manifolds. Namely, we construct a tubular neighborhood function and show that an embedded closed totally geodesic hypersurface in a hyperbolic manifold has a tubular neighborhood whose width only depends on the area of the hypersurface (and hence not on the geometry of the ambient manifold). The implications of this result for volumes of hyperbolic manifolds is discussed. We also derive a (hyperbolic) quantitative version of the KleinMaskit combination theorem (in all dimensions) for free products of fuchsian groups. Using this last theorem, we construct examples to illustrate the qualitative sharpness of the tubular neighborhood function.
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 Boris Apanasov, Discrete groups in space and uniformization problems, Kluwer Academic, Dordrecht, Netherlands, 1991. MR 1191903 (93h:57026)
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 Ara Basmajian, Constructing pairs of pants, Ann. Acad. Sci. Fenn. Ser. A I Math. 15 (1990), 6574. MR 1050782 (91g:57041)
 [Ba2]
 , The orthogonal spectrum of a hyperbolic manifold, Amer. J. Math., (to appear, 1992). MR 1246187 (94j:57014)
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 , The stable neighborhood theorem and lengths of closed geodesics, Proc. Amer. Math. Soc., (to appear). MR 1152271 (93k:30072)
 [Ba4]
 , Tubular neighborhoods of totally geodesic hypersurfaces in hyperbolic manifolds, preprint.
 [Be]
 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)
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 Peter Buser, The collar theorem and examples, Manuscripta Math. 25 (1978), 349357. MR 509590 (80h:53046)
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 N. 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)
 [Jo]
 Troels Jorgensen, on discrete groups of Möbius transformations, Amer. J. Math. 98 (1976), 739749. MR 0427627 (55:658)
 [Ke]
 Linda Keen, Collars on Riemann surfaces, Discontinuous Groups and Riemann Surfaces, Ann. of Math. Stud., vol. 79, Princeton Univ. Press, Princeton, NJ, 1974, pp. 263268. MR 0379833 (52:738)
 [KM]
 S. Kojima and Y. Miyamoto, The smallest hyperbolic 3manifolds with totally geodesic boundary, J. Differential Geometry 34 (1991), 175192. MR 1114459 (92f:57019)
 [Ma1]
 Bernard Maskit, Comparison of hyperbolic and extremal lengths, Ann. Acad. Sci. Fenn. Ser. A I Math. 10 (1985), 381386. MR 802500 (87c:30062)
 [Ma2]
 , Kleinian groups, SpringerVerlag, New York, 1988. MR 959135 (90a:30132)
 [Mt]
 Peter Matelski, A compactness theorem for Fuchsian groups of the second kind, Duke Math. J. 43 (1976), 829840. MR 0432921 (55:5900)
 [Me]
 Bob Meyerhoff, A lower bound for the volume of hyperbolic 3manifolds, Canad. J. Math. 39 (1987), 10381056. MR 918586 (88k:57049)
 [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)
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 Burton Randol, Cylinders in Riemann surfaces, Comment. Math. Helv. 54 (1979), 15. MR 522028 (80j:30065)
 [Th]
 William Thurston, The geometry and topology of 3manifolds, lecture notes, Princeton University, 1977.
 [Y1]
 Akira Yamada, On Marden's universal constant of Fuchsian groups, Kodai Math. J. 4 (1981), 266277. MR 630246 (83k:30047)
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 Akira Yamada, On Marden's universal constant of Fuchsian groups II, J. Analyse Math. 41 (1982), 234248. MR 687954 (84h:30073)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S027309791992002987
PII:
S 02730979(1992)002987
Keywords:
Collar,
geodesic,
hyperbolic manifold,
selfintersection,
totally geodesic hypersurface,
tubular neighborhood,
volume
Article copyright:
© Copyright 1992
American Mathematical Society
