Generalizing the hyperbolic collar lemma

Author:
Ara Basmajian

Journal:
Bull. Amer. Math. Soc. **27** (1992), 154-158

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 Klein-Maskit 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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Additional Information

DOI:
https://doi.org/10.1090/S0273-0979-1992-00298-7

Keywords:
Collar,
geodesic,
hyperbolic manifold,
self-intersection,
totally geodesic hypersurface,
tubular neighborhood,
volume

Article copyright:
© Copyright 1992
American Mathematical Society