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PACKING STRIPS IN THE HYPERBOLIC PLANE

Published online by Cambridge University Press:  27 January 2003

T. H. Marshall  and
G. J. Martin
T. H. Marshall
Affiliation:
Department of Mathematics, University of Auckland, Private Bag 92019, Auckland, New Zealand ([email protected]; [email protected])
G. J. Martin
Affiliation:
Department of Mathematics, University of Auckland, Private Bag 92019, Auckland, New Zealand ([email protected]; [email protected])
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Abstract

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A strip of radius $r$ in the hyperbolic plane is the set of points within distance $r$ of a given geodesic. Wedefine the density of a packing of strips of radius $r$ and prove that this density cannot exceed

$$ \mathcal{S}(r)=\frac{3}{\pi}\sinh r\mathrm{arccosh}\biggl(1+\frac{1}{2\sinh^2r}\biggr). $$

This bound is sharp for every value of $r$ and provides sharp bounds on collaring theorems for simple geodesics onsurfaces.

AMS 2000 Mathematics subject classification: Primary 51M09; 52C15. Secondary 51M04

Keywords

striphorodiscpackingdensityhyperbolic planeFuchsian group
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 46 , Issue 1 , February 2003 , pp. 67 - 73
Copyright
Copyright © Edinburgh Mathematical Society 2003