Packing strips in the hyperbolic plane

Abstract

A strip of radius r in the hyperbolic plane is the set of points within distance r of a given geodesic. We define the density of a packing of strips of radius r and prove that this density cannot exceed S(r) = 3/π sinh r arccosh (1 + 1/2sinh2r). This bound is sharp for every value of r and provides sharp bounds on collaring theorems for simple geodesics on surfaces.