A Note on Collars of Simple Closed Geodesics

Abstract

For compact hyperbolic Riemann surfaces, the collar theorem gives a lower bound on the distance between a simple closed geodesic and all other simple closed geodesics that do not intersect the initial geodesic. Here it is shown that there are two possible configurations, and in each configuration there is a natural collar width associated to a simple closed geodesic. If one extends the natural collar of a simple closed geodesic α by e >0, then the extended collar contains an infinity of simple closed geodesics that do not intersect α.