The stable neighborhood theorem and lengths of closed geodesics

Abstract

We derive a generalized collar lemma, called the stable neighborhood theorem, for nonsimple closed geodesics. As an application, we show that there is a lower bound for the length of a closed geodesic having crossing number k k on a hyperbolic surface. This lower bound only depends on k k and tends to infinity as k k goes to infinity. Also, we show that the shortest nonsimple closed geodesic on a closed hyperbolic surface has (geometric) crossing number bounded above by a constant which only depends on the genus.