Effective counting of simple closed geodesics on hyperbolic surfaces

Simple Closed Geodesics in Hyperbolic 3-Manifolds

This paper determines which orientable hyperbolic 3-manifolds contain simple closed geodesics. The Fuchsian group corresponding to the thrice-punctured sphere generates the only example of a complete nonelementary orientable hyperbolic 3-manifold that does not contain a simple closed geodesic. We do not assume that the manifold is geometrically finite or that it has finitely generated fundament...

Growth of the number of simple closed geodesics on hyperbolic surfaces

A norm on homology of surfaces and counting simple geodesics

Let S be a hyperbolic surface of finite volume, and define NS(L) to be the number of simple (that is, without self-intersections) closed geodesics on S whose length does not exceed L. This quantity arises naturally in a number of contexts, and hence it is useful to obtain some estimates on the order of growth of NS(L) as L grows large. Although a number of people have worked on this question, t...

Shortening all the simple closed geodesics on surfaces with boundary

We give a proof of an unpublished result of Thurston showing that given any hyperbolic metric on a surface of finite type with nonempty boundary, there exists another hyperbolic metric on the same surface for which the lengths of all simple closed geodesics have are shorter. Furthermore, we show that we can do the shortening in such a way that it is bounded below by a positive constant. This im...

Closed geodesics on incomplete surfaces

We consider the problem of finding embedded closed geodesics on the two-sphere with an incomplete metric defined outside a point. Various techniques including curve shortening methods are used.