Closed geodesics in hyperbolic manifolds

Hyperbolic surfaces are well known to have many
self-intersecting closed geodesics. In fact, in a compact hyperbolic
surface, while there are (countably) infinitely many closed geodesics,
both simple and non-simple, one can quantify the fact that most of them
are non-simple (see e.g. Selberg, Huber, Margulis, Mirzakhani). In
higher dimensions the situation is more mysterious. On one hand, any
hyperbolic manifold containing a totally geodesic surface will contain
all the geodesics of that surface. On the other hand, Chinburg and Reid
constructed in the early 1990's infinitely many compact hyperbolic
3-manifolds, all of whose closed geodesics are simple. In this
introductory talk we will review some of the highlights of the surface
case, discuss some of the tools involved in the Chinburg-Reid
construction, and discuss further related questions and directions.