Two problems on closed geodesics in hyperbolic 3 manifolds

This thesis is about two questions related to hyperbolic 3-manifolds.;The first question arises as an extension of a result of Adams and Reid [2]. Adams and Reid proved that the length of a shortest closed geodesic in a hyperbolic knot or link complement in a closed 3-manifold which does not admit any Riemannian metric of negative curvature, for example S 3, is bounded above by 7.171646.. (independent of the hyperbolic knot or link). As an extension, one can ask if there is an upper bound for the length of an nth shortest closed geodesic in a hyperbolic knot or link complement in a closed 3-manifold which does not admit any Riemannian metric of negative curvature (the bound being independent of the hyperbolic knot or link). Using techniques similar to the ones used by Adams and Reid, we answer this question in an affirmative. We produce an explicit upper bound for the length of an nth shortest closed geodesic in a hyperbolic knot or link complement in a closed 3-manifold which does not admit any Riemannian metric of negative curvature as a function of the length rank n.;The second question is about the presence of infinitely many simple closed geodesics in hyperbolic 3-manifolds. Adams, Hass and Scott [1] showed that every hyperbolic 3-manifold has a simple closed geodesic. Kuhlmann [7] showed that every finite volume cusped hyperbolic 3-manifold has infinitely many simple closed geodesics. Kuhlmann [8] also showed that a closed hyperbolic 3-manifold which satisfies certain algebraic and geometric conditions also contains infinitely many simple closed geodesics. But the question of exactly which closed hyperbolic 3-manifolds contain infinitely many simple closed geodesics is still unresolved. We show that if a hyperbolic 3-manifold satisfies certain geometric conditions, then it contains infinitely many simple closed geodesics.