Non-simple Geodesics in Hyperbolic 3-manifolds

Chinburg and Reid have recently constructed examples of hyperbolic 3manifolds in which every closed geodesic is simple. These examples are constructed in a highly non-generic way and it is of interest to understand in the general case the geometry of and structure of the set of closed geodesics in hyperbolic 3-manifolds. For hyperbolic 3-manifolds which contain an immersed totally geodesic surfaces there are always non-simple closed geodesics. Here we construct examples of manifolds with non-simple closed geodesics and no totally geodesic surfaces. Section 0 Introduction By a hyperbolic 3-manifold (resp. orbifold) we shall always mean a complete orientable hyperbolic 3-manifold (resp. orbifold) of finite volume. Let M be a hyperbolic 3-manifold, a closed geodesic in M is called simple if it has no selfintersections, and non-simple otherwise. As was recently shown in Chinburg and Reid [3], there are infinitely many commensurability classes of closed hyperbolic 3-manifolds all of whose closed geodesics are simple. It would seem, at least at the intuitive level, that this should be the “generic case” for hyperbolic 3-manifolds. However the examples in [3] are arithmetic, and their construction arises by interpreting the existence of a non-simple geodesic in terms of a quaternion algebra that is naturally associated to these groups. These examples are far from being understood in terms of hyperbolic geometry. On the other hand many hyperbolic 3-manifolds contain immersions of a totally geodesic surface, and therefore must contain non-simple closed geodesics, as any hyperbolic 2-manifold of finite volume contains such closed geodesics. The focus of this paper is manifolds “that arise between these two extremes.” Our main result shows that there are infinitely many commensurability classes of closed hyperbolic 3-manifolds that contain a non-simple closed geodesic, but have no immersed totally geodesic surface. Again the methods appeal to arithmetic hyperbolic 3-manifolds. This is not surprising, since at present the only methods known to show that a hyperbolic 3-manifold cannot contain an immersed totally geodesic surface are also arithmetic in nature, cf. Maclachlan and Reid [9] and Reid [11] and [13]. 1991 Mathematics Subject Classification. Primary: 57M50, Secondary: 30F40.