Geodesic intersections in arithmetic hyperbolic 3-manifolds

Abstract

It was shown by Chinburg and Reid that there exist closed hyperbolic 3-manifolds in which all closed geodesics are simple. Subsequently, Basmajian and Wolpert showed that almost all quasi-Fuchsian 3-manifolds have all closed geodesics simple and disjoint. The natural conjecture arose that the Chinburg-Reid examples also had disjoint geodesics. Here we show that this conjecture is both almost true (they have no geodesics that intersect except at right angles) and spectacularly false (any pair of closed geodesics admits infinitely many closed geodesics which intersects both geodesics of the pair perpendicularly). The latter statement is shown to be true for all closed arithmetic hyperbolic 3-manifolds. Section 0 Introduction By a hyperbolic n-manifold we shall mean a complete orientable n-dimensional Riemannian manifold all of whose sectional curvatures are −1. If M is a hyperbolic 3-manifold, the universal cover of M can be identified with H, the upper half-space model of hyperbolic 3-space, andM is realized asH/Γ for some Γ a discrete torsionfree subgroup of Isom(H3). Now Isom(H3) can be identified with PSL(2,C) ( which in turn is isomorphic to PGL(2,C)), and Γ is called a Kleinian group. In the sequel we will only be interested in the case when M is closed, in which case Γ is referred to as cocompact. Of interest to us is the structure of the set of closed geodesics in closed hyperbolic 3-manifolds. The motivation comes from the following. A closed geodesic in a (closed) hyperbolic n-manifold is simple if it has no self-intersections, and nonsimple otherwise. In dimension 2 every closed hyperbolic manifold has a non-simple closed geodesic. However in dimension 3 the situation is much more complex. Many closed hyperbolic 3-manifolds contain immersions of totally geodesic surfaces and so there are non-simple closed geodesics. In [JR] examples were given of closed hyperbolic 3-manifolds containing a non-simple closed geodesic but having no immersed totally geodesic surface. However it was shown in [CR] that there exist closed hyperbolic 3-manifolds in which all closed geodesics are simple. Subsequently, Basmajian and Wolpert [BW] showed that almost all 3-manifolds arising as the quotient of H 1991 Mathematics Subject Classification. Primary: 57M50, Secondary: 30F40.