Intersections of Closed Geodesics in Hyperbolic 3-manifolds
by
Kerry N. Jones
Ball State University
Coauthors: Alan W. Reid

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.

It is still an open question whether or not a closed hyperbolic 3-manifold exists in which all closed geodesics are simple and disjoint (even though one might reasonably conjecture that almost all closed hyperbolic 3-manifolds have this property), but this work shows that such a manifold, if it exists, cannot be arithmetic or commensurable with a 2-generator manifold.

Date received: February 6, 1998