DIVERGENCE IN 3-MANIFOLD GROUPS

The divergence of the fundamental group of compact irreducible 3-manif olds satisfying Thurston's geometrization conjecture is calculated. Fo r every closed Haken 3-manifold group, the divergence is either linear , quadratic or exponential, where quadratic divergence occurs precisel y for graph manifolds and exponential divergence occurs when a geometr ic piece has hyperbolic geometry. An example is given of a closed 3-ma nifold N with a Riemannian metric of nonpositive curvature such that t he divergence is quadratic and such that there are two geodesic rays i n the universal cover (N) over tilde whose divergence is precisely qua dratic, settling in the negative a question of Gromov's.