  Let M be a closed 3-manifold that supports a partially hyperbolic
diffeomorphism f. If $\pi_1(M)$ is nilpotent, the induced action of f on
$H_1(M, R)$ is partially hyperbolic. If $\pi_1(M)$ is almost nilpotent or if
$\pi_1(M)$ has subexponential growth, M is finitely covered by a circle bundle
over the torus. If $\pi_1(M)$ is almost solvable, M is finitely covered by a
torus bundle over the circle. Furthermore, there exist infinitely many
hyperbolic 3-manifolds that do not support dynamically coherent partially
hyperbolic diffeomorphisms; this list includes the Weeks manifold.
  If f is a strong partially hyperbolic diffeomorphism on a closed 3-manifold M
and if $\pi_1(M)$ is nilpotent, then the lifts of the stable and unstable
foliations are quasi-isometric in the universal of M. It then follows that f is
dynamically coherent.
  We also provide a sufficient condition for dynamical coherence in any
dimension. If f is center bunched and if the center-stable and center-unstable
distributions are Lipschitz, then the partially hyperbolic diffeomorphism f
must be dynamically coherent.
