  Let $M$ be a simple holomorphically symplectic manifold, that is, a simply
connected holomorphically symplectic manifold of Kahler type with $h^{2,0}=1$.
We prove that the group of holomorphic automorphisms of $M$ acts on the set of
faces of its Kahler cone with finitely many orbits, whenever $b_2(M)\neq 5$.
This is a version of the Morrison-Kawamata cone conjecture for hyperkahler
manifolds. The proof is based on the following observation, proven with ergodic
theory. Let $M$ be a complete Riemannian orbifold of dimension at least three,
constant negative curvature and finite volume, and $\{S_i\}$ an infinite set of
locally geodesic hypersurfaces. Then the union of $S_i$ is dense in $M$.
