Let M be a compact holomorphically symplectic manifold and K its Kahler cone. Morrison-Kawamata cone conjecture says that the automorphism group of M acts on polyhedral faces of K with finitely many orbits. I would explain the proof of this result (obtained jointly with Ekaterina Amerik), based on ergodic theory and hyperbolic geometry. It turns out that the Morrison-Kawamata cone conjecture can be interpreted as a result of hyperbolic geometry: the quotient of the projectivization of rational positive cone of M by the group of Hodge isometries is a hyperbolic manifold H of finite volume, and the ample cone of M corresponds to a finite polyhedron in H with piecewisely geodesic boundary. As an application, we obtain that M has only finitely many holomorphically symplectic birational models.