a projective manifold is algebraically hyperbolic if the degree of any curve is bounded from above by its genus times a constant, which is independent from the curve. this is a property which follows from kobayashi hyperbolicity. we prove that hyperkahler manifolds are not algebraically hyperbolic when the picard rank is at least 3, or if the picard rank is 2 and the syz conjecture on existence of lagrangian fibrations is true. we also prove that if the automorphism group of a hyperkahler manifold is infinite, then it is algebraically non-hyperbolic. \\ \\ these results are a joint work with misha verbitsky.