Groups of Automorphisms of Null-entropy of Hyperkähler Manifolds

The following two results are proven: The full automorphism group of any non-projective hyperkähler manifold M is almost abelian of rank at most maximum of ρ(M) − 1 and 1. Any groups of automorphisms of nullentropy of a projective hyperkähler manifold M is almost abelian of rank at most ρ(M) − 2. A few applications for K3 surfaces are also given.