Tits alternative in hypek\"ahler manifolds

We show an analogous result of the famous Tits alternative for a group G of birational automorphisms of a projective hyperk"aher manifold: Either G contains a non-commutative free group or G is an almost abelian group of finite rank. As an application, we show that the automorphism groups of the so-called singular K3 surfaces contain non-commutative free groups.