Automorphisms of hyperk\"ahler manifolds and groups acting on CAT(0) spaces

We study groups of bimeromorphic and biholomorphic automorphisms of projective hyperk"ahler manifolds. Using an action of these groups on some non-positively curved space, we deduce many of their properties, including finite presentation, strong form of Tits’ alternative and some structural results about groups consisting of transformations with infinite order.