Geometrical finiteness for automorphism groups via cone conjecture

Abstract: This paper aims to establish the geometrical finiteness for the natural isometric actions of (birational) automorphism groups on the hyperbolic spaces for K3 surfaces, Enriques surfaces, Coble surfaces, and irreducible symplectic varieties. As an application, it follows that such groups are non-positively curved: ${\rm CAT(0)}$ and relatively hyperbolic. In the case of K3 surfaces, we provide a dynamical characterization of relative hyperbolicity, and clarify the relationship between convex-cocompactness and genus one fibrations.