Geometrical finiteness for automorphism groups via cone conjecture

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: relatively hyperbolic and ${\rm CAT(0)}$. In the case of K3 surfaces, we clarify the relationship between Kleinian lattices and $(-2)$-curves, and between convex-cocompact Kleinian groups and genus-one fibrations.