Geometrical finiteness for automorphism groups via cone conjecture

TL;DR

This paper proves the geometrical finiteness of automorphism groups acting on hyperbolic spaces for certain algebraic surfaces, showing they are non-positively curved and exploring their geometric properties.

Contribution

It establishes geometrical finiteness for automorphism groups on hyperbolic spaces for K3, Enriques, Coble, and symplectic varieties, linking group actions to curvature properties.

Findings

Automorphism groups are non-positively curved: relatively hyperbolic and CAT(0).

Clarifies the relationship between Kleinian lattices, (-2)-curves, and genus-one fibrations for K3 surfaces.

Proves geometrical finiteness for automorphism groups on hyperbolic spaces.

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: 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.