Hyperbolicity for large automorphism groups of projective surfaces

TL;DR

This paper investigates the hyperbolic dynamics of large automorphism groups acting on complex surfaces like K3 and Enriques, revealing conditions for expansion, orbit structure, and ergodic properties.

Contribution

It provides new criteria for uniform expansion and analyzes the dynamics of automorphism groups with parabolic elements on complex surfaces.

Findings

Invariant measures have non-zero Lyapunov exponents when parabolic elements are present.

Criteria for uniform expansion on the entire surface are established.

Results have implications for orbit closures, equidistribution, and ergodicity.

Abstract

We study the hyperbolicity properties of the action of a non-elementary automorphism group on a compact complex surface, with an emphasis on K3 and Enriques surfaces. A first result is that when such a group contains parabolic elements, Zariski diffuse invariant measures automatically have non-zero Lyapunov exponents. In combination with our previous work, this leads to simple criteria for a uniform expansion property on the whole surface, for groups with and without parabolic elements. This, in turn, has strong consequences on the dynamics: description of orbit closures, equidistribution, ergodicity properties, etc. Along the way, we provide a reference discussion on uniform expansion of non-linear discrete group actions on compact (real) manifolds and the construction of Margulis functions under optimal moment conditions.