On Discrete groups of automorphisms of $$\mathbb P ^2_\mathbb{C }$$

Abstract

We study the geometry and dynamics of discrete subgroups \(\Gamma \) of \(\mathrm{PSL}(3,\mathbb C )\) with an open invariant set \(\Omega \subset \mathbb P _\mathbb{C }^2\) where the action is properly discontinuous and the quotient \(\Omega /\Gamma \) contains a connected component whicis compact. We call such groups quasi-cocompact. In this case \(\Omega /\Gamma \) is a compact complex projective orbifold and \(\Omega \) is a divisible set. Our first theorem refines classical work by Kobayashi–Ochiai and others about complex surfaces with a projective structure: We prove that every such group is either virtually affine or complex hyperbolic. We then classify the divisible sets that appear in this way, the corresponding quasi-cocompact groups and the orbifolds \(\Omega /\Gamma \). We also prove that excluding a few exceptional cases, the Kulkarni region of discontinuity coincides with the equicontinuity region and is the largest open invariant set where the action is properly discontinuous.