On Discrete Subgroups of automorphism of $P^2_C$

Abstract

We study the geometry and dynamics of discrete subgroups $Γ$ of $\PSL(3,\mathbb{C})$ with an open invariant set $Ω\subset \PC^2$ where the action is properly discontinuous and the quotient $Ω/Γ$ contains a connected component whicis compact. We call such groups {\it quasi-cocompact}. In this case $Ω/Γ$ is a compact complex projective orbifold and $Ω$ is a {\it 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 $Ω/Γ$. 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.