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

TL;DR

This paper investigates the structure and classification of discrete subgroups of a7(3,\u211d) acting on complex projective plane, identifying conditions under which their quotient spaces form compact orbifolds and describing their geometric properties.

Contribution

It classifies quasi-cocompact subgroups of a7(3,a7) and characterizes their associated divisible sets and orbifolds, extending classical results on complex surfaces with projective structures.

Findings

Every such group is either virtually affine or complex hyperbolic.

Classified the divisible sets and corresponding orbifolds.

Established the coincidence of Kulkarni region and equicontinuity region in most cases.

Abstract

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