2008/06/08 by Angel Cano, Cano, Angel, José Seade +1
Mathematics · #37F99 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric and Algebraic Topology #math.DS #msc:37F99
paper · pdf · doi:10.48550/arxiv.0806.1336
openalex publication_date 2008/06/08 · arxiv created 2012/09/06 · arxiv updated 2012/09/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the geometry and dynamics of discrete subgroups Γ of \PSL(3,ℂ) with an open invariant set Ω⊂ \PC2 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.