Regular Sets for Discrete Projective Group Actions on $$\mathbb{C}\mathbb{P}^2$$

Abstract We prove that, except for two explicit families, every discrete subgroup $$G \subset \textrm{PSL}(3,\mathbb {C})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>G</mml:mi> <mml:mo>⊂</mml:mo> <mml:mtext>PSL</mml:mtext> <mml:mo>(</mml:mo> <mml:mn>3</mml:mn> <mml:mo>,</mml:mo> <mml:mi>C</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> that acts properly discontinuously on a non-empty open subset of $$\mathbb{C}\mathbb{P}^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>C</mml:mi> <mml:msup> <mml:mrow> <mml:mi>P</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:math> admits a unique largest $$G$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>G</mml:mi> </mml:math> -invariant open set $$\Omega _G \subset \mathbb{C}\mathbb{P}^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>Ω</mml:mi> <mml:mi>G</mml:mi> </mml:msub> <mml:mo>⊂</mml:mo> <mml:mi>C</mml:mi> <mml:msup> <mml:mrow> <mml:mi>P</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:math> on which the action is properly discontinuous. We refer to $$\Omega _G$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Ω</mml:mi> <mml:mi>G</mml:mi> </mml:msub> </mml:math> as the regular set of $$G$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>G</mml:mi> </mml:math> . We also establish several geometric and analytic properties of $$\Omega _G$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Ω</mml:mi> <mml:mi>G</mml:mi> </mml:msub> </mml:math> and compare its complement with other natural limit sets associated to the action.