<p>We prove that, except for two explicit families, every discrete subgroup <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$G \subset \textrm{PSL}(3,\mathbb {C})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>G</mi> <mo>⊂</mo> <mtext>PSL</mtext> <mo stretchy="false">(</mo> <mn>3</mn> <mo>,</mo> <mi mathvariant="double-struck">C</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> that acts properly discontinuously on a non-empty open subset of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathbb{C}\mathbb{P}^2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="double-struck">C</mi> <msup> <mrow> <mi mathvariant="double-struck">P</mi> </mrow> <mn>2</mn> </msup> </mrow> </math></EquationSource> </InlineEquation> admits a unique largest <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$G$</EquationSource> <EquationSource Format="MATHML"><math> <mi>G</mi> </math></EquationSource> </InlineEquation>-invariant open set <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\Omega _G \subset \mathbb{C}\mathbb{P}^2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="normal">Ω</mi> <mi>G</mi> </msub> <mo>⊂</mo> <mi mathvariant="double-struck">C</mi> <msup> <mrow> <mi mathvariant="double-struck">P</mi> </mrow> <mn>2</mn> </msup> </mrow> </math></EquationSource> </InlineEquation> on which the action is properly discontinuous. We refer to <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\Omega _G$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="normal">Ω</mi> <mi>G</mi> </msub> </math></EquationSource> </InlineEquation> as the <i>regular set</i> of <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$G$</EquationSource> <EquationSource Format="MATHML"><math> <mi>G</mi> </math></EquationSource> </InlineEquation>. We also establish several geometric and analytic properties of <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\Omega _G$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="normal">Ω</mi> <mi>G</mi> </msub> </math></EquationSource> </InlineEquation> and compare its complement with other natural limit sets associated to the action.</p>

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

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2026年5月21日

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Angel Cano,
Laura A. Cano Cordero,
Luis Loeza

摘要

We prove that, except for two explicit families, every discrete subgroup $G \subset \textrm{PSL}(3,\mathbb {C})$ $G \subset PSL (3, C)$ that acts properly discontinuously on a non-empty open subset of $\mathbb{C}\mathbb{P}^2$ $C P^{2}$ admits a unique largest $G$ $G$ -invariant open set $\Omega _G \subset \mathbb{C}\mathbb{P}^2$ $Ω_{G} \subset C P^{2}$ on which the action is properly discontinuous. We refer to $\Omega _G$ $Ω_{G}$ as the regular set of $G$ $G$ . We also establish several geometric and analytic properties of $\Omega _G$ $Ω_{G}$ and compare its complement with other natural limit sets associated to the action.

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

摘要