The limit set of discrete subgroups of $PSL(3,\C)$

TL;DR

This paper characterizes the equicontinuity and discontinuity regions of discrete subgroups of PSL(3,C) acting on complex projective plane, linking these regions to the limit set and providing conditions for maximality and hyperbolicity.

Contribution

It establishes a detailed description of the equicontinuity region for such groups, relating it to Kulkarni's limit set and identifying conditions for maximal discontinuity domains and hyperbolic components.

Findings

Equicontinuity region equals the Kulkarni discontinuity set under certain conditions.

Connected components of the equicontinuity region are holomorphy domains.

Under specific hypotheses, the equicontinuity region is the largest discontinuous action domain.

Abstract

If $\Gamma$ is a discrete subgroup of $PSL(3,\Bbb{C})$, it is determined the equicontinuity region $Eq(\Gamma)$ of the natural action of $\Gamma$ on $\Bbb{P}^2_\Bbb{C}$. It is also proved that the action restricted to $Eq(\Gamma)$ is discontinuous, and $Eq(\Gamma)$ agrees with the discontinuity set in the sense of Kulkarni whenever the limit set of $\Gamma$ in the sense of Kulkarni, $\Lambda(\Gamma)$, contains at least three lines in general position. Under some additional hypothesis, it turns out to be the largest open set on which $\Gamma$ acts discontinuously. Moreover, if $\Lambda(\Gamma)$ contains at least four complex lines and $\Gamma$ acts on $\Bbb{P}^2_\Bbb{C}$ without fixed points nor invariant lines, then each connected component of $Eq(\Gamma)$ is a holomorphy domain and a complete Kobayashi hyperbolic space.