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

Abstract

If $\Gamma$ is a discrete subgroup of $PSL(3,\mathbb{C})$, it is determined the equicontinuity region $Eq(\Gamma )$ of the natural action of $\Gamma$ on ${\mathbb{P}}_{\mathbb{C}}^2$. 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 ${\mathbb{P}}_{\mathbb{C}}^2$ without fixed points nor invariant lines, then each connected component of $Eq(\Gamma )$ is a holomorphy domain and a complete Kobayashi hyperbolic space.