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

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.