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.
In this article we provide an algebraic characterization of those groups of $PSL(3,Bbb{C})$ whose limit set in the Kulkarni sense has, exactly, four lines in general position. Also we show that, for this class of groups, the equicontinuity set of the
group is the largest open set where the group acts discontinuously and agrees with the discontinuity set of the group.
