In this work we study and provide a full description, up to a finite index subgroup, of the dynamics of solvable complex Kleinian subgroups of \(\text {PSL}\left( 3,\mathbb {C}\right) \). These groups have simple dynamics, contrary to strongly irreducible groups. Because of this, we propose to define elementary subgroups of \(\text {PSL}\left( 3,\mathbb {C}\right) \) as solvable groups. We show that triangular groups can be decomposed in four layers, via the semi-direct product of four types of elements, with parabolic elements in the inner most layers and loxodromic elements in the outer layers. It is also shown that solvable groups, up to a finite index subgroup, act properly and discontinuously on the complement of either a line, two lines, a line and a point outside of the line, or a pencil of lines passing through a point. These results are another step towards the completion of the study of elementary subgroups of \(\text {PSL}\left( 3,\mathbb {C}\right) \).

在这项工作中，我们研究并提供了可解决的复杂Kleinian子组\（\ text {PSL} \ left（3，\ mathbb {C} \ right）\）的动力学的完整描述，直到有限索引子组为止。这些组具有简单的动力学，与强烈不可还原的组相反。因此，我们建议定义\（\ text {PSL} \ left（3，\ mathbb {C} \ right）\）的基本子组作为可解决的群体。我们显示，三角形组可以通过四种元素的半直接乘积分解为四层，最里面的层是抛物线型元素，最外面的层是氧碘化物元素。还显示出，直到有限索引子组的可解组，在一条直线，两条直线，一条直线和一条直线外的点或一条穿过点的直线笔的补码上正确且不连续地起作用。这些结果是完成\（\ text {PSL} \ left（3，\ mathbb {C} \ right）\）基本子组研究的又一步。