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 PSL(3,C). This groups have simple dynamics, contrary to strongly irreducible groups, and because of this, we propose to define elementary subgroups of PSL(3,C) 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 needed to complete the study of elementary subgroups of PSL(3,C).

This paper was published in Bulletin of the Brazilian Mathematics Society, New Series (2021). https://doi.org/10.1007/s00574-021-00254-9. A full version can be found here.