The Dynamics of Solvable Subgroups of $$\text {PSL}\left( 3,\mathbb {C}\right) $$

Abstract

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). These groups havesimpledynamics, contrary to strongly irre-ducible groups. Because of this, we propose to define elementary subgroups of PSL(3,C) as solvable groups. We show that triangular groups can be de-composed in four layers, via the semi-direct product of fourtypes of elements,with parabolic elements in the inner most layers and loxodromic elements inthe outer layers. It is also shown that solvable groups, up toa finite index sub-group, 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 oflines passingthrough a point. These results are another step towards the completion of thestudy of elementary subgroups of PSL(3,C)