The Dynamics of Solvable Subgroups of PSL(3,C)

TL;DR

This paper characterizes the dynamics of solvable subgroups of PSL(3,C), revealing their layered structure and proper discontinuous actions, advancing the understanding of elementary complex Kleinian groups.

Contribution

It provides a full description of solvable complex Kleinian subgroups of PSL(3,C), including their decomposition and action properties, up to finite index subgroups.

Findings

Triangular groups decompose into four layers with specific element types.

Solvable groups act properly discontinuously on certain geometric complements.

The results advance the classification of elementary subgroups in PSL(3,C).

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)