Representations of Solvable Subgroups of PSL(3,C)

TL;DR

This paper classifies and describes all upper triangular complex Kleinian subgroups of PSL(3,C), detailing how loxodromic elements can be combined with parabolic elements within these groups.

Contribution

It provides a complete classification of upper triangular Kleinian groups in PSL(3,C) with loxodromic elements, extending the understanding of their structure and restrictions.

Findings

16 types of upper triangular groups with loxodromic elements identified

Restrictions on loxodromic elements imposed by parabolic parts

Advances towards classifying elementary discrete subgroups of PSL(3,C)

Abstract

In this paper, we give a complete description of the representations of all upper triangular complex Kleinian subgroups of PSL(3,C). In https://doi.org/10.1007/s00574-021-00254-9 we show that any solvable group is virtually triangularizable and can be constructed as the semidirect product of two layers of parabolic elements and two layers of loxodromic elements. There are five families of purely parabolic discrete groups of PSL(3,C) https://doi.org/10.1016/j.laa.2022.08.027, therefore, the parabolic part of any upper triangular group belongs to one of these five families. In this paper we study which loxodromic elements can be combined with the elements of the parabolic part of an upper triangular discrete subgroup of PSL(3,C) in each of these five cases. These parabolic elements impose strong restrictions on the type, and number, of loxodromic elements that can be present in the group. We show that up to conjugation, there are 16 types of upper triangular complex Kleinian groups in PSL(3,C) containing loxodromic elements. These results are a further step towards the completion of the study of elementary discrete subgroups of PSL(3,C).