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

Abstract

In this paper, we give a complete description of the representations of all upper triangular complex Kleinian subgroups of $\text {PSL}\left( 3,\mathbb {C}\right)$. In Toledo-Acosta (Bull Braz Math Soc New Ser 20:1–45, 2021), 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 $\text {PSL}\left( 3,\mathbb {C}\right)$ (Barrera et al. in Linear Algebra Appl 653:430–500, 2022), 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 $\text {PSL}\left( 3,\mathbb {C}\right)$ 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 $\text {PSL}\left( 3,\mathbb {C}\right)$ containing loxodromic elements. These results are a further step towards the completion of the study of elementary discrete subgroups of $\text {PSL}\left( 3,\mathbb {C}\right)$.