Representations of solvable subgroups of \(\mathrm{PSL}\left( 3,\mathbb{C}\right)\) (Q6063021)

The basic subject of the present paper are complex Kleinian groups in dimension 2, i.e., discrete subgroups of \(\mathrm{PSL}(3, \mathbb{C})\) acting on \(\mathbb{CP}^2\), and in particular ``elementary subgroups'' with a simple dynamics. One possibility to define ``elementary'' here is solvability, and such solvable subgroups of \(\mathrm{PSL}(3,\mathbb{C})\) are considered in a previous paper by the author [Bull. Braz. Math. Soc. (N.S.) 53, No. 1, 127--171 (2022; Zbl 1494.37029)], where he shows that any solvable subgroup is virtually triangularizable and can be constructed as the semidirect product of two layers of parabolic elements and two layers of loxodromic elements. In the present sequal to that paper, the author provides a full description of the representations of solvable subgroups of \(\mathrm{PSL}(3,\mathbb{C})\), up to finite index subgroups, building upon work done by \textit{W. Barrera} et al. [Linear Algebra Appl. 653, 430--500 (2022; Zbl 1506.37067)], where the authors prove that, up to conjugation, there are five families of purely parabolic discrete subgroups of \(\mathrm{PSL}(3,\mathbb{C})\). ``In the present paper we study which loxodromic elements can be combined with the elements of the parabolic part of an upper triangular discrete subgroup of \(\mathrm{PSL}(3,\mathbb{C})\). We show that, up to conjugation, there are 16 types of upper triangular complex Kleinian groups in \(\mathrm{PSL}(3,\mathbb{C})\) containing loxodromic elements. These results are a further step towards the completion of the study of elementary discrete subgroups of \(\mathrm{PSL}(3, \mathbb{C})\).''