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

Representations of Solvable Subgroups of PSL(3,C)

“…Theorem 2.2 (Theorem 3.1 of [TA19b]). Let G ⊂ GL (3, C) be a discrete solvable subgroup then G is virtually triangularizable.…”

“…Corolary 2.3 (Corollary 4.8 of [TA19b]). Let Γ ⊂ U + be a non-commutative, torsion-free discrete subgroup, then Γ cannot contain a type III complex homothety.…”

“…In this paper we continue the study of discrete solvable subgroups of PSL (3, C). In the first part [TA19b], we studied the dynamics of such groups and proved that solvable groups are virtually triangular and that non-commutative solvable groups can be decomposed in four layers, via the semi-direct product of four types of elements. We also showed that solvable groups 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 of lines passing through a point.…”

“…The paper is organized as follows: In Section 2 we give a brief background necessary for the next sections, we also summarized the needed results from the fist part [TA19b]. In Section 3, we give a full description of the representation of upper triangular discrete triangular subgroups of PSL (3, C), we first deal with the case in which the parabolic part is commutative and then, when it is not commutative.…”

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This paper is the second part of the work started in [TA19b]. In this second part we provide a full description of the representations of all upper triangular complex Kleinian subgroups of PSL (3, C).

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