arXiv:1802.08360v1 [math.DS] 23 Feb 2018 TOWARDS A SULLIVAN DICTIONARY IN DIMENSION TWO, PART I: PURELY PARABOLIC COMPLEX KLEINIAN GROUPS WALDEMAR BARRERA, ANGEL CANO, JUAN PABLO NAVARRETE, AND JOS ´ E SEADE Abstract. In this article we provide a full description of all the complex kleinian groups of PSL(3, C) which contains only parabolic elements. Introduction In order to understand the structure of the limit set for groups of PSL(3, C) it is very important to understand the those subgroups of PSL(3, C) without loxo- dromic elements. The main purpose of this note is to provide a description of the representation of such groups as well as a precise description of its limit set. More precisely, in this article we show: Theorem 0.1. Let Γ 0 ⊂ PSL(3, C) be a complex Kleinian group without loxodromic elements, then there exist a subgroup Γ ⊂ Γ 0 of finite index such that Γ is conjugate to one of the following groups (1) The group: W µ =      µ(w) µ(w)w 0 0 µ(w) 0 0 0 µ(w) −2   : w ∈ W    . where W ⊂ C is a discrete additive subgroup and µ : W → (C, +) is a group morphism. (2) The group W =      1 0 x 0 1 y 0 0 1   :(x, y) ∈ Span Z (W )    . where W ⊂ C 2 is a set of R-linearly independent points. (3) The group WR L =      1 x L(x)+ x 2 /2+ w 0 1 x 0 0 1   : x ∈ R, w ∈ W    . where W ⊂ C is an additive discrete subgroup, R ⊂ C is an additive group and L : R → C is an additive function, subject to the following conditions (a) if R is discrete, then rank(W )+ rank(R) ≤ 4, 1991 Mathematics Subject Classification. Primary 37F99, Secondary 30F40, 20H10, 57M60. Partially supported by grants of project PAPPIT UNAM IN101816. 1