Discrete parabolic groups in [formula omitted

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We study and classify the purely parabolic discrete subgroups of PSL(3,C). This includes all discrete subgroups of the Heisenberg group Heis(3,C). While for PSL(2,C) every purely parabolic subgroup is Abelian and acts on PC1 with limit set a single point, the case of PSL(3,C) is far more subtle and intriguing. We show that there are five families of purely parabolic discrete groups in PSL(3,C), and some of these actually split into subfamilies. We classify all these by means of their limit set and the control group. We use first the Lie-Kolchin Theorem and Borel's fixed point theorem to show that all purely parabolic discrete groups in PSL(3,C) are virtually triangularizable. Then we prove that purely parabolic groups in PSL(3,C) are virtually solvable and polycyclic, hence finitely presented. We then prove a slight generalization of the Lie-Kolchin Theorem for these groups: they are either virtually unipotent or else Abelian of rank 2 and of a very special type. All the virtually unipotent ones turn out to be conjugate to subgroups of the Heisenberg group Heis(3,C). We classify these using the obstructor dimension introduced by Bestvina, Kapovich and Kleiner. We find that their Kulkarni limit set is either a projective line, a cone of lines with base a circle or else the whole PC2. We determine the relation with the Conze-Guivarc'h limit set of the action on the dual projective space PˇC2 and we show that in all cases the Kulkarni region of discontinuity is the largest open set where the group acts properly discontinuously.