Discrete parabolic groups in ${\rm PSL}(3, \Bbb{C})$

Abstract

We study and classify the purely parabolic discrete subgroups of $PSL(3,\Bbb{C})$. This includes all discrete subgroups of the Heisenberg group ${\rm Heis}(3,\Bbb{C})$. While for $PSL(2,\Bbb{C})$ every purely parabolic subgroup is Abelian and acts on $\Bbb{P}^1_\Bbb{C}$ with limit set a single point, the case of $PSL(3,\Bbb{C})$ is far more subtle and intriguing. We show that there are five families of purely parabolic discrete groups in $PSL(3,\Bbb{C})$, 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 $PSL(3,\Bbb{C})$ are virtually triangularizable. Then we prove that purely parabolic groups in $PSL(3,\Bbb{C})$ 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 ${\rm Heis}(3,\Bbb{C})$. 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 $\Bbb{P}^2_\Bbb{C}$. We determine the relation with the Conze-Guivarc'h limit set of the action on the dual projective space $\check{\Bbb{P}}^2_\Bbb{C}$ and we show that in all cases the Kulkarni region of discontinuity is the largest open set where the group acts properly discontinuously.