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

TL;DR

This paper classifies and analyzes purely parabolic discrete subgroups of PSL(3,C), revealing their structure, limit sets, and the conditions under which they are virtually unipotent or Abelian, extending understanding beyond the PSL(2,C) case.

Contribution

It provides a comprehensive classification of purely parabolic discrete groups in PSL(3,C), including their limit sets and structural properties, using advanced theorems and new generalizations.

Findings

Identified five families of purely parabolic discrete groups in PSL(3,C).

Showed all such groups are virtually triangularizable, solvable, and polycyclic.

Classified virtually unipotent groups via obstructor dimension and analyzed their limit sets.

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.