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• W3120362310 abstract "We study and classify the purely parabolic discrete subgroups of ${rm PSL}(3,Bbb{C})$. This includes all discrete subgroups of the Heisenberg group ${rm Heis}(3,Bbb{C})$. While for ${rm PSL}(2,Bbb{C})$ every purely parabolic subgroup is Abelian and acts on $Bbb{P}^1_{Bbb{C}}$ with limit set a single point, the case of ${rm PSL}(3,Bbb{C})$ is far more subtle and intriguing. We show that there are twelve classes of purely parabolic discrete groups in ${rm PSL}(3,Bbb{C})$, and we classify them. We use first the Tits Alternative and Borel's fixed point theorem to show that all purely parabolic discrete groups in ${rm PSL}(3,Bbb{C})$ are virtually triangularizable; this extends a Theorem by Lie-Kolchin. Then we prove that purely parabolic groups in ${rm PSL}(3,Bbb{C})$ are virtually solvable and cocyclic, hence finitely presented. We then prove a Tits-inspired alternative 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 an Euclidean circle, or else the whole $Bbb{P}^2_{Bbb{c}}$. We determine its relation with the Conze-Givarc'h limit set of the action on the dual projective space $(Bbb{P}^2)^*$. We show that in all cases the Kulkarni region of discontinuity is the largest open set where the group acts properly discontinuously, and that in all cases but one, this set coincides with the equicontinuity region." @default.
• W3120362310 created "2021-01-18" @default.
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• W3120362310 date "2018-02-23" @default.
• W3120362310 modified "2023-09-27" @default.
• W3120362310 title "Discrete subgroups of the Heisenberg group ${rm Heis}(3,Bbb{C})$ and purely parabolic groups in ${rm PSL}(3, Bbb{C})$" @default.
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