No abstract

It is well known that the elements of PSL(2, ℂ) are classified as elliptic, parabolic or loxodromic according to the dynamics and their fixed points; these three types are also distinguished by their trace. If we now look at the elements in PU(2,1), then there are the equivalent notions of elliptic, parabolic or loxodromic elements; Goldman classified these transformations by their trace. In this work we extend the classification of elements of PU(2,1) to all of PSL(3, ℂ); we also extend to this setting the theorem that classifies them according to their trace. We use the notion of limit set introduced by Kulkarni, and calculate the limit set of every cyclic subgroup of PSL(3, ℂ) acting on [Formula: see text]. Given a classical Kleinian group it is possible to "suspend" this group to a subgroup of PSL(3, ℂ); we also calculate the limit set of this suspended group.

Let G be a discrete subgroup of PU(2, 1); G acts on P 2 C preserving the unit ball H 2 C , equipped with the Bergman metric. Let L(G) ⊂ S 3 = ∂H 2 C be the limit set of G in the sense of Chen-Greenberg, and let (G) ⊂ P 2 C be the limit set of the G-action on P 2 C in the sense of Kulkarni. We prove that L(G) = (G) ∩ S 3 and (G) is the union of all complex projective lines in P 2 C which are tangent to S 3 at a point in L(G).

No abstract

Given a discret subgroup Γ ⊂ P SL(3, C), we determine the number of complex lines and complex lines in general position lying in the complement of: maximal regions on which Γ acts properly discontinuously, the Kularni's limit set of Γ and the equicontinuity set of Γ. We also provide sufficient conditions to ensure that the equicontinuity region agrees with the Kulkarni's discontinuity region and is the largest set where the group acts properly discontinuously and we provide a description of he respective limit set in terms of the elements of the group.