Abstract. Let G be a discrete subgroup of P U (1, n). Then G acts on P n C preserving the unit ball H n C , where it acts by isometries with respect to the Bergman metric. In this work we determine the equicontinuty region Eq(G) of G in P n C : It is the complement of the union of all complex projective hyperplanes in P n C which are tangent to ∂H n C at points in the Chen-Greenberg limit set ΛCG(G), a closed G-invariant subset of ∂H n C , which is minimal for non-elementary groups. We also prove that the action on Eq(G) is discontinuous.
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.
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