The limit set for discrete complex hyperbolic groups

Abstract

Given a discrete subgroup Γ of P U (1, n) that acts by isometries on the unit complex ball H n C . In this setting a lot of work has been done in order to understand the action of the group. However, when we look at the action of Γ on all of P n C little or nothing is known. In this paper, we study the action in the whole projective space and we are able to show that its equicontinuity agrees with its Kulkarni discontinuity set. Moreover, in the non-elementary case, this set turns out to be the largest open set on which the group acts properly and discontinuously. It can be described as 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 Λ CG (Γ).

A Family of Complex Kleinian Split Solvable Groups

“…[1] [3]) also every discontinuity region of an infinite complex Kleinian group contains a complex projective line while in the classical theory there are groups with only finitely many points in the complement of the discontinuity region. In dimension N > 2 there are a lot of technical difficulties to determine whether a given discrete subgroup of PSL(N + 1, C) is complex Kleinian, for example the Kulkarni limit set, one of the most important tools in dimension 2, is hard to compute [4,6,5]. In this article we aim to propose a new approach in order to construct a discontinuity region for a family of discrete subgroups of PSL(N + 1, C).…”

On Classical Uniformization Theorems for Higher Dimensional Complex Kleinian Groups

“…By Theorem 1.4, we deduce that G = Γ, up to projective conjugation. On the there hand, by the main theorem in [5], we know that H n C is the largest open set of P n C on which Γ acts properly discontinuously, which is a contradiction. Proof of theorem 0.2.…”

Comparison of Limit Sets for the Action of Kleinian Groups in $\mathbb{C}P^n$

“…The -−lemma describes the accumulation sets on CP 2 for a sequence on a purely loxodromic cyclic subgroup of PSL(3; C) . This lemma has been generalized for a more general type of (divergent) sequences of PSL(n + 1; C); see Lemma [6], where the authors use the KAK-decomposition by bloques. In the particular case of the Veronese groups, we can use the Singular Value Decomposition of divergent sequence in PSL(2; C) to provide a the KAK decomposition of the image, and describe its accumulation subsets on CP n for the action via the irreducible representation.…”

“…Complex Kleinian groups are examples of really rich holomorphic dynamics, and a way to generalize the Kleinian group theory into a (complex) higher dimensional setting. Some examples of complex Kleinian groups are complex hyperbolic groups, i.e., discrete subgroups of the isometries of the complex hyperbolic space Isom(H n C ); see [6,18], for such groups the complex hyperbolic space is part of the open set where the action is properly-discontinuous. Other examples are discrete subgroups of PSL(3; C) whose action on CP 2 is irreducible, see [4], or when the subgroup is virtually solvable, see [2].…”

“…Theorem 1.5 (Theorem 0.1, [6]). Let Γ ⊂ PU(n; 1) be a discrete subgroup; then, the Kulkarni limit set of Γ can be described as the hyperplanes tangent to @H n C at points in the Chen-Greenberg limit set of Γ, that is…”