Remarks on points of approximation of discrete subgroups of \(U(1,n;\mathbb{C})\). (English) Zbl 0814.30026

Let \(G\) be a discrete subgroup of \(U(1,n; \mathbb{C})\) acting on the complex unit ball \(B^ n\). Let \(\zeta= (\zeta_ 1, \zeta_ 2,\dots, \zeta_ n)\) be a limit point of \(G\). If there exists a sequence \(\{g_ k\}\) of distinct elements of \(G\) and a region \(D_ \alpha (\zeta)= \{z= (z_ 1, z_ 2,\dots, z_ n)\in B^ n\mid | 1- \sum_{i=1}^ n \overline {z}_ i \zeta_ i|< \alpha(1- \sum_{i=1}^ n | z_ i|^ 2 )\}\) such that \(g_ k(0)\in D_ \alpha (\zeta)\) and \(g_ k(0)\to \zeta\), then the point \(\zeta\) is called a point of approximation of \(G\). We show that a point of approximation of \(G\) has some similar properties as in Kleinian groups. But in the case where \(n\geq 2\), an approach to a point of approximation is not necessarily non-tangential. We give an example of a point of approximation to which some orbit converges in the tangential direction.
Reviewer: S.Kamiya (Okayama)


30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)
22E40 Discrete subgroups of Lie groups
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