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Discreteness criteria for subgroups in complex hyperbolic space. (English) Zbl 1009.32016

The authors study discreteness criteria for subgroups of \(U(1,n; \mathbb{C})\) in complex hyperbolic space \(H_n^\mathbb{C}\).
The main result in this paper is as follows: A nonelementary subgroup \(G\) of \(U(1,n;\mathbb{C})\) in complex hyperbolic space \(H_n^\mathbb{C}\) with Condition A is discrete if and only if every two generator subgroup is discrete.
Here we say that \(G\) satisfies Condition A provided that \(G\) has no sequence \(\{g_j\}\) of distinct elements of finite order such that \(\text{Card (fix}(g_j))= \infty\) in \(\partial H_n^\mathbb{C}\) and \(g_j\to I\) as \(j\to\infty\).
They also prove that if a nonelementary subgroup \(G\) of \(U(1,n;\mathbb{C})\) contains a sequence \(\{g_j\}\) of distinct elements with \(\text{Card (fix}(g_j)\cap \partial H_n^\mathbb{C}) \neq\infty\) and \(g_j \to I\) as \(j\to\infty\), then \(G\) contains a non-discrete, nonelementary two generator subgroup.

MSC:

32Q45 Hyperbolic and Kobayashi hyperbolic manifolds
32M99 Complex spaces with a group of automorphisms
22E99 Lie groups
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