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Discreteness criteria for subgroups in complex hyperbolic space. (English) Zbl 1009.32016
The authors study discreteness criteria for subgroups of $U(1,n; \bbfC)$ in complex hyperbolic space $H_n^\bbfC$. The main result in this paper is as follows: A nonelementary subgroup $G$ of $U(1,n;\bbfC)$ in complex hyperbolic space $H_n^\bbfC$ 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 $\partial H_n^\bbfC$ and $g_j\to I$ as $j\to\infty$. They also prove that if a nonelementary subgroup $G$ of $U(1,n;\bbfC)$ contains a sequence $\{g_j\}$ of distinct elements with $\text{Card (fix}(g_j)\cap \partial H_n^\bbfC) \ne\infty$ and $g_j \to I$ as $j\to\infty$, then $G$ contains a non-discrete, nonelementary two generator subgroup.

MSC:
32Q45Hyperbolic and Kobayashi hyperbolic manifolds
32M99Complex spaces with a group of automorphisms
22E99Lie groups
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References:
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