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

The authors study discreteness criteria for subgroups of $U\left(1,n;ℂ\right)$ in complex hyperbolic space ${H}_{n}^{ℂ}$.

The main result in this paper is as follows: A nonelementary subgroup $G$ of $U\left(1,n;ℂ\right)$ in complex hyperbolic space ${H}_{n}^{ℂ}$ 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 $\left\{{g}_{j}\right\}$ of distinct elements of finite order such that $\text{Card}\phantom{\rule{4.pt}{0ex}}\text{(fix}\left({g}_{j}\right)\right)=\infty$ in $\partial {H}_{n}^{ℂ}$ and ${g}_{j}\to I$ as $j\to \infty$.

They also prove that if a nonelementary subgroup $G$ of $U\left(1,n;ℂ\right)$ contains a sequence $\left\{{g}_{j}\right\}$ of distinct elements with $\text{Card}\phantom{\rule{4.pt}{0ex}}\text{(fix}\left({g}_{j}\right)\cap \partial {H}_{n}^{ℂ}\right)\ne \infty$ 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