Proceedings of the Japan Academy, Series A, Mathematical Sciences

Discreteness criteria for subgroups in complex hyperbolic space

Binlin Dai, Ainong Fang, and Bing Nai

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In this paper, we study the discreteness criteria for subgroups of $U(1, n; \mathbf{C})$ in complex hyperbolic space $H^n_{\mathbf{C}}$. We prove that a nonelementary subgroup $G$ of $U(1, n; \mathbf{C})$ with condition A is discrete if and only if every two generator subgroup of $G$ is discrete. We also prove that if a nonelementary subgroup $G$ of $U(1, n; \mathbf{C})$ contains a sequence of distinct elements $\{g_m\}$ with $\operatorname{Card}(\operatorname{fix}(g_m) \cap \partial H^n_{\mathbf{C}}) \ne \infty$ and $g_m \rightarrow I$ as $m \rightarrow \infty$, then $G$ contains a non-discrete, nonelementary two generator subgroup.

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Proc. Japan Acad. Ser. A Math. Sci., Volume 77, Number 10 (2001), 168-172.

First available in Project Euclid: 23 May 2006

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Zentralblatt MATH identifier

Primary: 30F40: Kleinian groups [See also 20H10] 30C62: Quasiconformal mappings in the plane 20H10: Fuchsian groups and their generalizations [See also 11F06, 22E40, 30F35, 32Nxx]

Complex hyperbolic space limit set elementary groups


Dai, Binlin; Fang, Ainong; Nai, Bing. Discreteness criteria for subgroups in complex hyperbolic space. Proc. Japan Acad. Ser. A Math. Sci. 77 (2001), no. 10, 168--172. doi:10.3792/pjaa.77.168.

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