Discreteness criteria for subgroups in complex hyperbolic space

Abstract

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.