Discreteness criteria for subgroups in complex hyperbolic space (Q1609942)

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.