On products of isometries of hyperbolic space (Q837640)

Let \({\mathbb F} = {\mathbb R}\), \({\mathbb C}\), or \({\mathbb Q}\), the real, complex, or quaternionic fields, and let \(PU(2, 1,{\mathbb F})\) denote the isometry group of the two-dimensional hyperbolic space over \({\mathbb F}\). The authors prove that: If \(C_1,\dots,C_k\), \(k\geq 3\), are arbitrary conjugacy classes of loxodromic elements of \(PU(2, 1,{\mathbb F})\), then (i) there exist \(g_1, \dots, g_k \in PU(2, 1, {\mathbb F})\), \(g_i \in C_i\), such that \(g_1\cdots g_k = I\) and (ii) the set \(\{(g_1, g_2, g_3): g_i\in C_i\), \(g_1 g_2 g_3 = I \}\) is compact modulo the diagonal action of conjugation by \(PU(2, 1,{\mathbb F})\). (i) could not be proved for the isometry group of higher-dimensional hyperbolic space.