Discreteness of hyperbolic isometries by test maps

Abstract

Let ${\mathbb F}={\mathbb R}$, ${\mathbb C}$ or the Hamilton's quaternions ${\mathbb H}$. Let ${\bf H}_{\mathbb F}^n$ denote the $n$-dimensional ${\mathbb F}$-hyperbolic space. Let ${\rm U}(n,1; {\mathbb F})$ be the linear group that acts by the isometries of ${\bf H}_{\mathbb F}^n$. A subgroup $G$ of ${\rm U}(n,1; {\mathbb F})$ is called \textit{Zariski dense} if it does not fix a point on ${\bf H}_{\mathbb F}^n \cup \partial {\bf H}_{\mathbb F}^n$ and neither it preserves a totally geodesic subspace of ${\bf H}_{\mathbb F}^n$. We prove that a Zariski dense subgroup $G$ of ${\rm U}(n,1; {\mathbb F})$ is discrete if for every loxodromic element $g \in G$, the two generator subgroup $\langle f, g \rangle$ is discrete, where $f \in {\rm U}(n,1; {\mathbb F})$ is a test map not necessarily from $G$.