Discreteness Of Hyperbolic Isometries by Test Maps

Abstract

Let $\mathbb F=\mathbb R$, $\mathbb C$ or $\mathbb H$. 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. A subgroup $G$ of ${\rm U}(n,1; \mathbb F)$ is called \emph{Zariski dense} if it does not fix a point on the closure of the $\mathbb F$-hyperbolic space, and neither it preserves a totally geodesic subspace of it. 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 $G$.