Discreteness Of Hyperbolic Isometries by Test Maps

Abstract

Let $\mathbb{F}=\mathbb{R}$, $\mathbb{C}$ or $\mathbb{H}$. Let ${\mathbf{H}}_{\mathbb{F}}^n$ denote the $n$-dimensional $\mathbb{F}$-hyperbolic space. Let $\mathrm{U}(n,1;\mathbb{F})$ be the linear group that acts by the isometries. A subgroup $G$ of $\mathrm{U}(n,1;\mathbb{F})$ 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 $\mathrm{U}(n,1;\mathbb{F})$ is discrete if for every loxodromic element $g\in G$, the two generator subgroup $⟨f,g⟩$ is discrete, where $f\in \mathrm{U}(n,1;\mathbb{F})$ is a test map not necessarily from $G$.

arXiv commentCorrected a typo in the Introduction