On discreteness of subgroups of quaternionic hyperbolic isometries

Abstract

Let ${{\mathbf{H}}_{\mathbb{H}}}^n$ denote the $n$-dimensional quaternionic hyperbolic space. The linear group $\mathrm{Sp}(n,1)$ acts by the isometries of ${{\mathbf{H}}_{\mathbb{H}}}^n$. A subgroup $G$ of $\mathrm{Sp}(n,1)$ is called \emph{Zariski dense} if it does not fix a point on ${{\mathbf{H}}_{\mathbb{H}}}^n\cup \partial {{\mathbf{H}}_{\mathbb{H}}}^n$ and neither it preserves a totally geodesic subspace of ${{\mathbf{H}}_{\mathbb{H}}}^n$. We prove that a Zariski dense subgroup $G$ of $\mathrm{Sp}(n,1)$ is discrete if for every loxodromic element $g\in G$ the two generator subgroup $⟨f,gf{g}^{-1}⟩$ is discrete, where the generator $f\in \mathrm{Sp}(n,1)$ is certain fixed element not necessarily from $G$.

Reformatted, adding new result, and removing some. Removed parts will be subsumed elsewhere