On discreteness of subgroups of quaternionic hyperbolic isometries

TL;DR

This paper establishes a criterion for the discreteness of Zariski dense subgroups of quaternionic hyperbolic isometries based on the discreteness of certain two-generator subgroups involving a fixed element.

Contribution

It proves that Zariski dense subgroups are discrete if all such two-generator subgroups with a fixed element are discrete, providing a new criterion for discreteness in quaternionic hyperbolic geometry.

Findings

Zariski dense subgroups are discrete under the given condition.

Discreteness of two-generator subgroups implies discreteness of the entire subgroup.

The criterion applies to subgroups of ${ m{Sp}}(n,1)$ acting on quaternionic hyperbolic space.

Abstract

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