# Discreteness Of Hyperbolic Isometries by Test Maps

## TL;DR

This paper establishes a criterion for the discreteness of Zariski dense subgroups of hyperbolic isometry groups over real, complex, or quaternionic fields, based on the discreteness of two-generator subgroups involving test maps.

## Contribution

It introduces a new test map criterion for determining the discreteness of Zariski dense subgroups in hyperbolic isometry groups.

## Key findings

- Discreteness of G follows from discreteness of all two-generator subgroups with test maps.
- The criterion applies to groups over real, complex, and quaternionic hyperbolic spaces.
- Provides a new approach to analyze the structure of hyperbolic isometry groups.

## 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 $G$.