# Reversible Quaternionic Hyperbolic Isometries

## TL;DR

This paper classifies reversible and strongly reversible elements in quaternionic hyperbolic isometry groups, showing that all elements in the isometry group are strongly reversible, advancing understanding of symmetries in quaternionic hyperbolic geometry.

## Contribution

It provides a complete classification of reversible and strongly reversible elements in quaternionic hyperbolic isometry groups, including the proof that all elements in the isometry group are strongly reversible.

## Key findings

- All elements of PSp(n,1) are strongly reversible.
- Classification of reversible elements in Sp(n) and Sp(n,1).
- New insights into symmetries of quaternionic hyperbolic spaces.

## Abstract

Let $G$ be a group. An element $g$ in $G$ is called reversible if it is conjugate to $g^{-1}$ within $G$, and called strongly reversible if it is conjugate to its inverse by an order two element of $G$. Let $\textbf{H}_{\mathbb H}^n$ be the $n$-dimensional quaternionic hyperbolic space. Let $\mathrm{PSp}(n,1)$ be the isometry group of $\textbf{H}_{\mathbb H}^n$. In this paper, we classify reversible and strongly reversible elements in $\mathrm{Sp}(n)$ and $\mathrm{Sp}(n,1)$. Also, we prove that all the elements of $\mathrm{PSp}(n,1)$ are strongly reversible.