Reversible Quaternionic Hyperbolic Isometries

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.