Strongly Doubly Reversible Pairs in Quaternionic Unitary Group of Signature $(n,1)$

Abstract

Let $\PSp(n,1)$ denote the isometry group of the quaternionic hyperbolic space $\mathbb{H}^n$. A pair $(g_1,g_2)$ $\PSp(n,1)$ is \emph{strongly doubly reversible} if $(g_1,g_2)$ and $(g_1^{-1},g_2^{-1})$ are simultaneously conjugate in $\PSp(n,1)$ by an involution. Equivalently, there exist involutions $i_1,i_2,i_3 \in \PSp(n,1)$ such that $g_1 = i_1 i_2$, $g_2 = i_1 i_3$. We prove that the set of such pairs has Haar measure zero in $\PSp(n,1) \times \PSp(n,1)$. The same result also holds for $\PSp(n) \times \PSp(n)$ for $n\geq 2$. In the special case $n=1$, we show that every pair of elements in $\PSp(1)$ is strongly doubly reversible. Applying this result, we give a shorter proof of a theorem of Basmajian and Maskit showing that every pair of elements in ${\rm SO}(4)$ is strongly doubly reversible. Furthermore, we derive the necessary conditions for a pair of hyperbolic elements in $\PSp(1,1)$ to be strongly doubly reversible and provide a quantitative characterization of such pairs.