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

Abstract

Let $\mathrm{PSp}(n,1)$ denote the isometry group of quaternionic hyperbolic $n$--space $\h^n$. A pair $(g_1,g_2)\in \mathrm{PSp}(n,1{)}^2$ is \emph{strongly doubly reversible} if $(g_1,g_2)$ and $(g_1^{-1},g_2^{-1})$ are simultaneously conjugate by an involution. Equivalently, there exist involutions $i_1,i_2,i_3\in \mathrm{PSp}(n,1)$ such that $g_1=i_1{i}_2$ and $g_2=i_1{i}_3$. We prove that the set of strongly doubly reversible pairs has Haar measure zero in $\mathrm{PSp}(n,1)\times \mathrm{PSp}(n,1)$. The same conclusion holds for $\mathrm{PSp}(n)\times \mathrm{PSp}(n)$ and $\mathrm{SU}(n,1)\times \mathrm{SU}(n,1)$ for $n\ge 2$, and for ${\mathrm{SO}}_0(n,1)\times {\mathrm{SO}}_0(n,1)$ and $\mathrm{SO}(n+1)\times \mathrm{SO}(n+1)$ for $n\ge 4$. In the compact rank-one case, every pair in $\mathrm{PSp}(1)$ is strongly doubly reversible, which gives a short proof of the theorem of Basmajian and Maskit that every pair in $\mathrm{SO}(4)$ is strongly doubly reversible. For hyperbolic pairs, we prove that double reversibility and strong double reversibility are equivalent in $\mathrm{PSp}(1,1)$. In higher dimension, we prove the same implication when one member of the pair is regular, with pairwise distinct non-real unit eigenvalue classes. Finally, in $\mathrm{PSp}(1,1)$, for a hyperbolic element in normal form, we give a complete explicit matrix criterion characterizing all elements that form a strongly doubly reversible pair with it.

Corrected a couple of errors, removed a vacuous statement, tightened some of the arguments