Strongly Doubly Reversible Pairs in Quaternionic Unitary Group of Signature $(n,1)$
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| Format: | Preprint |
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2025
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| _version_ | 1866911324813918208 |
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| author | Gongopadhyay, Krishnendu Kalane, Sagar B. |
| author_facet | Gongopadhyay, Krishnendu Kalane, Sagar B. |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_14735 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Strongly Doubly Reversible Pairs in Quaternionic Unitary Group of Signature $(n,1)$ Gongopadhyay, Krishnendu Kalane, Sagar B. Group Theory Geometric Topology Primary 51M10, Secondary: 15B33, 22E43 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. |
| title | Strongly Doubly Reversible Pairs in Quaternionic Unitary Group of Signature $(n,1)$ |
| topic | Group Theory Geometric Topology Primary 51M10, Secondary: 15B33, 22E43 |
| url | https://arxiv.org/abs/2510.14735 |