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

2025/10/16 by Gongopadhyay, Krishnendu, Kalane, Sagar B.
#22E43 #FOS: Mathematics #Geometric Topology (math.GT) #Group Theory (math.GR) #Primary 51M10 #Secondary: 15B33

paper · doi:10.48550/arxiv.2510.14735

Abstract

Let \PSp(n,1) denote the isometry group of the quaternionic hyperbolic space ℍⁿ. A pair (g₁,g₂) \PSp(n,1) is strongly doubly reversible if (g₁,g₂) and (g₁^-1,g₂^-1) are simultaneously conjugate in \PSp(n,1) by an involution. Equivalently, there exist involutions i₁,i₂,i₃ ∈ \PSp(n,1) such that g₁ = i₁ i₂, g₂ = i₁ i₃. We prove that the set of such pairs has Haar measure zero in \PSp(n,1) × \PSp(n,1). The same result also holds for \PSp(n) × \PSp(n) for n≥ 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.

Citations

• Kronecker Coefficients and Simultaneous Conjugacy Classes
• Reversible Quaternionic Hyperbolic Isometries
• Real reflections, commutators and cross-ratios in complex hyperbolic space

Related

• Reversible Quaternionic Hyperbolic Isometries
• On Conjugation orbits of semisimple pairs in rank one
• Reversibility and algebraic characterization of the quaternionic Möbius transformations
• Quaternionic Hyperbolic Fenchel-Nielsen Coordinates
• Strongly reversible classes in SL(n,ℂ)