On Conjugation orbits of semisimple pairs in rank one

2018/01/18 by Gongopadhyay, Krishnendu, Kalane, Sagar B.
#15B33 (Secondary) #20H10 #37C15 (Primary) #51M10 #FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.1801.06431

Abstract

We consider Lie groups \rm SU(n,1) and \rm Sp(n,1) that act as the isometries of the complex and quaternionic hyperbolic spaces respectively. We classify pairs of semisimple elements in \rm Sp(n,1) and \rm SU(n,1) up to conjugacy. This gives local parametrization of the representations ρ in Hom(F₂, G)/G such that both ρ(x) and ρ(y) are hyperbolics, where F₂=⟨ x, y⟩, G=\rm Sp(n,1) or \rm SU(n,1). We use the \rm PSp(n,1)-configuration space M(n,i,m-i) of ordered m-tuples of points on \bf H_{\mathbb H}ⁿ, where first i points in an m-tuple are null points, to classify the semisimple pairs. Further, we also classify points on M(n,i,m-i).

Related

• Conjugation Orbits of Loxodromic Pairs in SU(n,1)
• Quaternionic Hyperbolic Fenchel-Nielsen Coordinates
• Reversible Quaternionic Hyperbolic Isometries
• Symplectic geometry of semisimple orbits
• The L_p-dual space of a semisimple Lie group