Classification of Pairs of Quaternionic Hyperbolic Isometries

Abstract

We consider the Lie groups SU(n, 1) and Sp(n, 1) that act as isometries of the complex and the quaternionic hyperbolic spaces respectively. We classify pairs of semisimple el- ements in Sp(n, 1) and SU(n, 1) up to conjugacy. This gives local parametrization of the representations ρ in Hom(F 2 , G)/G such that both ρ(x) and ρ(y) are semisimple elements in G, where F 2 = hx, yi, G = Sp(n, 1) or SU(n, 1). We use the PSp(n, 1)-configuration space M(n, i, m − i) of ordered m-tuples of distinct points in H n H , where the first i points in an m-tuple are boundary points, to classify the semisimple pairs. Further, we also classify points on M(n, i, m − i). Particularly interesting coordinates occur for lower values of n. The conjugacy classification of pairs is then applied geomet- rically to obtain Quaternionic hyperbolic Fenchel-Nielsen type parameters for generic representations of surface groups into Sp(2, 1) and Sp(1, 1).