Local Coordinates for Complex and Quaternionic Hyperbolic Pairs

Abstract

Let G(n)=Sp(n,1) or SU(n,1) . We classify conjugation orbits of generic pairs of loxodromic elements in G(n) . Such pairs, called ‘nonsingular’, were introduced by Gongopadhyay and Parsad for SU(3,1) . We extend this notion and classify G(n) -conjugation orbits of such elements in arbitrary dimension. For n=3 , they give a subspace that can be parametrized using a set of coordinates whose local dimension equals the dimension of the underlying group. We further construct twist-bend parameters to glue such representations and obtain local parametrization for generic representations of the fundamental group of a closed (genus g≥2 ) oriented surface into G(3).