Quaternionic hyperbolic Fenchel-Nielsen coordinates

Let Sp(2,1) be the isometry group of the quaternionic hyperbolic plane HH2. An element g in Sp(2,1) is hyperbolic if it fixes exactly two points on the boundary of HH2. We classify pairs of hyperbolic elements in Sp(2,1) up to conjugation. A hyperbolic element of Sp(2,1) is called loxodromic if it has no real eigenvalue. We show that the set of Sp(2,1) conjugation orbits of irreducible loxodromic pairs is a (CP1)4 bundle over a topological space that is locally a semi-analytic subspace of R13. We use the above classification to show that conjugation orbits of geometric' representations of a closed surface group (of genus g2) into Sp(2,1) can be determined by a system of 42g-42 real parameters. Further, we consider the groups Sp(1,1) and GL(2,H). These groups also act by the orientation-preserving isometries of the four and five dimensional real hyperbolic spaces respectively. We classify conjugation orbits of pairs of hyperbolic elements in these groups. These classifications determine conjugation orbits of geometric' surface group representations into these groups.