Quaternionic hyperbolic Fenchel–Nielsen coordinates

Abstract

Let Sp(2,1)be the isometry group of the quaternionic hyperbolic planeH2H.Anelementgin Sp(2,1)ishyperbolicif it fixes exactly two points on the boundary ofH2H.Weclassify pairs of hyperbolic elements in Sp(2,1)up to conjugation. A hyperbolic element ofSp(2,1)is calledloxodromicif it has no real eigenvalue. We show that the set of Sp(2,1)conjugation orbits of irreducible loxodromic pairs is a(CP1)4bundle over a topologicalspace that is locally a semi-analytic subspace ofR13. We use the above classification to showthat conjugation orbits of ‘geometric’ representations of a closed surface group (of genusg≥2) 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 classifi-cations determine conjugation orbits of ‘geometric’ surface group representations into thesegroups.