Quaternionic Hyperbolic Fenchel-Nielsen Coordinates

Abstract

Let $Sp(2,1)$ be the isometry group of the quaternionic hyperbolic plane ${{\bf H}_{\mathbb H}}^2$. An element $g$ in $Sp(2,1)$ is `hyperbolic' if it fixes exactly two points on the boundary of ${{\bf H}_{\mathbb H}}^2$. 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 $(\mathbb C {\mathbb P}^1)^4$-bundle over a topological space that is locally a semi-analytic subspace of ${\mathbb R}^{13}$. We use the above classification to show that conjugation orbits of `geometric' representations of a closed surface group (of genus $g \geq 2$) into $Sp(2,1)$ can be determined by a system of $42g-42$ real parameters. Further, we consider the groups $Sp(1,1)$ and $GL(2, {\mathbb H})$. 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.