Quaternionic Hyperbolic Fenchel-Nielsen Coordinates

TL;DR

This paper classifies pairs of hyperbolic elements in quaternionic hyperbolic isometry groups and uses this to parameterize geometric surface group representations into these groups.

Contribution

It provides a detailed classification of hyperbolic element pairs in quaternionic hyperbolic isometry groups and introduces a parameterization method for surface group representations.

Findings

Classification of hyperbolic pairs in $Sp(2,1)$ up to conjugation.

Parameterization of surface group representations with $42g-42$ real parameters.

Extension of classification to groups $Sp(1,1)$ and $GL(2, extbf{H})$.

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.