Local coordinates for complex and quaternionic hyperbolic pairs

TL;DR

This paper classifies conjugation orbits of generic pairs of loxodromic elements in complex and quaternionic hyperbolic groups, extending previous work and providing local parametrizations for representations of surface groups.

Contribution

It extends the classification of non-singular pairs to arbitrary dimensions and constructs local coordinates for representations into G(3).

Findings

Non-singular pairs form a small set for n ≥ 4.

For n=3, non-singular pairs form a parametrizable subspace.

Constructed twist-bend parameters for local representation parametrization.

Abstract

Let $G(n)={\rm Sp}(n,1)$ or ${\rm SU}(n,1)$. We classify conjugation orbits of generic pairs of loxodromic elements in $G(n)$. Such pairs, called `non-singular', were introduced by Gongopadhyay and Parsad for ${\rm SU}(3,1)$. We extend this notion and classify $G(n)$-conjugation orbits of such elements in arbitrary dimension. We prove that the set given by non-singular pairs in $G(n)$ is `small' for $n \geq 4$. However, 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 oriented surface into $G(3)$.