Moduli of triples of points in quaternionic hyperbolic geometry

TL;DR

This paper characterizes the moduli space of point triples in quaternionic hyperbolic geometry using new invariants, extending complex hyperbolic invariants to the quaternionic setting.

Contribution

It introduces quaternionic analogues of Goldman invariants to classify triples of points in quaternionic hyperbolic space.

Findings

Defined basic invariants for triples of points

Extended Goldman invariants to quaternionic hyperbolic geometry

Provided a classification of congruence classes of triples

Abstract

In this work, we describe the moduli of triples of points in quaternionic projective space which define uniquely the congruence classes of such triples relative to the action of the isometry group of quaternionic hyperbolic space ${\rm H}^n_{\mathbb{Q}}$. To solve this problem, we introduce some basic invariants of triples of points in quaternionic hyperbolic geometry. In particular, we define quaternionic analogues of the Goldman invariants for mixed configurations of points introduced by him in complex hyperbolic geometry.