Congruence classes of points in quaternionic hyperbolic spaces

TL;DR

This paper classifies point configurations in quaternionic hyperbolic spaces using new invariants, providing a complete description of congruence classes for specific cases and constructing moduli spaces.

Contribution

It introduces new geometric invariants and distance formulas, enabling classification of point tuples in quaternionic hyperbolic geometry.

Findings

Complete classification of 3-point configurations in quaternionic hyperbolic space.

Construction of moduli space for 4-point configurations on the boundary.

Introduction of quaternionic Cartan's angular invariants and distance formulas.

Abstract

An important problem in quaternionic hyperbolic geometry is to classify ordered $m$-tuples of pairwise distinct points in the closure of quaternionic hyperbolic n-space, $\overline{{\bf H}_\bh^n}$, up to congruence in the holomorphic isometry group ${\rm PSp}(n,1)$ of ${\bf H}_\bh^n$. In this paper we concentrate on two cases: $m=3$ in $\overline{{\bf H}_\bh^n}$ and $m=4$ on $\partial{\bf H}_\bh^n$ for $n\geq 2$. New geometric invariants and several distance formulas in quaternionic hyperbolic geometry are introduced and studied for this problem. The congruence classes are completely described by quaternionic Cartan's angular invariants and the distances between some geometric objects for the first case. The moduli space is constructed for the second case.