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Moduli of triples of points in quaternionic hyperbolic geometry

Igor Almeida, Nikolay Gusevskii

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.

Moduli of triples of points in quaternionic hyperbolic geometry

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 . 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.
Paper Structure (18 sections, 43 theorems, 62 equations)

This paper contains 18 sections, 43 theorems, 62 equations.

Table of Contents

  1. Preliminaries
  2. Quaternions
  3. Hyperbolic spaces
  4. Projective Model
  5. The ball model
  6. Totally geodesic submanifolds
  7. A little more about the isometry group of the quaternionic hyperbolic space
  8. Moduli of triples of points in quaternionic projective space
  9. The Gram matrix
  10. Invariants of triangles in quaternionic hyperbolic geometry: inside and on the boudary
  11. Quaternionic Cartan's angular invariant
  12. Quaternionic Brehm's invariants
  13. Moduli of triples of positive points
  14. Moduli of triples of points: mixed configurations
  15. Triangles with two ideal vertices
  16. ...and 3 more sections

Key Result

Proposition 1.1

Two quaternions $a$ and $b$ are similar if and only if ${\rm Re} \ a ={\rm Re} \ b$ and $\vert a\vert = \vert b\vert$. Moreover, every similarity class contains a complex number, unique up to conjugation.

Theorems & Definitions (51)

  • Proposition 1.1
  • Corollary 1.1
  • Proposition 1.2
  • Proposition 1.3
  • Proposition 1.4
  • Proposition 1.5
  • Corollary 1.2
  • Proposition 1.6
  • Proposition 2.1
  • Proposition 2.2
  • ...and 41 more