Abstract

We classify isometries of the real hyperbolic 4-space by their conjugacy classes of centralizers. We use the representation of the isometries by quaternionic matrices to obtain this characterization. Another characterization in terms of conjugacy invariants is also given.

Highlights

  • Let Hn denote the n-dimensional real hyperbolic space

  • The isometries of H2 can be identified with the group P SL 2, R

  • P SL 2, C acts on H3 by complex Mobius transformations

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Summary

Introduction

Let Hn denote the n-dimensional real hyperbolic space. The isometries of Hn are always assumed to be orientation preserving unless stated otherwise. The group of invertible quaternionic 2 × 2 matrices can be identified with the isometries of H5 Using this identification, the author has obtained an algebraic characterization of the isometries of H5, see 8, Theorem-1.1. In all the above works, the authors obtained their characterizations using conjugacy invariants of the isometries Another approach which has been used recently to characterize the isometries algebraically is in terms of the centralizers, up to conjugacy. The characterization by z-classes is based purely in terms of the internal structure of the group alone, and this does not involve any conjugacy invariant This approach is useful in certain contexts, for example, see Remark 3.3 in this paper. The dynamical types of isometries are precisely classified by the isomorphism types of the centralizers, see Theorem 4.1 This demonstrates the usefulness of the z-classes in the classification problem of the isometries. After the conjugacy classes are known, the characterization by conjugacy invariants is obtained essentially as an appendix to the author’s earlier work 8, Theorem-1.1

Classification of Isometries
The Quaternions
The Upper Half-Space Model
The Ball Model
The Conjugacy Classes
Algebraic Characterization
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