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# 6 - Isometry groups

Published online by Cambridge University Press:  05 June 2015

## Summary

Isometries of spherical, Euclidean, and hyperbolic space

For each positive integer n, there are three simply-connected, complete Rie-mannian n-manifolds with constant curvature, namely n-dimensional spherical space Sn, n-dimensional Euclidean space ℝn, and n-dimensional hyperbolic space ℍn. Note that ℝ1 and ℍ1 are isometric, but otherwise there are no repetitions in this list. We denote the isometry groups of these manifolds by Isom(Sn), Isom(ℝn), and Isom(ℍn). These groups are each generated by reflections. We denote the three subgroups of these three isometry groups, comprised of orientation-preserving isometries, by Isom+ (Sn), Isom+ (ℝRn), and Isom+ (ℍn). A map in Isom(Sn) lies in Isom+ (Sn) if and only if it can be expressed as a composite of an even number of reflections. Similar comments apply to the groups Isom+ (ℝn) and Isom+ (ℍn).

We studied the orthogonal group O(n, ℝ) and the special orthogonal group SO(n, ℝ) in Chapter 4; these two groups are Isom(Sn−1) and Isom+ (Sn−1), respectively. In this chapter we consider reversibility in the remaining four isometry groups Isom(ℝn), Isom(ℍn), Isom+ (ℝn), and Isom+ (ℍn).

Hyperbolic geometry in two and three dimensions

In Chapter 1 we briefly discussed reversibility in the Euclidean isometry groups Isom+ (ℝ2) and Isom+ (ℝ3). We found that, in two dimensions, the only elements that are strongly reversible, other than involutions, are translations. In three dimensions we found that all isometries are strongly reversible. Before we tackle higher-dimensional isometry groups we first, in this section, consider isometry groups of two- and three-dimensional hyperbolic space.

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Publisher: Cambridge University Press
Print publication year: 2015

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• Isometry groups
• Book: Reversibility in Dynamics and Group Theory
• Online publication: 05 June 2015
• Chapter DOI: https://doi.org/10.1017/CBO9781139998321.007
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• Isometry groups
• Book: Reversibility in Dynamics and Group Theory
• Online publication: 05 June 2015
• Chapter DOI: https://doi.org/10.1017/CBO9781139998321.007
Available formats
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# Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

• Isometry groups
• Book: Reversibility in Dynamics and Group Theory
• Online publication: 05 June 2015
• Chapter DOI: https://doi.org/10.1017/CBO9781139998321.007
Available formats
×