Appendix 2 - Lie Transformation Groups

Summary

When dealing with the automorphism groups of geometries on surfaces as considered in this book we usually obtain Lie groups. Furthermore, these Lie groups are of dimension at most 8 except in the case of the tubular circle planes of rank greater than 3. In this appendix we compile some useful results on lower-dimensional Lie groups. For general information on Lie groups we refer to Freudenthal–de Vries [1969], Hochschild [1965], and Varadarajan [1974].

We begin with topological groups, which are the basic objects underlying Lie groups, and list some of their properties.

Topological Groups

A topological group G is a group G equipped with a Hausdorff topology such that the two basic group operations G × G → G : (x, y) ↦ xy and G → G : x ↦ x⁻¹ are continuous. Most of the groups we encounter in this book operate on some manifold. It then follows that such a group can be equipped with a metric and that it is separable; see Section A2.3. So, if you are unfamiliar with the more general setting of topological spaces, you can always assume you are dealing with a metric space.

A topological group G is called (locally) compact, (locally) connected, finite-dimensional, etc., if the underlying topological space G has the corresponding property.