**Journal of Lie Theory, Vol. 9, No. 2, pp. 321-349 (1999)
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Discrete groups of affine isometries

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Herbert Abels

Fakultät für Mathematik, Universität Bielefeld, Universitätsstrasse 25, D-33501 Bielefeld, Germany, abels@mathematik.uni-bielefeld.de

**Abstract:** Given a finite set $S$ of isometries of an affine Euclidean space. We ask when the group $\Gamma$ generated by $S$ is discrete. This includes as a special case the question when the group generated by a finite set of rotations is finite. This latter question is answered in an appendix. In the main part of the paper the general case is reduced to this special case. The result is phrased as a series of tests: $\Gamma$ is discrete iff $S$ passes all the tests. The testing procedure is algorithmic.

**Keywords:** affine Euclidean space, isometries, rotations, discrete groups

**Classification (MSC91):** 51N10; 20H15

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