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Discontinuous groups in some metric and nonmetric spaces. (English) Zbl 0592.51021

The author accepts the definition of a discontinuous group as given by Hilbert and Cohn-Vossen which comes to this: There are only finite many points equivalent under the map group in any bounded domain. In the first section of the paper conditions are described which all characterize the discontinuous groups of certain metric spaces. In the following sections the groups under consideration are essentially studied for the cases of the normed and the pseudo-Euclidean plane respectively. In particular, a criterion is given that a normed plane is Euclidean and a criterion of ”badly approximable irrational numbers”. The author emphasises that his work has many relations to that of Minkowski.
Reviewer: O.Bottema

MSC:

51N25 Analytic geometry with other transformation groups
51M10 Hyperbolic and elliptic geometries (general) and generalizations
57S30 Discontinuous groups of transformations
54E35 Metric spaces, metrizability
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