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The Classification of Non-Euclidean Plane Crystallographic Groups

Published online by Cambridge University Press:  20 November 2018

A. M. Macbeath
A. M. Macbeath*
Affiliation:
University of Birmingham, Birmingham, England
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This paper deals with the algebraic classification of non-euclidean plane crystallographic groups (NEC groups, for short) with compact quotient space. The groups considered are the discrete groups of motions of the Lobatschewsky or hyperbolic plane, including those which contain orientation-reversing reflections and glide-reflections. The corresponding problem for Fuchsian groups, which contain only orientable transformations, is essentially solved in the work of Fricke and Klein (6).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 19 , 1967 , pp. 1192 - 1205
Copyright
Copyright © Canadian Mathematical Society 1967

References

1. Ahlfors, L. V., On quasiconformal mappings, J. Anal. Math. 4 (1954), 1–58.Google Scholar
2. Bers, L., Quasiconformal mappings and Teichmüller's theorem, Princeton Conference on Analytic Functions (1960).Google Scholar
3. Bundgaard, S. and Nielsen, J., On normal subgroups with finite index in F-groups, Mat. Tidsskr., B (1951), 56–58.Google Scholar
4. Coxeter, H. S. M. and Moser, W. O. J., Generators and relations for discrete groups (1957).Google Scholar
5. Fox, R. H., On Fenchel's conjecture about F-groups, Mat. Tidsskr., B (1952), 177–195.Google Scholar
6. Fricke, R. and Klein, F., Vorlesungen über die Theorie der automorphen Funktionen (Leipzig, 1926).Google Scholar
7. Macbeath, A. M., Geometrical realisation of isomorphisms between plane groups, Bull. Amer. Math. Soc., 71 (1965), 629–630.Google Scholar
8. Nielsen, J., Untersuchungen zur Topologie der geschlossenen zweiseitigen Flachen, Acta Math., 50 (1927), 189–358.Google Scholar
9. Wilkie, H. C., On non-euclidean crystallographic groups, Math. Z., 91 (1966), 87–102.Google Scholar
10. Zieschang, H., Ueber Automorphismen ebener diskontinuierlichen Gruppen, Math. Ann., 166 (1966), 148–167.Google Scholar