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On the structure of non-Euclidean crystallographic groups

Published online by Cambridge University Press:  24 October 2008

David Singerman
David Singerman
Affiliation:
The University, Southampton
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Extract

Let denote the group of transformations of the upper-half complex plane U onto itself of the form

If, on U, we introduce the Riemannian metric ds = |dz| y−1 (z = x + iy), then U becomes a model of the hyperbolic plane and its group of isometries. The set of elements of type I, the orientation-preserving isometries form a subgroup of index two in , which we denote by .

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 76 , Issue 1 , July 1974 , pp. 233 - 240
Copyright
Copyright © Cambridge Philosophical Society 1974

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References

REFERENCES

(1)Macbeath, A. M. Discontinuous groups and birational transformations. Dundee Summer School (1961).Google Scholar
(2)Macbeath, A. M.The classification of non-Euclidean plane Crystallographic groups. Canad. J. Math. 19 (1967), 1192–1205.CrossRefGoogle Scholar
(3)Macbeath, A. M. and Singerman, D.Spaces of subgroups and Teichmüller space. To appear in Proc. London Math. Soc.Google Scholar
(4)Singerman, D.Symmetries of Riemann surfaces with large automorphism group. To appear in Math. Ann.Google Scholar
(5)Wilkie, H. C.On non-Euclidean crystallographic groups. Math. Z. 91 (1966), 87–102.CrossRefGoogle Scholar