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Groups of hyperbolic crystallography

Published online by Cambridge University Press:  24 October 2008

A. M. Macbeath  and
A. H. M. Hoare
A. M. Macbeath
Affiliation:
University of Birmingham
A. H. M. Hoare
Affiliation:
University of Birmingham
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Extract

The aim of this paper is to describe the possible structures for NEC (non-euclidean crystallographic) groups of the hyperbolic plane with non-compact quotient space. The case of compact quotient space was settled by Wilkie (6), and it has been shown by Hoare, Karrass and Solitar (3) that all subgroups of infinite index in a Wilkie group have a certain structure pattern. Their proof depends on showing that a subgroup of infinite index in a Wilkie group has a presentation satisfying certain restrictions on the occurrences of each generator. Our method here is to study the presentation of an arbitrary NEC group which is given by the classical construction of the Dirichlet region, and to prove that it satisfies the same restrictions as were found for the above subgroups of Wilkie groups.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 79 , Issue 2 , March 1976 , pp. 235 - 249
Copyright
Copyright © Cambridge Philosophical Society 1976

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References

REFERENCES

(1)Ford, L. R.Automorphic functions. Chelsea, New York, 1951.Google Scholar
(2)Hoare, A. H. M., Karrass, A. and Solitar, D.Subgroups of infinite index in Fuchsien groups. Math. Z. 125 (1972), 59–69.CrossRefGoogle Scholar
(3)Hoaree, A. H. M., Karrass, A. and Solitar, D.Subgroups of NEC groups. Comm. Pure Appl. Math. 26 (1973), 731–744.CrossRefGoogle Scholar
(4)Lehner, J.Discontinuous groups and automorphic functions, mathematical surveys No. VIII (Amer. Math. Soc. 1964).CrossRefGoogle Scholar
(5)Macbeath, A. M.Groups of homeomorphisms of a simply connected space. Ann. of Math. 79 (1964), 473–488.CrossRefGoogle Scholar
(6)Wilkie, H. C.On non-euclidean crystallographic groups. Math. Z. 91 (1966), 87–102.CrossRefGoogle Scholar