Groups of hyperbolic crystallography

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Abstract
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.

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