Non Euclidean crystallographic groups

Abstract

The term non Euclidean crystallographic croup (or NEC group) was introduced by Wilkie [1] for a discrete group of isometries of the hyperbolic plane with compact quotient space. If an NEC group contains no orientation reversing transformation it is a Fuchsian group; otherwise it contains a Fuchsian group as a subgroup of index two. Wilkie obtained presentations of all NEC groups, and a partial classification of these groups under isomorphism. Macbeath [8] completed the classification, using the theory of quasi-conformal mappings; Keller [5] established the same result by purely geometric methods.