Proc. Camb. Phil. Soc. (1974), 76, 233 233 PCPS 76-23 Printed in Great Britain On the structure of non-Euclidean crystallographic groups BY DAVID SINGERMAN The University, Southampton (Received 12 May 1972) 1. Introduction. Let <& denote the group of transformations of the upper-half complex plane U onto itself of the form (I) z->- -:, a, b, c, dreal, ad — be = 1, CZ -(- (h (II) z->- ——-5, a, 6, c, cZ real, od — be = — 1. CZ -p Ct If, on U, we introduce the Riemannian metric ds = \dz\y~ 1 (z = x + iy), then £7 be- comes 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 IS, which we denote by '&+. By a non-Euclidean crystallographic (NEC) group, we shall mean a discrete sub- group F of ^ for which J7/F is compact. An NEC group contained in ^+ is called a Fuchsian group (and thus all Fuchsian groups considered in this paper have com- pact quotient-space). An NEC group, which contains elements of ( & — '8 + , will be called a proper NEC group. Wilkie(5) showed that every NEC group has a presentation of a certain form and he exhibited a number of isomorphisms between groups with formally different presentations. Macbeath(2)foundnecessary and sufficient conditionsfortwoofWilkie's presentations to define isomorphic groups, his proof being based on his geometrical isomorphism theorem. This says that two NEC groups I\, F 2 are isomorphic if and only if they are conjugate in the group of all homeomorphisms of U. The proof is an application of Teichmuller's theorem on extremal quasiconformal mappings. The corresponding conditions for Fuchsian groups ((2), section 2) are quite elementary to prove and it might be hoped to give a straightforward proof for NEC groups. In this paper, we investigate some algebraic properties of NEC groups, in particular their reflexions, and prove some of Macbeath's results in a more elementary (but not so simple!) way. We also look at the structure of the 'canonical Fuchsian group' of an NEC group; i.e. the subgroup of an NEC group which consists of orientation pre- serving isometries. 2. Signatures of NEC groups. (This section is a short account of paragraphs 3, 4 and 5 in (2).) NEC groups are classified according to their signature. The signature of an NEC group F is either of the form (g; +;[m v ...,m r ];{(n ll ,...,n 1Si ),...,(n kl ,...,n kSt )}) (1) or (g; -;[m u ...,m r ];{(n xl ,...,n 1Si ),...,{n ku ...,n kSk )}). (2)