Mathematical Proceedings of the Cambridge Philosophical Society

Research Article

On the structure of non-Euclidean crystallographic groups

David Singermana1

a1 The University, Southampton

Let S0305004100048891_xs1D4D6 denote the group of transformations of the upper-half complex plane U onto itself of the form

S0305004100048891_eqn001

S0305004100048891_eqn002

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 S0305004100048891_xs1D4D6 its group of isometries. The set of elements of type I, the orientation-preserving isometries form a subgroup of index two in S0305004100048891_xs1D4D6, which we denote by S0305004100048891_inline001.

(Received May 12 1972)