11 - The orientability of subgroups of plane groups

Summary

INTRODUCTION

Let X be either the hyperbolic plane, the Euclidean plane or the sphere with its associated metric and let Γ be a group of isometries of X. Γ is properly discontinuous if for each compact set K ⊂ X, {γ ε Γ ∣ γ K ∩ K ≠ ø} is finite. It then follows that X/Γ is a surface, possibly with boundary, [1, p.52]. For example, if X is the Euclidean plane then the properly discontinuous groups Γ for which X/Γ is compact are the 17 crystallographic groups and X/Γ is either a torus, a sphere, a disc, an annulus, a Klein bottle, a Möbius band or a projective plane. If r is the hyperbolic plane we obtain infinitely many examples including the non-Euclidean crystallographic groups described in.

In the study of these groups the topological characteristics of the quotient surface X/Γ and the nature of the singularities of the natural projection p : X → X/Γ are often invariants of the group Γ. If Λ is a subgroup of Γ and if we know either the permutation representation of r on the Λ-cosets, or equivalently a Schreier coset graph for Λ in Γ, then these characteristics are computable for Λ.