Affine group actions on Euclidean space.

Abstract

Upto affine conjugacy, we describe properly discontinuous rank two affine groups of Euclidean space (primarily dimensions two and three) in terms of "coordinates" of generators. In dimension two, a chosen generator is put into a normal form. The commuting condition simplifies a second generator, which can be identified with a point of the plane. Thus, R2 can be viewed as a parameter space of groups (since the first normalized generator is common to each group). A homomorphism Res:R2->R (the residue) singles out properly discontinuous groups G isomorphic to Z+Z. Affine conjugacy of two groups is characterized by their residues and an element of GL (2, Z). As a consequence, we show (i) There are uncountably many conjugacy classes of properly discontinuous rank two groups, and (ii) Each point of Ker(Res) is the limit point of every conjugacy class. An analog of the residue is used to determine three dimensional properly discontinuous rank two groups, although this description does not cover all normalized forms.