Let G be a group of affine transformations of the plane R² acting freely and withdiscrete orbits (in the subspace topology). We give an elementary, self-contained proofthat G then acts properly discontinuously, so that the orbit space R²/G is a (Hausdorff, complete flat affine) surface. In general a free action with discrete orbits need not beproper, and whether the manifold-quotient version holds in arbitrary dimension is open (Kapovich, Question 14) ; for n=2 this is Charlap's Open Problem 1. 1, for which only anunpublished proof-sketch of Kuiper had been recorded. The classification of the resultinggroups into Kuiper's three models is classical; the contribution here is a complete, elementary proof of the passage from the weak hypothesis to proper discontinuity indimension two. The only external inputs are the existence of a recurrent point for ahomeomorphism of a compact metric space and Weyl's equidistribution theorem.
Maik Pickl (Sun,) studied this question.