Free affine actions on the plane with discrete orbits are properly discontinuous

Let G be a group of affine transformations of the plane R^2 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 R^2/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.