Properly discontinuous groups of affine transformations with orthogonal linear part

Let Gamma be a subgroup of the group of all affine transformations of a real affine space A of finite dimension. Suppose that Gamma acts properly discontinuously on A. We determine which orthogonal groups can occur as Zariski closures of the linear part of Gamma. Our methods yield a proof of Auslander's conjecture for affine spaces of dimension at most 6.