Properly discontinuous groups of affine transformations with orthogonal linear part

Proper affine actions and geodesic flows of hyperbolic surfaces

“…Proper actions with irreducible linear parts occur only when V has dimension 4k C3 (see Abels [1], Abels-Margulis-Soifer [2], [3] and Labourie [31]). In those dimensions, AbelsMargulis-Soifer [2], [3] constructed proper affine deformations of Fuchsian actions of free groups.…”

Crystallographic actions on contractible algebraic manifolds

“…Since Γ is a discrete subgroup of G 0 , by Corollary 5.4 in [2], it follows that Γ is a discrete subgroup of S. Let C be an arbitrary compact subset of X S . Let C ′ be a compact subset of S such that π(C ′ ) = C where π : S → X S is the quotient map.…”

“…It is not difficult to see that the isotropy group Gx 0 at the pointx 0 = G 1 x 0 of X is V H where V = U ∩ Gx 0 . We define a left action of G on the homogeneous space of left cosets U/V by (1) wh…”

“…Moreover, the Auslander conjecture is still open and is only known to hold up to dimension 6. (In fact, up to our knowledge, a full proof has only been given up to dimension 3 (see [21]) and a positive result up to dimension 6 has been announced in [1]). …”

Geodesics in Margulis spacetimes

“…Abels-Margulis-Soifer [2,3] proved that if a discrete group of Lorentz isometries acts properly on Minkowski space E, and L(Γ) is Zariski dense in O(n − 1, 1), then n = 3. Consequently every complete flat Lorentz manifold is a flat Euclidean affine fibration over a complete flat Lorentz 3-manifold.…”