Unlike Euclidean crystallographic groups, properly discontinuous groups of affine transformations need not be amenable. For example, a free group of rank two admits a properly discontinuous affine action on 3-space. Milnor imagined how one might construct such an action: deform a Schottky subgroup of O(2,1) inside the group of Lorentzian isometries of Minkowski space, although as he wrote in 1977, ``it seems difficult to decide whether the resulting group action is properly discontinuous.'' In 1983, Margulis, while trying to prove such groups don't exist, constructed the first examples. In his 1990 doctoral thesis, Drumm constructed explicit geometric examples from fundamental polyhedra, and showed that every non-cocompact Fuchsian subgroup of O(2,1) admits proper affine deformations. (Work of Fried-Goldman and Mess implies that these conditions are necessary.)
This talk will discuss the classification and construction of these manifolds, and the relation with deformations of hyperbolic structures.