Mathematical Conversations

Bieberbach's 1912 theory of Euclidean crystallographic groups provides a satisfying qualitative classification of flat Riemannian manifolds. In 1977 Milnor asked whether a similar picture could extend to flat affine manifolds, that is, when the transformations are affine but not necessarily isometries of Euclidean space. In particular he asked whether a nonabelian free group could act properly discontinuously by affine transformations. In the early 1980's Margulis surprised the world by showing that such groups do exist. In fact, such examples are quotients of Minkowski (2+1)-space by discrete groups of Lorentz-isometries, and are now called Margulis spacetimes. In his 1990 doctoral thesis Todd Drumm gave a geometric construction by fundamental domains built from surfaces called crooked planes. This completes the classification of geodesically complete affine 3-manifolds.