As curvature is one of the most important invariants in differential geometry, the classification of complete manifolds of zero curvature is of fundamental significance. Such a manifold M is the quotient E/G of an affine space E by a discrete group G of affine transformations of E acting properly and freely. When G is a group of Euclidean isometries, M is flat Riemannian, and when G is a group of Lorentz isometries of Minkowski space, then M is flat Lorentzian. The condition that the quotient E/G be a smooth manifold is equivalent to the action of G being properly discontinuous.

In 1912 Bieberbach proved that if M is flat Riemannian, then its fundamental group G is a finite extension of a free abelian group. This result provides a satisfying qualitative understanding of such structures and leads to an algebraic classification. However, if M is a manifold with a general flat affine connection (even if M is flat Lorentzian), such a classification is unknown, unless the fundamental group G is already known to be a finite extension of a solvable group. It was natural to conjecture that only such finite-by-solvable groups admit properly discontinuous actions by affine transformations.

In his influential 1977 survey paper, Milnor provided both ``evidence'' for this conjecture (based on the analogous question for connected Lie groups), and a strategy for constructing a counterexample. He suggested that one should start with a free discrete group of isometries acting on the hyperbolic plane (which is embedded as a "sphere of imaginary radius" in Minkowski space) and then add translational parts to make the action free. But as he stated, ``it seems difficult to decide whether the resulting group action is properly discontinuous.'' In 1983, Margulis, while trying to prove this conjecture, discovered counterexamples. In his 1990 doctoral thesis, Drumm constructed explicit geometric examples from fundamental polyhedra.

In this talk I hope to convince you these groups do actually exist and discuss relationship with 2-dimensional hyperbolic geometry and Teichmuller theory.