Around 1980 Margulis proved the existence of proper affine actions of nonabelian free groups on 3-space. Later Todd Drumm found polyhedra from which one can build fundamental domains for these properly discontinuous actions and give a satisfying geometric picture of these examples. The quotients are geodesically complete flat Lorentzian 3-manifolds and intimately relate to complete hyperbolic surfaces and deformations in which every geodesic lamination infinitesimally lengthens. In this talk I will describe the geometry of these manifolds, the dynamics of their geodesic flows and how this leads to a topological classification of the 3-manifolds as well as the determination of the moduli space. This represents joint work with Fried, Drumm, Margulis, Charette, Labourie and Choi, and closely relates to recent work by Danciger, Gueritaud and Kassel on constant curvature Lorentzian 3-manifolds.
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