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Organizers |
Flat Lorentz 3-Manifolds
by
Bill Goldman
University of Maryland
In 1977, Milnor asked whether a free group of rank 2 can act properly by affine transformations on R}3. He suggested that one might start with a Fuchsian group G_0 discretely embedded in SO(2,1) and deform it by "adding translations" to obtain a group G of affine transformations. However he said, "it seems difficult to determine whether the group acts properly." In 1983, Margulis showed that this is indeed possible, and in 1990 Drumm constructed examples using fundamental polyhedra. Drumm showed that every noncocompact torsionfree discrete subgroup of SO(2,1) admits proper affine deformations. Margulis introduced a class function A:G → R which is a ``signed marked Lorentzian length spectrum.'' Furthermore he showed that unless A(g) is positive (respectively negative) for all nontrivial elements of G, then the affine deformation is nonproper. It was natural to conjecture that this necessary condition is sufficient. In this talk I will describe an extension of this invariant to geodesic currents (joint work with Labourie and Margulis), relating this conjecture to the ergodic theory of the geodesic flow of convex cocompact hyperbolic surfaces, and will survey recent results on proper affine actions on R^3.
Date received: December 10, 2003
Copyright © 2003 by the author(s). The author(s) of this work and the organizers of the conference have granted their consent to include this abstract in Topology Atlas. Document # camw-13.