The topological and geometrical finiteness of complete flat Lorentzian 3-manifolds with free fundamental groups

Details The topological and geometrical finiteness of complete flat Lorentzian 3-manifolds with free fundamental groups The topological and geometrical finiteness of complete flat Lorentzian 3-manifolds with free fundamental groups

2012-04-24

arxiv.org/abs/1204.5308v1

Mathematics

Geometric Topology

40 pages, 6 figures

Scientific paper

We prove the topological tameness of a 3-manifold with a free fundamental group admitting a complete flat Lorentzian metric; i.e., a Margulis space-time isomorphic to the quotient of the complete flat Lorentzian space by the free and properly discontinuous isometric action of the free group of rank $\geq 2$. We will use our particular point of view that a Margulis space-time is a real projective manifold in an essential way. The basic tools are a bordification by a closed real projective surface with a free holonomy group, the important work of Goldman, Labourie, and Margulis on geodesics in the Margulis space-times and the 3-manifold topology. Finally, we show that Margulis space-times are geometrically finite under our definition.

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