, Volume 95, Issue 3, pp 615-628

Autour de la conjecture de L. Markus sur les variétés affines

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Summary

For any subgroupG of (ℝn), we introduce some integer discGn called thediscompacity ofG. This number measures to what extent the closure ofG is not compact. The Markus' conjecture says that a compact affinely flat unimodular manifold is complete. Our main result (called the ≪discompact theorem≫) is that this conjecture is true under the assumption that the linear holonomy i.e. the parallel transport has discompacity ≦1. Because discSO(n−1, 1)=1, this ensures that a compact flat Lorentz manifoldM is geodesically complete. Hence, by a previous result of W. Goldman and Y. Kamishima [GK], such aM is, up to finite covering, a solvmanifold. This achieves the proof of a Bieberbach's theorem for compact Lorentz flat manifolds.