, Volume 95, Issue 3, pp 615-628

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

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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.