Inventiones mathematicae

, Volume 95, Issue 3, pp 615–628

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

  • Yves Carrière

DOI: 10.1007/BF01393894

Cite this article as:
Carrière, Y. Invent Math (1989) 95: 615. doi:10.1007/BF01393894


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.

Copyright information

© Springer-Verlag 1989

Authors and Affiliations

  • Yves Carrière
    • 1
  1. 1.Institut FourierSaint Martin d'HèresFrance

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