Geodesic
Vanishing of the First Cohomologies for Lattices in Lie groups
Journal of Lie theory, Tome 12 (2002) no. 2, pp. 449-46

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We prove the following "maximal" theorem on vanishing of the first cohomologies. Let G be a connected semisimple Lie group with a lattice Γ. Assume that there is no epimorphism ϕ : G --> H onto a Lie group H locally isomorphic to SO(1, n) or SU(1, n) such that ϕ(Γ) is a lattice in H. Then H1(Γ, ρ) = 0 for any finite-dimensional representation ρ of Γ over R. This generalizes Margulis' Theorem on vanishing of the first cohomologies for lattices in higher rank semisimple Lie groups. Some applications for proving general results on the structure of lattices in arbitrary Lie groups, are given. 
@article{JLT_2002_12_2_JLT_2002_12_2_a7,
     author = {A. N. Starkov },
     title = {Vanishing of the {First} {Cohomologies} for {Lattices} in {Lie} groups},
     journal = {Journal of Lie theory},
     pages = {449--46},
     publisher = {mathdoc},
     volume = {12},
     number = {2},
     year = {2002},
     url = {http://geodesic.mathdoc.fr/item/JLT_2002_12_2_JLT_2002_12_2_a7/}
}
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A. N. Starkov . Vanishing of the First Cohomologies for Lattices in Lie groups. Journal of Lie theory, Tome 12 (2002) no. 2, pp. 449-46. http://geodesic.mathdoc.fr/item/JLT_2002_12_2_JLT_2002_12_2_a7/