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Journal of Lie Theory 12 (2002), No. 2, 449--460
Copyright Heldermann Verlag 2002

Vanishing of the First Cohomologies for Lattices in Lie groups

A. N. Starkov
Dept. of Mechanics and Mathematics, Moscow State University, 117234 Moscow, Russia

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.

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