EMIS ELibM Electronic Journals Journal of Lie Theory
Vol. 12, No. 2, pp. 449--460 (2002)

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Vanishing of the first cohomologies for lattices in Lie groups

A. N. Starkov

A. N. Starkov
All-Russian Institute of Electro\-technics
143500, Istra, Moscow Region


Dept. of Mechanics and Mathematics
Moscow State University,
117234 Moscow

Abstract: We prove the following ``maximal'' theorem on vanishing of the first cohomologies. Let $G$ be a connected semisimple Lie group with a lattice $\Gamma$. Assume that there is no epimorphism $\phi\colon G\to H$ onto a Lie group $H$ locally isomorphic to SO$\scriptstyle(1,n)$ or SU$\scriptstyle(1,n)$ such that $\phi(\Gamma)$ is a lattice in $H$. Then $H^1(\Gamma,\rho)=0$ for any finite-dimensional representation $\rho$ of $\Gamma$ over ${\bf R}$. 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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Electronic fulltext finalized on: 6 May 2002. This page was last modified: 21 May 2002.

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