Journal of Lie Theory Vol. 12, No. 2, pp. 449460 (2002) 

Vanishing of the first cohomologies for lattices in Lie groupsA. N. StarkovA. N. StarkovAllRussian Institute of Electro\technics 143500, Istra, Moscow Region Russia RDIEalex@istra.ru and Dept. of Mechanics and Mathematics Moscow State University, 117234 Moscow Russia. 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 finitedimensional 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. Full text of the article:
Electronic fulltext finalized on: 6 May 2002. This page was last modified: 21 May 2002.
© 2002 Heldermann Verlag
