Vanishing of the first cohomologies for lattices in Lie groups

A. N. Starkov

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.

Full text of the article:

• Compressed DVI file (26 kilobytes)
• Compressed PostScript file (74 kilobytes)
• PDF file (186 kilobytes)