On the first cohomology of cocompact arithmetic groups

Abstract

Results of Matsushima and Raghunathan imply that the first cohomology of a cocompact irreducible lattice in a semisimple Lie group G, with coefficients in an irreducible finite dimensional representation of G, vanishes unless the Lie group is SO(n, 1) or SU(n, 1) and the highest weight of the representation is an integral multiple of that of the standard representation. We show here that every cocompact arithmetic lattice in SO(n, 1) contains a subgroup of finite index whose first cohomology is non-zero when the representation is one of the exceptional types mentioned above.