We prove that the natural map Hb2(&)‘H2(&) from bounded to usual cohomology is injective if & is an irreducible cocompact lattice in a higher rank Lie group. This result holds also for nontrivial unitary coefficients, and implies finiteness results for &: the stable commutator length vanishes and any C1-action on the circle is almost trivial. We introduce the continuous bounded cohomology of a locally compact group and prove our statements by relating Hb”(&) to the continuous bounded cohomology of the ambient group with coefficients in some induction module.
Cite this article
Marc Burger, Nicolas Monod, Bounded cohomology of lattices in higher rank Lie groups. J. Eur. Math. Soc. 1 (1999), no. 2 pp. 199–235