Bounded cohomology of lattices in higher rank Lie groups

Abstract

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.