Bounded cohomology of lattices in higher rank Lie groups (Q1290362)

It is shown that the natural map \(H_b^2(\Gamma) \to H^2(\Gamma)\) from a bounded cohomology to the usual cohomology is injective if \(\Gamma\) is an irreducible cocompact lattice in a higher rank Lie group. The result holds also for nontrivial unitary coefficients. This implies that the stable length on the commutator \([\Gamma,\Gamma]\) vanishes and any \(C^1\)-action of \(\Gamma\) on the circle is almost trivial. Similar results are also obtained for the continuous bounded homologies of locally compact groups.