Bounded cohomology of lattices in higher rank Lie groups

Abstract

We prove that the natural map $H_b^2(\Gamma )\to H^2(\Gamma )$ from bounded to usual cohomology is injective if $\Gamma$ is an irreducible cocompact lattice in a higher rank Lie group. This result holds also for nontrivial unitary coefficients, and implies finiteness results for $\Gamma$: the stable commutator length vanishes and any $C^1$-action on the circle is almost trivial. We introduce the continuous bounded cohomology of a locally compact group and prove our statements by relating $H_b^{\bullet }(\Gamma )$ to the continuous bounded cohomology of the ambient group with coefficients in some induction module.

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