# Bounded cohomology of lattices in higher rank Lie groups

Bounded cohomology of lattices in higher rank Lie groups We prove that the natural map Hb 2(Γ)?H2(Γ) from bounded to usual cohomology is injective if Γ is an irreducible cocompact lattice in a higher rank Lie group. This result holds also for nontrivial unitary coefficients, and implies finiteness results for Γ: 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 •(Γ) to the continuous bounded cohomology of the ambient group with coefficients in some induction module. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of the European Mathematical Society Springer Journals

# Bounded cohomology of lattices in higher rank Lie groups

, Volume 1 (2) – Apr 1, 1999
37 pages

/lp/springer-journals/bounded-cohomology-of-lattices-in-higher-rank-lie-groups-tWLGt0EGjY
Publisher
Springer Journals
Subject
Mathematics; Mathematics, general
ISSN
1435-9855
DOI
10.1007/s100970050007
Publisher site
See Article on Publisher Site

### Abstract

We prove that the natural map Hb 2(Γ)?H2(Γ) from bounded to usual cohomology is injective if Γ is an irreducible cocompact lattice in a higher rank Lie group. This result holds also for nontrivial unitary coefficients, and implies finiteness results for Γ: 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 •(Γ) to the continuous bounded cohomology of the ambient group with coefficients in some induction module.

### Journal

Journal of the European Mathematical SocietySpringer Journals

Published: Apr 1, 1999