# Bounded cohomology of lattices in higher rank Lie groups

@article{Burger1999BoundedCO, title={Bounded cohomology of lattices in higher rank Lie groups }, author={M. Burger and Nicolas Monod}, journal={Journal of the European Mathematical Society}, year={1999}, volume={1}, pages={199-235} }

We prove that the natural map Hb2(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…

## 195 Citations

Surjectivity of the comparison map in bounded cohomology for Hermitian Lie groups

- Mathematics
- 2009

We prove surjectivity of the comparison map from continuous bounded cohomology to continuous cohomology for Hermitian Lie groups with finite center. For general semisimple Lie groups with finite…

On the bounded cohomology of semi-simple groups, S-arithmetic groups and products

- Mathematics
- 2007

Abstract We prove vanishing results for Lie groups and algebraic groups (over any local field) in bounded cohomology. The main result is a vanishing below twice the rank for semi-simple groups.…

Isometric Embeddings in Bounded Cohomology

- Mathematics
- 2013

This paper is devoted to the construction of norm-preserving maps between bounded cohomology groups. For a graph of groups with amenable edge groups we construct an isometric embedding of the direct…

Bounded cohomology of subgroups of mapping class groups (Hyperbolic Spaces and Discrete Groups)

- Mathematics
- 2001

We show that every subgroup of the mapping class group MCG(S )o f ac ompact surface S is either virtually abelian or it has innite dimensional second bounded cohomology. As an application, we give…

The cohomology of semi-simple Lie groups, viewed from infinity

- MathematicsTransactions of the American Mathematical Society, Series B
- 2022

We prove that the real cohomology of semi-simple Lie groups admits boundary values, which are measurable cocycles on the Furstenberg boundary. This generalises known invariants such as the Maslov…

Bounded Cohomology and Deformation Rigidity in Complex Hyperbolic Geometry

- Mathematics
- 2005

We develop further basic tools in the theory of continuous bounded cohomology of locally compact groups. We apply this tools to establish a Milnor-Wood type inequality in a very general context and…

Stabilization of Bounded Cohomology for Classical Groups

- Mathematics
- 2022

We show that bounded cohomology stabilizes along sequences of classical real and complex Lie groups, and along sequences of lattices in them. Our method is based on a stability criterion which adapts…

Continuous bounded cohomology and applications to rigidity theory

- Mathematics
- 2001

We present a theory of continuous bounded cohomology of locally compact groups with coefficients in Banach modules. A central role is played by amenable actions, as they give rise to relatively…

Transgression in bounded cohomology and a conjecture of Monod

- Mathematics
- 2018

We develop an algebro-analytic framework for the systematic study of the continuous bounded cohomology of Lie groups in large degree. As an application, we examine the continuous bounded cohomology…

ec 2 00 0 Bounded cohomology of subgroups of mapping class groups

- Mathematics
- 2009

We show that every subgroup of the mapping class group Mod(S) of a compact surface S is either virtually abelian or it has infinite dimensional second bounded cohomology. As an application, we give…

## References

SHOWING 1-10 OF 41 REFERENCES

The Second Bounded Cohomology of a Group Acting on a Gromov‐Hyperbolic Space

- Mathematics
- 1998

Suppose a group G acts on a Gromov‐hyperbolic space X properly discontinuously. If the limit set L(G) of the action has at least three points, then the second bounded cohomology group of G , Hb2(G;R)…

Bounded cohomology of group constructions

- Mathematics
- 1996

It is proved that the singular partHb2/(2)(G) of the second group of bounded homology of the discrete groupG is isomorphic to the space of 2-cocycles that vanish on the diagonal. For groupsG…

Bounded cohomology of certain groups of homeomorphisms

- Mathematics
- 1985

We consider the condition when bounded cohomology injects into ordinary cohomology and prove the vanishing of bounded cohomology of the group of all compactly supported homeomorphisms of RW.…

Foundations of the theory of bounded cohomology

- Mathematics
- 1987

In this paper we give a new approach to the theory of bounded cohomology. The ideas of relative homological algebra, modified so that they are based on a natural seminorm in the bounded cohomology,…

Straightening and bounded cohomology of hyperbolic groups

- Mathematics
- 2001

Abstract. It was stated by M. Gromov [Gr2] that, for any hyperbolic group G, the map from bounded cohomology
$ H^n_b(G,{\Bbb R}) $ to
$ H^n(G,{\Bbb R}) $ induced by inclusion is surjective for
$ n…

Arithmetic groups of higher -rank cannot act on 1-manifolds

- Mathematics
- 1994

Let r be a subgroup of finite index in SLn(Z) with n > 3. We show that every continuous action of r on the circle SI or on the real line R factors through an action of a finite quotient of r. This…

The gromov norm of the Kaehler class of symmetric domains

- Mathematics
- 1987

Let D be bounded symmetric domain of rank p, and let o9 be the KRhler form of its Bergmann metric, where the metric is normalized so that the minimum holomorphic sectional curvature is 1. I fX is a…

The second bounded cohomology of an amalgamated free product of groups

- Mathematics
- 1999

We study the second bounded cohomology of an amalgamated free product of groups, and an HNN extension of a group. As an application, we show that a group with infinitely many ends has infinite…

The zero-norm subspace of bounded cohomology

- Mathematics
- 1997

AbstractLet
$ \Sigma $ be a closed, orientable surface of genus > 1. In this paper, non-trivial elements
$\alpha $ of the third bounded cohomology
$H_b^3(\Sigma;\bf {R})$ with
$\Vert \alpha \Vert…

Products of similar matrices

- Mathematics
- 1998

Let A and B be n x n matrices of determinant 1 over a field K, with n > 2 or fKf > 3. We show that if A is not a scalar matrix, then B is a product of matrices similar to A. Analogously, we…