# Isometry Groups of Proper Hyperbolic Spaces

@article{Hamenstdt2005IsometryGO, title={Isometry Groups of Proper Hyperbolic Spaces}, author={Ursula Hamenst{\"a}dt}, journal={Geometric and Functional Analysis}, year={2005}, volume={19}, pages={170-205} }

Abstract.Let X be a proper hyperbolic geodesic metric space and let G be a closed subgroup of the isometry group Iso(X) of X. We show that if G is not elementary then for every p ∈ (1, ∞) the second continuous bounded cohomology group H2cb(G, Lp(G)) does not vanish. As an application, we derive some structure results for closed subgroups of Iso(X).

## 9 Citations

Isometry groups of proper CAT(0)-spaces of rank one

- Mathematics
- 2012

LetX be a proper CAT.0/-space and letG be a closed subgroup of the isometry group Iso.X/ of X . We show that if G is non-elementary and contains a rank-one element then its second continuous bounded…

Rank-one isometries of proper CAT(0)-spaces

- Mathematics
- 2008

Let G be a non-elementary group of isometries of a proper CAT(0)-space with limit set L. We survey properties of the action of G on L under the assumption that G contains a rank-one element. Among…

Bounded cohomology and isometry groups of hyperbolic spaces

- Mathematics
- 2005

Let $X$ be an arbitrary hyperbolic geodesic metric space and let $\Gamma$ be a countable subgroup of the isometry group ${\rm Iso}(X)$ of $X$. We show that if $\Gamma$ is non-elementary and weakly…

Bounded cohomology, cross ratios and cocycles

- Mathematics
- 2005

We use cross ratios to describe second real continuous bounded cohomology for locally compact topological groups. We also derive a rigidity result for cocycles with values in the isometry group of a…

Topology of open nonpositively curved manifolds

- Mathematics
- 2013

This is a survey of topological properties of open, complete nonpositively curved manifolds which may have infinite volume. Topics include topology of ends, restrictions on the fundamental group, as…

Bounded cohomology with coefficients in uniformly convex Banach spaces

- Mathematics
- 2013

We show that for acylindrically hyperbolic groups $\Gamma$ (with no nontrivial finite normal subgroups) and arbitrary unitary representation $\rho$ of $\Gamma$ in a (nonzero) uniformly convex Banach…

Rank-One Isometries of Buildings and Quasi-Morphisms of Kac–Moody Groups

- Mathematics
- 2008

Given an irreducible non-spherical non-affine (possibly non-proper) building X, we give sufficient conditions for a group G < Aut(X) to admit an infinite-dimensional space of non-trivial…

The Action homomorphism, quasimorphisms and moment maps on the space of compatible almost complex structures

- Mathematics
- 2011

We extend the definition of Weinstein's Action homomorphism to Hamiltonian actions with equivariant moment maps of (possibly infinite-dimensional) Lie groups on symplectic manifolds, and show that…

An invitation to bounded cohomology

- Mathematics
- 2006

A selection of aspects of the theory of bounded cohomology is presented. The emphasis
is on questions motivating the use of that theory as well as on some concrete issues suggested
by its study.…

## References

SHOWING 1-10 OF 17 REFERENCES

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 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)…

Reduction of cocycles with hyperbolic targets

- MathematicsErgodic Theory and Dynamical Systems
- 1996

Abstract We show that any cocycle from an ergodic, finite measure preserving action of a higher-rank group to a closed subgroup of the isometry group of a proper, geodesic hyperbolic, ‘at most…

Bounded cohomology of lattices in higher rank Lie groups

- Mathematics
- 1999

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…

Bounded cohomology and isometry groups of hyperbolic spaces

- Mathematics
- 2005

Let $X$ be an arbitrary hyperbolic geodesic metric space and let $\Gamma$ be a countable subgroup of the isometry group ${\rm Iso}(X)$ of $X$. We show that if $\Gamma$ is non-elementary and weakly…

Continuous Bounded Cohomology of Locally Compact Groups

- Mathematics
- 2001

The purpose of this monograph is (a) to lay the foundations for a conceptual approach to bounded cohomology; (b) to harvest the resulting applications in rigidity theory. Of central importance is the…

Cocycle superrigidity and bounded cohomology for negatively curved spaces

- Mathematics
- 2004

We introduce new techniques to extend superrigidity theory beyond the scope of Lie or algebraic groups. We construct a cohomological invariant which accounts for, and generalizes, all known…

Orbit equivalence rigidity and bounded cohomology

- Mathematics
- 2006

We establish new results and introduce new methods in the theory of measurable orbit equivalence, using bounded cohomology of group representations. Our rigidity statements hold for a wide…

Bounded cohomology, cross ratios and cocycles

- Mathematics
- 2005

We use cross ratios to describe second real continuous bounded cohomology for locally compact topological groups. We also derive a rigidity result for cocycles with values in the isometry group of a…