Integrable measure equivalence and rigidity of hyperbolic lattices

@article{Bader2010IntegrableME,
  title={Integrable measure equivalence and rigidity of hyperbolic lattices},
  author={Uri Bader and Alex Furman and Roman Sauer},
  journal={Inventiones mathematicae},
  year={2010},
  volume={194},
  pages={313-379}
}
We study rigidity properties of lattices in $\operatorname {Isom}(\mathbf {H}^{n})\simeq \mathrm {SO}_{n,1}({\mathbb{R}})$, n≥3, and of surface groups in $\operatorname {Isom}(\mathbf {H}^{2})\simeq \mathrm {SL}_{2}({\mathbb{R}})$ in the context of integrable measure equivalence. The results for lattices in $\operatorname {Isom}(\mathbf {H}^{n})$, n≥3, are generalizations of Mostow rigidity; they include a cocycle version of strong rigidity and an integrable measure equivalence classification… 

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References

SHOWING 1-10 OF 71 REFERENCES

Cocycle Superrigidity for Profinite Actions of Property (T) Groups

Let $\Gamma$ be an irreducible lattice in a product of two locally compact groups and assume that $\Gamma$ is densely embedded in a profinite group $K$. We give necessary conditions which imply that

Strong rigidity of II1 factors arising from malleable actions of w-rigid groups, I

We consider crossed product II1 factors $M = N\rtimes_{\sigma}G$, with G discrete ICC groups that contain infinite normal subgroups with the relative property (T) and σ trace preserving actions of G

Orbit equivalence rigidity

Consider a countable group acting ergodically by measure p reserving transformations on a probability space (X,µ), and let R be the corresponding orbit equivalence relation on X. The following

Cocycle superrigidity and harmonic maps with infinite-dimensional targets

We announce a generalization of Zimmer's cocycle superrigidity theorem proven using harmonic map techniques. This allows us to generalize many results concerning higher rank lattices to all lattices

On the superrigidity of malleable actions with spectral gap

Some of the most interesting aspects of the dynamics of measure preserving actions of countable groups on probability spaces, V rx (X, /?), are revealed by the study of group measure space von

Mostow-Margulis rigidity with locally compact targets

Abstract. Let $ \Gamma $ be a lattice in a simple higher rank Lie group G. We describe all locally compact (not necessarily Lie) groups H in which $ \Gamma $ (with the only exception of non-uniform

Convergence groups are Fuchsian groups

A group of homeomorphisms of the circle satisfying the "convergence property" is shown to be the restriction of a discrete group of Mobius transformations of the unit disk. This completes the proof

Discrete Subgroups of Semisimple Lie Groups

1. Statement of Main Results.- 2. Synopsis of the Chapters.- 3. Remarks on the Structure of the Book, References and Notation.- 1. Preliminaries.- 0. Notation, Terminology and Some Basic Facts.- 1.

Coût des relations d’équivalence et des groupes

Abstract.We study a new dynamical invariant for dicrete groups: the cost. It is a real number in {1−1/n}∪[1,∞], bounded by the number of generators of the group, and it is well behaved with respect

Amalgamated free products of weakly rigid factors and calculation of their symmetry groups

We consider amalgamated free product II1 factors M = M1*BM2*B… and use “deformation/rigidity” and “intertwining” techniques to prove that any relatively rigid von Neumann subalgebra Q ⊂ M can be
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