# 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…

## 48 Citations

### Borel invariant for Zimmer cocycles of 3-manifold groups

- Mathematics
- 2019

Let $\Gamma$ be a non-uniform lattice of $\text{PSL}(2,\mathbb{C})$. Given any representation $\rho:\Gamma \rightarrow \text{PSL}(n,\mathbb{C})$ we can define a numerical invariant $\beta_n(\rho)$,…

### Superrigidity of maximal measurable cocycles of complex hyperbolic lattices

- Materials ScienceMathematische Zeitschrift
- 2021

Let Γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek}…

### Invariance of conformal dimension under $L^p$-OE for hyperbolic coxeter groups with CLP

- Mathematics
- 2018

Conformal dimension is one of the important geometric quantities attached to a metric space. It is well known that if two Gromov hyperbolic groups are quasi-isometric, then conformal dimensions (of…

### Integrable tautness of isometries of complex hyperbolic spaces.

- Mathematics
- 2020

Consider $n \geq 2$. In this paper we prove that the group $\text{PU}(n,1)$ is $1$-taut. This result concludes the study of $1$-tautness of rank-one Lie groups of non-compact type. Additionally the…

### Integrable measure equivalence and the central extension of surface groups

- Mathematics
- 2014

Let $\Gamma_g$ be a surface group of genus $g\geq 2$. It is known that the canonical central extension $\tilde{\Gamma}_g$ and the direct product $\Gamma_g\times \mathbb{Z}$ are quasi-isometric. It is…

### Permanence properties of property A and coarse embeddability for locally compact groups

- Mathematics
- 2014

If $H$ is a lattice in a locally compact second countable group $G$, then we show that $G$ has property A (respectively is coarsely embeddable into Hilbert space) if and only if $H$ has property A…

### Superrigidity of maximal measurable cocycles of complex hyperbolic lattices

- MathematicsMathematische Zeitschrift
- 2021

Let $$\Gamma $$ Γ be a torsion-free lattice of $$PU (p,1)$$ P U ( p , 1 ) with $$p \ge 2$$ p ≥ 2 and let $$(X,\mu _X)$$ ( X , μ X ) be an ergodic standard Borel probability $$\Gamma $$ Γ -space. We…

### Quantitative measure equivalence

- Mathematics
- 2020

We initiate a quantitative study of measure equivalence (and orbit equivalence) between finitely generated groups that extends the classical setting of $\mathrm L^p$ measure equivalence. In…

### Arithmeticity, superrigidity, and totally geodesic submanifolds

- MathematicsAnnals of Mathematics
- 2021

Let $\Gamma$ be a lattice in $\mathrm{SO}_0(n, 1)$. We prove that if the associated locally symmetric space contains infinitely many maximal totally geodesic subspaces of dimension at least $2$, then…

### A Dual Interpretation of the Gromov–Thurston Proof of Mostow Rigidity and Volume Rigidity for Representations of Hyperbolic Lattices

- Mathematics
- 2013

We use bounded cohomology to define a notion of volume of an \(\operatorname{SO}(n,1)\)-valued representation of a lattice \(\varGamma<\operatorname{SO}(n,1)\) and, using this tool, we give a…

## References

SHOWING 1-10 OF 71 REFERENCES

### Cocycle Superrigidity for Profinite Actions of Property (T) Groups

- Mathematics
- 2008

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

- Mathematics
- 2006

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

- Mathematics
- 1999

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

- Mathematics
- 2005

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

- Mathematics
- 2006

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

- Mathematics
- 2001

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

- Mathematics
- 1991

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

- Mathematics
- 1991

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

- Mathematics
- 2000

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

- Mathematics
- 2005

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…