# Integrable measure equivalence and rigidity of hyperbolic lattices

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## Abstract

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. Despite the lack of Mostow rigidity for *n*=2 we show that cocompact lattices in \(\operatorname {Isom}(\mathbf {H}^{2})\) allow a similar integrable measure equivalence classification.

## Keywords

Mapping Class Group Uniform Lattice Cocompact Lattice Compact Metrizable Space Strong Rigidity## Notes

### Acknowledgements

U. Bader and A. Furman were supported in part by the BSF grant 2008267. U. Bader was also supported in part by the ISF grant 704/08 and the ERC grant 306706. A. Furman was partly supported by the NSF grants DMS 0604611 and 0905977. R. Sauer acknowledges support from the *Deutsche Forschungsgemeinschaft*, made through grant SA 1661/1-2.

We thank the referee for his detailed and careful report, especially for his recommendations that led to a restructuring of Sect. 3 in a previous version.

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