Integrable measure equivalence and rigidity of hyperbolic lattices

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.