Inventiones mathematicae

, Volume 194, Issue 2, pp 313–379 | Cite as

Integrable measure equivalence and rigidity of hyperbolic lattices

  • Uri BaderEmail author
  • Alex Furman
  • Roman Sauer


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.


Mapping Class Group Uniform Lattice Cocompact Lattice Compact Metrizable Space Strong Rigidity 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



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.


  1. 1.
    Bader, U., Furman, A., Sauer, R.: Efficient subdivision in hyperbolic groups and applications. Groups Geom. Dyn. (2013, to appear). arXiv:1003.1562
  2. 2.
    Bridson, M.R., Tweedale, M., Wilton, H.: Limit groups, positive-genus towers and measure-equivalence. Ergod. Theory Dyn. Syst. 27(3), 703–712 (2007) MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bucher, M., Burger, M., Iozzi, A.: A dual interpretation of the Gromov–Thurston proof of Mostow rigidity and volume rigidity for representations of hyperbolic lattices (2012). arXiv:1205.1018
  4. 4.
    Burger, M., Iozzi, A.: Boundary maps in bounded cohomology. Appendix to: “Continuous bounded cohomology and applications to rigidity theory” [Geom. Funct. Anal. 12(2), 219–280 (2002)] by Burger, and N. Monod. Geom. Funct. Anal. 12(2), 281–292 (2002) MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Burger, M., Iozzi, A.: A useful formula in bounded cohomology. In Seminaires et Congres, nr. 18, to appear. Available at
  6. 6.
    Burger, M., Monod, N.: Continuous bounded cohomology and applications to rigidity theory. Geom. Funct. Anal. 12(2), 219–280 (2002) MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Burger, M., Mozes, S.: \({\rm CAT}\)(-1)-spaces, divergence groups and their commensurators. J. Am. Math. Soc. 9(1), 57–93 (1996) MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Casson, A., Jungreis, D.: Convergence groups and Seifert fibered 3-manifolds. Invent. Math. 118(3), 441–456 (1994) MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Corlette, K.: Archimedean superrigidity and hyperbolic geometry. Ann. Math. (2) 135(1), 165–182 (1992) MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Corlette, K., Zimmer, R.J.: Superrigidity for cocycles and hyperbolic geometry. Int. J. Math. 5(3), 273–290 (1994) MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Fell, J.M.G., Doran, R.S.: Representations of -Algebras, Locally Compact Groups, and Banach -Algebraic Bundles. Vol. 1. Pure and Applied Mathematics, vol. 125. Academic Press, San Diego (1988). Basic representation theory of groups and algebras zbMATHGoogle Scholar
  12. 12.
    Fisher, D., Hitchman, T.: Cocycle superrigidity and harmonic maps with infinite-dimensional targets. Int. Math. Res. Not. (2006). doi: 10.1155/IMRN/2006/72405 MathSciNetGoogle Scholar
  13. 13.
    Fisher, D., Morris, D.W., Witte, K.: Nonergodic actions, cocycles and superrigidity. N.Y. J. Math. 10, 249–269 (2004) (electronic) zbMATHGoogle Scholar
  14. 14.
    Furman, A.: Orbit equivalence rigidity. Ann. Math. (2) 150(3), 1083–1108 (1999) MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Furman, A.: Gromov’s measure equivalence and rigidity of higher rank lattices. Ann. Math. (2) 150(3), 1059–1081 (1999) MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Furman, A.: Mostow-Margulis rigidity with locally compact targets. Geom. Funct. Anal. 11(1), 30–59 (2001) MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Furman, A.: A survey of measured group theory. In: Geometry, Rigidity, and Group Actions. Chicago Lectures in Math., pp. 296–374. Univ. Chicago Press, Chicago (2011) Google Scholar
  18. 18.
    Furstenberg, H.: Boundary theory and stochastic processes on homogeneous spaces. In: Harmonic Analysis on Homogeneous Spaces, Williams Coll., Williamstown, Mass., 1972. Proc. Sympos. Pure Math., vol. XXVI, pp. 193–229. Am. Math. Soc., Providence (1973) CrossRefGoogle Scholar
  19. 19.
    Furstenberg, H.: Rigidity and Cocycles for Ergodic Actions of Semisimple Lie Groups (after G.A. Margulis and R. Zimmer), Bourbaki Seminar, Vol. 1979/80. Lecture Notes in Math., vol. 842, pp. 273–292. Springer, Berlin (1981) Google Scholar
  20. 20.
    Gabai, D.: Convergence groups are Fuchsian groups. Ann. Math. (2) 136(3), 447–510 (1992) MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Gaboriau, D.: Coût des relations d’équivalence et des groupes. Invent. Math. 139(1), 41–98 (2000) MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Gaboriau, D.: Invariants l 2 de relations d’équivalence et de groupes. Publ. Math. Inst. Hautes Études Sci. 95, 93–150 (2002) (French) MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Gaboriau, D.: Examples of groups that are measure equivalent to the free group. Ergod. Theory Dyn. Syst. 25(6), 1809–1827 (2005) MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Ghys, É.: Groups acting on the circle. Enseign. Math. (2) 47(3–4), 329–407 (2001) MathSciNetzbMATHGoogle Scholar
  25. 25.
    Gromov, M.: Asymptotic invariants of infinite groups. In: Geometric Group Theory, Vol. 2, Sussex, 1991. London Math. Soc. Lecture Note Ser., vol. 182, pp. 1–295. Cambridge Univ. Press, Cambridge (1993) Google Scholar
  26. 26.
    Hjorth, G.: A converse to Dye’s theorem. Trans. Am. Math. Soc. 357(8), 3083–3103 (2005) (electronic) MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Haagerup, U., Munkholm, H.J.: Simplices of maximal volume in hyperbolic n-space. Acta Math. 147(1–2), 1–11 (1981) MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Ioana, A.: Cocycle superrigidity for profinite actions of property (T) groups. Duke Math. J. 157(2), 337–367 (2011) MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Ioana, A., Peterson, J., Popa, S.: Amalgamated free products of weakly rigid factors and calculation of their symmetry groups. Acta Math. 200(1), 85–153 (2008) MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Iozzi, A.: Bounded cohomology, boundary maps, and rigidity of representations into \({\rm Homeo}_{+}(S^{1})\) and \({\rm SU}(1,n)\). In: Rigidity in Dynamics and Geometry, Cambridge, 2000, pp. 237–260. Springer, Berlin (2002) CrossRefGoogle Scholar
  31. 31.
    Ivanov, N.V.: Foundations of the theory of bounded cohomology. Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova (LOMI) 143, 69–109 (1985), 177–178 (Russian, with English summary) zbMATHGoogle Scholar
  32. 32.
    Kechris, A.S.: Classical Descriptive Set Theory. Graduate Texts in Mathematics, vol. 156. Springer, Berlin (1995) CrossRefzbMATHGoogle Scholar
  33. 33.
    Kida, Y.: The mapping class group from the viewpoint of measure equivalence theory. Mem. Am. Math. Soc. 196(916), vii+190 (2008) MathSciNetGoogle Scholar
  34. 34.
    Kida, Y.: Orbit equivalence rigidity for ergodic actions of the mapping class group. Geom. Dedic. 131, 99–109 (2008) MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Kida, Y.: Measure equivalence rigidity of the mapping class group. Ann. Math. (2) 171(3), 1851–1901 (2010) MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Kida, Y.: Rigidity of amalgamated free products in measure equivalence. J. Topol. 4(3), 687–735 (2011) MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Löh, C.: Measure homology and singular homology are isometrically isomorphic. Math. Z. 253(1), 197–218 (2006) MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Margulis, G.A.: Non-uniform lattices in semisimple algebraic groups. In: Lie Groups and Their Representations, Proc. Summer School on Group Representations of the Bolyai János Math. Soc., Budapest, 1971, pp. 371–553. Halsted, New York (1975) Google Scholar
  39. 39.
    Margulis, G.A.: Discrete groups of motions of manifolds of nonpositive curvature. In: Proceedings of the International Congress of Mathematicians, Vancouver, B.C., 1974, vol. 2, pp. 21–34. Canad. Math. Congress, Montreal (1975) (Russian) Google Scholar
  40. 40.
    Margulis, G.A.: Discrete Subgroups of Semisimple Lie Groups. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 17. Springer, Berlin (1991) CrossRefzbMATHGoogle Scholar
  41. 41.
    Monod, N.: Continuous Bounded Cohomology of Locally Compact Groups. Lecture Notes in Mathematics, vol. 1758. Springer, Berlin (2001) CrossRefzbMATHGoogle Scholar
  42. 42.
    Monod, N., Shalom, Y.: Cocycle superrigidity and bounded cohomology for negatively curved spaces. J. Differ. Geom. 67(3), 395–455 (2004) MathSciNetzbMATHGoogle Scholar
  43. 43.
    Monod, N., Shalom, Y.: Orbit equivalence rigidity and bounded cohomology. Ann. Math. (2) 164(3), 825–878 (2006) MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Mostow, G.D.: Strong Rigidity of Locally Symmetric Spaces. Annals of Mathematics Studies, vol. 78. Princeton University Press, Princeton (1973) zbMATHGoogle Scholar
  45. 45.
    Pansu, P., Zimmer, R.J.: Rigidity of locally homogeneous metrics of negative curvature on the leaves of a foliation. Isr. J. Math. 68(1), 56–62 (1989) MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Popa, S.: On a class of type \({\rm II}\sb{1}\) factors with Betti numbers invariants. Ann. Math. (2) 163(3), 809–899 (2006) CrossRefzbMATHGoogle Scholar
  47. 47.
    Popa, S.: Strong rigidity of \(\rm II\sb{1}\) factors arising from malleable actions of w-rigid groups. I. Invent. Math. 165(2), 369–408 (2006) MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Popa, S.: Strong rigidity of \(\rm II\sb{1}\) factors arising from malleable actions of w-rigid groups. II. Invent. Math. 165(2), 409–451 (2006) MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Popa, S.: Cocycle and orbit equivalence superrigidity for malleable actions of w-rigid groups. Invent. Math. 170(2), 243–295 (2007) MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Popa, S.: Deformation and Rigidity for Group Actions and von Neumann Algebras. International Congress of Mathematicians, vol. I, pp. 445–477. Eur. Math. Soc., Zürich (2007) Google Scholar
  51. 51.
    Popa, S.: On the superrigidity of malleable actions with spectral gap. J. Am. Math. Soc. 21(4), 981–1000 (2008) CrossRefzbMATHGoogle Scholar
  52. 52.
    Prasad, G.: Strong rigidity of \({\bf Q}\)-rank 1 lattices. Invent. Math. 21, 255–286 (1973) MathSciNetCrossRefzbMATHGoogle Scholar
  53. 53.
    Ratcliffe, J.G.: Foundations of Hyperbolic Manifolds. Graduate Texts in Mathematics, vol. 149. Springer, New York (1994) CrossRefzbMATHGoogle Scholar
  54. 54.
    Ratner, M.: Interactions Between Ergodic Theory, Lie Groups, and Number Theory, Zürich, 1994, vol. 2, pp. 157–182. Birkhäuser, Basel (1995) Google Scholar
  55. 55.
    Shalom, Y.: Rigidity, unitary representations of semisimple groups, and fundamental groups of manifolds with rank one transformation group. Ann. Math. (2) 152(1), 113–182 (2000) MathSciNetCrossRefzbMATHGoogle Scholar
  56. 56.
    Shalom, Y.: Measurable Group Theory. European Congress of Mathematics, pp. 391–423. Eur. Math. Soc., Zürich (2005) Google Scholar
  57. 57.
    Stroppel, M.: Locally Compact Groups. EMS Textbooks in Mathematics. Eur. Math. Soc., Zürich (2006) CrossRefzbMATHGoogle Scholar
  58. 58.
    Tits, J.: Sur le groupe des automorphismes d’un arbre. In: Essays on Topology and Related Topics (Mémoires dédiés à Georges de Rham). Springer, Berlin (1970) Google Scholar
  59. 59.
    Thurston, W.P.: The geometry and topology of three-manifolds (1978). Available at
  60. 60.
    Zastrow, A.: On the (non)-coincidence of Milnor-Thurston homology theory with singular homology theory. Pac. J. Math. 186(2), 369–396 (1998) MathSciNetCrossRefzbMATHGoogle Scholar
  61. 61.
    Zimmer, R.J.: Strong rigidity for ergodic actions of semisimple Lie groups. Ann. Math. (2) 112(3), 511–529 (1980) MathSciNetCrossRefzbMATHGoogle Scholar
  62. 62.
    Zimmer, R.J.: On the Mostow rigidity theorem and measurable foliations by hyperbolic space. Isr. J. Math. 43(4), 281–290 (1982) MathSciNetCrossRefzbMATHGoogle Scholar
  63. 63.
    Zimmer, R.J.: Ergodic Theory and Semisimple Groups. Monographs in Mathematics, vol. 81. Birkhäuser, Basel (1984) CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Mathematics DepartmentTechnion—Israel Institute of TechnologyHaifaIsrael
  2. 2.Mathematics, Statistics and Computer ScienceUniversity of Illinois at ChicagoChicagoUSA
  3. 3.Department of MathematicsUniversity of ChicagoChicagoUSA
  4. 4.Department of Mathematics, Institute for Algebra and GeometryKarlsruhe Institute of TechnologyKarlsruheGermany

Personalised recommendations