Duke Mathematical Journal

Product groups acting on manifolds

Alex Furman and Nicolas Monod

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We analyse volume-preserving actions of product groups on Riemannian manifolds. To this end, we establish a new superrigidity theorem for ergodic cocycles of product groups ranging in linear groups. There are no a priori assumptions on the acting groups except a spectral gap assumption on their action.

Our main application to manifolds concerns irreducible actions of Kazhdan product groups. We prove the following dichotomy: Either the action is infinitesimally linear, which means that the derivative cocycle arises from unbounded linear representations of all factors, or, otherwise, the action is measurably isometric, in which case there are at most two factors in the product group.

As a first application, this provides lower bounds on the dimension of the manifold in terms of the number of factors in the acting group. Another application is a strong restriction for actions of nonlinear groups

Article information

Duke Math. J. Volume 148, Number 1 (2009), 1-39.

First available in Project Euclid: 22 April 2009

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Zentralblatt MATH identifier

Primary: 22F10: Measurable group actions [See also 22D40, 28Dxx, 37Axx] 57S30: Discontinuous groups of transformations 58F11
Secondary: 22D40: Ergodic theory on groups [See also 28Dxx] 37A15: General groups of measure-preserving transformations [See mainly 22Fxx]


Furman, Alex; Monod, Nicolas. Product groups acting on manifolds. Duke Math. J. 148 (2009), no. 1, 1--39. doi:10.1215/00127094-2009-018. https://projecteuclid.org/euclid.dmj/1240432189

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  • U. Bader, A. Furman, T. Gelander, and N. Monod, Property $(T)$ and rigidity for actions on Banach spaces, Acta Math. 198 (2007), 57--105.
  • U. Bader, A. Furman, and A. Shaker. Superrigidity via Weyl groups: actions on the circle, preprint.
  • U. Bader and Y. Shalom, Factor and normal subgroup theorems for lattices in products of groups, Invent. Math. 163 (2006), 415--454.
  • M. E. B. Bekka, P. De La Harpe, and A. Valette, Kazhdan's Property $(T)$, New Math. Monogr. 11, Cambridge Univ. Press, Cambridge, 2008.
  • A. Borel and Harish-Chandra, Arithmetic subgroups of algebraic groups, Ann. of Math. (2) 75 (1962), 485--535.
  • M. Burger and N. Monod, Continuous bounded cohomology and applications to rigidity theory, Geom. Funct. Anal. 12 (2002), 219--280.
  • M. Burger and S. Mozes, Lattices in product of trees, Inst. Hautes Études Sci. Publ. Math. 92 (2000), 151--194.
  • P.-E. Caprace and B. RéMy, Simplicité abstraite des groupes de Kac-Moody non affines, C. R. Math. Acad. Sci. Paris 342 (2006), 539--544.
  • P.-A. Cherix, M. Cowling, P. Jolissaint, P. Julg, and A. Valette, Groups with the Haagerup Property; Gromov's a-T-menability, Progr. Math. 197, Birkhäuser, Basel, 2001.
  • K. Corlette, Archimedean superrigidity and hyperbolic geometry, Ann. of Math. (2) 135 (1992), 165--182.
  • P. De La Harpe and A. Valette, La propriété $(T)$ de Kazhdan pour les groupes localement compacts (avec un appendice de Marc Burger), with an appendix by M. Burger, Astérisque 175, Soc. Math. France, Montrouge, 1989.
  • J. Dixmier. Sur les représentations unitaires des groupes de Lie algébriques, Ann. Inst. Fourier (Grenoble) 7 (1957), 315--328.
  • —, Représentations induites holomorphes des groupes résolubles algébriques, Bull. Soc. Math. France 94 (1966), 181--206.
  • J. Dymara and T. Januszkiewicz, Cohomology of buildings and their automorphism groups, Invent. Math. 150 (2002), 579--627.
  • D. Fisher and T. Hitchman, Cocycle superrigidity and harmonic maps with infinite-dimensional targets, Int. Math. Res. Not. 2006, art. ID 72405.
  • D. Fisher and G. A. Margulis, ``Local rigidity for cocycles'' in Surveys in Differential Geometry, Vol. 8 (Boston, 2002), Surv. Differ. Geom. 8, Int. Press, Somerville, Mass., 2003, 191--234.
  • D. Fisher and L. Silberman, Groups not acting on manifolds, Int. Math. Res. Not. IMRN 2008, no. 16, art. ID rnn060.
  • D. Fisher and R. J. Zimmer, Geometric lattice actions, entropy and fundamental groups, Comment. Math. Helv. 77 (2002), 326--338.
  • A. Furman, ``Random walks on groups and random transformations'' in Handbook of Dynamical Systems, Vol. 1A, North-Holland, Amsterdam, 2002, 931--1014.
  • —, Outer automorphism groups of some ergodic equivalence relations, Comment. Math. Helv. 80 (2005), 157--196.
  • H. Furstenberg, A Poisson formula for semi-simple Lie groups, Ann. of Math. (2) 77 (1963), 335--386.
  • —, ``Rigidity and cocycles for ergodic actions of semisimple Lie groups (after G. A. Margulis and R. Zimmer)'' in Seminar Bourbaki, 1979/80, Lecture Notes in Math. 842, Springer, Berlin, 1981, 273--292.
  • T. Gelander, A. Karlsson, and G. A. Margulis, Superrigidity, generalized harmonic maps and uniformly convex spaces, Geom. Funct. Anal. 17 (2008), 1524--1550.
  • M. Gromov, ``Hyperbolic groups'' in Essays in Group Theory, Math. Sci. Res. Inst. Publ. 8, Springer, New York, 1987, 75--263.
  • —, Random walk in random groups, Geom. Funct. Anal. 13 (2003), 73--146.
  • P. R. Halmos and J. Von Neumann, Operator methods in classical mechanics, II, Ann. of Math. (2) 43 (1942), 332--350.
  • G. Hjorth and A. S. Kechris, Rigidity theorems for actions of product groups and countable Borel equivalence relations, Mem. Amer. Math. Soc. 177 (2005), no. 833.
  • S. Kakutani, ``Random ergodic theorems and Markoff processes with a stable distribution'' in Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, 1950, Univ. of California Press, Berkeley, 1951, 247--261.
  • D. A. Kazhdan [KažDan], On the connection of the dual space of a group with the structure of its closed subgroups (in Russian), Funkcional. Anal. i Priložen. 1 (1967), 71--74.
  • A. Lubotzky and R. J. Zimmer, Arithmetic structure of fundamental groups and actions of semisimple Lie groups, Topology 40 (2001), 851--869.
  • G. W. Mackey, Ergodic transformation groups with a pure point spectrum, Illinois J. Math. 8 (1964), 593--600.
  • A. I. Malcev, On isomorphic matrix representations of infinite groups (in Russian), Rec. Math. [Mat. Sbornik] N.S. 8 (50) (1940), 405--422.
  • —, On a class of homogeneous spaces, Amer. Math. Soc. Translation 1951, no. 39.
  • G. A. Margulis, Arithmeticity and finite-dimensional representations of uniform lattices (in Russian), Funkcional. Anal. i Priložen. 8, no. 3 (1974), 77--78.
  • —, ``Discrete groups of motions of manifolds of nonpositive curvature'' (in Russian) in Proceedings of the International Congress of Mathematicians (Vancouver, 1974), Vol. 2, Canad. Math. Congress, Montreal, 1975, 21--34.
  • —, ``Discrete groups of motions of manifolds of nonpositive curvature'' in 20 Lectures Delivered at the International Congress of Mathematicians (Vancouver, 1974), vol. 2, Amer. Math. Soc. Translations Ser. 2 109 Amer. Math. Soc., Providence, 1977, 33--45.
  • —, Discrete Subgroups of Semisimple Lie Groups, Ergeb. Math. Grenzgeb (3) 17 Springer, Berlin, 1991.
  • N. Monod, Superrigidity for irreducible lattices and geometric splitting, J. Amer. Math. Soc. 19 (2006), 781--814.
  • N. Monod and Y. Shalom, Cocycle superrigidity and bounded cohomology for negatively curved spaces, J. Differential Geom. 67 (2004), 395--455.
  • —, Orbit equivalence rigidity and bounded cohomology, Ann. of Math. (2) 164 (2006), 825--878.
  • V. I. Oseledets [Oseledec], A multiplicative ergodic theorem: Characteristic Ljapunov, exponents of dynamical systems (in Russian), Trudy Moskov. Mat. Obšč. 19 (1968), 179--210.
  • Ja. B. [Y. B.] Pesin, Characteristic Ljapunov exponents, and smooth ergodic theory (in Russian), Uspekhi Mat. Nauk 32, no. 4 (1977), 55--112.
  • S. Popa, On the superrigidity of malleable actions with spectral gap, J. Amer. Math. Soc. 21 (2008), 981--1000.
  • M. S. Raghunathan, Discrete Subgroups of Lie Groups, Ergeb. Math. Grenzgeb. 68, Springer, New York, 1972.
  • —, A proof of Oseledec's multiplicative ergodic theorem, Israel J. Math. 32 (1979), 356--362.
  • B. RéMy, ``Classical and non-linearity properties of Kac-Moody lattices'' in Rigidity in Dynamics and Geometry (Cambridge 2000), Springer, Berlin, 2002, 391--406.
  • —, Topological simplicity, commensurator super-rigidity and non-linearities of Kac-Moody groups, with an appendix by P. Bonvin, Geom. Funct. Anal. 14 (2004), 810--852.
  • —, Integrability of induction cocycles for Kac-Moody groups, Math. Ann. 333 (2005), 29--43.
  • D. Ruelle, Ergodic theory of differentiable dynamical systems, Inst. Hautes Études Sci. Publ. Math. 50 (1979), 27--58.
  • Y. Shalom, Rigidity of commensurators and irreducible lattices, Invent. Math. 141 (2000), 1--54.
  • G. Stuck, Cocycles of ergodic group actions and vanishing of first cohomology for $S$-arithmetic groups. Amer. J. Math. 113 (1991), 1--23.
  • J. Von Neumann, Zur Operatorenmethode in der klassischen Mechanik, Ann. of Math. (2) 33 (1932), 587--642.
  • H. Zassenhaus, Beweis eines Satzes über diskrete Gruppen, Abh. Math. Semin. Hansische Univ. 12 (1938), 289--312.
  • R. J. Zimmer, Strong rigidity for ergodic actions of semisimple Lie groups, Ann. of Math. (2), 112 (1980), 511--529.
  • —, Ergodic Theory and Semisimple Groups, Monogr. Math. 81, Birkhäuser, Basel, 1984.
  • —, Kazhdan groups acting on compact manifolds, Invent. Math. 75 (1984), 425--436.
  • —, Volume preserving actions of lattices in semisimple groups on compact manifolds, Inst. Hautes Études Sci. Publ. Math. 59 (1984), 5--33.
  • —, ``Actions of semisimple groups and discrete subgroups'' in Proceedings of the International Congress of Mathematicians, Vols. 1, 2 (Berkeley, 1986), Amer. Math. Soc., Providence, 1987, 1247--1258.
  • —, Spectrum, entropy, and geometric structures for smooth actions of Kazhdan groups, Israel J. Math. 75 (1991), 65--80.