Zimmer’s conjecture: Subexponential growth, measure rigidity, and strong property (T)

Abstract

We prove several cases of Zimmer’s conjecture for actions of higher-rank, cocompact lattices on low-dimensional manifolds. For example, if $\Gamma$ is a cocompact lattice in $\mathrm{SL}(n,\mathbb{R})$, $M$ is a compact manifold, and $\omega$ a volume form on $M$, we show that any homomorphism $\alpha : \Gamma \rightarrow \mathrm{Diff}(M)$ has finite image if the dimension of $M$ is less than $n-1$ and that any homomorphism $\alpha : \Gamma \rightarrow \mathrm{Diff}(M,\omega)$ has finite image if the dimension of $M$ is less than $n$. The key step in the proof is to show that any such action has uniform subexponential growth of derivatives. This is established using ideas from the smooth ergodic theory of higher-rank abelian groups, structure theory of semisimple groups, and results from homogeneous dynamics. Having established uniform subexponential growth of derivatives, we apply Lafforgue’s strong property \rm (T) to establish the existence of an invariant Riemannian metric.

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      author = {Labourie, François},
      title = {Large groups actions on manifolds},
      booktitle = {Proceedings of the {I}nternational {C}ongress of {M}athematicians, {V}ol. {II}},
      venue = {{B}erlin, 1998},
      journal = {Doc. Math.},
      fjournal = {Documenta Mathematica},
      year = {1998},
      number = {Extra Vol. II},
      pages = {371--380},
      issn = {1431-0635},
      mrclass = {53C20 (22E40 57S30 58F15)},
      mrnumber = {1648087},
      mrreviewer = {Dave Witte Morris},
      zblnumber = {0904.57014},
      }
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      author = {Lafforgue, Vincent},
      title = {Un renforcement de la propriété ({T})},
      journal = {Duke Math. J.},
      fjournal = {Duke Mathematical Journal},
      volume = {143},
      year = {2008},
      number = {3},
      pages = {559--602},
      issn = {0012-7094},
      mrclass = {22D20 (20E08 20F67 22E35 22E46 46B04)},
      mrnumber = {2423763},
      mrreviewer = {Alain Valette},
      doi = {10.1215/00127094-2008-029},
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      author = {Ledrappier, F. and Young, L.-S.},
      title = {The metric entropy of diffeomorphisms. {I}. {C}haracterization of measures satisfying {P}esin's entropy formula},
      journal = {Ann. of Math. (2)},
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      volume = {122},
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      mrclass = {58F11 (58F15)},
      mrnumber = {0819556},
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      mrclass = {22D12 (46B85)},
      mrnumber = {3190138},
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      title = {Non-uniform lattices in semisimple algebraic groups},
      booktitle = {Lie {G}roups and their {R}epresentations ({P}roc. {S}ummer {S}chool on {G}roup {R}epresentations of the {B}olyai {J}\'{a}nos {M}ath. {S}oc., {B}udapest, 1971)},
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      year = {1975},
      mrclass = {22E40},
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      volume = {17},
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      pages = {x+388},
      isbn = {3-540-12179-X},
      mrclass = {22E40 (20Hxx 22-02 22D40)},
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      }
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      mrnumber = {1503467},
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      booktitle = {Geometry, Rigidity, and Group Actions},
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      title = {Arithmetic groups acting on compact manifolds},
      journal = {Bull. Amer. Math. Soc. (N.S.)},
      fjournal = {Amer. Math. Soc.. Bulletin. New Series},
      volume = {8},
      year = {1983},
      number = {1},
      pages = {90--92},
      issn = {0273-0979},
      mrclass = {22E40 (20H99 57S99)},
      mrnumber = {0682830},
      doi = {10.1090/S0273-0979-1983-15093-0},
      url = {https://doi.org/10.1090/S0273-0979-1983-15093-0},
      zblnumber = {0532.22012},
      }
  • [MR0776417] Go to document R. J. Zimmer, Ergodic Theory and Semisimple Groups, Birkhäuser Verlag, Basel, 1984, vol. 81.
    @BOOK{MR0776417,
      author = {Zimmer, Robert J.},
      title = {Ergodic Theory and Semisimple Groups},
      series = {Monogr. Math.},
      volume = {81},
      publisher = {Birkhäuser Verlag, Basel},
      year = {1984},
      pages = {x+209},
      isbn = {3-7643-3184-4},
      mrclass = {22E40 (22D40 28D15)},
      mrnumber = {0776417},
      mrreviewer = {S. G. Dani},
      doi = {10.1007/978-1-4684-9488-4},
      url = {https://doi.org/10.1007/978-1-4684-9488-4},
      zblnumber = {0571.58015},
      }
  • [MR743815] Go to document R. J. Zimmer, "Volume preserving actions of lattices in semisimple groups on compact manifolds," Inst. Hautes Études Sci. Publ. Math., iss. 59, pp. 5-33, 1984.
    @ARTICLE{MR743815,
      author = {Zimmer, Robert J.},
      title = {Volume preserving actions of lattices in semisimple groups on compact manifolds},
      journal = {Inst. Hautes \'{E}tudes Sci. Publ. Math.},
      fjournal = {Institut des Hautes \'{E}tudes Scientifiques. Publications Mathématiques},
      number = {59},
      year = {1984},
      pages = {5--33},
      issn = {0073-8301},
      mrclass = {22E40 (28D15 57S99)},
      mrnumber = {0743815},
      mrreviewer = {G. A. Soĭfer},
      url = {http://www.numdam.org/item?id=PMIHES_1984__59__5_0},
      zblnumber = {0576.22013},
      }
  • [MR934329] R. J. Zimmer, "Actions of semisimple groups and discrete subgroups," in Proceedings of the International Congress of Mathematicians, Vol. 1, 2, 1987, pp. 1247-1258.
    @INPROCEEDINGS{MR934329,
      author = {Zimmer, Robert J.},
      title = {Actions of semisimple groups and discrete subgroups},
      booktitle = {Proceedings of the {I}nternational {C}ongress of {M}athematicians, {V}ol. 1, 2},
      venue = {{B}erkeley, {C}alif., 1986},
      pages = {1247--1258},
      publisher = {Amer. Math. Soc., Providence, RI},
      year = {1987},
      mrclass = {22E40 (57S15 57S20)},
      mrnumber = {0934329},
      mrreviewer = {S. G. Dani},
      zblnumber = {0671.57028},
      }
  • [MR900826] Go to document R. J. Zimmer, "Lattices in semisimple groups and invariant geometric structures on compact manifolds," in Discrete Groups in Geometry and Analysis (New Haven, Conn., 1984), Birkhäuser Boston, Boston, MA, 1987, vol. 67, pp. 152-210.
    @INCOLLECTION{MR900826,
      author = {Zimmer, Robert J.},
      title = {Lattices in semisimple groups and invariant geometric structures on compact manifolds},
      booktitle = {Discrete {G}roups in {G}eometry and {A}nalysis ({N}ew {H}aven, {C}onn., 1984)},
      series = {Progr. Math.},
      volume = {67},
      pages = {152--210},
      publisher = {Birkhäuser Boston, Boston, MA},
      year = {1987},
      mrclass = {22E40 (53C10)},
      mrnumber = {0900826},
      mrreviewer = {K. H. Hofmann},
      doi = {10.1007/978-1-4899-6664-3_6},
      url = {https://doi.org/10.1007/978-1-4899-6664-3_6},
      zblnumber = {0663.22008},
      }
  • [MR1147291] Go to document R. J. Zimmer, "Spectrum, entropy, and geometric structures for smooth actions of Kazhdan groups," Israel J. Math., vol. 75, iss. 1, pp. 65-80, 1991.
    @ARTICLE{MR1147291,
      author = {Zimmer, Robert J.},
      title = {Spectrum, entropy, and geometric structures for smooth actions of {K}azhdan groups},
      journal = {Israel J. Math.},
      fjournal = {Israel Journal of Mathematics},
      volume = {75},
      year = {1991},
      number = {1},
      pages = {65--80},
      issn = {0021-2172},
      mrclass = {22E40 (22D40)},
      mrnumber = {1147291},
      doi = {10.1007/BF02787182},
      url = {https://doi.org/10.1007/BF02787182},
      zblnumber = {0763.22011},
      }
  • [MR2457556] Go to document R. J. Zimmer and D. W. Morris, Ergodic Theory, Groups, and Geometry, published for the Conference Board of the Mathematical Sciences, Washington, DC; by the Amer. Math. Soc., Providence, RI, 2008, vol. 109.
    @BOOK{MR2457556,
      author = {Zimmer, Robert J. and Morris, Dave Witte},
      title = {Ergodic Theory, Groups, and Geometry},
      series = {CBMS Regional Conf. Ser. Math.},
      volume = {109},
      publisher = {published for the Conference Board of the Mathematical Sciences, Washington, DC; by the Amer. Math. Soc., Providence, RI},
      year = {2008},
      pages = {x+87},
      isbn = {978-0-8218-0980-8},
      mrclass = {37C85 (22F10 28D15 37A15 53C24 57S20)},
      mrnumber = {2457556},
      mrreviewer = {David Michael Fisher},
      doi = {10.1090/cbms/109},
      url = {https://doi.org/10.1090/cbms/109},
      zblnumber = {1177.37002},
      }

Authors

Aaron Brown

Northwestern University, Evanston, IL

David Fisher

Indiana University, Bloomington, Bloomington, IN

Current address:

Rice University, Houston, TX Sebastian Hurtado

University of Chicago, Chicago, IL

Current address:

Yale University, New Haven, CT