Invariant measures and measurable projective factors for actions of higher-rank lattices on manifolds

Aaron Brown; Federico Rodriguez Hertz; Zhiren Wang

doi:https://doi.org/10.4007/annals.2022.196.3.2

Abstract

We consider smooth actions of lattices in higher-rank semisimple Lie groups on manifolds. We define two numbers $r(G)$ and $m(G)$ associated with the roots system of the Lie algebra of a Lie group $G$. If the dimension of the manifold is smaller than $r(G)$, then we show the action preserves a Borel probability measure. If the dimension of the manifold is at most $m(G)$, we show there is a quasi-invariant measure on the manifold such that the action is measurably isomorphic to a relatively measure-preserving action over a standard boundary action.

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@ARTICLE{MR1253197, author = {Margulis, G. A. and Tomanov, G. M.}, title = {Invariant measures for actions of unipotent groups over local fields on homogeneous spaces}, journal = {Invent. Math.}, fjournal = {Inventiones Mathematicae}, volume = {116}, year = {1994}, number = {1-3}, pages = {347--392}, issn = {0020-9910}, mrclass = {22E40 (11E57 20G25 22D40)}, mrnumber = {1253197}, mrreviewer = {Nimish A. Shah}, doi = {10.1007/BF01231565}, url = {https://doi.org/10.1007/BF01231565}, zblnumber = {0816.22004}, }
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@ARTICLE{MR1720183, author = {Nevo, Amos and Zimmer, Robert J.}, title = {Homogenous projective factors for actions of semi-simple {L}ie groups}, journal = {Invent. Math.}, fjournal = {Inventiones Mathematicae}, volume = {138}, year = {1999}, number = {2}, pages = {229--252}, issn = {0020-9910}, mrclass = {22D40 (22F10 57S20)}, mrnumber = {1720183}, mrreviewer = {Scot Adams}, doi = {10.1007/s002220050377}, url = {https://doi.org/10.1007/s002220050377}, zblnumber = {0936.22007}, }
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@ARTICLE{MR1933077, author = {Nevo, Amos and Zimmer, Robert J.}, title = {A structure theorem for actions of semisimple {L}ie groups}, journal = {Ann. of Math. (2)}, fjournal = {Annals of Mathematics. Second Series}, volume = {156}, year = {2002}, number = {2}, pages = {565--594}, issn = {0003-486X}, mrclass = {22F10 (22E46 37A15)}, mrnumber = {1933077}, mrreviewer = {Douglas P. Dokken}, doi = {10.2307/3597198}, url = {https://doi.org/10.2307/3597198}, zblnumber = {1012.22038}, }
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@ARTICLE{MR1946555, author = {Polterovich, Leonid}, title = {Growth of maps, distortion in groups and symplectic geometry}, journal = {Invent. Math.}, fjournal = {Inventiones Mathematicae}, volume = {150}, year = {2002}, number = {3}, pages = {655--686}, issn = {0020-9910}, mrclass = {53D35 (53D40)}, mrnumber = {1946555}, mrreviewer = {Karl Friedrich Siburg}, doi = {10.1007/s00222-002-0251-x}, url = {https://doi.org/10.1007/s00222-002-0251-x}, zblnumber = {1036.53064}, }
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@ARTICLE{MR1198459, author = {Witte, Dave}, title = {Arithmetic groups of higher {${\bf Q}$}-rank cannot act on {$1$}-manifolds}, journal = {Proc. Amer. Math. Soc.}, fjournal = {Proceedings of the American Mathematical Society}, volume = {122}, year = {1994}, number = {2}, pages = {333--340}, issn = {0002-9939}, mrclass = {22E40 (20F60 20H05 57S25)}, mrnumber = {1198459}, mrreviewer = {Nimish A. Shah}, doi = {10.2307/2161021}, url = {https://doi.org/10.2307/2161021}, zblnumber = {0818.22006}, }
[MR776417] $Go to document$ R. J. Zimmer, Ergodic Theory and Semisimple Groups, Birkhäuser Verlag, Basel, 1984, vol. 81.

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@BOOK{MR776417, 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}, }

Invariant measures and measurable projective factors for actions of higher-rank lattices on manifolds

Abstract

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