Smooth and analytic actions of SL(n,R) and SL(n,Z) on closed n-dimensional manifolds

Fisher, David; MELNICK, Karin

doi:10.1215/21562261-2024-0009

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Smooth and analytic actions of SL(n,R) and SL(n,Z) on closed n-dimensional manifolds

Fisher, David; MELNICK, Karin

2024 • In Journal of Mathematics of Kyoto University, 64 (4), p. 873-904

Peer reviewed

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https://hdl.handle.net/10993/61812

DOI
10.1215/21562261-2024-0009

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Keywords :

rigid structures , transformation groups , Zimmer program

Abstract :

[en] The main theorem is a classification of smooth actions of $\SL(n,\BR)$, $n \geq 3$, or connected groups locally isomorphic to it, on closed $n$-manifolds, extending a theorem of Uchida \cite{uchida.slnr.sn}. We also construct new exotic actions of $\SL(n,\BZ)$ on the $n$-torus and connected sums of $n$-tori, and we formulate a conjectural classification of actions of lattices in $\SL(n,\BR)$ on closed $n$-manifolds. We prove some related results about invariant rigid geometric structures for $\SL(n,\BR)$-actions.

Disciplines :

Mathematics

Author, co-author :

Fisher, David; Department of Mathematics, Rice University, Houston, Texas, USA

MELNICK, Karin ; University of Luxembourg > Faculty of Science, Technology and Medicine (FSTM) > Department of Mathematics (DMATH)

External co-authors :

yes

Language :

English

Title :

Smooth and analytic actions of SL(n,R) and SL(n,Z) on closed n-dimensional manifolds

Publication date :

04 August 2024

Journal title :

Journal of Mathematics of Kyoto University

ISSN :

0023-608X

Publisher :

Duke University Press

Volume :

Issue :

Pages :

873-904

Peer reviewed :

Peer reviewed

Additional URL :

http://arxiv.org/abs/2210.09516

Funders :

NSF - National Science Foundation

Funding number :

DMS-2109347; DMS-1906107; DMS-2208430

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since 09 August 2024

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Bibliography

W. Ballmann, Geometric structures, preprint, 2000, http://people.mpim-bonn.mpg.de/hwbllmnn/archiv/geostr00.pdf.
E. J. Benveniste and D. Fisher, Nonexistence of invariant rigid structures and invariant almost rigid structures, Comm. Anal. Geom. 13 (2005), no. 1, 89–111. MR 2154667.
A. Brown, D. Fisher, and S. Hurtado, Zimmer’s conjecture: Subexponential growth, measure rigidity, and strong property (T), Ann. of Math. (2) 196 (2022), no. 3, 891–940, MR 4502593.
A. Brown, D. Fisher, and S. Hurtado, Zimmer’s conjecture for SL(m,Z), Inv. Math. 221 (2020), no. 3, 1001–1060. MR 4132960. DOI 10.1007/s00222-020-00962-x.
A. Brown, D. Fisher, and S. Hurtado, Zimmer’s conjecture for non-uniform lattices and escape of mass and growth of cocycles, preprint, arXiv:2105.14541 [math.DS].
A.Brown, F. Rodrigez Hertz, Z. Wang Boundary actions by higher-rank lattices: Classification and embedding in low dimensions, local rigidity, smooth factors, preprint, 2024, arXiv:2405.16202 [math.DS].
G. Cairns and É. Ghys, The local linearization problem for smooth SL(n)-actions, Enseign. Math. 43 (1997), no. 1-2, 133–171. MR 1460126.
A. Čap and K. Melnick, Essential killing fields of parabolic geometries: Projective and conformal structures, Cent. Eur. J. Math. 11 (2013), no. 12, 2053–2061. MR 3111705. DOI 10.2478/s11533-013-0317-6.
G. D’Ambra and M. Gromov, “Lectures on transformation groups: Geometry and dynamics” in Surveys in Differential Geometry (Lehigh University, Bethlehem, PA, 1990), 1991, 19–111. MR 1144526.
B. Farb and P. Shalen, Lattice actions, 3-manifolds and homology, Topology 39 (2000), no. 3, 573–587. MR 1746910. DOI 10.1016/S0040-9383(99)00020-8.
R. Feres, “Rigid geometric structures and actions of semisimple Lie groups” in Rigidité, Groupe Fondamentale et Dynamique, Panor. Synthéses, Vol. 13, Société Mathématique de France, 2002, 121–166. MR 1993149.
D. Fisher, “Groups acting on manifolds: Around the Zimmer program” in Geometry, Rigidity, and Group Actions, Chicago Lect. Math., Univ. Chicago Press, 2011, pp. 74–93. MR 2807830. DOI 10.7208/chicago/9780226237909.001.0001.
D. Fisher and K. Whyte, Continuous quotients for lattice actions on compact spaces, Geom. Dedicata 87 (2001), no. 1-3, 181–189. MR 1866848. DOI 10.1023/A:1012041230518.
M. Foreman, “The complexity of the structure and classification of dynamical systems” in Ergodic Theory, Encycl. Complex Syst. Sci., Springer, New York, 2023, 529–576. MR 4647089. DOI 10.1007/978-1-0716-2388-6_726.
M. Gromov, “Rigid transformations groups” in Géométrie Différentielle (Paris, 1986), vol. 33, Hermann, Paris, 1988, 65–139. MR 0955852.
V. W. Guillemin and S. Sternberg, Remarks on a paper of Hermann, Trans. Amer. Math. Soc. 130 (1968), 110–116. MR 0217226. DOI 10.2307/1994774.
N. Hitchin, “Vector fields on the circle” in Mechanics, Analysis and Geometry: 200 Years after Lagrange, North-Holland Delta Ser., North-Holland, Amsterdam, 1991, 359–378. MR 1098524.
A. Katok and J. Lewis, Global rigidity results for lattice actions on tori and new examples of volume-preserving actions, Israel J. Math. 93 (1996), 253–280. MR 1380646. DOI 10.1007/BF02761106.
A. W. Knapp, Lie Groups: Beyond an Introduction, Prog. Math., Vol. 40, Birkhäuser, Boston, MA, 1996. MR 1399083. DOI 10.1007/978-1-4757-2453-0.
S. Kobayashi, Transformation Groups in Differential Geometry, reprint of the 1972 edition, Classics in Mathematics, Springer-Verlag, Berlin, 1995. viii+182 pp. ISBN: 3-540-58659-8, MR 1336823.
A. G. Kušnirenko, An analytic action of a semisimple Lie group in a neighborhood of a fixed point is equivalent to a linear one, Funkcional. Anal. i Priložen. 1 (1967), 103–104. MR 0210833.
K. Lin, The real Grassmannian Gr(2,4), https://math.harvard.edu/-klin/Gr24.pdf.
K. Melnick and K. Neusser, Strongly essential flows for irreducible parabolic geometries, Trans. Amer. Math. Soc. 368 (2016), no. 11, 8079–8110. MR 3546794. DOI 10.1090/tran/6814.
T. Nagano and T. Ochiai, On compact Riemannian manifolds admitting essential projective transformations, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 33 (1986), no. 2, 233–246. MR 0866391.
V. Ovsienko and S. Tabachnikov, Projective Geometry Old and New, Cambridge Tracts in Math., vol. 165, Cambridge Univ. Press, Cambridge, 2005. MR 2177471.
V. Pecastaing, Projective and conformal closed manifolds with a higher-rank lattice action, Math Ann. 388 (2024), no. 1, 939–986. MR 4693950. DOI 10.1007/s00208-022-02528-z.
G. Prasad and A. S. Rapinchuk, Irreducible tori in semisimple groups, Internat. Math. Res. Notices (2001), no. 23, 1229–1242. MR 1866442. DOI 10.1155/S1073792801000587.
C. Rosendal, Descriptive classification theory and separable Banach spaces, Notices Amer. Math. Soc. 58 (2011), no. 9, 1251–1262. MR 2868100.
C. R. Schneider, SL(2,R)-actions on surfaces, Amer. J. Math. 96 (1974), no. 4, 511–528. MR 0368056. DOI 10.2307/2373705.
R. W. Sharpe, Differential Geometry: Cartan’s Generalization of Klein’s Erlangen Program, Grad. Texts in Math., vol. 166, Springer-Verlag, New York, 1996. MR 1453120.
D. C. Stowe, Real analytic actions of SL(2,R) on a surface, Ergodic Theory Dynam. Systems 3 (1983), 447–499. MR 0741396. DOI 10.1017/S0143385700002066.
W. P. Thurston, Three-Dimensional Geometry and Topology, Vol. 1, Princeton Mathematical Series, Princeton Univ. Press, Princeton, NJ, 1997. MR 1435975.
F. Uchida, Classification of real analytic SL(n,R) actions on n-sphere, Osaka J. Math. 16 (1979), no. 3, 561–579. MR 0551576.
F. Uchida, On smooth SO0(p,q)-actions on Sp+q−1, Osaka J. Math. 26 (1989), no. 4, 775–787. MR 1040424.
F. Uchida, Smooth SL(n, C) actions on (2n − 1)-manifolds, Hokkaido Math. J. 21 (1992), no. 1, 79–86. MR 1153753. DOI 10.14492/hokmj/1381413266.
F. Uchida, On smooth SO0(p,q)-actions on Sp+q−1. II, Tohoku Math. J. (2) 49 (1997), no. 2, 185–202. MR 1447181. DOI 10.2748/tmj/1178225146.
F. Uchida, On smooth Sp(p,q)-actions on S4p+4q−1, Osaka J. Math. 39 (2002), no. 2, 293–314. MR 1914295.
F. Uchida and K. Mukoyama, “Smooth actions of non-compact semi-simple Lie groups” in Current Trends in Transformation Groups, K-Monogr. Math., vol. 7, Kluwer Acad. Publ., Dordrecht, 2002, 201–215. MR 1941644. DOI 10.1007/978-94-009-0003-5_13.
R. J. Zimmer, Ergodic Theory and Semisimple Groups, Monogr. Math., Vol. 81, Birkhäuser Verlag, Basel, 1984. MR 0776417. DOI 10.1007/978-1-4684-9488-4.
R. J. Zimmer, “Actions of semisimple groups and discrete subgroups” in Proceedings of the ICM (Berkeley, California, 1986), Amer. Math. Soc., Providence, RI, 1987, 1247–1258.
R. Zimmer, Group Actions in Ergodic Theory, Geometry, and Toplogy—Selected Papers, Univ. Chicago Press, Chicago, 2020. MR 4521466. DOI 10.7208/chicago/9780226568270.001.0001.