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On effective equidistribution for quotients of SL(d,ℝ)

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Abstract

We prove the first case of polynomially effective equidistribution of closed orbits of semisimple groups with nontrivial centralizer. The proof relies on uniform pectral gap, builds on, and extends work of Einsiedler, Margulis, and Venkatesh.

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References

  1. A. Borel, Linear Algebraic Groups, Graduate Texts in Mathematics, Vol. 126, Springer, New York, 1991.

    Google Scholar 

  2. A. Borel and J. Tits, Groupes réductifs, Institut des Hautes Etudes Scientifiques. Publications Mathématiques 27 (1965), 55–150.

    Article  Google Scholar 

  3. M. Burger and P. Sarnak, Ramanujan duals. II, Inventiones mathematicae 106 (1991), 1–11.

    Article  MathSciNet  Google Scholar 

  4. L. Clozel, Démonstration de la conjecture t, Inventiones mathematicae 151 (2003), 297–328.

    Article  MathSciNet  Google Scholar 

  5. S. G. Dani and G. A. Margulis, Asymptotic behaviour of trajectories of unipotent flows on homogeneous spaces, Indian Academy of Sciences. Proceedings. Mathematical Sciences 101 (1991), 1–17.

    MathSciNet  MATH  Google Scholar 

  6. M. Einsiedler, G. Margulis and A. Venkatesh, Effective equidistribution for closed orbits of semisimple groups on homogeneous spaces, Inventiones Mathematicae 177 (2009), 137–212.

    Article  MathSciNet  Google Scholar 

  7. M. Einsiedler, G. Margulis, A. Mohammadi and A. Venkatesh, Effective equidistribution and property (t), Journal of the American Mathematical Society, to appear, https://doi.org/10.1090/jams/930.

  8. A. Gorodnik, F. Maucourant and H. Oh, Manin’s and Peyre’s conjectures on rational points and adelic mixing, Annales Scientifiques de l’École Normale Supérieure 41 (2008), 383–435.

    MathSciNet  MATH  Google Scholar 

  9. H. Jacquet and R. P. Langlands, Automorphic Forms on GL (2), Lecture Notes in Mathematics, Vol. 114, Springer, Berlin-New York, 1970.

    Google Scholar 

  10. D. A. Kazhdan, Connection of the dual space of a group with the structure of its close subgroups, Functional Analysis and its Applications 1 (1967), 63–65.

    Article  Google Scholar 

  11. D. Y. Kleinbock and G. A. Margulis, Bounded orbits of nonquasiunipotent flows on homogeneous spaces, in Sinaį’s Moscow Seminar on Dynamical Systems, American Mathematical Society Translations, Series 2, Vol. 171, American Mathematical Society, Providence, RI, 1996, pp. 141–172.

    Google Scholar 

  12. D. Y. Kleinbock and G. A. Margulis, Flows on homogeneous spaces and diophantine approximation on manifolds, Annals of mathematics 148 (1998), 339–360.

    Article  MathSciNet  Google Scholar 

  13. D. Y. Kleinbock and G. A. Margulis, On effective equidistribution of expanding translates of certain orbits in the space of lattices, in Number Theory, Analysis and Geometry, Springer, New York, 2012, pp. 385–396.

    Chapter  Google Scholar 

  14. G. Margulis, Discrete subgroups and ergodic theory, in Number Theory, Trace Formulas and Discrete Groups (Oslo, 1987), Academic Press, Boston, MA, 1989, pp. 377–398.

    Google Scholar 

  15. H. Oh, Uniform pointwise bounds for matrix coefficients of unitary representations and applications to Kazhdan constants, Duke Mathematical Journal 113 (2002), 133–192.

    Article  MathSciNet  Google Scholar 

  16. M. Ratner, Invariant measures for unipotent translations on homogeneous spaces, Proceedings of the National Academy of Sciences of the United States of America 87 (1990), 4309–4311.

    Article  MathSciNet  Google Scholar 

  17. M. Ratner, On measure rigidity of unipotent subgroups of semisimple groups, ActaMathematica 165 (1990), 229–309.

    MathSciNet  MATH  Google Scholar 

  18. M. Ratner, Strict measure rigidity for unipotent subgroups of solvable groups, Inventiones Mathematicae 101 (1990), 449–482.

    Article  MathSciNet  Google Scholar 

  19. M. Ratner, Distribution rigidity for unipotent actions on homogeneous spaces, Bulletin of the American Mathematical Society 24 (1991), 321–325.

    Article  MathSciNet  Google Scholar 

  20. M. Ratner, On Raghunathan’s measure conjecture, Annals of Mathematics 134 (1991), 545–607.

    Article  MathSciNet  Google Scholar 

  21. A. Selberg, On the estimation of fourier coefficients of modular forms, in Theory of Numbers, Proceedings of Symposia in Pure Mathematics, Vol. 8, American Mathematical Society, Providence, RI, 1965, pp. 1–15.

    Google Scholar 

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Authors and Affiliations

  1. Departement Mathematik, ETH Zürich, Rämistrasse 101, 8092, Zurich, Switzerland

    Menny Aka & Manfred Einsiedler

  2. Department of Mathematics and Computer Science, Wesleyan University, 265 Church Street, Middletown, CT, 06457, USA

    Han Li

  3. Department of Mathematics, University of California, San Diego, CA, 92093, USA

    Amir Mohammadi

Authors
  1. Menny Aka
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  2. Manfred Einsiedler
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  3. Han Li
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  4. Amir Mohammadi
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Corresponding author

Correspondence to Manfred Einsiedler.

Additional information

M. A. acknowledges the support of ISEF, and SNF Grant 200021-152819.

M. E. acknowledges the support of the SNF Grant 200021-152819 and 200020-178958.

H. L. acknowledges support by Simons Foundation (426090) and NSF (DMS 1700109).

A. M. acknowledges support by the NSF (DMS 1724316, 1764246, 1128155) and Alfred P. Sloan Research Fellowship.

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Aka, M., Einsiedler, M., Li, H. et al. On effective equidistribution for quotients of SL(d,ℝ). Isr. J. Math. 236, 365–391 (2020). https://doi.org/10.1007/s11856-020-1978-z

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  • DOI: https://doi.org/10.1007/s11856-020-1978-z

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