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A new lattice invariant for lattices in totally disconnected locally compact groups

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Abstract

We introduce and explore a natural rank for totally disconnected locally compact groups called the bounded conjugacy rank. This rank is shown to be a lattice invariant for lattices in sigma compact totally disconnected locally compact groups; that is to say, for a given sigma compact totally disconnected locally compact group, some lattice has bounded conjugacy rank n if and only if every lattice has bounded conjugacy rank n. Several examples are then presented.

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References

  1. U. Baumgartner and G. Willis, Contraction groups and scales of automorphisms of totally disconnected locally compact groups, Israel Journal of Mathematics 142 (2004), 221–248.

    Article  MathSciNet  Google Scholar 

  2. B. Bekka, P. de la Harpe and A. Valette, Kazhdan’s property (T), New Mathematical Monographs, Vol. 11, Cambridge University Press, Cambridge, 2008.

    Google Scholar 

  3. J. Braconnier, Sur les groupes topologiques localement compacts, Journal de Mathématiques Pures et Appliquées 27 (1948), 1–85.

    MathSciNet  MATH  Google Scholar 

  4. P.-E. Caprace, Amenable groups and Hadamard spaces with a totally disconnected isometry group, Commentarii Mathematici Helvetici 84 (2009), 437–455.

    Article  MathSciNet  Google Scholar 

  5. P.-E. Caprace and N. Monod, Decomposing locally compact groups into simple pieces, Mathematical Proceedings of the Cambridge Philosophical Society 150 (2011), 97–128.

    Article  MathSciNet  Google Scholar 

  6. P.-E. Caprace and N. Monod, Future directions in locally compact groups: a tentative problem list, in New Directions in Locally Compact Groups, London Mathematical Society Lecture Note Series, Vol. 447, Cambridge University Press, Cambridge, 2018, pp. 3434–355.

    Chapter  Google Scholar 

  7. P.-E. Caprace, C. D. Reid and P. R. Wesolek, Approximating simple locally compact groups by their dense locally compact subgroups, International Mathematics Research Notices, to appear, https://doi.org/10.1093/imrn/rny298.

  8. P.-E. Caprace, C. D. Reid and G. A. Willis, Locally normal subgroups of totally disconnected groups. Part II: Compactly generated simple groups, Forum of Mathematics. Sigma 5 (2017), Article no. e12.

  9. Y. Cornulier, Commability and focal locally compact groups Indiana University Mathematics Journal 64 (2015), 115–150.

    Article  MathSciNet  Google Scholar 

  10. A. Furman, A survey of measured group theory, in Geometry, Rigidity, and Group Actions, Chicago Lectures in mathematics, University of Chicago Press, Chicago, 2011, pp. 296–374.

    MATH  Google Scholar 

  11. T. Giordano and P. de La Harpe, Groupes de tresses et moyennabilité intérieure, Arkiv för Matematik 29 (1991), 63–72.

    Article  MathSciNet  Google Scholar 

  12. E. Hewitt and K. A. Ross, Abstract Harmonic Analysis. Vol. I, Grundlehren der Mathematischen Wissenschaften, Vol. 115, Springer, Berlin-New York, 1979.

    Google Scholar 

  13. J. I. Merzljakov, Locally solvable groups of finite rank. II, Algebra i Logika 8 (1969), 686–690.

    MathSciNet  Google Scholar 

  14. J. I. Merzljakov, On the theory of locally polycyclic groups, Journal of the London Mathematical Society 30 (1984), 67–72.

    Article  MathSciNet  Google Scholar 

  15. V. P. Platonov, Locally projectively nilpotent radical in topological groups, Doklady Akademii Nauk BSSR 9 (1965), 573–577.

    MathSciNet  MATH  Google Scholar 

  16. M. S. Raghunathan, Discrete Subgroups of Lie Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 68, Springer, New York-Heidelberg, 1972.

    Google Scholar 

  17. C. D. Reid and P. R. Wesolek, The essentially chief series of a compactly generated locally compact group, Mathematische Annalen 370 (2018), 841–861.

    Article  MathSciNet  Google Scholar 

  18. Y. Shalom and G. A. Willis, Commensurated subgroups of arithmetic groups, totally disconnected groups and adelic rigidity, Geometric and Functional Analysis 23 (2013), 1631–1683.

    Article  MathSciNet  Google Scholar 

  19. V. I. Usakov, Topological \(\overline {FC} \)-groups, Sibirskiĭ Matematičeskiĭ Žurnal 4 (1963), 1162–1174.

    MathSciNet  Google Scholar 

  20. S. P. Wang, Compactness properties of topological groups, Transactions of the American Mathematical Society 154 (1971), 301–314.

    Article  MathSciNet  Google Scholar 

  21. G. Willis, The structure of totally disconnected, locally compact groups, Mathematische Annalen 300 (1994), 341–363.

    Article  MathSciNet  Google Scholar 

  22. G. A. Willis, Tidy subgroups for commuting automorphisms of totally disconnected groups: an analogue of simultaneous triangularisation of matrices, New York Journal of Mathematics 10 (2004), 1–35.

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Institut Élie Cartan de Lorraine CNRS and Université de Lorraine, F-54052, Nancy, France

    Bruno Duchesne

  2. Department of Mathematocs, Texas A&M University, College Station, TX, 77843-3368, USA

    Robin Tucker-Drob

  3. Department of Mathematics and Computer Science, Wesleyan University, Middletown, CT, 06459, USA

    Phillip Wesolek

Authors
  1. Bruno Duchesne
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  2. Robin Tucker-Drob
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  3. Phillip Wesolek
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Corresponding author

Correspondence to Bruno Duchesne.

Additional information

B.D. is supported in part by French projects ANR-14-CE25-0004 GAMME and ANR-16-CE40-0022-01 AGIRA.

R.T.D. is supported in part by NSF grants DMS 1600904 and DMS 1855825.

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Duchesne, B., Tucker-Drob, R. & Wesolek, P. A new lattice invariant for lattices in totally disconnected locally compact groups. Isr. J. Math. 240, 539–565 (2020). https://doi.org/10.1007/s11856-020-2072-2

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  • DOI: https://doi.org/10.1007/s11856-020-2072-2

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