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Strong boundedness of split Chevalley groups

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Abstract

This paper is concerned with the diameter of certain word norms on S-arithmetic split Chevalley groups. Such groups are well known to be boundedly generated by root elements. We prove that word metrics given by conjugacy classes on S-arithmetic split Chevalley groups have an upper bound only depending on the number of conjugacy classes. This property, called strong boundedness, was introduced by Kędra, Libmann and Martin and proven for SLn(R), assuming R is a principal ideal domain and n ≥ 3. We also provide examples of normal generating sets for S-arithmetic split Chevalley groups proving our bounds are sharp in an appropriate sense and give a complete account of the existence of small normally generating sets of Sp4(R) and G2(R). For instance, we prove that \({\rm{S}}{{\rm{p}}_4}(\mathbb{Z}[{{1 + \sqrt { - 7} } \over 2}])\) cannot be generated by a single conjugacy class.

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References

  1. E. Abe, Chevalley groups over local rings, Tohoku Mathematical Journal 21 (1969), 474–494.

    Article  MathSciNet  MATH  Google Scholar 

  2. E. Abe, Normal subgroups of Chevalley groups over commutative rings, in Algebraic K-Theory and Algebraic Number Theory (Honolulu, HI, 1987), Contemporary Mathematics, Vol. 83, American Mathematical Society, Providence, RI, 1989, pp. 1–17.

    Chapter  Google Scholar 

  3. E. Abe and K. Suzuki, On normal subgroups of Chevalley groups over commutative rings, Tohoku Mathematical Journal 28 (1976), 185–198.

    Article  MathSciNet  MATH  Google Scholar 

  4. H. Bass, K-theory and stable algebra, Institut des Hautes Études Scientifiques. Publications Mathématiques 22 (1964), 5–60.

    Article  MathSciNet  MATH  Google Scholar 

  5. H. Bass, J. Milnor and J.-P. Serre, Solution of the congruence subgroup problem for SLn (n ≥ 3) and Sp2n (n ≥ 2), Institut des Hautes Études Scientifiques. Publications Mathématiques 33 (1967), 59–137.

    Article  Google Scholar 

  6. C. Chevalley, Certains schémas de groupes semi-simples, in Séminaire Bourbaki, Vol. 6, Sociétémathématique de France, Paris, 1995, pp. 219–234.

    Google Scholar 

  7. D. L. Costa and G. E. Keller, Radix redux: normal subgroups of symplectic groups, Journal für die Reine und Angewandte Mathematik 427 (1992), 51–105.

    MathSciNet  MATH  Google Scholar 

  8. D. L. Costa and G. E. Keller, On the normal subgroups of G2 (A), Transactions of the American Mathematical Society 351 (1999), 5051–5088.

    Article  MathSciNet  MATH  Google Scholar 

  9. R. K. Dennis and L. N. Vaserstein, On a question of M. Newman on the number of commutators, Journal of Algebra 118 (1988), 150–161.

    Article  MathSciNet  MATH  Google Scholar 

  10. Ś. R. Gal, J. Kędra and A. Trost, Finite index subgroups in Chevalley groups are bounded: an addendum to “On bi-invariant word metrics”, Journal of Topology and Analysis, to appear, https://arxiv.org/abs/1808.06376.

  11. J. E. Humphreys, Linear Algebraic Groups, Graduate Texts in Mathematics, Vol. 21, Springer, New York—Heidelberg, 1975.

    MATH  Google Scholar 

  12. J. Kędra, A. Libman and B. Martin, Strong and uniform boundedness of groups, Journal of Topology and Analysis, https://doi.org/10.1142/S1793525321500497.

  13. D. A. Marcus, Number Fields, Universitext, Springer, Cham, 2018.

    Book  MATH  Google Scholar 

  14. A. V. Morgan, A. S. Rapinchuk and B. Sury, Bounded generation of SL2over rings of S-integers with infinitely many units, Algebra & Number Theory 12 (2018), 1949–1974.

    Article  MathSciNet  MATH  Google Scholar 

  15. D. W. Morris, Bounded generation of SL(n, A) (after D. Carter, G. Keller, and E. Paige), New York Journal of Mathematics 13 (2007), 383–421.

    MathSciNet  MATH  Google Scholar 

  16. J. Neukirch, Algebraic Number Theory, Grundlehren der Mathematischen Wissenschaften, Vol. 322, Springer, Berlin, 1999.

    Google Scholar 

  17. B. Nica, On bounded elementary generation for SLnover polynomial rings, Israel Journal of Mathematics 225 (2018), 403–410.

    Article  MathSciNet  MATH  Google Scholar 

  18. W. Rautenberg. A Concise Introduction to Mathematical Logic, Universitext, Springer, New York, 2010.

    Book  MATH  Google Scholar 

  19. R. Steinberg, Lectures on Chevalley Groups, University Lecture Series, Vol. 66, American Mathematical Society, Providence, RI, 2016.

    MATH  Google Scholar 

  20. O. I. Tavgen, Bounded generability of Chevalley groups over rings of S-integer algebraic numbers, Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya 54 (1990), 97–122, 221–222.

    MathSciNet  MATH  Google Scholar 

  21. A. Trost, Quantitative aspects of normal generation of split Chevalley groups, Ph.D. thesis, University of Aberdeen, 2020.

  22. A. trost, Bounded generation by root elements for Chevalley groups defined over rings of integers of function fields with an application in strong boundedness, https://arxiv.org/abs/2108.12254.

  23. A. Trost, Explicit strong boundedness for higher rank symplectic groups, Journal of Algebra 604 (2022), 694–726.

    Article  MathSciNet  MATH  Google Scholar 

  24. P. Vámos, 2-good rings, Quarterly Journal of Mathematics 56 (2005), 417–430.

    Article  MathSciNet  MATH  Google Scholar 

  25. L. N. Vaserstein, On normal subgroups of Chevalley groups over commutative rings, Tohoku Mathematical Journal 38 (1986), 219–230.

    Article  MathSciNet  MATH  Google Scholar 

  26. N. A. Vavilov, A note on the subnormal structure of general linear groups, Mathematical Proceedings of the Cambridge Philosophical Society 107 (1990), 193–196.

    Article  MathSciNet  MATH  Google Scholar 

  27. N. A. Vavilov, A. V. Smolenskiĭ and B. Sury, Unitriangular factorizations of Chevalley groups, Sankt-Peterburgskoe Otdelenie. Matematicheskiĭ Institut im. V. A. Steklova. Zapiski Nauchnykh Seminarov (POMI) 388 (2011), 17–47, 309–310.

    Google Scholar 

  28. H. You, Subgroups of classical groups normalized by relative elementary groups, Journal of Pure and Applied Algebra 216 (2012), 1040–1051.

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

I want to thank Bastien Karlhofer for pointing out Proposition 5.5 to me and for always being willing to listen and talk about mathematics. Further, I want to thank Ehud Meir for helpful comments regarding how to write a paper, Ben Martin for being available if I had questions, and him and Jarek Kędra for tirelessly reading several iterations of this paper. I also want to thank the anonymous referee for their helpful comments and the suggestion of a strategy to potentially sharpen the statement of Theorem 5.13. This work was funded by Leverhulme Trust Research Project Grant RPG-2017-159.

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  1. Institut für Algebra und Geometrie, Fakultät für Mathematik, Otto-von-Guericke University Magdeburg, 39016, Magdeburg, Germany

    Alexander A. Trost

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  1. Alexander A. Trost
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Trost, A.A. Strong boundedness of split Chevalley groups. Isr. J. Math. 252, 1–46 (2022). https://doi.org/10.1007/s11856-022-2344-0

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