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Bounded reduction for Chevalley groups of types \(E_6\) and \(E_7\)

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Abstract

We prove that an element from the Chevalley group of type \(E_6\) or \(E_7\) over a polynomial ring with coefficients in a small-dimensional ring can be reduced to an element of certain proper subsystem subgroup by a bounded number of elementary root elements. The bound is given explicitly. This result is an effective version of the early stabilisation of the corresponding \(K_1\)-functor. We also give a part of the proof of similar hypothesis for \(E_8\).

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References

  1. Abe, E.: Whitehead groups of Chevalley groups over polynomial rings. Commun. Algebra 11(12), 1271–1307 (1983)

    MathSciNet  Google Scholar 

  2. Bak, A.: Nonabelian K-theory: the nilpotent class of \(\rm K _{1}\) and general stability. K-Theory 4(4), 363–397 (1991)

    MathSciNet  Google Scholar 

  3. Bass, H.: \(K\)-theory and stable algebra. Inst. Hautes Études Sci. Publ. Math. 22, 5–60 (1964)

    MathSciNet  Google Scholar 

  4. Bourbaki, N.: Elements of Mathematics: Lie Groups and Lie Algebras. Chapters 7–9. Springer, Berlin (2004)

  5. Bourbaki, N.: Elements of Mathematics: Lie Groups and Lie Algebras. Chapters 4–6. Springer, Berlin (2008)

  6. Carter, D., Keller, G.: Bounded elementary generation of \(\operatorname{SL}_n({\cal{O} })\). Amer. J. Math. 105(3), 673–687 (1983)

    MathSciNet  Google Scholar 

  7. Carter, D., Keller, G.: Elementary expressions for unimodular matrices. Commun. Algebra 12(3–4), 379–389 (1984)

    MathSciNet  Google Scholar 

  8. Chevalley, C.: Certains schémas de groupes semi-simples. Séminaire Bourbaki 6(219), 219–234 (1995)

    Google Scholar 

  9. Erovenko, I.V., Rapinchuk, A.S.: Bounded generation of some \(S\)-arithmetic orthogonal groups. C. R. Acad. Sci. Paris Sér. I Math. 333(5), 395–398 (2001)

  10. Erovenko, I.V., Rapinchuk, A.S.: Bounded generation of \(S\)-arithmetic subgroups of isotropic orthogonal groups over number fields. J. Number Theory 119(1), 28–48 (2006)

    MathSciNet  Google Scholar 

  11. Ershov, M., Jaikin-Zapirain, A.: Property \((T)\) for noncommutative universal lattices. Invent. Math. 179(2), 303–347 (2010)

    MathSciNet  Google Scholar 

  12. Ershov, M., Jaikin-Zapirain, A., Kassabov, M.: Property \((T)\) for Groups Graded by Root Systems. Memoirs of the American Mathematical Society, vol. 249(1186). American Mathematical Society, Providence (2017)

  13. Estes, D., Ohm, J.: Stable range in commutative rings. J. Algebra 7, 343–362 (1967)

    MathSciNet  Google Scholar 

  14. Grunewald, F., Mennicke, J., Vaserstein, L.: On symplectic groups over polynomial rings. Math. Z. 206(1), 35–56 (1991)

    MathSciNet  Google Scholar 

  15. Gvozdevsky, P.: Improved \({K}_1\)-stability for the embedding \({D}_5\) into \({E}_6\). Commun. Algebra 48(11), 4922–4931 (2020)

    MathSciNet  Google Scholar 

  16. Gvozdevsky, P.: Bounded reduction of orthogonal matrices over polynomial rings. J. Algebra 602, 300–321 (2022)

    MathSciNet  Google Scholar 

  17. Hazrat, R., Stepanov, A., Vavilov, N., Zhang, Z.: Commutator width in Chevalley groups. Note Mat. 33(1), 139–170 (2013)

    MathSciNet  Google Scholar 

  18. Hermida, J.A., Sánchez-Giralda, T.: Linear equations over commutative rings and determinantal ideals. J. Algebra 99(1), 72–79 (1986)

    MathSciNet  Google Scholar 

  19. Kassabov, M.: Kazhdan constants for \({\rm SL}_n({\mathbb{Z} })\). Int. J. Algebra Comput. 15(5–6), 971–995 (2005)

    Google Scholar 

  20. Kopeiko, V.I.: The stabilization of symplectic groups over a polynomial ring. Math. USSR-Sb. 34(5), 655–669 (1978)

    Google Scholar 

  21. Kopeiko, V.I.: On the structure of the symplectic group of polynomial rings over regular rings of dimension \(\leqslant 1\). Russian Math. Surveys 47(4), 210–211 (1992)

    MathSciNet  Google Scholar 

  22. Kopeiko, V.I.: On the structure of the special linear groups over Laurent polynomial rings. Fundam. Prikl. Mat. 1(4), 1111–1114 (1995)

    MathSciNet  Google Scholar 

  23. Kopeiko, V.I.: Symplectic groups over Laurent polynomial rings and patching diagrams. Fundam. Prikl. Mat. 5(3), 943–945 (1999)

    MathSciNet  Google Scholar 

  24. Kunyavskii, B., Plotkin, E., Vavilov, N.: Bounded generation and commutator width of Chevalley groups: function case. Eur. J. Math. 9(3), Art. No. 53 (2023)

  25. Lichtenstein, W.: A system of quadrics describing the orbit of the highest weight vector. Proc. Amer. Math. Soc. 84(4), 605–608 (1982)

    MathSciNet  Google Scholar 

  26. Lubotzky, A.: Subgroup growth and congruence subgroups. Invent. Math. 119(2), 267–295 (1995)

    MathSciNet  Google Scholar 

  27. Magurn, B.A., van der Kallen, W., Vaserstein, L.N.: Absolute stable rank and Witt cancellation for noncommutative rings. Invent. Math. 91(3), 525–542 (1988)

    MathSciNet  Google Scholar 

  28. Matsumoto, H.: Sur les sous-groupes arithmétiques des groupes semi-simples déployés. Ann. Sci. École Norm. Sup. 2(1), 1–62 (1969)

    MathSciNet  Google Scholar 

  29. Morgan, A.V., Rapinchuk, A.S., Sury, B.: Bounded generation of \({{\rm SL}}_{2}\) over rings of \(S\)-integers with infinitely many units. Algebra Number Theory 12(8), 1949–1974 (2018)

    MathSciNet  Google Scholar 

  30. Morris, D.W.: Bounded generation of \({{\rm SL}}(n,A)\) (after D. Carter, G. Keller, and E. Paige). New York J. Math. 13, 383–421 (2007)

  31. Platonov, V.P., Rapinchuk, A.S.: Abstract properties of \(S\)-arithmetic groups and the congruence subgroup problem. Izv. Ross. Akad. Nauk. Ser. Mat. 56(3), 483–508 (1992)

    Google Scholar 

  32. Plotkin, E.B.: Stability theorems of \(K_1\)-functor for Chevalley groups.. In: Yamaguti, K., Kawamoto, N. (eds.) Nonassociative Algebras and Related Topics, pp. 203–217. World Sientific, River Edge (1991) (1991)

  33. Plotkin, E.B.: Surjective stabilization for \({K}_1\)-functor for some exceptional Chevalley groups. J. Soviet Math. 64(1), 751–767 (1993)

    MathSciNet  Google Scholar 

  34. Plotkin, E.: On the stability of the \(K_1\)-functor for Chevalley groups of type \(E_7\). J. Algebra 210(1), 67–85 (1998)

    MathSciNet  Google Scholar 

  35. Plotkin, E., Semenov, A., Vavilov, N.: Visual basic representations: an atlas. Int. J. Algebra Comput. 8(1), 61–95 (1998)

    MathSciNet  Google Scholar 

  36. Shalom, Y.: Bounded generation and Kazhdan’s property (T). Publ. Math. Inst. Hautes Études Sci. Publ. Math. 90, 145–168 (1999)

  37. Shalom, Y.: The algebraization of Kazhdan’s property (T). In: Proceedings of the International Congress of Mathematicians, Vol. II, pp. 1283–1310. European Mathematical Society, Zürich (2006)

  38. Shalom, Y., Willis, G.A.: Commensurated subgroups of arithmetic groups, totally disconnected groups and adelic rigidity. Geom. Funct. Anal 23(5), 1631–1683 (2013)

    MathSciNet  Google Scholar 

  39. Sivatski, A.S., Stepanov, A.V.: On the word length of commutators in \({{\rm GL}}_n(R)\). K-Theory 17(4), 295–302 (1999)

    MathSciNet  Google Scholar 

  40. Smolensky, A., Sury, B., Vavilov, N.: Gauss decomposition for Chevalley groups, revisited. Int. J. Group Theory 1(1), 3–16 (2012)

    MathSciNet  Google Scholar 

  41. Stavrova, A.K.: Homotopy invariance of non-stable \(K_1\)-functors. J. K-Theory 13(2), 199–248 (2014)

    MathSciNet  Google Scholar 

  42. Stavrova, A.K.: Chevalley groups of polynomial rings over Dedekind domains. J. Group Theory 23(1), 121–132 (2020)

    MathSciNet  Google Scholar 

  43. Stein, M.R.: Stability theorems for \(K_1,K_2\) and related functors modeled on Chevalley groups. Japan J. Math. (N.S.) 4(1), 77–108 (1978)

  44. Stein, M.R.: Matsumoto’s solution of the congruence subgroup problem and stability theorems in algebraic K-theory. In: Proceedings of the 19th Meeting Algebra Section, pp. 32–44. Mathematical Society of Japan, Tokyo (1983)

  45. Steinberg, R.G.: Some consequences of the elementary relations in \({{\rm SL}_n}\). In: McKay, J. (ed.) Finite Groups-Coming of Age. Contemporary Mathematics, vol. 45, pp. 335–350. American Mathematical Society, Providence (1985)

  46. Stepanov, A., Vavilov, N.: Length of commutators in Chevalley groups. Israel J. Math. 185, 253–276 (2011)

    MathSciNet  Google Scholar 

  47. Stepanov, A., Vavilov, N.: Decomposition of transvections: a theme with variations. K-Theory 19(2), 109–153 (2000)

    MathSciNet  Google Scholar 

  48. Suslin, A.A.: On the structure of the special linear group over polynomial rings. Math. USSR-Izv. 11, 221–238 (1977)

    Google Scholar 

  49. Suslin, A.A., Kopeiko, V.I.: Quadratic modules and orthogonal group over polynomial rings. J. Soviet Math. 20(6), 2665–2691 (1982)

    Google Scholar 

  50. Tavgen, O.I.: Bounded generation of Chevalley groups over rings of algebraic \(S\)-integers. Math. USSR-Izv. 36(1), 101–128 (1991)

    MathSciNet  Google Scholar 

  51. Tavgen, O.I.: Bounded generation of normal and twisted Chevalley groups over the rings of \(S\)-integers. In: Bokut’, L.A. et al. (eds.) Proceedings of the International Conference on Algebra, Part 1. Contemporary Mathematics, vol. 131.1, pp. 409–421. American Mathematical Society, Providence (1992)

  52. Trost, A.A.: Bounded generation by root elements for Chevalley groups defined over rings of integers of function fields with an application in strong boundedness (2021). arXiv:2108.12254

  53. Trost, A.A.: Explicit strong boundedness for higher rank symplectic groups. J. Algebra 604, 694–726 (2022)

    MathSciNet  Google Scholar 

  54. Trost, A.A.: Strong boundedness of split Chevalley groups. Israel J. Math. 252(1), 1–46 (2022)

    MathSciNet  Google Scholar 

  55. Trost, A.: Conjugation-invariant norms on \({{\rm SL}}_2(R)\) for rings of S-algebraic integers with infinitely many units. Commun. Algebra 51(10), 4329–4346 (2023)

    Google Scholar 

  56. Trost, A.: Stability, bounded generation and strong boundedness (2023). arXiv:2305.11562

  57. Vaserstein, L.N.: On the stabilization of the general linear group over a ring. Math. USSR-Sb. 8(3), 383–400 (1969)

    MathSciNet  Google Scholar 

  58. Vaserstein, L.N.: Bounded reduction of invertible matrices over polynomial rings by addition operations (2006). http://www.personal.psu.edu/lxv1/pm2.pdf

  59. Vavilov, N.: A third look at weight diagrams. Rend. Semin. Mat. Univ. Padova 104, 201–250 (2000)

    MathSciNet  Google Scholar 

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  1. Department of Mathematics, Bar-Ilan University, 5290002, Ramat Gan, Israel

    Pavel Gvozdevsky

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Dedicated to the memory of the brilliant mathematician Irina Suprunenko.

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The paper is written as part of the author’s post-doctoral fellowship at Bar-Ilan University, Department of Mathematics. Research is supported by the Russian Science Foundation grant (project No. 22-21-00257).

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Gvozdevsky, P. Bounded reduction for Chevalley groups of types \(E_6\) and \(E_7\). European Journal of Mathematics 9, 102 (2023). https://doi.org/10.1007/s40879-023-00698-x

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