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On the groups SL2(ℤ[x]) and SL2(k[x, y])

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Abstract

This paper studies free quotients of the groups SL2(ℤ[x]) and SL2(k[x, y]),k a finite field. These quotients give information about the relation of the above groups to their subgroups generated by elementary or unipotent elements.

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References

  1. H. Bass,K-theory and stable algebra, Publ. Math., Inst. Hautes Étud. Sci.22 (1964), 485–544.

    Google Scholar 

  2. H. Bass,Algebraic K-theory, W. A. Benjamin, New York, Amsterdam, 1968.

    MATH  Google Scholar 

  3. H. Bass, J. Milnor and J. P. Serre,Solution of the congruence subgroup problem for SL n (A) (n ≥ 3)and Sp2n (A) (n ≥ 2), Publ. Math. Inst. Hautes Étud. Sci.33 (1967), 59–137.

    Article  MATH  MathSciNet  Google Scholar 

  4. P. M. Cohn,On the structure of the GL2 of a ring, Publ. I. H. E. S.30 (1966), 5–53.

    Google Scholar 

  5. G. Cooke and P. Weinberger,On the construction of division chains in algebraic number rings, with application to SL2, Commun. in Algebra3 (1975), 481–524.

    Article  MATH  MathSciNet  Google Scholar 

  6. J. Dieudonné,La géometrie des groupes classiques, Springer-Verlag, Berlin, Heidelberg, New York, 1971.

    MATH  Google Scholar 

  7. D. Estes and J. Ohm,Stable range in commutative rings, J. Algebra7 (1967), 343–362.

    Article  MATH  MathSciNet  Google Scholar 

  8. F. Grunewald, H. Helling and J. Mennicke, SL2 over complex quadratic numberfields, I, Algebra i Logica17 (1978), 512–580.

    MathSciNet  Google Scholar 

  9. F. Grunewald, J. Mennicke and L. Vaserstein,On symplectic groups over polynomial rings, Math. Z.206 (1990), 35–56.

    Article  MathSciNet  Google Scholar 

  10. F. Grunewald and J. Schwermer,Free non-abelian quotients of SL2 over orders of imaginary quadratic numberfields, J. Algebra69 (1981), 298–304.

    Article  MATH  MathSciNet  Google Scholar 

  11. F. Grunewald and J. Schwermer,Arithmetic quotients of hyperbolic 3-space, cups forms, and link complements, Duke Math. J.48 (1981), 351–358.

    Article  MATH  MathSciNet  Google Scholar 

  12. E.-U. Gekeler,Drinfeld Modular Curves, Lecture Notes in Math., Vol. 1231, Springer-Verlag, Berlin, Heidelberg, New York, 1987.

    Google Scholar 

  13. E.-U. Gekeler,Le genre des courbes modulaires de Drinfeld, C. R. Acad. Sci. Paris300 (1985), 647–650.

    MATH  MathSciNet  Google Scholar 

  14. B. Liehl,On the group SL2 over orders of arithmetic type, J. Reine Angew. Math.323 (1981), 153–171.

    MATH  MathSciNet  Google Scholar 

  15. R. Maschinsky,Die Operation der Gruppen SL2 und GL2 auf dem Baum der Gitterklassen einer elliptischen Kurve, Diplomarbeit, Wuppertal, 1989.

    Google Scholar 

  16. C. Queen,Some arithmetic properties of subrings of function fields over finite fields, Archiv der Mathematik26 (1975), 51–56.

    Article  MATH  MathSciNet  Google Scholar 

  17. J. P. Serre,Trees, Springer-Verlag, Berlin, Heidelberg, New York, 1980.

    MATH  Google Scholar 

  18. J. P. Serre,Local Fields, Springer-Verlag, Berlin, Heidelberg, New York, 1980.

    Google Scholar 

  19. J. P. Serre,Le problème des groupes de congruence pour SL2, Annals of Math.92 (1970), 489–527.

    Article  MathSciNet  Google Scholar 

  20. R. Swan,Generators and relations for certain linear groups, Advances in Math.6 (1971), 1–77.

    Article  MATH  MathSciNet  Google Scholar 

  21. U. Stuhler,Über die Faktorkommutatorgruppe der Gruppen SL2(O)im Funktionenkörperfall, Archiv der Mathematik42 (1984), 314–324.

    Article  MATH  MathSciNet  Google Scholar 

  22. A. Suslin,On the structure of the special linear group over polynomial rings, Math. USSR, Izv11 (1977), 221–238.

    Article  MATH  Google Scholar 

  23. W. Terhalle,Zur Gruppe SL2(ℤ[x]), Diplomarbeit, Bielefeld, 1989.

    Google Scholar 

  24. L. Tavgen,On bounded generation of Chevalley groups over S-integer algebraic numbers, Izv. Nauk, 1990.

  25. L. Vaserstein,Stabilization for classical groups over rings, Mat. Sbornik81 (1970), 328–351 (Translated as: Math. USSR, Sb.10 (1970), 307–326).

    MathSciNet  Google Scholar 

  26. L. Vaserstein,On the SL2 over Dedekind rings of arithmetic type, Math. USSR Sb.18 (1972), 321–332.

    Article  Google Scholar 

  27. L. Vaserstein,Structure of the classical arithmetic groups of rank greater than 1, Math USSR Sb.20 (1973), 465–492.

    Article  Google Scholar 

  28. L. Vaserstein,On the normal subgroups of GL n over a ring, Lecture Notes in Math., Vol. 854, Springer-Verlag, Berlin, Heidelberg, New York, 1972, pp. 456–465.

    Google Scholar 

  29. L. Vaserstein,Stabilization of unitary and orthogonal groups over a ring with involution, Mat. Sbornik81 (1970), 307–326.

    Article  MathSciNet  Google Scholar 

  30. L. Vaserstein and A. Suslin,Serre’s problem on projective modules over polynomial rings and algebraic K-theory, Izv. Akad. Nauk SSSR10 (1976), 937–1000.

    Google Scholar 

  31. A. Weil,Basic Number Theory, Springer-Verlag, Berlin, Heidelberg, New York, 1967.

    MATH  Google Scholar 

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Authors and Affiliations

  1. Fakultät für Mathematik, Universität Bonn, Beringstrasse 4, W-5300, Bonn 1, Germany

    Fritz Grunewald

  2. Fakultät für Mathematik, Universität Bielefeld, W-4800, Bielefeld, Germany

    Jens Mennicke

  3. Department of Mathematics, The Pennsylvania State University, 16802, University Park, PA, USA

    Leonid Vaserstein

Authors
  1. Fritz Grunewald
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  2. Jens Mennicke
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  3. Leonid Vaserstein
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Grunewald, F., Mennicke, J. & Vaserstein, L. On the groups SL2(ℤ[x]) and SL2(k[x, y]). Israel J. Math. 86, 157–193 (1994). https://doi.org/10.1007/BF02773676

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  • DOI: https://doi.org/10.1007/BF02773676

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