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On graphs and valuations

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Abstract

In the last two decades new techniques emerged to construct valuations on an infinite division ring D, given a normal subgroup \({N \subseteq D^\times}\) of finite index. These techniques were based on the commuting graph of D ×/N in the case where D is non-commutative, and on the Milnor K-graph on D ×/N, in the case where D is commutative. In this paper we unify these two approaches and consider V-graphs on D ×/N and how they lead to valuations. We furthermore generalize previous results to situations of finitely many valuations.

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References

  1. Arason J.K., Elman R., Jacob B.: Rigid elements, valuations, and realization of Witt rings. J. Algebra 110, 449–467 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  2. Becker E.: Summen n-ter Potenzen in Körpern. J. Reine Angew. Math. 307(308), 8–30 (1979)

    MathSciNet  Google Scholar 

  3. Bergelson V., Shapiro D.B.: Multiplicative subgroups of finite index in a ring. Proc. Am. Math. Soc. 116, 885–896 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bogomolov F., Tschinkel Y.: Reconstruction of function fields. GAFA 18, 400–462 (2008)

    MATH  MathSciNet  Google Scholar 

  5. Bogomolov F., Tschinkel Y.: Introduction to birational anabelian geometry. In: Caporaso, L., McKernan, J., Mustata, M., Popa, M. (eds.) Current Developments in Algebraic Geometry, vol. 59, pp. 17–63. Cambridge University Press, Cambridge (2012)

    Google Scholar 

  6. Bogomolov F., Tschinkel Y.: Galois theory and projective geometry. Commun. Pure Appl. Math. 66, 1335–1359 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  7. Bourbaki N.: Commutative Algebra, ch. 1–7. Springer, Berlin (1989)

    Google Scholar 

  8. Cohn P.M.: On extending valuations in division algebras. Stud. Sci. Math. Hungar. 16, 65–70 (1981)

    MATH  Google Scholar 

  9. Efrat I.: Construction of valuations from K-theory. Math. Res. Lett. 6, 335–343 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  10. Efrat I.: Compatible valuations and generalized Milnor K-theory. Trans. Am. Math. Soc. 359, 4695–4709 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  11. Efrat I.: Valuations, Orderings, and Milnor K-Theory. Mathematical Surveys and Monographs, vol. 124. American Mathematical Society, Providence (2006)

    Book  Google Scholar 

  12. Efrat I.: Valuations and diameters of Milnor K-rings. Isr. J. Math. 172, 75–92 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  13. Endler O.: Valuation Theory. Springer, Berlin (1972)

    Book  MATH  Google Scholar 

  14. Engler A.J., Prestel A.: Valued Fields. Springer, Berlin (2005)

    MATH  Google Scholar 

  15. Hwang Y.S., Jacob B.: Brauer group analogues of results relating the Witt ring to valuations and Galois theory. Can. J. Math. 47, 527–543 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  16. Jacob B.: On the structure of Pythagorean fields. J. Algebra 68, 247–267 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  17. Jacob B.: Fans, real valuations, and hereditarily-pythagorean fields. Pac. J. Math. 93, 95–105 (1981)

    Article  MATH  Google Scholar 

  18. Jacob B., Wadsworth A.R.: A new construction of noncrossed division algebras. Tran. Am. Math. Soc. 293, 693–721 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  19. Koenigsmann J.: From p-rigid elements to valuations (with a Galois-characterization of p-adic fields). J. Reine Angew. Math. 465, 165–182 (1995)

    MATH  MathSciNet  Google Scholar 

  20. Lang S.: Algebra, 3rd edn, GTM 211. Springer, Berlin (2002)

    Google Scholar 

  21. Marshall M.A.: An approximation theorem for coarse V-topologies on rings. Can. Math. Bull. 37, 527–533 (1994)

    Article  MATH  Google Scholar 

  22. Marubayashi H., Miyamoto H., Ueda A.: Non-Commutative Valuation Rings and Semi-Hereditary Orders. Kluwer, Dordrecht (1997)

    Book  MATH  Google Scholar 

  23. Pierce R.S.: Associative Algebras, GTM 88. Springer, Berlin (1982)

    Book  Google Scholar 

  24. Platonov V.P., Rapinchuk A.S.: Algebraic Groups and Number Theory. Academic Press, New York (1993)

    Google Scholar 

  25. Rapinchuk A.S.: The Margulis–Platonov conjecture for SL1 , D and 2-generation of finite simple groups. Math. Z. 252, 295–313 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  26. Rapinchuk A., Potapchik A.: Normal subgroups of SL 1, D and the classification of finite simple groups. Proc. Indian Acad. Sci. 106, 329–336 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  27. Rapinchuk A.S., Segev Y.: Valuation-like maps and the congruence subgroup property. Invent. Math. 144, 571–607 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  28. Rapinchuk A.S., Segev Y., Seitz G.: Finite quotients of the multiplicative group of a finite dimensional division algebra are solvable. J. Am. Math. Soc. 15, 929–978 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  29. Rehmann U.: Zentrale Erweiterungen der speziellen linearen Gruppe eines Schiefkörpers. J. Reine Angew. Math. 301, 77–104 (1978)

    MATH  MathSciNet  Google Scholar 

  30. Rehmann, U., Stuhler, U.: On K 2 of finite-dimensional division algebras over arithmetical fields. Invent. Math. 50, 75–90 (1978/79)

  31. Ribenboim P.: The Theory of Classical Valuations. Springer, Berlin (1999)

    Book  MATH  Google Scholar 

  32. Schilling O.F.G.: The Theory of Valuations. American Mathematical Society, Providence (1950)

    Book  MATH  Google Scholar 

  33. Segev Y.: The commuting graph of minimal nonsolvable groups. Geom. Dedic. 88(1–3), 55–66 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  34. Segev Y.: On finite homomorphic images of the multiplicative group of a division algebra. Ann. Math. 149, 219–251 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  35. Segev Y., Seitz G.M.: Anisotropic groups of type A n and the commuting graph of finite simple groups. Pac. J. Math. 202, 125–226 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  36. Topaz, A.: Almost-commuting-liftable subgroups of Galois groups. Preprint, available at arXiv:1202.1786

  37. Topaz, A.: Commuting-liftable subgroups of Galois groups II. Preprint, available at arXiv:1208.0583

  38. Turnwald G.: Multiplicative subgroups of finite index in rings. Proc. Am. Math. Soc. 120, 377–381 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  39. Wadsworth, A.R.: (2002) Valuation theory on finite dimensional division algebras. In: Saskatoon, S.K. (eds) Valuation Theory and Its Applications, vol. I. Fields Inst. Commun., vol. 32, pp. 385–449. American Mathematical Society, Providence, RI (1999)

  40. Ware R.: Valuation rings and rigid elements in fields. Can. J. Math. 33, 1338–1355 (1981)

    Article  MATH  MathSciNet  Google Scholar 

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Correspondence to Ido Efrat.

Additional information

Ido Efrat: This research was supported by the Israel Science Foundation (Grant No. 152/13). Andrei S. Rapinchuk: Partially supported by NSF Grant DMS-1301800 and BSF Grant 201049.

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Efrat, I., Rapinchuk, A.S. & Segev, Y. On graphs and valuations. manuscripta math. 146, 395–432 (2015). https://doi.org/10.1007/s00229-014-0699-1

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