Advertisement

Mathematische Zeitschrift

, Volume 291, Issue 3–4, pp 1389–1417 | Cite as

Algebraic entropy of amenable group actions

  • Simone ViriliEmail author
Article

Abstract

Let R be a ring, let G be an amenable group and let \(R{*}G\) be a crossed product. The goal of this paper is to construct, starting with a suitable additive function L on the category of left modules over R, an additive function on a subcategory of the category of left modules over \(R{*}G\), which coincides with the whole category if \(L({}_RR) <\infty \). This construction can be performed using a dynamical invariant associated with the original function L, called algebraic L-entropy. We apply our results to two classical problems on group rings: the stable finiteness and the zero-divisors conjectures.

Keywords

Length functions Gabriel dimension Algebraic entropy Amenable groups Zero divisors Stable finiteness 

Mathematics Subject Classification

16S35 43A07 16D10 18E35 

Notes

Acknowledgements

It is a pleasure for me to thank Peter Vámos for giving me a copy of his Ph.D. thesis and for useful discussions started in Padova in 2010. I am also sincerely grateful to my Ph.D. advisor Dolors Herbera for her encouragement and trust: she gave me time, freedom and several suggestions that helped me to work independently on this project. Finally, I would like to thank Pere Ara, Ferran Cedó, Hanfeng Li, Bingbing Liang and Nhan-Phu Chung for some useful discussions on preliminary versions of this paper.

References

  1. 1.
    Ara, P., O’Meara, K.C., Perera, F.: Stable finiteness of group rings in arbitrary characteristic. Adv. Math. 170(2), 224–238 (2002)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Cohn, P.M.: Free Ideal Rings and Localization in General Rings. New Mathematical Monographs, vol. 3. Cambridge University Press, Cambridge (2006)Google Scholar
  3. 3.
    Ceccherini-Silberstein, T., Coornaert, M.: The Garden of Eden theorem for linear cellular automata. Ergod. Theory Dyn. Syst. 26(1), 53–68 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Ceccherini-Silberstein, T., Coornaert, M.: Cellular Automata and Groups, Springer Monographs in Mathematics. Springer, Berlin (2010)CrossRefzbMATHGoogle Scholar
  5. 5.
    Chung, N.-P., Thom, A.: Some remarks on the entropy for algebraic actions of amenable groups. arXiv:1302.5813 (2013)
  6. 6.
    Dikranjan, D., Goldsmith, B., Salce, L., Zanardo, P.: Algebraic entropy of endomorphisms of abelian groups. Trans. AMS 361, 3401–3434 (2009)CrossRefzbMATHGoogle Scholar
  7. 7.
    Elek, G.: The Euler characteristic of discrete groups and Yuzvinskii’s entropy addition formula. Bull. Lond. Math. Soc. 31(6), 661–664 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Elek, G.: The rank of finitely generated modules over group algebras. Proc. Am. Math. Soc. 131(11), 3477–3485 (2003). (electronic)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Elek, G., Szabó, E.: Sofic groups and direct finiteness. J. Algebra 280(2), 426–434 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Facchini, A.: Module Theory. Modern Birkhäuser Classics. Endomorphism Rings and Direct Sum Decompositions in Some Classes of Modules. Birkhäuser/Springer Basel AG, Basel (1998) (2012 reprint of the 1998 original)Google Scholar
  11. 11.
    Følner, E.: On groups with full Banach mean value. Math. Scand. 3, 243–254 (1955)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Gabriel, P.: Des catégories abéliennes. Bull. Soc. Math. France 90, 323–448 (1962)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Gordon, R., Chris Robson, J.: The Gabriel dimension of a module. J. Algebra 29, 459–473 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Huang, W., Ye, X., Zhang, G.: Local entropy theory for a countable discrete amenable group action. J. Funct. Anal. 261(4), 1028–1082 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Kaplansky, I.: Fields and Rings. Chicago Lectures in Mathematics. University of Chicago Press, Chicago (1995) (Reprint of the second (1972) edition)Google Scholar
  16. 16.
    Krieger, F.: Le lemme d’Ornstein-Weiss d’après Gromov. In: Dynamics, Ergodic Theory, and Geometry, vol. 54 of Math. Sci. Res. Inst. Publ., pp 99–111. Cambridge Univ. Press, Cambridge (2007)Google Scholar
  17. 17.
    Li, H., Liang, B.: Sofic mean length. arXiv:1510.07655
  18. 18.
    Lück, W.: \(L^2\)-Invariants: Theory and Applications to Geometry and \(K\)-Theory, vol. 44 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics. Springer, Berlin (2002)Google Scholar
  19. 19.
    McConnell, J.C., Robson, J.C.: Noncommutative Noetherian Rings, vol. 30 of Graduate Studies in Mathematics. American Mathematical Society, Providence (2001) (With the cooperation of L. W. Small (revised edition))Google Scholar
  20. 20.
    Northcott, D.G., Reufel, M.: A generalization of the concept of length. Q. J. Math. Oxford Ser. 2(16), 297–321 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Năstăsescu, C., van Oystaeyen, F.: Dimensions of ring theory. Mathematics and its Applications, vol. 36. Dordrecht etc. D. Reidel Publishing Company, a member of the KluwerAcademic Publishers Group. XI, p. 360 (1987)Google Scholar
  22. 22.
    Ornstein, D.S., Weiss, B.: Entropy and isomorphism theorems for actions of amenable groups. J. Anal. Math. 48, 1–141 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Passman, D.S.: The Algebraic Structure of Group Rings. Robert E. Krieger Publishing Co. Inc., Melbourne (1985) (Reprint of the 1977 original)Google Scholar
  24. 24.
    Passman, D.S.: Infinite Crossed Products. Pure and Applied Mathematics, vol. 135. Academic Press Inc., Boston (1989)Google Scholar
  25. 25.
    Pete, G.: Probability and geometry on groups (2013). http://math.bme.hu/~gabor/PGG.pdf. Accessed 28 Nov 2018
  26. 26.
    Ranicki, A. (ed.): Non-Commutative Localization in Algebra and Topology, vol. 330 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge (2006)Google Scholar
  27. 27.
    Salce, L., Zanardo, P.: A general notion of algebraic entropy and the rank entropy. Forum Math. 21(4), 579–599 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Salce, L., Vámos, P., Virili, S.: Length functions, multiplicities and algebraic entropy. Forum Math. 25, 255–282 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Stenström, B.: Rings and Modules of Quotients. Lecture Notes in Mathematics, vol. 237. Springer, Berlin (1971)zbMATHGoogle Scholar
  30. 30.
    Vámos, P.: Length Functions on Modules. PhD thesis, University of Sheffield (1968)Google Scholar
  31. 31.
    Ward, T., Zhang, Q.: The Abramov–Rokhlin entropy addition formula for amenable group actions. Monatsh. Math. 114(3–4), 317–329 (1992)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Facultad de matemáticasUniversidad de MurciaMurciaSpain

Personalised recommendations