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From acyclic groups to the Bass conjecture for amenable groups

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Abstract.

We prove that the Bost Conjecture on the ℓ1-assembly map for countable discrete groups implies the Bass Conjecture. It follows that all amenable groups satisfy the Bass Conjecture.

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References

  1. Bass, H.: Euler characteristics and characters of discrete groups. Invent. Math. 35, 155–196 (1976)

    MATH  Google Scholar 

  2. Baum, P., Connes, A., Higson, N.: Classifying spaces for proper actions and K-theory of group C *-algebras. Contemp. Math. 167, 241–291 (1994)

    MATH  Google Scholar 

  3. Baumslag, G., Dyer, E., Heller, A.: The topology of discrete groups. J. Pure Appl. Algebra 16, 1–47 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bekka, M.E.B., Cherix, P.-A., Valette, A.: Proper affine isometric actions of amenable groups. Novikov Conjectures, Index Theorems and Rigidity, Vol. 2 (Oberwolfach, 1993), London Math. Soc. Lect. Note Ser., 227, Cambridge Univ. Press Cambridge, 1995, pp. 1–4

  5. Berrick, A.J.: An Approach to Algebraic K-Theory. Research Notes in Math. 56, Pitman, London, 1982

  6. Berrick, A.J.: Two functors from abelian groups to perfect groups. J. Pure Appl. Algebra 44, 35–43 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  7. Berrick, A.J.: Universal groups, binate groups and acyclicity. Group Theory, Proc. Singapore Conf., de Gruyter, Berlin, 1989, pp. 253–266

  8. Berrick, A.J., Varadarajan, K.: Binate towers of groups. Arch. Math. 62, 97–111 (1994)

    MathSciNet  MATH  Google Scholar 

  9. Berrick, A.J.: The acyclic group dichotomy. Preprint in preparation

  10. Bridson, M., Haefliger, A.: Metric Spaces of Non-Positive Curvature. Grundlehren der Math. Wissen. 319, Springer, Berlin, 1999

  11. Britton, J.L.: The word problem. Ann. Math. 77, 16–32 (1963)

    MATH  Google Scholar 

  12. Brown, K.S.: Cohomology of Groups. Graduate Texts in Math. 87, Springer, New York, 1994

  13. Chatterji, I., Mislin, G.: Atiyah’s L 2-index theorem. Enseign. Math. 49, 85–93 (2003)

    MathSciNet  MATH  Google Scholar 

  14. Cherix, P.-A., Cowling, M., Jolissaint, P., Julg, P., Valette, P.: Groups with the Haagerup Property (Gromov’s A-T-menability). Progress in Math. 197, Birkhäuser, Basel, 2001

  15. Davis, J.F., Lück, W.: Spaces over a category and assembly maps in isomorphism conjectures in K-and L-theory. K-Theory 15, 201–252 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  16. Eckmann, B.: Cyclic homology of groups and the Bass conjecture. Comment. Math. Helv. 61, 193–202 (1986)

    MathSciNet  MATH  Google Scholar 

  17. Eckmann, B.: Idempotents in a complex group algebra, projective modules, and the von Neumann algebra. Arch. Math. (Basel) 76, 241–249 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  18. Emmanouil, I.: On a class of groups satisfying Bass’ conjecture. Invent. Math. 132, 307–330 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  19. Farrell, T., Linnell, P.: Whitehead groups and the Bass conjecture. Math. Annalen 326, 723–757 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  20. Grigorchuk, R.I.: The growth rate of finitely generated groups and the theory of invariant means. Invest. Acad. Nauk USSR 45, 939–986 (1984) (in Russian)

    Google Scholar 

  21. Ji, R.: Nilpotency of Connes’ periodicity operator and the idempotent conjectures. K-Theory 9, 59–76 (1995)

    MathSciNet  MATH  Google Scholar 

  22. Kan, D.M., Thurston, W.P.: Every connected space has the homology of a K(π,1). Topology 15, 253–258 (1976)

    Article  MATH  Google Scholar 

  23. Karoubi, M.: K-Theory. An introduction. Grundlehren der Math. Wissen. 226, Springer, Berlin, 1978

  24. Kasparov, G.: Equivariant KK-theory and the Novikov conjecture. Invent. Math. 91, 147–201 (1988)

    MathSciNet  MATH  Google Scholar 

  25. Kasparov, G., Skandalis, G.: Groups acting properly on ‘‘bolic’’ spaces and the Novikov conjecture. Ann. of Math. (2) 158, 165–206 (2003)

    Google Scholar 

  26. Kropholler, P., Mislin, G.: Groups acting on finite-dimensional spaces with finite stabilizers. Comment. Math. Helv. 73, 122–136 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  27. Lafforgue, V.: K-théorie bivariante pour les algèbres de Banach et conjecture de Baum-Connes. Invent. Math. 149, 1–95 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  28. Linnell, P.A.: Decomposition of augmentation ideals and relation modules. Proc. London Math. Soc. (3) 47, 83–127 (1983)

    Google Scholar 

  29. Lück, W.: The relation between the Baum-Connes Conjecture and the Trace Conjecture. Invent. Math. 149, 123–152 (2002)

    Article  MathSciNet  Google Scholar 

  30. Lück, W.: Chern characters for proper equivariant homology theories and applications to K- and L-theory. J. reine angew. Math. 543, 193–243 (2002)

    MathSciNet  Google Scholar 

  31. Lyndon, R.C., Schupp, P.E.: Combinatorial Group Theory. Ergeb. 89, Springer, Berlin, 1977

  32. Mineyev, I., Yu, G.: The Baum-Connes conjecture for hyperbolic groups. Invent. Math. 149, 97–122 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  33. Mislin, G., Valette, A.: Proper Group Actions and the Baum-Connes Conjecture. Birkhäuser, Basel, 2003

  34. Puschnigg, M.: The Kadison-Kaplansky conjecture for word hyperbolic groups. Invent. Math. 149, 153–194 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  35. Skandalis, G.: Progrès récents sur la conjecture de Baum-Connes. Contribution de Vincent Lafforgue. Sém. Bourbaki, nov. 1999, exposé 869.

  36. Wagoner, J.B.: Delooping classifying spaces in algebraic K-theory. Topology 11, 349–370 (1972)

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, National University of Singapore, Singapore

    A. J. Berrick

  2. Department of Mathematics, Cornell University, White Hall, Ithaca NY, 14853-7901, USA

    I. Chatterji

  3. Department of Mathematics, ETH Zürich, 8092, Zürich, Switzerland

    G. Mislin

Authors
  1. A. J. Berrick
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  2. I. Chatterji
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  3. G. Mislin
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Correspondence to A. J. Berrick.

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Berrick, A., Chatterji, I. & Mislin, G. From acyclic groups to the Bass conjecture for amenable groups. Math. Ann. 329, 597–621 (2004). https://doi.org/10.1007/s00208-004-0521-6

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