Advertisement

Inventiones mathematicae

, Volume 172, Issue 1, pp 29–70 | Cite as

The K-theoretic Farrell–Jones conjecture for hyperbolic groups

  • Arthur Bartels
  • Wolfgang LückEmail author
  • Holger Reich
Article

Abstract

We prove the K-theoretic Farrell–Jones conjecture for hyperbolic groups with (twisted) coefficients in any associative ring with unit.

Keywords

Simplicial Complex Chain Complex Cayley Graph Hyperbolic Group Whitehead Group 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Anderson, D.R.: The Whitehead torsion of a fiber-homotopy equivalence. Mich. Math. J. 21, 171–180 (1974)zbMATHCrossRefGoogle Scholar
  2. 2.
    Anderson, D.R., Munkholm, H.J.: Boundedly Controlled Topology. Springer, Berlin, (1988) Foundations of algebraic topology and simple homotopy theory.Google Scholar
  3. 3.
    Bartels, A., Farrell, T., Jones, L., Reich, H.: On the isomorphism conjecture in algebraic K-theory. Topology 43(1), 157–213 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Bartels, A., Lück, W., Echterhoff, S.: Inheritance of isomorphism conjectures under colimits. In preparation (2007)Google Scholar
  5. 5.
    Bartels, A., Lück, W., Reich, H.: Equivariant covers of hyperbolic groups. Preprintreihe SFB 478 – Geometrische Strukturen in der Mathematik, Heft 434, Münster, arXiv:math.GT/0609685 (2006)Google Scholar
  6. 6.
    Bartels, A., Lück, W., Reich, H.: On the Farrell–Jones conjecture and its applications. In preparation (2007)Google Scholar
  7. 7.
    Bartels, A., Reich, H.: Coefficients for the Farrell–Jones conjecture. Preprintreihe SFB 478 – Geometrische Strukturen in der Mathematik, Heft 402, Münster, arXiv:math.KT/0510602 (2005)Google Scholar
  8. 8.
    Bartels, A., Reich, H.: On the Farrell–Jones conjecture for higher algebraic K-theory. J. Am. Math. Soc. 18(3), 501–545 (2005) (electronic)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Bartels, A.C.: On the domain of the assembly map in algebraic K-theory. Algebr. Geom. Topol. 3, 1037–1050 (2003) (electronic)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Bartels, A.C.: Squeezing and higher algebraic K-theory. K-Theory 28(1), 19–37 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Bestvina, M., Mess, G.: The boundary of negatively curved groups. J. Am. Math. Soc. 4(3), 469–481 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Bowditch, B.H.: Notes on Gromov’s hyperbolicity criterion for path-metric spaces. In: Group Theory from a Geometrical Viewpoint (Trieste, 1990), pp. 64–167. World Sci. Publishing, River Edge, NJ (1991)Google Scholar
  13. 13.
    Bridson, M.: Non-positive curvature and complexity for finitely presented groups. Lecture on the ICM 2006 in Madrid, to appear in the ICM-Proceedings (2006)Google Scholar
  14. 14.
    Bridson, M.R., Haefliger, A.: Metric spaces of non-positive curvature. Grundlehren Math. Wiss., vol. 319. Springer, Berlin (1999)zbMATHGoogle Scholar
  15. 15.
    Bryant, J., Ferry, S., Mio, W., Weinberger, S.: Topology of homology manifolds. Ann. Math. (2) 143(3), 435–467 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Cárdenas, M., Pedersen, E.K.: On the Karoubi filtration of a category. K-Theory 12(2), 165–191 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Carlsson, G., Pedersen, E.K.: Controlled algebra and the Novikov conjectures for K- and L-theory. Topology 34(3), 731–758 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Chapman, T.A., Ferry, S.: Approximating homotopy equivalences by homeomorphisms. Am. J. Math. 101(3), 583–607 (1979)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Davis, J.F., Lück, W.: Spaces over a category and assembly maps in isomorphism conjectures in K- and L-theory. K-Theory 15(3), 201–252 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Dwyer, W., Weiss, M., Williams, B.: A parametrized index theorem for the algebraic K-theory Euler class. Acta Math. 190(1), 1–104 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Elek, G., Szabó, E.: On sofic groups. J. Group Theory 9(2), 161–171 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Farrell, F.T., Hsiang, W.C.: On Novikov’s conjecture for nonpositively curved manifolds. I. Ann. Math. (2) 113(1), 199–209 (1981)CrossRefMathSciNetGoogle Scholar
  23. 23.
    Farrell, F.T., Hsiang, W.C.: The Whitehead group of poly-(finite or cyclic) groups. J. Lond. Math. Soc., II Ser. 24(2), 308–324 (1981)zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Farrell, F.T., Jones, L.E.: K-theory and dynamics. I. Ann. Math. (2) 124(3), 531–569 (1986)CrossRefMathSciNetGoogle Scholar
  25. 25.
    Farrell, F.T., Jones, L.E.: Algebraic K-theory of discrete subgroups of Lie groups. Proc. Natl. Acad. Sci. USA 84(10), 3095–3096 (1987)zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Farrell, F.T., Jones, L.E.: The surgery L-groups of poly-(finite or cyclic) groups. Invent. Math. 91(3), 559–586 (1988)zbMATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Farrell, F.T., Jones, L.E.: Rigidity in geometry and topology. In: Proceedings of the International Congress of Mathematicians (Kyoto, 1990)., vol. I, II, pp. 653–663. Math. Soc. Japan., Tokyo (1991)Google Scholar
  28. 28.
    Farrell, F.T., Jones, L.E.: Isomorphism conjectures in algebraic K-theory. J. Am. Math. Soc. 6(2), 249–297 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Farrell, F.T., Jones, L.E.: Topological rigidity for compact non-positively curved manifolds. In: Differential geometry: Riemannian geometry (Los Angeles, CA, 1990). Proc. Sympos. Pure Math., vol. 54, pp. 229–274. Am. Math. Soc., Providence, RI (1993)Google Scholar
  30. 30.
    Farrell, F.T., Jones, L.E.: Rigidity for aspherical manifolds with π1GL m(ℝ). Asian J. Math. 2(2), 215–262 (1998)zbMATHMathSciNetGoogle Scholar
  31. 31.
    Ferry, S.: Homotoping ε-maps to homeomorphisms. Am. J. Math. 101(3), 567–582 (1979)zbMATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    Ferry, S.C., Weinberger, S.: Curvature, tangentiality, and controlled topology. Invent. Math. 105(2), 401–414 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    Ghys, É., de la Harpe, P. (eds.): Sur les groupes hyperboliques d’après Mikhael Gromov. Birkhäuser, Boston, MA (1990)Google Scholar
  34. 34.
    Gromov, M.: Hyperbolic groups. In: Essays in Group Theory, pp. 75–263. Springer, New York (1987)Google Scholar
  35. 35.
    Gromov, M.: Asymptotic invariants of infinite groups. In: Geometric Group Theory, vol. 2, pp. 1–295. Cambridge Univ. Press, Cambridge (1993)Google Scholar
  36. 36.
    Gromov, M.: Spaces and questions. Geom. Funct. Anal. Special Volume, Part I, 118–161 (2000)MathSciNetGoogle Scholar
  37. 37.
    Hambleton, I., Pedersen, E.K.: Identifying assembly maps in K- and L-theory. Math. Ann. 328(1–2), 27–57 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  38. 38.
    Hambleton, I., Pedersen, E.K.: Topological equivalence of linear representations of cyclic groups. I. Ann. Math. (2) 161(1), 61–104 (2005)zbMATHMathSciNetCrossRefGoogle Scholar
  39. 39.
    Higson, N., Lafforgue, V., Skandalis, G.: Counterexamples to the Baum–Connes conjecture. Geom. Funct. Anal. 12(2), 330–354 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  40. 40.
    Higson, N., Pedersen, E.K., Roe, J.: C *-algebras and controlled topology. K-Theory 11(3), 209–239 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  41. 41.
    Hu, B.: Retractions of closed manifolds with nonpositive curvature. In: Geometric Group Theory (Columbus, OH, 1992). Ohio State Univ. Math. Res. Inst. Publ., vol. 3, pp. 135–147. de Gruyter, Berlin, (1995)Google Scholar
  42. 42.
    Hu, B.Z.: Whitehead groups of finite polyhedra with nonpositive curvature. J. Differ. Geom. 38(3), 501–517 (1993)zbMATHGoogle Scholar
  43. 43.
    Lück, W.: The transfer maps induced in the algebraic K 0- and K 1-groups by a fibration. I. Math. Scand. 59(1), 93–121 (1986)zbMATHMathSciNetGoogle Scholar
  44. 44.
    Lück, W.: Transformation Groups and Algebraic K-Theory. Springer, Berlin, (1989)Google Scholar
  45. 45.
    Lück, W.: Survey on classifying spaces for families of subgroups. In: Infinite Groups: Geometric, Combinatorial and Dynamical Aspects. Prog. Math., vol. 248, pp. 269–322. Birkhäuser, Basel (2005)CrossRefGoogle Scholar
  46. 46.
    Lück, W., Reich, H.: The Baum–Connes and the Farrell–Jones conjectures in K- and L-theory. In: Handbook of K-Theory, vol. 1, 2, pp. 703–842. Springer, Berlin (2005)Google Scholar
  47. 47.
    Meintrup, D., Schick, T.: A model for the universal space for proper actions of a hyperbolic group. New York J. Math. 8, 1–7 (2002)zbMATHMathSciNetGoogle Scholar
  48. 48.
    Mineyev, I.: Flows and joins of metric spaces. Geom. Topol. 9, 403–482 (2005) (electronic)zbMATHCrossRefMathSciNetGoogle Scholar
  49. 49.
    Pedersen, E.K., Weibel, C.A.: A non-connective delooping of algebraic K-theory. In: Algebraic and Geometric Topology, Proc. Conf. Rutgers Univ., (New Brunswick, 1983). Lect. Notes Math., vol. 1126, pp. 166–181. Springer, Berlin (1985)Google Scholar
  50. 50.
    Pedersen, E.K., Weibel, C.A.: K-theory homology of spaces. In: Algebraic Topology (Arcata, CA, 1986), pp. 346–361. Springer, Berlin (1989)CrossRefGoogle Scholar
  51. 51.
    Quillen, D.: Higher algebraic K-theory. I. In: Algebraic K-theory. I: Higher K-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972). Lect. Notes Math., vol. 341, pp. 85–147. Springer, Berlin (1973)Google Scholar
  52. 52.
    Quinn, F.: Ends of maps. I. Ann. Math. (2) 110(2), 275–331 (1979)CrossRefMathSciNetGoogle Scholar
  53. 53.
    Quinn, F.: Ends of maps. II. Invent. Math. 68(3), 353–424 (1982)zbMATHCrossRefMathSciNetGoogle Scholar
  54. 54.
    Quinn, F.: Applications of topology with control. In: Proceedings of the International Congress of Mathematicians (Berkeley, Calif., 1986), vol. 1, 2, pp. 598–606. Am. Math. Soc., Providence, RI (1987)Google Scholar
  55. 55.
    Staffeldt, R.E.: On fundamental theorems of algebraic K-theory. K-Theory 2(4), 511–532 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  56. 56.
    Thomason, R.W., Trobaugh, T.: Higher algebraic K-theory of schemes and of derived categories. In: The Grothendieck Festschrift III. Prog. Math., vol. 88, pp. 247–435. Birkhäuser, Boston, MA (1990)CrossRefGoogle Scholar
  57. 57.
    Waldhausen, F.: Algebraic K-theory of spaces. In: Algebraic and Geometric Topology (New Brunswick, N.J., 1983), pp. 318–419. Springer, Berlin, (1985)CrossRefGoogle Scholar
  58. 58.
    Weibel, C.A.: Homotopy algebraic K-theory. In: Algebraic K-theory and Algebraic Number Theory (Honolulu, HI, 1987). Contemp. Math., vol. 83, pp. 461–488. Am. Math. Soc., Providence, RI (1989)Google Scholar
  59. 59.
    Yu, G.: The Novikov conjecture for groups with finite asymptotic dimension. Ann. Math. (2) 147(2), 325–355 (1998)zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Mathematisches InstitutWestfälische Wilhelms-Universität MünsterMünsterGermany
  2. 2.Mathematisches InstitutHeinrich-Heine-Universität DüsseldorfDüsseldorfGermany

Personalised recommendations