Approximation Properties for Discrete Groups

  • Jacek Brodzki
  • Graham A. Niblo
Part of the Trends in Mathematics book series (TM)


We provide an illustration of an interesting and nontrivial interaction between analytic and geometric properties of a group. We provide a short survey of approximation properties of operator algebras associated with discrete groups. We then demonstrate directly that groups that satisfy the property RD with respect to a conditionally negative length function have the metric approximation property.


Rapid decay metric approximation property 


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  1. [1]
    M.R. Bridson, A. Haefliger, Metric spaces of non-positive curvature. Springer Verlag, Berlin 1999.zbMATHGoogle Scholar
  2. [2]
    I. Chatterji, Property (RD) for cocompact lattices in a finite product of rank one Lie groups with some rank two Lie groups. Geom. Dedicata 96 (2003), 161–177.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    I. Chatterji, K. Ruane, Some geometric groups with rapid decay. Geom. Funct. Anal. 15 (2005), no. 2, 311–339.zbMATHMathSciNetCrossRefGoogle Scholar
  4. [4]
    J. de Canniere, U. Haagerup, Multipliers of the Fourier algebra of some simple Lie groups and their discrete subgroups. Amer. J. Math. 107 (1984), 455–500.CrossRefGoogle Scholar
  5. [5]
    P. Eymard, L’algèbre de Fourier d’un groupe localement compact. (French) Bull. Soc. Math. France 92 (1964), 181–236.zbMATHMathSciNetGoogle Scholar
  6. [6]
    J. Faraut and K. Harzallah, Distances hilbertiennes invariantes sur un espace homogène. Ann. Inst. Fourier (Grenoble) 24 (1974), no. 3, xiv, 171–217.zbMATHMathSciNetGoogle Scholar
  7. [7]
    A. Grothendieck, Produits tensoriels topologiques et espaces nucléaires. Mem. Amer. Math. Soc. (1955), no. 16, 140 pp.Google Scholar
  8. [8]
    U. Haagerup, An example of a non-nuclear C*-algebra which has the metric approximation property. Inventiones Math. 50 (1979), 279–293.zbMATHMathSciNetCrossRefGoogle Scholar
  9. [9]
    U. Haagerup, J. Kraus, Approximation properties for group C*-algebras and group von Neumann algebras. Trans. Amer. Math. Soc. 344 (1994), no. 2, 667–699.zbMATHMathSciNetCrossRefGoogle Scholar
  10. [10]
    P. de la Harpe, Groupes hyperboliques, algèbres d’opérateurs et un théorème de Jolissaint. C. R. Acad. Sci. Paris Ser. I 307 (1988), 771–774zbMATHGoogle Scholar
  11. [11]
    P. de la Harpe, A. Valette, La propriété (T) de Kazhdan pour les groupes localement compacts. Astérisque 175 (1989), Soc. Mathématique de France.Google Scholar
  12. [12]
    P. Jolissaint, Rapidly decreasing functions in reduced C*-algebras of groups. Trans. Amer. Math. Soc. 317 (1990), 167–196.zbMATHMathSciNetCrossRefGoogle Scholar
  13. [13]
    P. Jolissaint, A. Valette, Normes de Sobolev et convoluteurs bornés sur L 2(G). Ann. Inst. Fourier (Grenoble) 41 (1991), no. 4, 797–822.MathSciNetGoogle Scholar
  14. [14]
    E. Kirchberg, S. Wassermann, Exact groups and continuous bundles of C*-algebras. Math. Ann. 315 (1999), no. 2, 169–203.zbMATHMathSciNetCrossRefGoogle Scholar
  15. [15]
    V. Lafforgue, A proof of property (RD) for cocompact lattices of SL(3,R) and SL(3,C). J. Lie Theory 10 (2000), no. 2, 255–267zbMATHMathSciNetGoogle Scholar
  16. [16]
    E.C. Lance, On nuclear C*-algebras. J. Functional Analysis 12 (1973), 157–176zbMATHMathSciNetCrossRefGoogle Scholar
  17. [17]
    H. Leptin, Sur l’algèbre de Fourier d’un groupe localement compact. (French) C.R. Acad. Sci. Paris Sér. A-B 266 (1968) A1180–A1182.MathSciNetGoogle Scholar
  18. [18]
    V. Mathai, Heat kernels and the range of the trace on completions of twisted group algebras. Contemporary Mathematics 398 (2006), 321–346.MathSciNetGoogle Scholar
  19. [19]
    G.A.Niblo, L.D. Reeves, The geometry of cube complexes and the complexity of their fundamental groups. Topology, Vol. 37, No 3, (1988) 621–633.MathSciNetCrossRefGoogle Scholar
  20. [20]
    G.A. Niblo and L.D. Reeves, Coxeter groups act on CAT(0) cube complexes. Journal of Group Theory, 6, (2003), 309–413.MathSciNetCrossRefGoogle Scholar
  21. [21]
    V. Paulsen, Completely bounded maps and dilations. Pitman Research Notes in Mathematics Series, 146 Longman, New York, 1986.zbMATHGoogle Scholar
  22. [22]
    G. Robertson, Crofton formulae and geodesic distance in hyperbolic spaces. J. Lie Theory 8 (1998), no. 1, 163–172.zbMATHMathSciNetGoogle Scholar
  23. [23]
    S. Wassermann, Exact C*-algebras and related topics. Lecture Notes Series, 19. Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1994.Google Scholar
  24. [24]
    Daniel T. Wise, Cubulating Small Cancellation Groups. Geometric and Functional Analysis 14, no. 1, 150–214.Google Scholar
  25. [25]
    G. Yu, The coarse Baum-Connes conjecture for spaces which admit a uniform embedding into Hilbert space. Invent. Math. 139 (2000), no. 1, 201–240.zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2006

Authors and Affiliations

  • Jacek Brodzki
    • 1
  • Graham A. Niblo
    • 1
  1. 1.School of Mathematical SciencesUniversity of SouthamptonSouthamptonUK

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