Acta Mathematica

, 192:119 | Cite as

Harmonic analysis, cohomology, and the large-scale geometry of amenable groups

  • Yehuda Shalom
Article

References

  1. [1]
    Assouad, P., Plongements lipschitziens dansR n.Bull. Soc. Math. France, 111 (1983), 429–448.MATHMathSciNetGoogle Scholar
  2. [2]
    Baumslag, G., Wreath products and finitely presented groups.Math. Z., 75 (1960/1961). 22–28.CrossRefMathSciNetGoogle Scholar
  3. [3]
    Benjamini, I. &Schramm, O., Every graph with a positive Cheeger constant contains a tree with a positive Cheeger constant.Geom. Funct. Anal., 7 (1997), 403–419.MATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    Benoist, Y., Private communication (available upon request).Google Scholar
  5. [5]
    Bergelson, V., Weakly mixing PET.Ergodic Theory Dynam Systems, 7 (1987), 337–349.MATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    Bergelson, V. &Rosenblatt, J., Mixing actions of groups.Illinois J. Math., 32 (1988), 65–80.MATHMathSciNetGoogle Scholar
  7. [7]
    Bestvina, M. &Feighin, M., Proper actions of lattices on contractible manifolds.Invent. Math., 150 (2002), 237–256.MATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    Bestvina, M., Kapovich, M. &Kleiner, B., van Kampen's embedding obstruction for discrete groups.Invent. Math., 150 (2002), 219–235.MATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    Bieri, R.,Homological Dimensibn of Discrete Groups. Queen Mary College Mathematics Notes, Queen Mary College, London, 1976.Google Scholar
  10. [10]
    Bieri, R. &Strebel, R., Almost finitely presented soluble groups.Comment. Math. Helv., 53 (1978), 258–278.MATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    Block, J. &Weinberger, S., Large scale homology theories and geometry, inGeometric Topology (Athens, GA, 1993), pp. 522–569. AMS/IP Stud. Adv. Math., 2.1., Amer. Math. Soc., Providence, RI, 1997.Google Scholar
  12. [12]
    Borel, A. &Serre, J.-P., Corners and arithmetic groups.Comment. Math. Helv., 48 (1973), 436–491.MATHCrossRefMathSciNetGoogle Scholar
  13. [13]
    Bourdon, M. &Pajot, H., Rigidity of quasi-isometries for some hyperbolic buildings.Comment. Math. Helv., 75 (2000), 701–736.MATHCrossRefMathSciNetGoogle Scholar
  14. [14]
    Bridson, M. R. &Gersten, S. M., The optimal isoperimetric inequality for torus bundles over the circle.Q. J. Math., 47 (1996), 1–23.MATHCrossRefMathSciNetGoogle Scholar
  15. [15]
    Chou, C., Elementary amenable groups.Illinois J. Math., 24 (1980), 396–407.MATHMathSciNetGoogle Scholar
  16. [16]
    Delorme, P., 1-cohomologie des représentations unitaires des groupes de Lie semi-simples et résolubles. Produits tensoriels continus de représentations.Bull. Soc. Math. France, 105 (1977), 281–336.MATHMathSciNetGoogle Scholar
  17. [17]
    Drutu, C., Quasi-isometry invariants and asymptotic cones.Internat. J. Algebra Comput., 12 (2002), 99–135.MATHCrossRefMathSciNetGoogle Scholar
  18. [18]
    Dunwoody, M. J., Accessibility and groups of cohomological dimension one.Proc. London Math. Soc., 38 (1979), 193–215.MATHCrossRefMathSciNetGoogle Scholar
  19. [19]
    Dyubina, A., Instability of the virtual solvability and the property of being virtually torsion-free for quasi-isometric groups.Int. Math. Res. Not., 2000 (2000), 1997–1101.CrossRefGoogle Scholar
  20. [20]
    Erschler, A., On isoperimetric profiles of finitely generated groups.Geom. Dedicata, 100 (2003), 151–171.CrossRefMathSciNetGoogle Scholar
  21. [21]
    Farb, B., The quasi-isometry classification of lattices in semisimple Lie groups.Math. Res. Lett., 4 (1997), 705–717.MATHMathSciNetGoogle Scholar
  22. [22]
    Farb, B. &Mosher, L., A rigidity theorem for the solvable Baumslag-Solitar groups.Invent. Math., 131 (1998), 419–451.MATHCrossRefMathSciNetGoogle Scholar
  23. [23]
    —, Quasi-isometric rigidity for the solvable Baumslag-Solitar groups, II.Invent. Math., 137 (1999), 613–649.MATHCrossRefMathSciNetGoogle Scholar
  24. [24]
    —, On the asymptotic geometry of abelian-by-cyclic groups.Acta Math., 184 (2000), 145–202.MATHCrossRefMathSciNetGoogle Scholar
  25. [25]
    —, Problems on the geometry of finitely generated solvable groups, inCrystallographic Groups and their Generalizations (Kortrijk, 1999), pp. 121–134. Contemp. Math., 262. Amer. Math. Soc., Providence, RI, 2000.Google Scholar
  26. [26]
    —, The geometry of surface-by-free groups.Geom. Funct. Anal., 12 (2002), 915–963.MATHCrossRefMathSciNetGoogle Scholar
  27. [27]
    Gersten, S. M., Quasi-isometry invariance of cohomological dimension.C. R. Acad. Sci. Paris Sér. I. Math., 316 (1993), 411–416.MATHMathSciNetGoogle Scholar
  28. [28]
    Ghys, É. &de la Harpe, P. (editors),Sur les groupes hyperboliques d'après Mikhael Gromov. Papers from the Swiss Seminar on Hyperbolic Groups held in Bern, 1988. Progr. Math., 83. Birkhäuser, Boston, MA, 1990.MATHGoogle Scholar
  29. [29]
    Gildenhuys, D. &Strebel, R., On the cohomology of soluble groups, II.J. Pure Appl. Algebra, 26 (1982), 293–323.MATHCrossRefMathSciNetGoogle Scholar
  30. [30]
    Gromov, M., Groups of polynomial growth and expanding maps.Inst. Hautes Études Sci. Publ. Math., 53 (1981), 53–73.MATHCrossRefMathSciNetGoogle Scholar
  31. [31]
    —, Asymptotic invariants of infinite groups, inGeometric Group Theory, Vol. 2 (Sussex, 1991), pp. 1–295. London Math. Soc. Lecture Note Ser., 182. Cambridge Univ. Press, Cambridge, 1993.Google Scholar
  32. [32]
    Guichardet, A.,Cohomologie des groupes topologiques et des algèbres de Lie. Textes Mathématiques, 2. CEDIC, Paris, 1980.MATHGoogle Scholar
  33. [33]
    de la Harpe, P.,Topics in Geometric Group Theory. Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 2000.MATHGoogle Scholar
  34. [34]
    de la Harpe, P. &Valette, A.,La propriété (T) de Kazhdan pour les groupes localement compacts. Astérisque, 175. Soc. Math. France, Paris, 1989.MATHGoogle Scholar
  35. [35]
    Kapovich, M. & Kleiner, B., Coarse Alexander duality and duality groups. To appear inJ. Differential Geom.Google Scholar
  36. [36]
    Korevaar, N. J. &Schoen, R. M., Global existence theorems for harmonic maps to non-locally compact spaces.Comm. Anal. Geom., 5 (1997), 333–387.MATHMathSciNetGoogle Scholar
  37. [37]
    Kropholler, P. H., Cohomological dimension of soluble groups.J. Pure Appl. Algebra, 43 (1986), 281–287.MATHCrossRefMathSciNetGoogle Scholar
  38. [38]
    Macdonald, I. D.,The Theory of Groups. Reprint of the 1968 original. Krieger, Malabar, FL, 1988.Google Scholar
  39. [39]
    Merzlyakov, Yu. I., Locally solvable groups of finite rank.Algebra i Logika Sem., 3 (1964), 5–16 (Russian).Google Scholar
  40. [40]
    Milnor, J., Growth of finitely generated solvable groups.J. Differential Geom., 2 (1968), 447–449.MATHMathSciNetGoogle Scholar
  41. [41]
    Monod, N. & Shalom, Y., Orbit equivalence rigidity and bounded cohomology. To appear inAnn. of Math. Google Scholar
  42. [42]
    Montgomery, D. &Zippin, L.,Topological Transformation Groups. Reprint of the 1955 original. Krieger, Huntington, NY, 1974.MATHGoogle Scholar
  43. [43]
    Mosher, L., Sageev, M. &Whyte, K., Quasi-actions on trees, I: Bounded valence.Ann. of Math., 158 (2003), 115–164.MATHCrossRefMathSciNetGoogle Scholar
  44. [44]
    Pansu, P., Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un.Ann. of Math., 129 (1989), 1–60.CrossRefMathSciNetGoogle Scholar
  45. [45]
    Raghunathan, M. S.,Discrete Subgroups of Lie Groups. Ergeb. Math. Grenzgeb., 68. Springer-Verlag, New York-Heidelberg, 1972.MATHGoogle Scholar
  46. [46]
    Reiter Ahlin, A., The large scale geometry of nilpotent-by-cyclic groups. Ph.D. Thesis, University of Chicago, 2002.Google Scholar
  47. [47]
    Rosenblatt, J. M., Invariant measures and growth conditions.Trans. Amer. Math. Soc., 193 (1974), 33–53.MATHCrossRefMathSciNetGoogle Scholar
  48. [48]
    Sela, Z., Uniform embeddings of hyperbolic groups in Hilbert spaces.Israel J. Math., 80 (1992), 171–181.MATHCrossRefMathSciNetGoogle Scholar
  49. [49]
    Shalom, Y., The growth of linear groups.J. Algebra, 199 (1998), 169–174.MATHCrossRefMathSciNetGoogle Scholar
  50. [50]
    —, Rigldity of commensurators and irreducible lattices.Invent. Math., 141 (2000), 1–54.MATHCrossRefMathSciNetGoogle Scholar
  51. [51]
    Stammbach, U., On the weak homological dimension of the group algebra of solvable groups.J. London Math. Soc., 2 (1970), 567–570.MATHMathSciNetGoogle Scholar
  52. [52]
    Tits, J., Appendix to “Groups of polynomial growth and expanding maps” by M. Gromov.Inst. Hautes Études Sci. Publ. Math., 53 (1981), 74–78.MathSciNetGoogle Scholar
  53. [53]
    Vershik, A. M. &Karpushev, S. I., Cohomology of groups in unitary representations, neighborhood of the identity and conditionally positive definite functions.Mat. Sb., 119 (161), (1982), 521–533. (Russian); English translation inMath. USSR-Sb., 47 (1984), 513–526.MathSciNetGoogle Scholar
  54. [54]
    Wolf, J. A., Growth of finitely generated solvable groups and curvature of Riemannian manifolds.J. Differential Geom., 2 (1968), 421–446.MATHGoogle Scholar
  55. [35]
    Zimmer, R. J.,Ergodic Theory and Semisimple Groups. Monographs Math., 81. Birkhäuser, Basel, 1984.MATHGoogle Scholar

Copyright information

© Institut Mittag-Leffler 2004

Authors and Affiliations

  • Yehuda Shalom
    • 1
  1. 1.School of Mathematical SciencesTel Aviv UniversityTel AvivIsrael

Personalised recommendations