Skip to main content
SpringerLink
Log in

Asymptotic invariants of finitely generated algebras. A generalization of Gromov's quasi-isometric viewpoint

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

Abstract

We generalize Gromov's quasi-isometric technique for finitely generated algebras.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. S. I. Adyan, “Random walks on free periodic groups,” Izv. Akad. Nauk SSSR, Ser. Mat., 46, 1139–1149 (1982).

    MATH  MathSciNet  Google Scholar 

  2. M. A. Alekseev, L. Yu. Glebski, and E. I. Gordon, “On approximations of groups, group actions and Hopf algebras,” J. Math. Sci., 107, 4305–4332 (2001).

    Google Scholar 

  3. J. M. Alonso, “Inégalités isopérimétriques et quasi-isométries,” C. R. Acad. Sci. Paris Sér. I Math., 311, 761–764 (1990).

    MATH  MathSciNet  Google Scholar 

  4. J. Alonso, T. Brady, D. Cooper, V. Ferlini, M. Lustig, M. Mihalik, M. Shapiro, and H. Short, “Notes on word hyperbolic groups,” in Group Theory from a Geometrical Viewpoint, Trieste (1990), World Sci. Publishing, River Edge, New Jersey (1991), pp. 3–63.

    Google Scholar 

  5. C. Anantharaman-Delaroche and J. Renault, Amenable Goupoids, Monographies de L'Enseignement Mathématique, 36, L'Enseignement Mathématique, Geneva (2000).

    Google Scholar 

  6. P. Ara, K. C. O'Meara, and F. Perera, “Gromov translation algebras over discrete trees are exchange rings,” Trans. Amer. Math. Soc., 356, 2067–2079 (2004).

    MATH  MathSciNet  Google Scholar 

  7. G. N. Arzhantseva, “Generic properties of finitely presented groups and Howson's theorem,” Comm. Algebra, 26, 3783–3792 (1998).

    MATH  MathSciNet  Google Scholar 

  8. G. N. Arzhantseva, and P. A. Cherix, “On the Cayley graph of a generic finitely presented group,” Bull. Soc. Math. Belgique, 11, 589–601 (2004).

    MATH  MathSciNet  Google Scholar 

  9. G. N. Arzhantseva and A. Yu. Ol'shanski, “Generality of the class of groups in which subgroups with a lesser number of generators are free,” Math. Notes, 59, 350–355 (1996).

    MATH  MathSciNet  Google Scholar 

  10. O. Attie, “Quasi-isometry classification of some manifolds of bounded geometry,” Math. Z., 216, 501–527 (1994).

    MATH  MathSciNet  Google Scholar 

  11. I. Babenko, “Problems of growth and rationality in algebra and topology,” Russian Math. Surveys, 41, 117–175 (1986).

    MATH  Google Scholar 

  12. I. K. Babenko, “Asymptotic invariants of smooth manifolds,” Russian Acad. Sci. Izv. Math., 41, 1–38 (1993).

    MathSciNet  Google Scholar 

  13. Yu. A. Bakhturin and A. Yu. Ol'shanski, “Identities,” in: Algebra II, Noncommutative Rings, Identities, Encyclopaedia of Math. Sciences, 18, Springer-Verlag, Berlin (1991), pp. 107–221.

    Google Scholar 

  14. V. G. Bardakov, “Construction of a regularly exhaustive sequence in groups of subexponential growth,” Algebra Logic 40, 12–16 (2001).

    MathSciNet  Google Scholar 

  15. L. Bartholdi, private communication.

  16. G. Baumslag, A. Myasnikov, and V. Shpilrain, “Open problems in combinatorial group theory (Second edition),” in: Combinatorial and Geometric Group Theory, New York (2000) — Hoboken, New Jersey (2001), Contemp. Math., 296, Amer. Math. Soc., Providence, Rhode Island (2002), pp. 1–38.

    Google Scholar 

  17. B. Baumslag and S. J. Pride, “Groups with two more generators than relators,” J. London Math. Soc., 17, 425–426 (1978).

    MATH  MathSciNet  Google Scholar 

  18. E. Bédos, “Notes on hypertraces and C*-algebras,” J. Operator Theory, 34, 285–306 (1995).

    MATH  MathSciNet  Google Scholar 

  19. M. B. Bekka, “Amenable unitary representations of locally compact groups,” Invent. Math., 100, 383–401 (1990).

    MATH  MathSciNet  Google Scholar 

  20. M. B. Bekka, “On the full C*-algebras of arithmetic groups and the congruence subgroup problem,” Forum Math., 11, 705–715 (1999).

    MATH  MathSciNet  Google Scholar 

  21. B. Bekka, P. de la Harpe, and A. Valette, “Kazhdan's property” (T), book in preparation, preprint Geneva (2003); http://www.unige.ch/math/biblio/preprint/2003/pp2003.html; http://www.mmas.univ-metz.fr/~bekka/.

  22. A. Ya. Belov, V. V. Borisenko, and V. N. Latyshev, “Monomial algebras,” J. Math. Sci., 87, 3463–3575 (1997).

    MATH  MathSciNet  Google Scholar 

  23. B. Blackadar, Shape theory for C*-algebras, Math. Scandinavica, 56, 249–275 (1985).

    MATH  MathSciNet  Google Scholar 

  24. J. Block and S. Weinberger, “Aperiodic tilings, positive scalar curvature, and amenability of spaces,” J. Amer. Math. Soc., 5, 907–918 (1992).

    MATH  MathSciNet  Google Scholar 

  25. J. Block and S. Weinberger, “Large scale homology theories and geometry,” in: Geometric Topology, Athens, GA (1993), AMS/IP Stud. Adv. Math., 2.1, Amer. Math. Soc., Providence, Rhode Island (1997), pp. 522–569.

    Google Scholar 

  26. B. H. Bowditch, “Notes on Gromov's hyperbolicity for path-metric spaces,” in: Group Theory from a Geometrical Viewpoint, Trieste, (1990), World Sci. Publishing, River Edge, New Jersey (1991), pp. 64–167.

    Google Scholar 

  27. S. Brick, “Quasi-isometries and ends of groups,” J. Pure Appl. Algebra, 86, 23–33 (1993).

    MATH  MathSciNet  Google Scholar 

  28. M. Bridson and A. Haefliger, Metric Spaces of Non-Positive Curvature, Grundlehren der Mathematischen Wissenschaften, 319, Springer-Verlag, Berlin (1999).

    MATH  Google Scholar 

  29. R. Brooks, “Amenability and the spectrum of the Laplacian,” Bull. Amer. Math. Soc., 6, 87–89 (1982).

    MATH  MathSciNet  Google Scholar 

  30. R. Brooks, “Combinatorial problems in spectral geometry,” in: Curvature and Topology of Riemannian Manifolds, Springer Lecture Notes in Mathematics, 1201 (1987), pp. 14–32.

  31. T. G. Ceccherini-Silberstein, R. Grigorchuk, and P. de la Harpe, “Amenability and paradoxes for pseudogroups and for discrete metric spaces,” Proc. Steklov Inst., 224, 57–97 (1999).

    Google Scholar 

  32. T. G. Ceccherini-Silberstein and A. Y. Samet-Vaillant, Gromov's Translation Algebras, Growth and Amenability of Operator Algebras, preprint.

  33. T. G. Ceccherini-Silberstein and A. Y. Samet-Vaillant, Ends of Operator Algebras, preprint.

  34. C. Champetier, “Propriétés statistiques des groupes de présentation finie,” Adv. Math., 116, 197–262 (1995).

    MATH  MathSciNet  Google Scholar 

  35. C. Champetier, “L'espace des groupes de type fini,” Topology, 39, 657–680 (2000).

    MATH  MathSciNet  Google Scholar 

  36. C. Champetier and V. Guirardel, “Limit groups as limits of free groups: compactifying. The set of free groups,” Israel J. Math., 146, 1 76 (2005).

    MathSciNet  Google Scholar 

  37. B. Chandler and W. Magnus, The History of Combinatorial Group Theory, A Case Study in the History of Ideas, Studies in the History of Mathematics and Physical Sciences, 9, Springer-Verlag, New York (1982).

    MATH  Google Scholar 

  38. M. D. Choi and E. Effros, “Nuclear C*-algebras and the approximation property,” Amer. J. Math., 100, 61–79 (1978).

    MATH  MathSciNet  Google Scholar 

  39. R. Chow, “Groups quasi-isometric to complex hyperbolic space,” Trans. Amer. Math. Soc., 348, 1757–1769 (1996).

    MATH  MathSciNet  Google Scholar 

  40. J. Cohen, “Cogrowth and amenability of discrete groups,” J. Funct. Anal., 48, 301–309 (1982).

    MATH  MathSciNet  Google Scholar 

  41. D. Collins and H. Zieschang, “Combinatorial group theory and fundamental groups,” in: Encyclopedia of Math. Sciences, Algebra VII, 58, Springer, Berlin (1993), pp. 1–166.

    Google Scholar 

  42. A. Connes, “Classiffication of injective factors,” Annals of Math., 104, 73–115 (1976).

    MathSciNet  Google Scholar 

  43. A. Connes, “On cohomology of operator algebras,” J. Funct. Anal., 28, 248–253 (1978).

    MATH  MathSciNet  Google Scholar 

  44. A. Connes, “C*-algèbres et géométrie differentielle,” C. R. Acad. Sci. Paris Sér. I Math, 290, 599–604 (1980).

    MATH  MathSciNet  Google Scholar 

  45. A. Connes, “A survey of foliations and operator algebras,” in: Operator Algebras and Applications, Part I, Kingston, Ont. (1980), Proc. Sympos. Pure Math., 38, Amer. Math. Soc., Providence, Rhode Island (1982), pp. 521–628.

    Google Scholar 

  46. A. Connes, “Noncommutative differential geometry,” Publ. Math. Inst. Hautes Études Sci., 62, 41–144 (1985).

    MATH  Google Scholar 

  47. A. Connes, “Compact metric spaces, Fredholm modules and hyperfiniteness,” J. of Ergodic Theory and Dynam. Systems, 9, 207–220 (1989).

    MATH  MathSciNet  Google Scholar 

  48. A. Connes, Noncommutative Geometry, Academic Press, San Diego, California (1994).

    MATH  Google Scholar 

  49. A. Connes and H. Moscovici, “Conjecture de Novikov et groupes hyperboliques,” C. R. Acad. Sci. Paris Sér. I Math., 307, 475–480 (1988).

    MATH  MathSciNet  Google Scholar 

  50. A. Connes and H. Moscovici, “Cyclic cohomology, the Novikov conjecture and hyperbolic groups,” Topology, 29, 345–388 (1990).

    MATH  MathSciNet  Google Scholar 

  51. A. Connes, M. Gromov, and H. Moscovici, “Group cohomology with Lipschitz control and higher signatures,” Geom. Funct. Anal., 3, 1–78 (1993).

    MATH  MathSciNet  Google Scholar 

  52. M. Coornaert, T. Delzant, and A. Papadopoulos, Géométrie et théorie des groupes — Les groupes hyperboliques de Gromov (with an English summary), Springer Lecture Notes in Mathematics, 1441, Springer-Verlag, Berlin (1990).

    Google Scholar 

  53. M. Day, “Semigroups and amenability,” in Semigroups, Proc. Sympos., Wayne State Univ., Detroit, Mich. (1968), Academic Press, New York (1969), pp. 5–53.

    Google Scholar 

  54. A. N. Dranishnikov, “Asymptotic topology,” Russian Math. Surveys, 55, 1085–1129 (2000).

    MATH  MathSciNet  Google Scholar 

  55. L. van der Dries and A. Wilkie, “Gromov's theorem on groups of polynomial growth and elementary logic,” J. Algebra, 89, 349–374 (1984).

    MATH  MathSciNet  Google Scholar 

  56. C. Druţu, “Quasi-isometric classiffication of nonuniform lattices in semisimple groups of higher rank,” Geom. Funct. Anal., 10, 327–388 (2000).

    MATH  MathSciNet  Google Scholar 

  57. C. Druţu, “Cônes asymptotiques et invariants de quasi-isométrie pour des espaces métriques hyperboliques,” Ann. Inst. Fourier, 51, 81–97 (2001).

    MATH  Google Scholar 

  58. C. Druţu, “Quasi-isometry invariants and asymptotic cones,” in: International Conference on Geometric and Combinatorial Methods in Group Theory and Semigroup Theory, 12, Lincoln, NE (2000), Internat. J. Algebra Comput., (2002), pp. 99–135.

    MATH  MathSciNet  Google Scholar 

  59. A. Dyubina, “Instability of the virtual solvability and the property of being virtually torsion-free for quasi-isometric groups,” Internat. Math. Res. Notices, 2000, No. 21, 1097–1101.

  60. V. Efremovitch, “The proximity geometry of Riemannian manifolds,” Usp. Math. Nauk., 8, 189 (1953).

    Google Scholar 

  61. G. Elek, “The K-theory of Gromov's translation algebras and the amenability of discrete groups,” Proc. Amer. Math. Soc., 125, 2551–2553 (1997).

    MATH  MathSciNet  Google Scholar 

  62. G. Elek, The Hochschild Cohomologies of Translation Algebras and Group Cohomologies, preprint of the Mathematical Institute of the Hungarian Academy of Science (1997); http://www.math.uiuc.edu/K-theory/0190/.

  63. G. Elek, “The amenability of affine algebras,” J. Algebra, 264, 469–478 (2003).

    MATH  MathSciNet  Google Scholar 

  64. G. Elek and A. Y. Samet-Vaillant, “The ends of algebras,” Comm. Algebra, to appear.

  65. A. Erschler, “On isoperimetric profiles of finitely generated groups,” Geom. Dedicata, 100, 157–171 (2003).

    MATH  MathSciNet  Google Scholar 

  66. D. Epstein, “Ends,” in Topology of 3-manifolds and Related Results (M. Ford, Ed.), Prentice Hall (1962), pp. 110–117.

  67. A. Eskin, “Quasi-isometric rigidity of nonuniform lattices in higher rank symmetric spaces,” J. Amer. Math. Soc., 11, 321–361 (1998).

    MATH  MathSciNet  Google Scholar 

  68. A. Eskin and B. Farb, “Quasi-flats and rigidity in higher rank symmetric spaces,” J. Amer. Math. Soc., 10, 653–692 (1997).

    MATH  MathSciNet  Google Scholar 

  69. A. Eskin and B. Farb, “Quasi-flats in H 2 × H 2,” in: Lie Groups and Ergodic Theory, Mumbai (1996), Tata Inst. Fund. Res. Stud. Math., 14, Tata Inst. Fund. Res., Bombay (1998), pp. 75–103.

    Google Scholar 

  70. B. Farb, “The quasi-isometry classification of lattices in semisimple Lie groups,” Math. Res. Lett., 4, 705–717 (1997).

    MATH  MathSciNet  Google Scholar 

  71. B. Farb and L. Mosher, “A rigidity theorem for the solvable Baumslag-Solitar groups,” Invent. Math., 131, 419–451 (1998).

    MATH  MathSciNet  Google Scholar 

  72. B. Farb and L. Mosher, “Quasi-isometric rigidity for the solvable Baumslag-Solitar groups. II,” Invent. Math., 137, 613–649 (1999).

    MATH  MathSciNet  Google Scholar 

  73. B. Farb and L. Mosher, “Problems on the geometry of finitely generated solvable groups,” in Crystallographic Groups and Their Generalizations, Kortrijk (1999), Contemp. Math., 262, Amer. Math. Soc., Providence, Rhode Island (2000), pp. 121–134.

    Google Scholar 

  74. B. Farb and L. Mosher, “On the asymptotic geometry of abelian-by-cyclic groups,” Acta Math., 184, 145–202 (2000).

    MATH  MathSciNet  Google Scholar 

  75. B. Farb and L. Mosher, “The geometry of surface-by-free groups,” Geom. Funct. Anal., 12, 915–963 (2002).

    MATH  MathSciNet  Google Scholar 

  76. B. Farb and R. Schwartz, “The large-scale geometry of Hilbert modular groups,” J. Differential Geom., 44, 435–478 (1996).

    MATH  MathSciNet  Google Scholar 

  77. C. Farsi, “Soft C*-algebras,” Proc. Edinb. Math. Soc., 45, 59–65 (2002).

    MATH  MathSciNet  Google Scholar 

  78. H. Freudenthal, “Über die Enden diskreter Räume und Gruppen,” Comment. Math. Helv., 17, 1–38 (1945).

    MATH  MathSciNet  Google Scholar 

  79. T. Gateva-Ivanova and V. Latyshev, “On recognisable properties of associative algebras,” J. Symb. Comput., 6, 371–388 (1988).

    MATH  MathSciNet  Google Scholar 

  80. I. M. Gelfand and A. A. Kirillov, “Sur les corps liés aux algèbres enveloppantes des algèbres de Lie,” Publ. Math. Inst. Hautes Études Sci., 31, 5–19 (1966).

    MathSciNet  Google Scholar 

  81. S. M. Gersten, “Quasi-isometry invariance of cohomological dimension,” C. R. Acad. Sci. Paris Sér. I Math., 316, 411–416 (1993).

    MATH  MathSciNet  Google Scholar 

  82. S. M. Gersten and H. B. Short, “Small cancellation theory and automatic groups,” Invent. Math., 102, 305–334 (1990).

    MATH  MathSciNet  Google Scholar 

  83. E. Ghys, “Les groupes hyperboliques,” in: Séminaire Bourbaki, 1989/90, Exposé No. 722, 189–190 (1990), pp. 203–238.

    MathSciNet  Google Scholar 

  84. É. Ghys, “Topologie des feuilles génériques,” Ann. Math., 141, 387–422 (1995).

    MATH  MathSciNet  Google Scholar 

  85. É. Ghys and P. de la Harpe, Sur les groupes hyperboliques d'aprés Mikhael Gromov, Swiss Seminar on Hyperbolic Groups held in Bern (1988), Progress in Math., 83, Birkhäuser Boston Inc., Boston, Massachusetts (1990).

    Google Scholar 

  86. É. Ghys, P. de la Harpe, “Infinite groups as geometric objects (after Gromov),” in: Ergodic Theory, Symbolic Dynamics, and Hyperbolic Spaces, Trieste (1989), Oxford Sci. Publ., Oxford Univ. Press, New York (1991), pp. 299–314.

    Google Scholar 

  87. É. Ghys, R. Langevin, and P. Walczak, “Entropie géométrique des feuilletages,” Acta Math., 160, 105–142 (1988).

    MATH  MathSciNet  Google Scholar 

  88. L. Yu. Glebsky and E. I. Gordon, “On approximation of locally compact groups by finite algebraic systems,” Electron. Res. Announc. Amer. Math. Soc., 10, 21–28 (2004).

    MATH  MathSciNet  Google Scholar 

  89. C. Godbillon, Feuilletages — Études géométriques, Progress in Math., 98, Birkhäuser Verlag, Basel (1991).

    MATH  Google Scholar 

  90. R. I. Grigorchuk, “Symmetrical random walks on discrete groups,” in: Multicomponent Random Systems, Advances in Probability and Related Topics, 6, Dekker, New York (1980), pp. 285–325.

    Google Scholar 

  91. R. I. Grigorchuk, “Degrees of growth of finitely generated groups and the theory of invariant means,” Math. USSR-Izv., 25, 259–300 (1985).

    MATH  Google Scholar 

  92. R. I. Grigorchuk, “Growth and amenability of a semigroup and its group of quotients,” in: Proceedings of the International Symposium on the Semigroup Theory and Its Related Fields, Kyoto (1990), Shimane Univ., Matsue (1990), pp. 103–108.

    Google Scholar 

  93. R. I. Grigorchuk, “On growth in group theory,” in: Proceedings of the International Congress of Mathematicians, Mathematical Society of Japan, 1, Springer Verlag, Tokyo (1991), pp. 325–338.

    Google Scholar 

  94. R. I. Grigorchuk and P. de la Harpe, “On problems related to growth, entropy and spectrum in group theory,” J. Dynam. Control Systems, 3, 51–89 (1997).

    MATH  MathSciNet  Google Scholar 

  95. R. I. Grigorchuk and P. de la Harpe, “Limit behaviour of exponential growth rates for finitely generated groups,” in: Essays on Geometry and Related Topics, 1, 2, Monographies de L'Enseignement Mathématique, 38, L'Enseignement Mathématique, Geneva (2001), 351–370.

    Google Scholar 

  96. R. I. Grigorchuk and P. Kurchanov, Some Questions of Group Theory Related to Geometry, Encycl. of Math. Sciences, 58, Algebra VII, Springer, Berlin (1993).

    Google Scholar 

  97. R. I. Grigorchuk and A. Zuk, “On the asymptotic spectrum of random walk on infinite families of graphs,” in: Random Walks and Discrete Potential Theory, Cortona (1997), Sympos. Math., XXXIX, Cambridge Univ. Press, Cambridge (1999), pp. 188–204.

    Google Scholar 

  98. M. Gromov, Structures Métriques pour les Variétés Riemanniennes (J. Lafontaine and P. Pansu Eds.), Textes Mathématiques, 1, CEDIC, Paris (1981).

    Google Scholar 

  99. M. Gromov, “Groups of polynomial growth and expanding maps,” Publ. Math. Inst. Hautes Études Sci., 53, 53–78 (1981).

    MATH  MathSciNet  Google Scholar 

  100. M. Gromov, “Hyperbolic manifolds, groups and actions,” in: Riemann Surfaces and Related Topics, Proc. of the 1978 Stony Brook Conference, State Univ. New York, Stony Brook, New York (1978), Ann. of Math. Stud., 97, Princeton Univ. Press, Princeton (1981), pp. 183–213.

    Google Scholar 

  101. M. Gromov, “Volume and bounded cohomology,” Publ. Math. Inst. Hautes Études Sci., 56, 5–100 (1982).

    MATH  MathSciNet  Google Scholar 

  102. M. Gromov, “Infinite groups as geometric objects,” in: Proceedings of the International Congress of Mathematicians, Warszawa (1984), pp. 53–73.

  103. M. Gromov, “Hyperbolic groups,” in: Essays in Group Theory (S. M. Gersten Ed.), MSRI Pub., 8, Springer (1987), pp. 75–263.

  104. M. Gromov, “Asymptotic invariants of infinite groups,” in: Geometric Group Theory (Graham A. Niblo and Martin A. Roller, Eds.), Vol. 2, Sussex (1991), London Mathematical Society Lecture Note Series, 182, Cambridge Univ. Press, Cambridge, UK (1993), pp. 1–295.

    Google Scholar 

  105. M. Gromov, “Spaces and questions,” in: GAFA 2000, Tel Aviv (1999), Special Volume, Part I, Geom. Funct. Anal. (2000), pp. 118–161.

  106. M. Gromov, “Random walk in random groups,” Geom. Funct. Anal., 13, 73–146 (2003).

    MATH  MathSciNet  Google Scholar 

  107. M. Gromov and P. Pansu, “Rigidity of lattices: An introduction,” in: Geometric Topology: Recent Developments, Montecatini Terme (1990), Lect. Notes Math., 1504, Springer-Verlag, Berlin (1991), pp. 39–137.

    Google Scholar 

  108. U. Haagerup, “Amenable C*-algebras are nuclear,” Invent. Math. 74, 305–319 (1983).

    MATH  MathSciNet  Google Scholar 

  109. P. de la Harpe, “Moyennabilité du groupe unitaire et propriété P de Schwartz des algèbres de von Neumann,” in: Algèbres d'opérateurs, Sém., Les Plans-sur-Bex (1978), Lecture Notes in Math., 725, Springer, Berlin (1979), pp. 220–227.

    Google Scholar 

  110. P. de la Harpe, “Operator algebras free groups and other groups,” in: Recent Advances in Operator Algebras, Orléans (1992), 232, (1995), pp. 121–153.

    Google Scholar 

  111. P. de la Harpe, Topics in Geometric Group Theory, University of Chicago Press, Chicago, Illinois (2000).

    MATH  Google Scholar 

  112. P. de la Harpe, G. Robertson, and A. Valette, “On the spectrum of the sum of generators for a finitely generated group,” Israel J. Math., 81, 65–96 (1993).

    MATH  MathSciNet  Google Scholar 

  113. J.-C. Hausmann, “On the Vietoris-Rips complexes and a cohomology theory for metric spaces,” in: Prospects in Topology, Princeton, New Jersey (1994), Ann. of Math. Stud., 138, Princeton Univ. Press, Princeton, New Jersey (1995), pp. 175–188.

    Google Scholar 

  114. A. Ya. Helemskii, The Homology of Banach and Topological Algebras, Mathematics and Its Applications, 41, Kluwer Academic Publishers, Dordrecht-Boston-London (1986).

    Google Scholar 

  115. N. Higson and J. Roe, Analytic K-Homology, Oxford Math. Monogr., Oxford Science Publ., Oxford Univ. Press, Oxford (2000).

    Google Scholar 

  116. N. Hitchin, “Global differential geometry,” in: Mathematics Unlimited — 2001 and Beyond, Springer, Berlin (2001), pp. 577–591.

    Google Scholar 

  117. C. Houghton, “Ends of groups and the associated first cohomology group,” J. London Math. Soc., 6, 81–92 (1972).

    MATH  MathSciNet  Google Scholar 

  118. S. Hurder, “Coarse geometry of foliations,” in: Geometric Study of Foliations, Tokyo (1993), World Sci. Publishing, River Edge, New Jersey (1994), pp. 35–96.

    Google Scholar 

  119. S. Hurder and A. Katok, “Ergodic theory and Weil measures for foliations,” Ann. Math., 126, 221–275 (1987).

    MathSciNet  Google Scholar 

  120. W. Imrich and N. Seifter, “A survey on graphs with polynomial growth,” in: Directions in Infinite Graph Theory and Combinatorics, Cambridge (1989), Discrete Math., 95 (1991), pp. 101–117.

  121. R. Ji, “Smooth dense subalgebras of reduced group C*-algebras, Schwartz cohomology of groups, and cyclic cohomology,” J. Funct. Anal., 107, 1–33 (1992).

    MATH  MathSciNet  Google Scholar 

  122. R. Ji, L. B. Schweitzer, “Spectral invariance of smooth crossed products, and rapid decay locally compact groups,” K-Theory, 10, 283–305 (1996).

    MATH  MathSciNet  Google Scholar 

  123. B. E. Johnson, “Cohomology in Banach algebras,” Memoirs of the American Mathematical Society, 127, Amer. Math. Soc., Providence, Rhode Island (1972).

    Google Scholar 

  124. P. Jolissaint, “K-theory of reduced C*-algebras and rapidly decreasing functions on groups,” K-Theory, 2, 723–735 (1989).

    MATH  MathSciNet  Google Scholar 

  125. P. Jolissaint, “Rapidly decreasing functions in reduced C*-algebras of groups,” Trans. Amer. Math. Soc., 317, 167–196 (1990).

    MATH  MathSciNet  Google Scholar 

  126. V. F. R. Jones, “Index for subfactors,” Invent. Math., 72, 1–25 (1983).

    MATH  MathSciNet  Google Scholar 

  127. V. F. R. Jones, “Index for subrings of rings,” in: Group Actions on Rings, Brunswick, Maine (1984), Contemp. Math., 43, Amer. Math. Soc., Providence, Rhode Island (1985), pp. 181–190.

    Google Scholar 

  128. V. A. Kaimanovich and A. M. Vershik, “Random walks on discrete groups: boundary and entropy,” Annals Prob., 11, 457–490 (1983).

    MATH  MathSciNet  Google Scholar 

  129. W. M. Kantor, “Some topics in asymptotic group theory,” in: Groups, Combinatorics & Geometry, Durham (1990), London Math. Soc. Lecture Note Ser., 165, Cambridge Univ. Press, Cambridge (1992), pp. 403–421.

    Google Scholar 

  130. M. Kapovich, “Hyperbolic manifolds and discrete groups,” Progress in Mathematics, 183, Birkhäuser Boston Inc., Boston, Massachusetts (2001).

    Google Scholar 

  131. M. Kapovich and B. Leeb, “On asymptotic cones and quasi-isometry classes of fundamental groups of 3-manifolds,” Geom. Funct. Anal., 5, 582–603 (1995).

    MATH  MathSciNet  Google Scholar 

  132. E. Kirchberg and G. Vaillant, “On C*-algebras having subexponential, polynomial and linear growth,” Invent. Math., 108, 635–652 (1992).

    MATH  MathSciNet  Google Scholar 

  133. B. Kleiner and B. Leeb, “Rigidity of quasi-isometries for symmetric spaces and Euclidean buildings,” Publ. Math. Inst. Hautes Études Sci., 86, 115–197 (1997).

    MATH  MathSciNet  Google Scholar 

  134. B. Kleiner and B. Leeb, “Groups quasi-isometric to symmetric spaces,” Comm. Anal. Geom., 9, 239–260 (2001).

    MATH  MathSciNet  Google Scholar 

  135. D. Kouksov, “On rationality of the cogrowth series,” Proc. Amer. Math. Soc., 126, 2845–2847 (1998).

    MATH  MathSciNet  Google Scholar 

  136. G. Krause and T. Lenagan, Growth of Algebras and the Gelfand-Kirillov Dimension, Graduate Studies in Math., 22, Amer. Math. Soc., Providence, Rhode Island (2000).

    Google Scholar 

  137. E. C. Lance, “On nuclear C*-algebras,” J. Funct. Anal., 12, 151–176 (1973).

    MathSciNet  Google Scholar 

  138. E. C. Lance, “Tensor products and nuclear C*-algebras,” Proc. Symp. in Pure Math., 38, 379–399 (1982).

    MathSciNet  Google Scholar 

  139. E. C. Lance, “Finitely-presented C*-algebras,” in: Operator Algebras and Applications, Samos (1996), NATO Adv. Sci. Inst., Ser. C Math. Phys. Sci., 495, Kluwer Acad. Publ., Dordrecht (1997), pp. (255–266).

    Google Scholar 

  140. T. A. Loring, “C*-algebras generated by stable relations,” J. Funct. Anal., 112, 159–201 (1993).

    MATH  MathSciNet  Google Scholar 

  141. T. A. Loring, Lifting Solutions to Perturbing Problems in C*-Algebras, Fields Institute Monographs, 8, Amer. Math. Soc., Providence, Rhode Island (1997).

    MATH  Google Scholar 

  142. W. Lück, “L 2-invariants and their applications to geometry, group theory and spectral theory,” in: Mathematics Unlimited — 2001 and Beyond, Springer, Berlin (2001), pp. 859–871.

    Google Scholar 

  143. W. Lück, L 2- invariants: Theory and Applications to Geometry and K-Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, A Series of Modern Surveys in Math., 44, Springer-Verlag, Berlin (2002).

    Google Scholar 

  144. Yu. I. Manin, Quantum Groups and Noncommutative Geometry, Université de Montréal, Centre de Recherches Mathématiques, Montreal, QC (1988).

    Google Scholar 

  145. G. A. Margulis, Discrete Subgroups of Semisimple Lie Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 17, Springer-Verlag, Berlin (1991).

    MATH  Google Scholar 

  146. J. Milnor, “A note on curvature and fundamental group,” J. Differential Geometry, 2, 1–7 (1968).

    MATH  MathSciNet  Google Scholar 

  147. J. Milnor, “Growth of finitely generated solvable groups,” J. Differential Geometry, 2, 447–449 (1968).

    MATH  MathSciNet  Google Scholar 

  148. I. Moerdijk, “Foliations, groupoids and Grothendieck étendues,” Rev. Acad. Cienc. Zaragoza, 48, 5–33 (1993).

    MATH  MathSciNet  Google Scholar 

  149. B. Mohar and W. Woess, “A survey on spectra of infinite graphs,” Bull. London Math. Soc., 21, 209–234 (1989).

    MATH  MathSciNet  Google Scholar 

  150. L. Mosher, M. Sageev, and K. Whyte, “Quasi-actions on trees. I. Bounded valence,” Ann. Math., 158, 115–164 (2003).

    Article  MATH  MathSciNet  Google Scholar 

  151. G. D. Mostow, “Strong rigidity of locally symmetric spaces,” Ann. Math. Stud., 78, Princeton University Press, Princeton, New Jersey, University of Tokyo Press, Tokyo (1973).

    Google Scholar 

  152. J. von Neumann, Zur allgemeinen Theorie des Masses,” Fund. Math., 13, 73–116 (1929).

    MATH  Google Scholar 

  153. A. Nevo, “On discrete groups and pointwise ergodic theory,” in: Random Walks and Discrete Potential Theory, Cortona (1997), Sympos. Math., XXXIX, Cambridge Univ. Press, Cambridge (1999), pp. 279–305.

    Google Scholar 

  154. S. Northshield, “Cogrowth of regular graphs,” Proc. Amer. Math. Soc., 116, 203–205 (1992).

    MATH  MathSciNet  Google Scholar 

  155. S. Northshield, Cogrowth of Arbitrary Graphs, Preprint ESI 1053 (2001).

  156. S. Northshield, “Quasi-regular graphs, cogrowth, and amenability,” in: Dynamical Systems and Differential Equations, Wilmington, NC (2002), Discrete Contin. Dyn. Syst., 2003, pp. 678–687.

  157. A. Yu. Ol'shanski, “On the question of the existence of an invariant mean on a group,” Usp. Mat. Nauk., 35, 199–200 (1980).

    Google Scholar 

  158. A. Yu. Ol'shanski, “Hyperbolicity of groups with subquadratic isoperimetric inequality,” Internat. J. Algebra Comput., 1, 281–289 (1991).

    MathSciNet  Google Scholar 

  159. A. Yu. Ol'shanski, “Almost every group is hyperbolic,” Internat. J. Algebra Comput., 2, 1–17 (1992).

    MathSciNet  Google Scholar 

  160. A. Yu. Ol'shanski and M. V. Sapir, “Non-amenable finitely presented torsion-by-cyclic groups,” Publ. Math. Inst. Hautes Études Sci., 96, 43–169 (2002).

    Google Scholar 

  161. A. Yu. Ol'shanski and A. L. Shmel'kin, “Infinite groups,” in: Algebra IV, Encyclopaedia of Math. Sciences, 37, Springer-Verlag, Berlin (1993), pp. 1–95.

    Google Scholar 

  162. D. V. Osin, “Kazhdan constants of hyperbolic groups,” Funct. Anal. Appl., 36, 290–297 (2002).

    MATH  MathSciNet  Google Scholar 

  163. P. Pansu, “Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un,” Ann. Math., 129, 1–60 (1989).

    MathSciNet  Google Scholar 

  164. P. Pansu, “Sous-groupes discrets des groupes de Lie: rigidité, arithméticité,” in: Séminaire Bourbaki, (1993/94), 227, Exposé No. 778, 3 (1995), pp. 69–105.

    MathSciNet  Google Scholar 

  165. P. Papasoglu and K. Whyte, “Quasi-isometries between groups with infinitely many ends,” Comment. Math. Helv., 77, 133–144 (2002).

    MATH  MathSciNet  Google Scholar 

  166. A. L. T. Paterson, Amenability, Mathematical Surveys and Monographs, 29, Amer. Math. Soc., Providence, Rhode Island (1988).

    MATH  Google Scholar 

  167. F. Paulin, “Un groupe hyperbolique est déterminé par son bord,” J. London Math. Soc., 54, 50–74 (1996).

    MATH  MathSciNet  Google Scholar 

  168. N. C. Phillips, “Inverse limits of C*-algebras and applications,” in: Operator Algebras and Applications, Vol. 1, London Math. Soc. Lecture Note Ser., 135, Cambridge Univ. Press, Cambridge (1988), pp. 127–185.

    Google Scholar 

  169. C. Pittet, “Folner sequences in polycyclic groups,” Revista Matemática Iberoamericana, 11, 675–685 (1995).

    MATH  MathSciNet  Google Scholar 

  170. C. Pittet and L. Saloff-Coste, “Amenable groups, isoperimetric profiles and random walks,” in: Geometric Group Theory Down Under, Canberra (1996), de Gruyter, Berlin (1999), pp. 293–316.

    Google Scholar 

  171. F. Point, “Group of polynomial growth and their associated metric spaces,” J. Algebra, 175, 105–121 (1995).

    MATH  MathSciNet  Google Scholar 

  172. S. Popa, “Amenability in the theory of subfactors,” in: Operator Algebras and Quantum Field Theory, Rome (1996), Internat. Press, Cambridge, Massachusetts (1997), pp. 199–211.

    Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  174. E. G. Rieffel, “Groups quasi-isometric to H 2 × R,” J. London Math. Soc., 64, 44–60 (2001).

    MATH  MathSciNet  Google Scholar 

  175. J. Roe, “Finite propagation speed and Connes' foliation algebra,” Math. Proc. Cambridge Philos. Soc., 102, 459–466 (1987).

    MATH  MathSciNet  Google Scholar 

  176. J. Roe, “An index theorem on open manifolds I–II,” J. Differential Geometry, 27, 87–136 (1988).

    MathSciNet  Google Scholar 

  177. J. Roe, Exotic Cohomology and Index Theory on Complete Riemannian Manifolds, preprint (1990).

  178. J. Roe, “Coarse cohomology and index theory on complete Riemannian manifolds,” Mem. Amer. Math. Soc., 104, No. 497 (1993).

  179. J. Roe, “From foliations to coarse geometry and back,” in: Analysis and Geometry in Foliated Manifolds, Santiago de Compostela (1994), World Sci. Publishing, River Edge, New Jersey (1995), pp. 195–205.

    Google Scholar 

  180. J. Roe, “Index theory, coarse geometry, and topology of manifolds,” in: CBMS Regional Conference Series in Mathematics, 90, Amer. Math. Soc., Providence, Rhode Island (1996).

    Google Scholar 

  181. J. Roe, Lectures on Coarse Geometry, University Lecture Series, 31, Amer. Math. Soc., Providence, Rhode Island (2003).

    MATH  Google Scholar 

  182. M. Rørdam, “Classification of nuclear, simple C*-algebras,” in Classification of Nuclear C*-algebras. Entropy in Operator Algebras, Encyclopaedia Math. Sci., 126, Springer, Berlin (2002), pp. 1–145.

    Google Scholar 

  183. A. Rosenmann, “Cogrowth and essentiality in groups and algebras,” in: Combinatorial and Geometric Group Theory, Edinburgh (1993), London Math. Soc. Lecture Note Ser., 204, Cambridge Univ. Press, Cambridge (1995), pp. 284–293.

    Google Scholar 

  184. A. Y. Samet-Vaillant, “C*-algebras, Gelfand-Kirillov dimension and Folner sets,” J. Funct. Anal., 171, 346–365 (2000).

    MATH  MathSciNet  Google Scholar 

  185. A. Y. Samet-Vaillant, “Free *-subalgebras of C*-algebras,” J. Funct. Anal., 171, 432–448 (2000).

    MATH  MathSciNet  Google Scholar 

  186. A. Y. Samet-Vaillant, “Algebras of linear growth, the Kurosh-Leviztky problem and large independent sets,” Europ. J. Combin., 23, 345–354 (2002).

    MATH  MathSciNet  Google Scholar 

  187. A. Schmidt, “Coarse geometry via Grothendieck topologies,” Math. Nachr., 203, 159–173 (1999).

    MATH  MathSciNet  Google Scholar 

  188. R. E. Schwartz, “Quasi-isometric classification of rank one lattices,” Publ. Math. Inst. Hautes Études Sci., 82, 133–168 (1995).

    MATH  Google Scholar 

  189. R. E. Schwartz, “Quasi-isometry rigidity and Diophantine approximation,” Acta Math., 177, 75–112 (1996).

    MATH  MathSciNet  Google Scholar 

  190. L. B. Schweitzer, “Spectral invariance of dense subalgebras of operator algebras,” Internat. J. Math., 4, 289–317 (1993).

    MATH  MathSciNet  Google Scholar 

  191. Z. Sela, “Uniform embeddings of hyperbolic groups in Hilbert spaces,” Israel J. Math., 80, 171–181 (1992).

    MATH  MathSciNet  Google Scholar 

  192. H. Short, (Ed.), Group Theory from a Geometrical Viewpoint, Trieste (1990), World Sci. Publishing, River Edge, New Jersey (1991).

    Google Scholar 

  193. J. T. Stafford, “Noncommutative projective geometry,” in: Proceedings of the International Congress of Mathematicians, Vol. II, Beijing (2002), Higher Ed. Press, Beijing (2002), pp. 93–103.

    Google Scholar 

  194. J. T. Stafford, M. van den Bergh, “Noncommutative curves and noncommutative surfaces,” Bull. Amer. Math. Soc., 38, 171–216 (2001).

    MATH  MathSciNet  Google Scholar 

  195. J. Stallings, “Group theory and three-dimensional manifolds,” A James K. Whittemore Lecture in Mathematics given at Yale University, Yale Mathematical Monographs (1969), 4, Yale Univ. Press, London (1971).

    Google Scholar 

  196. A. M. Stepin, “Approximation of groups and group actions, the Cayley topology,” in: Ergodic Theory of Z d Actions, Warwick (1993–1994), London Math. Soc. Lecture Note Ser., 228, Cambridge Univ. Press, Cambridge (1996), pp. 475–484.

    Google Scholar 

  197. A. Shvarts, “A volume invariant of coverings,” Dokl. Akad. Nauk. USSR, 105, 32–34 (1955).

    MATH  Google Scholar 

  198. R. Szwarc, “A short proof of the Grigorchuk-Cohen cogrowth theorem,” Proc. Amer. Math. Soc., 106, 663–665 (1989).

    MATH  MathSciNet  Google Scholar 

  199. R. Szwarc, “The ratio and generating function of cogrowth coefficients of finitely generated groups,” Studia Math., 131, 89–94 (1998).

    MATH  MathSciNet  Google Scholar 

  200. J. Taback, “Quasi-isometric rigidity for PSL(2, ℤp),” Duke Math. J., 101, 335–357 (2000).

    MATH  MathSciNet  Google Scholar 

  201. J. Tapia, “Le topos étale associé au quotient d'un feuilletage,” C. R. Acad. Sci. Paris Sér. I Math., 303, 753–755 (1986).

    MATH  MathSciNet  Google Scholar 

  202. J. Tapia, “Sur la pente du feuilletage de Kronecker et la cohomologie étale de l'espace des feuilles,” C. R. Acad. Sci. Paris Sér. I Math., 305, 427–429 (1987).

    MathSciNet  Google Scholar 

  203. P. Tapper, “Embedding *-algebras into C*-algebras,” Ph.D. Thesis, Univ. Leeds (1996).

  204. P. Tapper, “C*-ideals generated by polynomials,” Proc. Edinburgh Math. Soc., 42, 305–309 (1999).

    MATH  MathSciNet  Google Scholar 

  205. P. Tapper, “Embedding *-algebras into C*-algebras and C*-ideals generated by words,” J. Operator Theory, 41, 351–364 (1999).

    MATH  MathSciNet  Google Scholar 

  206. The Kourovka notebook, Unsolved Problems in Group Theory — Fifteenth Augmented Edition (V. D. Mazurov and E. I. Khukhro, Eds.), Rossiiskaya Akademiya Nauk Sibirskoe Otdelenie, Institut Matematiki, Novosibirsk (2002).

  207. J. Tits, “Free subgroups in linear groups,” J. Algebra, 20, 250–270 (1972).

    MATH  MathSciNet  Google Scholar 

  208. V. I. Trofimov, “The growth functions of finitely generated semigroups,” Semigroup Forum, 21, 351–360 (1980).

    MATH  MathSciNet  Google Scholar 

  209. V. I. Trofimov, “Graphs with polynomial growth,” Math. USSR-Sb., 51, 405–417 (1985).

    MATH  Google Scholar 

  210. V. A. Ufnarovski, “Combinatorial and asymptotic methods in algebra,” in: Encyclopedia of Math. Sciences, Algebra VI, 57, Springer, Berlin (1995), pp. 1–196.

    Google Scholar 

  211. G. Vaillant, “Følner conditions, nuclearity and subexponential growth in C*-algebras,” J. Funct. Anal., 141, 435–448 (1996).

    MATH  MathSciNet  Google Scholar 

  212. A. M. Vershik, “Amenability and approximation of infinite groups,” Selecta Math. Sovietica, 2, 311–330 (1982).

    MATH  MathSciNet  Google Scholar 

  213. A. M. Vershik, “Dynamic theory of growth in groups: entropy, boundaries, examples,” Russian Math. Surveys, 55, 667–733 (2000).

    MATH  MathSciNet  Google Scholar 

  214. A. M. Vershik and E. I. Gordon, “Groups that are locally embeddable in the class of finite groups,” St. Petersburg Math. J., 9, 49–67 (1998).

    MathSciNet  Google Scholar 

  215. D. Voiculescu, “On the existence of quasi-central approximate units relative to normed ideals,” J. Funct. Anal., 91, 1–36 (1990).

    MATH  MathSciNet  Google Scholar 

  216. D. Voiculescu, “On quasidiagonal operators,” Integral Equations and Operator Theory, 17, 137–148 (1993).

    MATH  MathSciNet  Google Scholar 

  217. S. Wagon, The Banach-Tarski Paradox, Cambridge University Press, Cambridge (1993).

    Google Scholar 

  218. K. Whyte, “Amenability, bi-Lipschitz equivalence, and the von Neumann conjecture,” Duke Math. J., 99, 93–112 (1999).

    MATH  MathSciNet  Google Scholar 

  219. K. Whyte, “The large scale geometry of the higher Baumslag-Solitar groups,” Geom. Funct. Anal., 11, 1327–1353 (2002).

    MathSciNet  Google Scholar 

  220. W. Woess, “Cogrowth of groups and simple random walks,” Arch. Math., 41, 363–370 (1983).

    MATH  MathSciNet  Google Scholar 

  221. W. Woess, “Random walks on infinite graphs and groups — a survey on selected topics,” Bull. London Math. Soc., 26, 1–60 (1994).

    MATH  MathSciNet  Google Scholar 

  222. W. Woess, “Random walks on infinite graphs and groups,” Cambridge Tracts in Mathematics, 138, Cambridge University Press, Cambridge (2000).

    Google Scholar 

  223. G. Yu, “Zero-in-the-spectrum conjecture, positive scalar curvature and asymptotic dimension,” Invent. Math., 127, 99–126 (1997).

    MATH  MathSciNet  Google Scholar 

  224. G. Yu, “The Novikov conjecture for groups with finite asymptotic dimension,” Ann. Math., 147, 325–355 (1998).

    MATH  Google Scholar 

  225. G. Yu, “The coarse Baum-Connes conjecture for spaces which admit a uniform embedding into Hilbert space,” Invent. Math., 139, 201–240 (2000).

    Article  MATH  MathSciNet  Google Scholar 

  226. E. Zelmanov, “On the restricted Burnside problem,” in: Proceedings of the International Congress of Mathematicians, 1, Math. Soc. of Japan, Springer Verlag, Tokyo (1991), pp. 395–402.

    Google Scholar 

  227. M. Ziman, “On finite approximations of groups and algebras,” Illinois J. Math., 46, 837–839 (2002).

    MATH  MathSciNet  Google Scholar 

  228. R. J. Zimmer, Ergodic Theory and Semisimple Groups, Monographs in Math., 81, Birkhäuser Verlag, Basel (1984).

    MATH  Google Scholar 

  229. A. Zuk, “Property (T) and Kazhdan constants for discrete groups,” Geom. Funct. Anal., 13, 643–670 (2003).

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Ingegneria, Università del Sannio, Benevento, Italy

    T. G. Ceccherini-Silberstein

  2. Yeshivat Nahar Shalom, Rechov Shilo 6, Jerusalem, Israel

    A. Y. Samet-Vaillant

Authors
  1. T. G. Ceccherini-Silberstein
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. Y. Samet-Vaillant
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to T. G. Ceccherini-Silberstein.

Additional information

Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 50, Functional Analysis, 2007.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ceccherini-Silberstein, T.G., Samet-Vaillant, A.Y. Asymptotic invariants of finitely generated algebras. A generalization of Gromov's quasi-isometric viewpoint. J Math Sci 156, 56–108 (2009). https://doi.org/10.1007/s10958-008-9257-2

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-008-9257-2

Keywords

Search

Navigation