Advertisement

Lück’s Approximation Theorem

  • Holger Kammeyer
Chapter
Part of the Lecture Notes in Mathematics book series (LNM, volume 2247)

Abstract

After motivating and presenting the statement of Lück’s approximation theorem, we prepare the proof with a detailed synopsis of spectral calculus for Banach-, C-, and von Neumann algebras. The proof is then presented in a manner that emphasizes the two main ingredients: weak convergence of spectral measures and the logarithmic bound on spectral distribution functions. We discuss in detail in how far some of the assumptions in the theorem can be relaxed. We explain the relation of the first 2-Betti number to rank gradient and cost and present a proof of the Abért–Nikolov theorem which builds a bridge between the two latter concepts. Next we discuss the approximation conjecture and show in what sense it is a consequence of the determinant conjecture. After proving the determinant conjecture for residually finite groups, we explain how the approximation conjecture gives further insight on the Atiyah conjecture and hence on the Kaplansky conjecture.

References

  1. 1.
    M. Abért, N. Nikolov, Rank gradient, cost of groups and the rank versus Heegaard genus problem. J. Eur. Math. Soc. 14(5), 1657–1677 (2012). MR 2966663MathSciNetCrossRefGoogle Scholar
  2. 2.
    M. Abért, N. Bergeron, I. Biringer, T. Gelander, N. Nikolov, J. Raimbault, I. Samet, On the growth of L 2-invariants for sequences of lattices in Lie groups. Ann. Math. (2) 185(3), 711–790 (2017). MR 3664810MathSciNetCrossRefGoogle Scholar
  3. 12.
    H. Bauer, Maß- und Integrationstheorie, 2nd edn. (de Gruyter Lehrbuch. [de Gruyter Textbook], Walter de Gruyter, Berlin, 1992) (German). MR 1181881Google Scholar
  4. 13.
    B. Bekka, A. Valette, Group cohomology, harmonic functions and the first L 2-Betti number. Potential Anal. 6(4), 313–326 (1997). MR 1452785Google Scholar
  5. 14.
    B. Bekka, P. de la Harpe, A. Valette, Kazhdan’s Property (T). New Mathematical Monographs, vol. 11 (Cambridge University Press, Cambridge, 2008). MR 2415834Google Scholar
  6. 17.
    N. Bergeron, D. Gaboriau, Asymptotique des nombres de Betti, invariants l 2 et laminations. Comment. Math. Helv. 79(2), 362–395 (2004) (French, with English summary). MR 2059438Google Scholar
  7. 21.
    A. Borel, The L 2-cohomology of negatively curved Riemannian symmetric spaces. Ann. Acad. Sci. Fenn. Ser. A I Math. 10, 95–105 (1985). MR 802471MathSciNetCrossRefGoogle Scholar
  8. 27.
    S.D. Brodskiı̆, Equations over groups, and groups with one defining relation. Sibirsk. Mat. Zh. 25(2), 84–103 (1984) (Russian). MR 741011Google Scholar
  9. 29.
    J. Cheeger, M. Gromov, L 2-cohomology and group cohomology. Topology 25(2), 189–215 (1986). MR 837621MathSciNetCrossRefGoogle Scholar
  10. 33.
    J. Dixmier, von Neumann Algebras. North-Holland Mathematical Library, vol. 27 (North-Holland, Amsterdam, 1981). With a preface by E.C. Lance; Translated from the second French edition by F. Jellett. MR 641217Google Scholar
  11. 36.
    J. Dodziuk, V. Mathai, Approximating L 2 invariants of amenable covering spaces: a combinatorial approach. J. Funct. Anal. 154(2), 359–378 (1998). MR 1612713MathSciNetCrossRefGoogle Scholar
  12. 37.
    J. Dodziuk, P. Linnell, V. Mathai, T. Schick, S. Yates, Approximating L 2-invariants and the Atiyah conjecture. Commun. Pure Appl. Math. 56(7), 839–873 (2003). Dedicated to the memory of Jürgen K. Moser. MR 1990479Google Scholar
  13. 41.
    G. Elek, The strong approximation conjecture holds for amenable groups. J. Funct. Anal. 239(1), 345–355 (2006). MR 2258227MathSciNetCrossRefGoogle Scholar
  14. 42.
    G. Elek, E. Szabó, Hyperlinearity, essentially free actions and L 2-invariants. The sofic property. Math. Ann. 332(2), 421–441 (2005). MR 2178069MathSciNetCrossRefGoogle Scholar
  15. 43.
    J. Elstrodt, Maß- und Integrationstheorie, 4th edn. Springer Textbook (Springer-Lehrbuch; Springer, Berlin, 2005) (German). Grundwissen Mathematik. [Basic Knowledge in Mathematics]. MR 2257838Google Scholar
  16. 44.
    M. Farber, Geometry of growth: approximation theorems for L 2 invariants. Math. Ann. 311(2), 335–375 (1998). MR 1625742Google Scholar
  17. 45.
    D.R. Farkas, P.A. Linnell, Congruence subgroups and the Atiyah conjecture, in Groups, Rings and Algebras. Contemporary Mathematics, vol. 420 (American Mathematical Society, Providence, 2006), pp. 89–102. MR 2279234Google Scholar
  18. 53.
    A. Furman, A survey of measured group theory, in Geometry, Rigidity, and Group Actions. Chicago Lectures in Mathematics (University Chicago Press, Chicago, 2011), pp. 296–374. MR 2807836Google Scholar
  19. 54.
    D. Gaboriau, Coût des relations d’équivalence et des groupes. Invent. Math. 139(1), 41–98 (2000) (French, with English summary). MR 1728876MathSciNetCrossRefGoogle Scholar
  20. 55.
    D. Gaboriau, Invariants l 2 de relations d’équivalence et de groupes. Publ. Math. Inst. Hautes Études Sci. 95, 93–150 (2002) (French). MR 1953191CrossRefGoogle Scholar
  21. 56.
    D. Gaboriau, Orbit equivalence and measured group theory, in Proceedings of the International Congress of Mathematicians, vol. III (Hindustan Book Agency, New Delhi, 2010), pp. 1501–1527. MR 2827853Google Scholar
  22. 57.
    D. Gaboriau, What is … cost? Notices Am. Math. Soc. 57(10), 1295–1296 (2010). MR 2761803Google Scholar
  23. 61.
    R.I. Grigorčuk, On Burnside’s problem on periodic groups. Funktsional. Anal. i Prilozhen. 14(1), 53–54 (1980) (Russian). MR 565099Google Scholar
  24. 65.
    M. Gromov, Endomorphisms of symbolic algebraic varieties. J. Eur. Math. Soc. 1(2), 109–197 (1999). MR 1694588MathSciNetCrossRefGoogle Scholar
  25. 66.
    M. Gromov, Metric Structures for Riemannian and Non-Riemannian Spaces. Progress in Mathematics, vol. 152 (Birkhäuser, Boston, 1999). Based on the 1981 French original [MR0682063 (85e:53051)]; With appendices by M. Katz, P. Pansu, S. Semmes; Translated from the French by Sean Michael Bates. MR 1699320Google Scholar
  26. 67.
    M. Hall Jr., The Theory of Groups (Macmillan, New York, 1959). MR 0103215Google Scholar
  27. 70.
    J. Hempel, Residual finiteness for 3-manifolds, in Combinatorial Group Theory and Topology (Alta, UT, 1984). Annals of Mathematics Studies, vol. 111 (Princeton University Press, Princeton, 1987), pp. 379–396. MR 895623CrossRefGoogle Scholar
  28. 74.
    G. Higman, A finitely generated infinite simple group. J. Lond. Math. Soc. 26, 61–64 (1951). MR 0038348MathSciNetCrossRefGoogle Scholar
  29. 75.
    K.A. Hirsch, On infinite soluble groups. IV. J. Lond. Math. Soc. 27, 81–85 (1952). MR 0044526Google Scholar
  30. 77.
    T. Hutchcroft, P. Gabor, Kazhdan groups have cost 1 (2018). arXiv:1810.11015Google Scholar
  31. 79.
    A. Jaikin-Zapirain, Approximation by subgroups of finite index and the Hanna Neumann conjecture. Duke Math. J. 166(10), 1955–1987 (2017). MR 3679885MathSciNetCrossRefGoogle Scholar
  32. 80.
    A. Jaikin-Zapirain, 2-Betti numbers and their analogues in positive characteristic, in Proceedings of St. Andrews (2017). http://verso.mat.uam.es/~andrei.jaikin/preprints/surveyl2.pdf
  33. 81.
    A. Jaikin-Zapirain, The base change in the Atiyah and the Lück approximation conjectures (2018). http://verso.mat.uam.es/~andrei.jaikin/preprints/sac.pdf
  34. 82.
    A. Jaikin-Zapirain, D. López-Álvarez, The strong Atiyah conjecture for one-relator groups (2018). arXiv:1810.12135Google Scholar
  35. 86.
    H. Kammeyer, Notes on the Abért–Nikolov Theorem on Rank Gradient and Cost (2016). Blog post. https://perso.math.univ-toulouse.fr/jraimbau/2016/11/28/
  36. 87.
    H. Kammeyer, The shrinkage type of knots. Bull. Lond. Math. Soc. 49(3), 428–442 (2017)MathSciNetCrossRefGoogle Scholar
  37. 92.
    A. Kar, N. Nikolov, Rank gradient and cost of Artin groups and their relatives. Groups Geom. Dyn. 8(4), 1195–1205 (2014). MR 3314944MathSciNetCrossRefGoogle Scholar
  38. 93.
    A.S. Kechris, Classical Descriptive Set Theory. Graduate Texts in Mathematics, vol. 156 (Springer, New York, 1995). MR 1321597Google Scholar
  39. 94.
    S. Kionke, On p-adic limits of topological invariants (2008). Preprint. arXiv:1811.00356Google Scholar
  40. 96.
    S. Kionke, The growth of Betti numbers and approximation theorems (2017). arXiv:1709.00769Google Scholar
  41. 97.
    S. Kionke, Characters, L 2-Betti numbers and an equivariant approximation theorem. Math. Ann. 371(1–2), 405–444 (2018). MR 3788853MathSciNetCrossRefGoogle Scholar
  42. 100.
    M. Lackenby, Expanders, rank and graphs of groups. Israel J. Math. 146, 357–370 (2005). MR 2151608MathSciNetCrossRefGoogle Scholar
  43. 101.
    G. Levitt, On the cost of generating an equivalence relation. Ergodic Theory Dyn. Syst. 15(6), 1173–1181 (1995). MR 1366313MathSciNetCrossRefGoogle Scholar
  44. 102.
    T. Li, Rank and genus of 3-manifolds. J. Am. Math. Soc. 26(3), 777–829 (2013). MR 3037787MathSciNetCrossRefGoogle Scholar
  45. 103.
    H. Li, A. Thom, Entropy, determinants, and L 2-torsion. J. Am. Math. Soc. 27(1), 239–292 (2014). MR 3110799Google Scholar
  46. 106.
    P. Linnell, T. Schick, Finite group extensions and the Atiyah conjecture. J. Am. Math. Soc. 20(4), 1003–1051 (2007). MR 2328714MathSciNetCrossRefGoogle Scholar
  47. 107.
    P. Linnell, W. Lück, R. Sauer, The limit of F p-Betti numbers of a tower of finite covers with amenable fundamental groups. Proc. Am. Math. Soc. 139(2), 421–434 (2011). MR 2736326Google Scholar
  48. 115.
    W. Lück, Approximating L 2-invariants by their finite-dimensional analogues. Geom. Funct. Anal. 4(4), 455–481 (1994). MR 1280122MathSciNetCrossRefGoogle Scholar
  49. 117.
    W. Lück, L 2-Invariants: Theory and Applications to Geometry and K-Theory. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, vol. 44 (Springer, Berlin, 2002). MR 1926649Google Scholar
  50. 127.
    S. Meskin, Nonresidually finite one-relator groups. Trans. Am Math. Soc. 164, 105–114 (1972). MR 0285589MathSciNetCrossRefGoogle Scholar
  51. 135.
    B. Nica, Linear groups—Malcev’s theorem and Selberg’s lemma (2013). arXiv:1306.2385Google Scholar
  52. 140.
    D.S. Ornstein, B. Weiss, Ergodic theory of amenable group actions. I. The Rohlin lemma. Bull. Am. Math. Soc. (N.S.) 2(1), 161–164 (1980). MR 551753MathSciNetCrossRefGoogle Scholar
  53. 141.
    D. Pape, A short proof of the approximation conjecture for amenable groups. J. Funct. Anal. 255(5), 1102–1106 (2008). MR 2455493MathSciNetCrossRefGoogle Scholar
  54. 153.
    W. Rudin, Real and Complex Analysis, 3rd edn. (McGraw-Hill, New York, 1987). MR 924157Google Scholar
  55. 155.
    R. Sauer, Amenable covers, volume and L 2-Betti numbers of aspherical manifolds. J. Reine Angew. Math. 636, 47–92 (2009). MR 2572246Google Scholar
  56. 156.
    T. Schick, Integrality of L 2-Betti numbers. Math. Ann. 317(4), 727–75 (2000). MR 1777117Google Scholar
  57. 157.
    T. Schick, L 2-determinant class and approximation of L 2-Betti numbers. Trans. Am. Math. Soc. 353(8), 3247–3265 (2001). MR 1828605Google Scholar
  58. 158.
    T. Schick, Erratum: “Integrality of L 2-Betti numbers”. Math. Ann. 322(2), 421–422 (2002). MR 1894160Google Scholar
  59. 159.
    K. Schmidt, Dynamical Systems of Algebraic Origin. Progress in Mathematics, vol. 128 (Birkhäuser, Basel, 1995). MR 1345152Google Scholar
  60. 161.
    K. Schreve, The strong Atiyah conjecture for virtually cocompact special groups. Math. Ann. 359(3–4), 629–636 (2014). MR 3231009MathSciNetCrossRefGoogle Scholar
  61. 164.
    B. Sury, T.N. Venkataramana, Generators for all principal congruence subgroups of SL(nZ) with n ≥ 3. Proc. Am. Math. Soc. 122(2), 355–358 (1994). MR 1239806Google Scholar
  62. 171.
    B. Weiss, Sofic groups and dynamical systems. Sankhyā Ser. A 62(3), 350–359 (2000). Ergodic theory and harmonic analysis (Mumbai, 1999). MR 1803462Google Scholar
  63. 173.
    D.T. Wise, Research announcement: the structure of groups with a quasiconvex hierarchy. Electron. Res. Announc. Math. Sci. 16, 44–55 (2009). MR 2558631MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Holger Kammeyer
    • 1
  1. 1.Institute for Algebra and GeometryKarlsruhe Institute of TechnologyKarlsruheGermany

Personalised recommendations