Countingp-groups and nilpotent groups

  • Marcus du Sautoy
Article

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    N. Blackburn, On a special class ofp-groups,Acta Math. 100 (1958), 49–92.CrossRefMathSciNetGoogle Scholar
  2. [2]
    R. M. Bryant andJ. R. J. Groves, Algebraic groups of automorphisms of nilpotent groups and Lie algebras,J. London Math. Soc. 33 (1986), 453–466.MATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    J. Denef, The rationality of the Poincaré series associated to thep-adic points on a variety,Invent. Math. 77 (1984), 1–23.MATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    J. Denef andL. van den Dries,p-adic and real subanalytic sets,Annals of Math. 128 (1988), 79–138.CrossRefGoogle Scholar
  5. [5]
    J. Denef andF. Loeser, Motivic Igusa zeta functions,J. Algebraic Geom.,7 (1998), 505–537.MATHMathSciNetGoogle Scholar
  6. [6]
    J. Denef andF. Loeser, Germs of arcs on singular algebraic varieties and motivic integration,Invent. Math.,135 (1999), 201–232.MATHCrossRefMathSciNetGoogle Scholar
  7. [7]
    J. D. Dixon, M. P. F. du Sautoy, A. Mann andD. Segal, Analytic pro-p groups, Second Edition,Cambridge Studies in Advanced Mathematics,61, Cambridge, CUP, 1999.Google Scholar
  8. [8]
    M. P. F. du Sautoy, Finitely generated groups,p-adic analytic groups and Poincaré series,Annals of Math. 137 (1993), 639–670.MATHCrossRefGoogle Scholar
  9. [9]
    M. P. F. du Sautoy, Zeta functions and counting finitep-groups,Electronic Research Announcements of the American Math. Soc.,5 (1999), 112–122.MATHCrossRefGoogle Scholar
  10. [10]
    M. P. F. duSautoy, A nilpotent group and its elliptic curve: non-uniformity of local zeta functions of groups, MPI preprint 2000-85. To appear inIsrael J. of Math. 126.Google Scholar
  11. [11]
    M. P. F. duSautoy, Counting subgroups in nilpotent groups and points on elliptic curves, MPI preprint 2000-86.Google Scholar
  12. [12]
    M. P. F. duSautoy, Natural boundaries for zeta functions of groups, preprint.Google Scholar
  13. [13]
    M. P. F. du Sautoy andF. J. Grunewald, Analytic properties of Euler products of Igusa-type zeta functions and subgroup growth of nilpotent groups,C. R. Acad. Sci. Paris 329, Série 1 (1999), 351–356.MATHGoogle Scholar
  14. [14]
    M. P. F. du Sautoy andF. J. Grunewald, Analytic properties of zeta functions and subgroup growth,Annals of Math. 152 (2000), 793–833.MATHCrossRefGoogle Scholar
  15. [15]
    M. P. F. duSautoy andF. J. Grunewald, Uniformity for 2-generator free nilpotent groups, in preparation.Google Scholar
  16. [16]
    M. P. F. duSautoy andF. Loeser, Motivic zeta functions of infinite dimensional Lie algebras, École polytechnique, preprint series 2000-12.Google Scholar
  17. [17]
    M. P. F. du Sautoy andA. Lubotzky, Functional equations and uniformity for local zeta functions of nilpotent groups,Amer. J. Math. 118 (1996), 39–90.MATHCrossRefMathSciNetGoogle Scholar
  18. [18]
    M. P. F. du Sautoy, J. J. McDermott andG. C. Smith, Zeta functions of crystallographic groups and analytic continuation,Proc. London Math. Soc. 79 (1999), 511–534.MATHCrossRefMathSciNetGoogle Scholar
  19. [19]
    M. P. F. du Sautoy andD. Segal, Zeta functions of groups, inNew horizons in pro-p groups.Progress in Mathematics, vol.184 (ed M. P. F. du Sautoy, D. Segal and A. Shalev), p. 249–286. Boston, Birkhäuser (2000).Google Scholar
  20. [20]
    M. D. Fried andM. Jarden,Field Arithmetic, Springer-Verlag, Berlin, Heidelberg, New York, 1986.MATHGoogle Scholar
  21. [21]
    F. J. Grunewald, D. Segal andG. C. Smith, Subgroups of finite index in nilpotent groups,Invent. Math. 93 (1988), 185–223.MATHCrossRefMathSciNetGoogle Scholar
  22. [22]
    G. Higman, Enumeratingp-groups, I,Proc. London Math. Soc. 10 (1960), 24–30.MATHCrossRefMathSciNetGoogle Scholar
  23. [23]
    G. Higman, Enumeratingp-groups, II,Proc. London Math. Soc. 10 (1960), 566–582.MATHCrossRefMathSciNetGoogle Scholar
  24. [24]
    K. Ireland andM. Rosen, A classical introduction to modern number theory, Second Edition,Graduate texts in mathematics 84, Springer-Verlag, New York, Berlin, Heidelberg, 1993.Google Scholar
  25. [25]
    S. Lang,Algebra, Addison-Wesley, Reading, MA, 1965.MATHGoogle Scholar
  26. [26]
    C. R. Leedham-Green andS. McKay, Onp-groups of maximal class II,Quart. J. Math. Oxford (2)29 (1978), 175–186.MATHCrossRefMathSciNetGoogle Scholar
  27. [27]
    C. R. Leedham-Green andS. McKay, Onp-groups of maximal class III,Quart. J. Math. Oxford (2)29 (1978), 281–299.MATHCrossRefMathSciNetGoogle Scholar
  28. [28]
    C. R. Leedham-Green andM. F. Newman, Space groups and groups of prime-power order I,Arch. Math. (Basel)35 (1980), 193–202.MATHCrossRefMathSciNetGoogle Scholar
  29. [29]
    C. R. Leedham-Green, The structure of finitep-groups,J. London Math. Soc. 50 (1994), 49–67.MATHMathSciNetGoogle Scholar
  30. [30]
    W. Magnus, A. Karras andD. Solitar,Combinatorial Group Theory, Wiley, Chichester, UK, 1966.MATHGoogle Scholar
  31. [31]
    M. F. Newman, Groups of prime-power order, Groups-Canberra 1989,Lecture Notes in Math.,1456 Springer-Verlag (1990), 49–62.CrossRefGoogle Scholar
  32. [32]
    M. F. Newman andE. A. O’Brien, Classifying 2-groups by coclass,Trans. Amer. Math. Soc. 351 (1999), 131–169.MATHCrossRefMathSciNetGoogle Scholar
  33. [33]
    V. P. Platonov, The problem of strong approximation and the Kneser-Tits conjecture for algebraic groups,Math. USSR-Izv. 3 (1969), 1139–1147.CrossRefGoogle Scholar
  34. [34]
    V. P. Platonov, Addendum,Math. USSR-Izv. 4 (1970), 784–786.MATHCrossRefGoogle Scholar
  35. [35]
    V. P. Platonov andA. S. Rapinchuk,Algebraic Groups and Number Theory,Pure and Applied Mathematics 139, London, Academic Press, 1994.MATHGoogle Scholar
  36. [36]
    D. Segal, Polycyclic Groups,Cambridge tracts in mathematics,82, CUP (1983).Google Scholar
  37. [37]
    A. Shalev, The structure of finitep-groups: effective proof of the coclass conjectures,Invent. Math. 115 (1994), 315–345.MATHCrossRefMathSciNetGoogle Scholar
  38. [38]
    C. C. Sims, Enumeratingp-groups,Proc. London Math. Soc. 15 (1965), 151–166.MATHCrossRefMathSciNetGoogle Scholar
  39. [39]
    R. P. Stanley, Enumerative Combinatorics, vol. 1,Cambridge Studies in Advanced Mathematics,49, CUP, 1997.Google Scholar

Copyright information

© Publications Mathematiques de L’I.H.E.S 2000

Authors and Affiliations

  • Marcus du Sautoy
    • 1
  1. 1.DPMMS, Centre for Mathematical SciencesCambridgeUK

Personalised recommendations