Skip to main content
Log in

Finitely generated groups of polynomial subgroup growth

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

We determine the structure of finitely generated residually finite groups in which the number of subgroups of each finite indexn is bounded by a fixed power ofn.

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

Access this article

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. J.D. Dixon, M.P.F. duSautoy, A. Mann and D. Segal,Analytic pro-p Groups, London Math. Soc. LNS 157, Cambridge University Press, 1991.

  2. M.P.F. DuSautoy,Finitely generated groups, p-adic analytic groups and Poincaré series, Bull. Am. Math. Soc.23(1990), 121–126 (also Appendix C of [DDMS]).

    MathSciNet  Google Scholar 

  3. M.P.F. duSautoy,Applications of p-adic methods to group theory, inp-Adic Methods and Their Application (A. Baker and R. Plyman, eds.), Oxford University Press, 1992.

  4. F.J. Grunewald, D. Segal and G.C. Smith,Subgroups of finite index in nilpotent groups, Invent. Math.93 (1988), 185–223.

    Article  MATH  MathSciNet  Google Scholar 

  5. I. Ilani,Counting finite index subgroups and the P. Hall enumeration principle, Isr. J. Math.68 (1989), 18–26.

    Article  MATH  MathSciNet  Google Scholar 

  6. M. Lazard,Groupes analytiques p-adiques, Publ. Math. IHES26 (1965), 389–603.

    MATH  MathSciNet  Google Scholar 

  7. A. Lubotzky and A. Mann,Powerful p-group, I & II, J. Algebra105 (1987), 484–515.

    Article  MATH  MathSciNet  Google Scholar 

  8. A. Lubotzky and A. Mann,Residually finite groups of finite rank, Math. Proc. Camb. Phil. Soc.106 (1989), 385–388.

    Article  MATH  MathSciNet  Google Scholar 

  9. A. Lubotzky and A. Mann,On groups of polynomial subgroup growth, Invent. Math.104 (1991), 521–533.

    Article  MATH  MathSciNet  Google Scholar 

  10. A. Lubotzky,A group-theoretic characterization in linear groups, J. Algebra113 (1988), 207–214.

    Article  MATH  MathSciNet  Google Scholar 

  11. A. Mann,Some applications of powerful p-groups, inGroups—St. Andrews 1989, Vol. 2, London Math. Soc. LNS 160, Cambridge University Press, 1991, pp. 370–385.

  12. A. Mann,Some properties of polynomial subgroup growth groups, Israel J. Math.82 (1993), 373–380.

    MATH  MathSciNet  Google Scholar 

  13. A. Mann and D. Segal,Uniform finiteness conditions in residually finite groups, Proc. London Math. Soc. (3)61 (1990), 529–545.

    Article  MATH  MathSciNet  Google Scholar 

  14. C.R. Matthews, L.N. Vaserstein and B. Weisfeller,Congruence properties of Zariski-dense subgroups I, Proc. London Math. Soc.48 (1984), 514–532.

    Article  MATH  MathSciNet  Google Scholar 

  15. M. Newman,Asymptotic formulas related to free products of cyclic groups, Math. Comp.30 (1976), 838–846.

    Article  MATH  MathSciNet  Google Scholar 

  16. M. Nori,On subgroups of GL n (F p ), Invent. Math.88 (1987), 257–275.

    Article  MATH  MathSciNet  Google Scholar 

  17. D.J.S. Robinson,Finiteness Conditions and Generalized Soluble Groups II, Springer-Verlag, Berlin, 1972.

    Google Scholar 

  18. D. Segal,Subgroups of finite index in soluble groups I, inGroups—St. Andrews 1985, London Math. Soc. LNS 121, Cambridge University Press, 1986, pp. 307–314.

  19. D. Segal,Residually finite groups, inGroups—Canberra 1989, Lecture Notes in Math.1456, Springer-Verlag, Berlin, 1990, pp. 85–95.

    Chapter  Google Scholar 

  20. A. Shalev,Growth functions, p-adic analytic groups, and groups of finite coclass, J. London Math. Soc., to appear.

  21. B.A.F. Wehrfritz,Infinite Linear Groups, Springer-Verlag, Berlin, 1973.

    MATH  Google Scholar 

  22. J.S. Wilson,Two-generator conditions for residually finite groups, Bull. London Math. Soc.104 (1991), 239–248.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

To John Thompson, an inspiration to group theory, on his being awarded the Wolf Prize

Partially supported by BSF and GIF grants.

Partially supported by a BSF grant.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Lubotzky, A., Mann, A. & Segal, D. Finitely generated groups of polynomial subgroup growth. Israel J. Math. 82, 363–371 (1993). https://doi.org/10.1007/BF02808118

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02808118

Keywords

Navigation