Inventiones mathematicae

, Volume 119, Issue 1, pp 267–295 | Cite as

Subgroup growth and congruence subgroups

  • Alexander Lubotzky
Article

Summary

Letk be a global field,O its ring of integers,G an almost simple, simply connected, connected algebraic subgroups ofGLm, defined overk and Γ=G(O) which is assumed to be infinite. Let σn(Г) (resp. γn(Г) be the number of all (resp. congruence) subgroups of index at mostn in Γ. We show:(a) If char(k)=0 then:
  1. (i)

    \(C_1 \frac{{\log ^2 n}}{{\log \log n}} \leqq \log \gamma _n (\Gamma ) \leqq C_2 \frac{{\log ^2 n}}{{\log \log n}}\) for suitable constantsC1 andC2.

    Under some mild assumptions we also have:

     
  2. (ii)

    Γ has the congruence subgroup property if and only if log σn(Г)=o(log2n).

     
  3. (iii)

    If Γ is boundedly generated the Γ has the congruence subgroup property. (This confirms a conjecture of Rapinchuk [R1] which was also proved by Platonov and Rapinchuk [PR3].) (b) If char(k)>0 (and under somewhat stronger conditions onG) then for suitable constantsC3 andC4,C3 log2n≦log γn(Γ)≦C4log3n.

     

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References

  1. [AG] M. Aschbacher, R. Guralnick: Solvable generation of groups and Sylow subgroups of the lower central series. J. Algebra77 (1982), 189–201Google Scholar
  2. [BCP] L. Babai, P.J. Cameron, P.P. Palfy: On the orders of primitive groups with restricted non-abelian composition factors. J. Algebra79 (1982) 161–168Google Scholar
  3. [BP] A. Borovik, L. Pyber: On the number of maximal solvable subgroups of a finite group (in preparation)Google Scholar
  4. [CF] R. Carter, P. Fong: The sylow 2-subgroups of the classical groups. J. Algebra (1964) 139–151Google Scholar
  5. [CK] D. Carter, G. Keller: Bounded elementary generation ofSL n(O). Am. J. Math.105 (1983) 673–687Google Scholar
  6. [DDMS] J.D. Dixon, M.P.F. du Sautoy, A. Mann, D. Segal: Analytic Pro-p Groups, LMS Lecture Notes Series 157, Cambridge University Press, 1991Google Scholar
  7. [dS] M.P.F. du Sautoy: Counting congruence subgroups in arithmetic subgroups. Bull. LMS26 (1994) (to appear)Google Scholar
  8. [El] W. and F. Ellison: Prime Numbers, Wiley, New York, 1985Google Scholar
  9. [Gr] M. Gromov: Hyperbolic groups. In: Essays in Group Theory, M.S.R.I. Series, Vol. 8 (Ed: M. Gersten), Springer-Verlag: Berlin Heidelberg 1987Google Scholar
  10. [Gu] B.M. Guralnick: On the number of generators of a finite group. Arch. Math.53 (1989) 521–523Google Scholar
  11. [Jo] G.A. Jones: Congruence and non-congruence subgroups of the modular group: a survey. Proc. of Groups-St. Andrews 1985 (E.F. Robertson, C.M. Campbell (eds.)) 223–234, LMS Lecture Notes Series 121, Cambridge University Press, 1986Google Scholar
  12. [Li] J.S. Li: Non vanishing theorems for the cohomology of certain arithmetic quotients. J. Reine angew. Math.428 (1992) 177–217Google Scholar
  13. [L1] A. Lubotzky: Pro-finite groups and the congruence subgroup problem. Thesis, Bar-Ilan University, 1980 (in Hebrew)Google Scholar
  14. [L2] A. Lubotzky: Group presentation,p-adic analytic groups and lattices inSL 2(ℂ). Ann. Math.118 (1983) 115–130Google Scholar
  15. [L3] A. Lubotzky: Lattices in rank one Lie groups over local fields. Geometric and Functional Analysis (GAFA)1 (1991) 405–431Google Scholar
  16. [L4] A. Lubotzky: On finite index subgroups of linear groups. Bull. London Math. Soc.19 (1987) 325–328Google Scholar
  17. [LM] A. Lubotzky A. Mann: On groups of polynomial subgroup growth. Invent. Math.104 (1991) 521–533Google Scholar
  18. [LMS] A. Lubotzky, A. Mann, D. Segal, Finitely generated groups of polynomial subgroup growth. Israel J. Math.82 (1993) 363–371Google Scholar
  19. [LS] A. Lubotzky, A. Shalev: On some Λ-analytic pro-p groups. Israel J. Math.85 (1994) 307–337Google Scholar
  20. [Luc] A. Lucchini: A bound on the number of generators of a finite group. Arch. Math.53 (1989) 313–317Google Scholar
  21. [MS] A. Mann, D. Segal: Uniform finiteness conditions in residually finite groups. Proc. Lond. Math. Soc.61 (1990) 529–545Google Scholar
  22. [Mi] J.J. Millson: On the first Betti number of a constant negatively curved manifold. Ann. Math.104 (1976) 235–247Google Scholar
  23. [Pat] A.R. Patterson: The minimal number of generators forp-subgroups ofGL(n,p). J. Algebra32 (1974) 132–40Google Scholar
  24. [P1] V.P. Platnov: The problem of strong approximation and the Kneser-Tits conjecture for algebraic groups. Math. USSR Izvestiya33 (1969), 1211–1219 (in Russian)Google Scholar
  25. [PR1] V.P. Platonov, A.S. Rapinchuk: Algebraic Groups and Number Theory. Academic Press, Boston, 1994Google Scholar
  26. [PR2] V.P. Platonov, A.S. Rapinchuk: Abstract characterization of arithmetic groups with the congruence subgroup property. Dokl. Akad. USSR319 (1991) 1322–1327Google Scholar
  27. [PR3] V.P. Platonov, A.S. Rapinchuk: Abstract properties ofS-arithmetic subgroups and the congruence subgroup problem, Izvestiya Rossüskoi Acad. Sci. Ser. Math. (former Math. USSR Izvestiya)56 (1992) 483–508Google Scholar
  28. [Pr] G. Prasad: Strong approximation for semi-simple groups over function fields, Ann. Math.105 (1977) 553–572Google Scholar
  29. [PR] G. Prasad, M.S. Raghunathan: On the congruence subgroup problem: Determination of the “metaplectic kernel”. Inven. Math.71 (1983) 21–42Google Scholar
  30. [Py1] L. Pyber: Enumerating finite groups of given order. Ann. Math.137 (1993) 203–220Google Scholar
  31. [Py2] L. Pyber: Asymptotic results for permutation groups, DIMACS Series in Disc. Math. and Theor. Comp. Sci11 (1993) 197–219Google Scholar
  32. [Py3] L. Pyber: Dixon's type theorems (in preparation)Google Scholar
  33. [Ra] M.S. Raghunathan: On the congruence subgroup problem. Publ. Math. IHES46 (1976) 107–161Google Scholar
  34. [RV] M.S. Raghunathan, T.N. Venkataramana: The first Betti number of arithmetic groups and the congruence subgroup problem, Proc. of conference held in honour of R. Steinberg, UCLA, 1992, Contemp. Math.153 (1993) 95–107Google Scholar
  35. [R1] A.S. Rapinchuk: Combinatorial theory of arithmetic groups. Preprint no. 20 of the Inst. of Math., Minsk (1990)Google Scholar
  36. [R2] A.S. Rapinchuk: The congruence problem for arithmetic groups of finite width. Sov. Math. Dokl.42 (1991) 664–668Google Scholar
  37. [R3] A.S. Rapinchuk: Congruence subgroup problem for algebraic groups: old and new. Astérisque209 (1992) 73–84Google Scholar
  38. [SS] D. Segal, A. Shalev: Groups with fractionally exponential subgroup growth. J. Pure App. Alg.88 (1993) 205–223Google Scholar
  39. [Se] J.P. Serre: Le probleme des groupes de congruence pour SL2. Ann. Math.92 (1970) 489–527Google Scholar
  40. [Sh] A. Shalev: Growth functions,p-adic analytic groups and groups of finite co-class. J. London Math. Soc.,46 (1992) 111–122Google Scholar
  41. [St1] W.W. Stothers: The number of subgroups of given index in the modular groups. Proc. Royal Soc. Edinbugh78A (1977) 105–112Google Scholar
  42. [St2] W.W. Stothers: Level and index in the modular group. Proc. Royal Soc. Edinburgh99A (1984) 115–126Google Scholar
  43. [Ta] O.I. Tavgen: Bounded generation of Chevalley groups over rings of algebraicS-integers. Math. USSR Izvestiya36 (1991) 101–128Google Scholar
  44. [V] T.N. Venakatraramana: On the first cohomology of arithmetic groups (preprint)Google Scholar
  45. [W] T.S. Weigel: On the profinite completion of arithmetic groups of split type (preprint)Google Scholar
  46. [We] A. Weir: Sylowp-subgroups of the classical groups over finite field with characteristic prime top. Proc. AMS6 (1955) 529–533Google Scholar
  47. [Wr] B. Weisfeiler: Strong approximation for Zariski dense subgroups of semi-simple algebraic grups. Ann. Math.120 (1984) 271–315Google Scholar

Copyright information

© Springer-Verlag 1995

Authors and Affiliations

  • Alexander Lubotzky
    • 1
  1. 1.Institute of MathematicsHebrew UniversityJerusalemIsrael

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