Inventiones mathematicae

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

Subgroup growth and congruence subgroups

  • Alexander Lubotzky


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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Copyright information

© Springer-Verlag 1995

Authors and Affiliations

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

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