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Subgroup growth and congruence subgroups

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Summary

Letk be a global field,O its ring of integers,G an almost simple, simply connected, connected algebraic subgroups ofGL m , defined overk and Γ=G(O) which is assumed to be infinite. Let σ n (Г) (resp. γ n (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 constantsC 1 andC 2.

    Under some mild assumptions we also have:

  2. (ii)

    Γ has the congruence subgroup property if and only if log σ n (Г)=o(log2 n).

  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 constantsC 3 andC 4,C 3 log2 n≦log γn(Γ)≦C 4log3 n.

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  1. Institute of Mathematics, Hebrew University, 91904, Jerusalem, Israel

    Alexander Lubotzky

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  1. Alexander Lubotzky
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Lubotzky, A. Subgroup growth and congruence subgroups. Invent Math 119, 267–295 (1995). https://doi.org/10.1007/BF01245183

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