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- W1996422350 abstract "Let us call a profinite group D adelic, if D is a subgroup of the product Ž . Ł SL Z for some n G 2. One of the main cases of interest is when D p n p is the profinite completion of an S-arithmetic subgroup of a semisimple algebraic group. Let k be an algebraic number field, S be a finite set of places of k containing the infinite ones, G be a semisimple, connected algebraic group over k, and G be an S-arithmetic group. We say that G satisfies the congruence subgroup property if the S-congruence kernel SŽ . Ž w x. 4 C G is a finite group see PR2, Chap. 9 . If n G 3, and S s ` , the SŽ . S-congruence kernel C SL is trivial, and so we can identify the profinite n $ Ž . Ž . completion SL Z of SL Z with the product Ł SL Z . Ž . n n p n p The purpose of this note is twofold. Firstly, we prove some results on bounded generation for an arbitrary finitely generated adelic, profinite group. Then, we apply these results to the case of arithmetic groups to prove the following conjecture of A. Lubotzky. ˆ Conjecture. If the profinite completion G of an S-arithmetic group G is adelic, then G satisfies the congruence subgroup property." @default.
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- W1996422350 date "1997-07-01" @default.
- W1996422350 modified "2023-09-30" @default.
- W1996422350 title "Adelic Profinite Groups" @default.
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- W1996422350 doi "https://doi.org/10.1006/jabr.1996.7011" @default.
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