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Adelic Profinite Groups
193, 757]763 Ž1997. JA967011
JOURNAL OF ALGEBRA ARTICLE NO.
Adelic Profinite Groups V. P. Platonov* Department of Pure Mathematics, Uni¨ ersity of Waterloo, Ontario, N2L 3G1, Canada
and B. Sury School of Mathematics, Tata Institute of Fundamental Research, Bombay, 400005, India Communicated by Efim Zelmano¨ Received December 4, 1996
INTRODUCTION Let us call a profinite group D adelic, if D is a subgroup of the product Ł p SL nŽZ p . for some n G 2. One of the main cases of interest is when D 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 C S Ž G . is a finite group Žsee wPR2, Chap. 9x.. If n G 3, and S s `4 , the S-congruence$ kernel C S Ž SL n . is trivial, and so we can identify the profinite completion SL n Ž Z . of SL nŽZ. with the product Ł p SL nŽZ 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 of an S-arithmetic group G is Conjecture. If the profinite completion G adelic, then G satisfies the congruence subgroup property. * Supported in part by an NSERC grant. 757 0021-8693r97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.
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It is easy to observe that conversely, the congruence subgroup property ˆ as for an S-arithmetic group G implies the existence of an embedding of G an adelic group for some SL n Žsee Section 1.. Hence this property can be thought of as an abstract topological characterisation of S-arithmetic groups with of as an abstract topological characterisation of S-arithmetic groups with the congruence subgroup property. We may assume throughout the paper that S-arithmetic groups are infinite, i.e., that the group Ł ¨ g S GŽ k ¨ . is not compact. 1. GENERALITIES ON ADELIC GROUPS Recall that a profinite group D is said to have bounded generation Žof degree F r ., if there are elements g 1 , . . . , g r Žnot necessarily distinct. such that D s ² g 1 : ??? ² g r :. It is well known Žsee wPR1, L1x. that for profinite completions of arithmetic groups, this property characterises those with the congruence subgroup property. Moreover, for a pro-p group D, this is equivalent Žloc.cit. to D being a p-adic analytic group. The following lemma is probably well known but we could not find an explicit reference. LEMMA 1. Let D be a finitely generated, profinite group. If D is nilpotent, then any subgroup of D has bounded generation. Proof. This can be proved by induction on the nilpotency class. If D is abelian, it is evident. Let us now assume that D is not abelian and that bounded generation holds for finitely generated profinite groups of nilpotency class smaller than that of D. Let us write D s A, where A is a finitely generated Žnilpotent. group. It is well known that any subgroup Žin particular w A, A x. of A is finitely generated. But, clearly, w A, A x is dense in DŽ D . s w D , D x . Therefore, DŽ D . s w D , D x is finitely generated Žnotice that the discussion shows that any subgroup of D is topologically finitely generated.. But the induction hypothesis, it follows that DŽ D . has bounded generation. Since the abelian quotient DrDŽ D . obviously has bounded generation, it is evident that D itself has bounded generation. Since any subgroup of D is finitely generated, it also has bounded generation. COROLLARY 1. A finitely generated, pro-p group D which is nilpotent, is a p-adic analytic group. Remark. This result is not true for solvable groups. For instance, for any prime p, the group P s C p X Z p which is the wreath product of a cyclic group of order p with Z p , is a solvable pro-p group which is not analytic. The reason is that P has as subquotients the elementary abelian groups C pn for arbitrary n.
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ˆ LEMMA 2. Suppose G satisfies the congruence subgroup property. Then, G ˆ is adelic, i.e., the profinite completion G ¨ Ł p SL nŽZ 0 . for some n. Proof. We are given that the S-congruence kernel C s C S Ž G . is finite and central in the exact sequence p
ˆ ª G ª 1, 1ªCªG where G is the completion of G in the S-congruence topology. Now, since ˆ which intersects C trivially. In C is finite, there is an open subgroup of G other words, there is $ a subgroup G0 of finite index in G such that the profinite completion G0( G0. Now, the S-congruence completion G0 is isomorphic to Ł ¨ f S U¨ , where the U¨ are open compact in GŽ O¨ . and U¨ s GŽ O¨ . for almost all ¨ . Clearly, the product Ł al l ¨ GŽ O¨ . can be embedded in Ł p SL nŽZ p . for some large n Žwe can first go to the Q-group R k r Q G, and then go to GLn via a Q-representation, and finally put GLn in $ SL nq 1 .. We have shown that G0 is an adelic group. Since it is of finite index ˆ the latter can be embedded as an adelic group by means of the in G, induced representation. Remark. Ži. It is not difficult to show Žsee wL2x. that the profinite completion of a free nonabelian group cannot be embedded in the product P p GLnŽZ p . for any n G 1. In particular, the profinite completion of SL 2 ŽZ. is not adelic. Žii. It is known Žsee wPR1, theorem 2x. that for a simple, simply connected Q-algebraic group M, the adelic group Ł p M ŽZ p . has bounded generation. It seems natural to pose: QUESTION. Does e¨ ery finitely generated, profinite, adelic group ha¨ e bounded generation? We show that this is true for pro-p groups. In fact, we prove the following general result. Before stating it, we recall that for any set of P of primes, a finite P-group is a group whose order is a product of primes from P. Define a pro-P group to be a projective limit of finite P-groups. THEOREM 1. Let P be a finite set of primes. Then, any finitely generated pro-P group H which is adelic has bounded generation. Proof. Write H F Ł p SL nŽZ p . or some n G 2. Let p : Ł p SL nŽZ p . ª Ł p f P SL nŽZ p . be the natural projection. Then, KerŽp . ( Ł p g P SL nŽZ p .. Also, HrH l KerŽp . ( p Ž H . F Ł p f P SL nŽZ p .. Now, since p Ž H . is a pro-P group, while SL nŽZ p . is virtually a pro-p group, p Ž H . F Ł p f P S p , where S p are finite. Moreover, by Jordan’s theorem, there is an abelian
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normal subgroup A p and S p of index at the most Ž49n. n Žsee wD, p. 98x.. Let us call p Ž H . as P for simplicity; then P is a finitely generated profinite group. Also PrP l Ł p f P A p F Ł p f P S prA p is a finitely generated profinite group of finite exponent. By Zelmanov’s solution of the restricted Burnside problem Žsee wZx., such a group is actually finite. Hence P l Ł p f P A p , being of finite index in P, is also finitely generated. But, this intersection is an abelian group. Hence, this intersection has bounded generation. Hence P has bounded generation as well. If we show that H l KerŽp . F KerŽp . ( Ł p g P SL nŽZ p . has bounded generation, it would follow that H itself has bounded generation. Now, H l KerŽp . is a pro-P subgroup of Ł p g P SL nŽZ p .. Hence, H l KerŽp . has an open subgroup D of finite index such that D F Ł p g P SL nŽ p, Z p .. Here SL nŽ p, Z p . denotes the subgroup consisting of matrices congruent to the identity modulo p, and is, therefore, a pro-p group. Hence, it suffices to show that D has bounded generation. Now, D is pronilpotent. So, it is a direct product of its Sylow pro-p subgroups D p and D p F SL nŽZ p .. Therefore D p , p g P are analytic; hence, they have bounded generation, and so does the finite product. 2
Remark. The above proof works without using the theorem of Zelmanov if all the primes in P are greater than n. The reason is as follows. As before, we have P [ p Ž H . F Ł p f P S p . We claim that the finite groups S p are actually abelian. For this consider S p F GLŽ n, Z p . F GLŽ n, Q p.. This representation of S p is completely reducible. If an irreducible component has degree d, then d divides the order of S p , so d is of the form Ł p g P p a p . But, d F n - p for each p g P, which means that d s 1. In other words, for each p f P, the group S p is conjugate Žin GLŽ n, Q p.. to the group of diagonal matrices. This shows that every S p , p f P is abelian. Now, P has bounded generation, as it is a finitely generated abelian profinite group. Therefore, H also has bounded generation as before.
2. CONGRUENCE SUBGROUP PROPERTY In this section, we first show that for any set P of primes, any pro-P quotient of a finitely generated profinite group D is the quotient of a pro-P subgroup of D. This is an easy consequence of a result of Huppert wH, Satz 11x for finite groups Žsee also wCKKx1 .. We use the above results along with results of wPR1x to show that adelic arithmetic groups are precisely the arithmetic groups with the congruence subgroup property 1
Thanks are due to A. Lubotzky for this reference.
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Žabbreviated CSP.. Recall that the Frattini subgroup F Ž D . of a profinite group D is the intersection of all maximal open subgroups. It is pronilpotent. PROPOSITION 1. Let D be a finite generated profinite group. Let f : D ª P be a continuous, surjecti¨ e homomorphism with kernel K. Then, there exists a subgroup H of D such that f Ž H . s P and K l H F F Ž H .. Further, H has the property that if P is a set of primes such that P is pro-P, then H is also pro-P. Proof. We first choose a closed subgroup H of D that is minimal among closed subgroups with the property f Ž H . s P. The existence of such an H is guaranteed by Zorn’s lemma as follows. Let C be a chain of closed subgroups with the above property and if the intersection C0 s F C does not satisfy f Ž C0 . s P, there exists x g P such that the inverse image fy1 Ž x . misses C0 . But, since D is compact, this inverse image misses a finite intersection. But, this is a contradiction. Thus, we do have a minimal closed subgroup H of D with the property f Ž H . s P. Now, we show that K l H F F Ž H .. Suppose not. Then, some maximal open subgroup M of H does not contain K l H. By maximality of M, the subgroup Ž K l H . M s H. But, then f Ž M . s P and M is a closed subgroup smaller than H. This contradiction proves the first part of the proposition. ˜ for some set P ˜ Assume now that P is pro-P. Obviously, H is pro-P ˜ containing P. Suppose, if possible, that there is a prime p in P but not in P. Let S p be a Sylow pro-p subgroup of H. So, S p F K l H. We know that K l H F F Ž H . and is, therefore, pro-nilpotent. Since S p is the unique Sylow pro-p subgroup of K l H, it is normal in H. By the analogue of the Schur]Zassenhaus theorem for profinite groups, S p has a complement L in H. But, then f Ž L. s f Ž H . s P, which contradicts the minimality of H. Hence we must have S p s e4 . This proves the proposition. COROLLARY 2. Let D be a finitely generated profinite group. Let P be any set of primes. Then, any pro-P quotient of D is a quotient of a pro-P subgroup of D. Putting together Theorem 1 and Corollary 2 we have: COROLLARY 3. If H is a finitely generated profinite group which is adelic, and if P is a finite set of primes, then pro-P quotients of H ha¨ e bounded generation. Let k be a number field, S be a finite set of places containing the archimedean ones, and G be a simple, simply connected algebraic k-group. Let T be the Žfinite. set of places of k, defined by T s ¨
finite : G Ž K ¨ . is compact 4 .
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ˆ is Let G be an S-arithmetic group. Assume that the profinite completion G ˆ adelic, i.e., for some n G 2, G ¨ Ł p SL nŽZ p .. Then, a special case of Corollary 3 is: ˆ ha¨ e COROLLARY 4. For any finite set P of primes, pro-P quotients of G bounded generation. In particular, for any prime p, the pro-p completion of G is an analytic pro-p group. Let us recall the so-called standard description of normal subgroups of GŽ k .. Let T be, as before, the finite set of places of k, defined by T s ¨ finite : GŽ k ¨ . is compact4 . Then, we say that GŽ k . has the standard description of normal subgroups if each non-central normal subgroup of GŽ k . is open in the T-adic topology given by diagonally embedding GŽ k . in Ł ¨ g T GŽ k ¨ .. This is the analogue of the congruence subgroup property for the group GŽ k . itself. This was conjectured by Platonov and Margulis and has been proved for all the groups except for those of type A n where it is known for some n and open for all odd n G 5 Žsee wPR2, Chap. 9x.. It should be noted Žloc.cit.. that for the CSP to hold for S-arithmetic groups, it is necessary that GŽ k . has a standard description of normal subgroups, and S l T s B. THEOREM 2. Let G be an S-arithmetic subgroup of G. Suppose the ˆ ¨ Ł p SL nŽZ p . for some n G 2. Assume that GŽ k . profinite completion G has the standard description of normal subgroups and also that S l T s B. Then, G has the congruence subgroup property. Proof. Look at the S-congruence kernel C. We have exact sequences $ p
1 ª C ª GŽ k . ª GŽ k . s GŽA S . ª 1 $ p
1 ª C ª G Ž OS . ª G Ž OS . s
Ł G Ž O¨ . ª 1,
¨ fS
where, for any subgroup H of GŽ k ., Hˆ and H denote its S-arithmetic and S-congruence completions, respectively. Now, it is well known from the work of Prasad-Raghunathan Žsee wPR2, Theorem 9.15x. that the finiteness of C is equivalent to its centrality in the above sequences. Further, it is easy to see wPR1, Proposition 2x that if C is not central, it is not of type ŽF. Žrecall that a profinite group is of type ŽF. if the number of continuous homomorphisms to any finite group of finite.. Let us suppose, if possible, that C is not of type ŽF.. Then, using this and the congruence sequence above wPR1x, construct an exact sequence of the type 1ª
d
Ł F ª P ª G¨ ª 1. I
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Here F is a finite simple group, I is an infinite set, ¨ is some place of k, G¨ is a finite central extension of GŽ k ¨ ., and, most importantly, the inverse image dy1 ŽW . of any open compact subgroup of G¨ is a quotient ˆ Note also that dy1 ŽW . is a pro-P group, where of an open subgroup of G. P is the finite set consisting of the primes dividing the order of F and the prime q corresponding to ¨ . By our assumption, G is adelic, and by Corollary 3, dy1 ŽW . has bounded generation for any open compact subgroup W of G¨ . But, as proved at the end of Sections 4 and 5 of wPR1x, if W is deep enough, the inverse image dy1 ŽW . does not have bounded generation. This contradiction proves the theorem. COROLLARY 5. With the assumptions as in the theorem, the profinite completion of G has bounded generation.
ACKNOWLEDGMENTS This work was done during the second author’s visit to the University of Waterloo during October]December 1996. He acknowledges its hospitality and excellent work atmosphere. He also thanks A. Lubotzky for some stimulating conversations.
REFERENCES wCKKx J. Cossey, O. H. Kegel, and L. G. Kovacs, Maximal Frattini extensions, Arch. Math. 35 Ž1980., 210]217. wDx J. Dixon, ‘‘The Structure of Linear Groups,’’ Van Nostrand, Princeton, NJ, 1971. wHx B. Huppert, Normalteiler und maximale untergruppen endlicher gruppen. Math. Z. 60 Ž1954., 409]434. wL1x A. Lubotzky, Subgroup growth and congruence subgroups, In¨ ent. Math. 119 Ž1995., 267]295. wL2x A. Lubotzky, Free quotients and the congruence kernel of SL 2 , J. Algebra 77 Ž1982., 411]418. wPR1x V. P. Platonov and A. S. Rapinchuk, Abstract properties of arithmetic groups and the congruence subgroup problem, Russian Acad. Sci., Iz¨ . 56 Ž1992., 483]508. wPR2x V. P. Platonov and A. S. Rapinchuk, Algebraic Groups and Number Theory,’’ Academic Press, San Diego, 1994. wZx E. I. Zelmanov, On the restricted Burnside problem, in ‘‘Proceedings of the International Congress of Mathematicians,’’ Kyoto, 1991,’’ Vol. I, pp. 395]402.
JOURNAL OF ALGEBRA ARTICLE NO.
Adelic Profinite Groups V. P. Platonov* Department of Pure Mathematics, Uni¨ ersity of Waterloo, Ontario, N2L 3G1, Canada
and B. Sury School of Mathematics, Tata Institute of Fundamental Research, Bombay, 400005, India Communicated by Efim Zelmano¨ Received December 4, 1996
INTRODUCTION Let us call a profinite group D adelic, if D is a subgroup of the product Ł p SL nŽZ p . for some n G 2. One of the main cases of interest is when D 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 C S Ž G . is a finite group Žsee wPR2, Chap. 9x.. If n G 3, and S s `4 , the S-congruence$ kernel C S Ž SL n . is trivial, and so we can identify the profinite completion SL n Ž Z . of SL nŽZ. with the product Ł p SL nŽZ 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 of an S-arithmetic group G is Conjecture. If the profinite completion G adelic, then G satisfies the congruence subgroup property. * Supported in part by an NSERC grant. 757 0021-8693r97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.
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It is easy to observe that conversely, the congruence subgroup property ˆ as for an S-arithmetic group G implies the existence of an embedding of G an adelic group for some SL n Žsee Section 1.. Hence this property can be thought of as an abstract topological characterisation of S-arithmetic groups with of as an abstract topological characterisation of S-arithmetic groups with the congruence subgroup property. We may assume throughout the paper that S-arithmetic groups are infinite, i.e., that the group Ł ¨ g S GŽ k ¨ . is not compact. 1. GENERALITIES ON ADELIC GROUPS Recall that a profinite group D is said to have bounded generation Žof degree F r ., if there are elements g 1 , . . . , g r Žnot necessarily distinct. such that D s ² g 1 : ??? ² g r :. It is well known Žsee wPR1, L1x. that for profinite completions of arithmetic groups, this property characterises those with the congruence subgroup property. Moreover, for a pro-p group D, this is equivalent Žloc.cit. to D being a p-adic analytic group. The following lemma is probably well known but we could not find an explicit reference. LEMMA 1. Let D be a finitely generated, profinite group. If D is nilpotent, then any subgroup of D has bounded generation. Proof. This can be proved by induction on the nilpotency class. If D is abelian, it is evident. Let us now assume that D is not abelian and that bounded generation holds for finitely generated profinite groups of nilpotency class smaller than that of D. Let us write D s A, where A is a finitely generated Žnilpotent. group. It is well known that any subgroup Žin particular w A, A x. of A is finitely generated. But, clearly, w A, A x is dense in DŽ D . s w D , D x . Therefore, DŽ D . s w D , D x is finitely generated Žnotice that the discussion shows that any subgroup of D is topologically finitely generated.. But the induction hypothesis, it follows that DŽ D . has bounded generation. Since the abelian quotient DrDŽ D . obviously has bounded generation, it is evident that D itself has bounded generation. Since any subgroup of D is finitely generated, it also has bounded generation. COROLLARY 1. A finitely generated, pro-p group D which is nilpotent, is a p-adic analytic group. Remark. This result is not true for solvable groups. For instance, for any prime p, the group P s C p X Z p which is the wreath product of a cyclic group of order p with Z p , is a solvable pro-p group which is not analytic. The reason is that P has as subquotients the elementary abelian groups C pn for arbitrary n.
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ˆ LEMMA 2. Suppose G satisfies the congruence subgroup property. Then, G ˆ is adelic, i.e., the profinite completion G ¨ Ł p SL nŽZ 0 . for some n. Proof. We are given that the S-congruence kernel C s C S Ž G . is finite and central in the exact sequence p
ˆ ª G ª 1, 1ªCªG where G is the completion of G in the S-congruence topology. Now, since ˆ which intersects C trivially. In C is finite, there is an open subgroup of G other words, there is $ a subgroup G0 of finite index in G such that the profinite completion G0( G0. Now, the S-congruence completion G0 is isomorphic to Ł ¨ f S U¨ , where the U¨ are open compact in GŽ O¨ . and U¨ s GŽ O¨ . for almost all ¨ . Clearly, the product Ł al l ¨ GŽ O¨ . can be embedded in Ł p SL nŽZ p . for some large n Žwe can first go to the Q-group R k r Q G, and then go to GLn via a Q-representation, and finally put GLn in $ SL nq 1 .. We have shown that G0 is an adelic group. Since it is of finite index ˆ the latter can be embedded as an adelic group by means of the in G, induced representation. Remark. Ži. It is not difficult to show Žsee wL2x. that the profinite completion of a free nonabelian group cannot be embedded in the product P p GLnŽZ p . for any n G 1. In particular, the profinite completion of SL 2 ŽZ. is not adelic. Žii. It is known Žsee wPR1, theorem 2x. that for a simple, simply connected Q-algebraic group M, the adelic group Ł p M ŽZ p . has bounded generation. It seems natural to pose: QUESTION. Does e¨ ery finitely generated, profinite, adelic group ha¨ e bounded generation? We show that this is true for pro-p groups. In fact, we prove the following general result. Before stating it, we recall that for any set of P of primes, a finite P-group is a group whose order is a product of primes from P. Define a pro-P group to be a projective limit of finite P-groups. THEOREM 1. Let P be a finite set of primes. Then, any finitely generated pro-P group H which is adelic has bounded generation. Proof. Write H F Ł p SL nŽZ p . or some n G 2. Let p : Ł p SL nŽZ p . ª Ł p f P SL nŽZ p . be the natural projection. Then, KerŽp . ( Ł p g P SL nŽZ p .. Also, HrH l KerŽp . ( p Ž H . F Ł p f P SL nŽZ p .. Now, since p Ž H . is a pro-P group, while SL nŽZ p . is virtually a pro-p group, p Ž H . F Ł p f P S p , where S p are finite. Moreover, by Jordan’s theorem, there is an abelian
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normal subgroup A p and S p of index at the most Ž49n. n Žsee wD, p. 98x.. Let us call p Ž H . as P for simplicity; then P is a finitely generated profinite group. Also PrP l Ł p f P A p F Ł p f P S prA p is a finitely generated profinite group of finite exponent. By Zelmanov’s solution of the restricted Burnside problem Žsee wZx., such a group is actually finite. Hence P l Ł p f P A p , being of finite index in P, is also finitely generated. But, this intersection is an abelian group. Hence, this intersection has bounded generation. Hence P has bounded generation as well. If we show that H l KerŽp . F KerŽp . ( Ł p g P SL nŽZ p . has bounded generation, it would follow that H itself has bounded generation. Now, H l KerŽp . is a pro-P subgroup of Ł p g P SL nŽZ p .. Hence, H l KerŽp . has an open subgroup D of finite index such that D F Ł p g P SL nŽ p, Z p .. Here SL nŽ p, Z p . denotes the subgroup consisting of matrices congruent to the identity modulo p, and is, therefore, a pro-p group. Hence, it suffices to show that D has bounded generation. Now, D is pronilpotent. So, it is a direct product of its Sylow pro-p subgroups D p and D p F SL nŽZ p .. Therefore D p , p g P are analytic; hence, they have bounded generation, and so does the finite product. 2
Remark. The above proof works without using the theorem of Zelmanov if all the primes in P are greater than n. The reason is as follows. As before, we have P [ p Ž H . F Ł p f P S p . We claim that the finite groups S p are actually abelian. For this consider S p F GLŽ n, Z p . F GLŽ n, Q p.. This representation of S p is completely reducible. If an irreducible component has degree d, then d divides the order of S p , so d is of the form Ł p g P p a p . But, d F n - p for each p g P, which means that d s 1. In other words, for each p f P, the group S p is conjugate Žin GLŽ n, Q p.. to the group of diagonal matrices. This shows that every S p , p f P is abelian. Now, P has bounded generation, as it is a finitely generated abelian profinite group. Therefore, H also has bounded generation as before.
2. CONGRUENCE SUBGROUP PROPERTY In this section, we first show that for any set P of primes, any pro-P quotient of a finitely generated profinite group D is the quotient of a pro-P subgroup of D. This is an easy consequence of a result of Huppert wH, Satz 11x for finite groups Žsee also wCKKx1 .. We use the above results along with results of wPR1x to show that adelic arithmetic groups are precisely the arithmetic groups with the congruence subgroup property 1
Thanks are due to A. Lubotzky for this reference.
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Žabbreviated CSP.. Recall that the Frattini subgroup F Ž D . of a profinite group D is the intersection of all maximal open subgroups. It is pronilpotent. PROPOSITION 1. Let D be a finite generated profinite group. Let f : D ª P be a continuous, surjecti¨ e homomorphism with kernel K. Then, there exists a subgroup H of D such that f Ž H . s P and K l H F F Ž H .. Further, H has the property that if P is a set of primes such that P is pro-P, then H is also pro-P. Proof. We first choose a closed subgroup H of D that is minimal among closed subgroups with the property f Ž H . s P. The existence of such an H is guaranteed by Zorn’s lemma as follows. Let C be a chain of closed subgroups with the above property and if the intersection C0 s F C does not satisfy f Ž C0 . s P, there exists x g P such that the inverse image fy1 Ž x . misses C0 . But, since D is compact, this inverse image misses a finite intersection. But, this is a contradiction. Thus, we do have a minimal closed subgroup H of D with the property f Ž H . s P. Now, we show that K l H F F Ž H .. Suppose not. Then, some maximal open subgroup M of H does not contain K l H. By maximality of M, the subgroup Ž K l H . M s H. But, then f Ž M . s P and M is a closed subgroup smaller than H. This contradiction proves the first part of the proposition. ˜ for some set P ˜ Assume now that P is pro-P. Obviously, H is pro-P ˜ containing P. Suppose, if possible, that there is a prime p in P but not in P. Let S p be a Sylow pro-p subgroup of H. So, S p F K l H. We know that K l H F F Ž H . and is, therefore, pro-nilpotent. Since S p is the unique Sylow pro-p subgroup of K l H, it is normal in H. By the analogue of the Schur]Zassenhaus theorem for profinite groups, S p has a complement L in H. But, then f Ž L. s f Ž H . s P, which contradicts the minimality of H. Hence we must have S p s e4 . This proves the proposition. COROLLARY 2. Let D be a finitely generated profinite group. Let P be any set of primes. Then, any pro-P quotient of D is a quotient of a pro-P subgroup of D. Putting together Theorem 1 and Corollary 2 we have: COROLLARY 3. If H is a finitely generated profinite group which is adelic, and if P is a finite set of primes, then pro-P quotients of H ha¨ e bounded generation. Let k be a number field, S be a finite set of places containing the archimedean ones, and G be a simple, simply connected algebraic k-group. Let T be the Žfinite. set of places of k, defined by T s ¨
finite : G Ž K ¨ . is compact 4 .
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ˆ is Let G be an S-arithmetic group. Assume that the profinite completion G ˆ adelic, i.e., for some n G 2, G ¨ Ł p SL nŽZ p .. Then, a special case of Corollary 3 is: ˆ ha¨ e COROLLARY 4. For any finite set P of primes, pro-P quotients of G bounded generation. In particular, for any prime p, the pro-p completion of G is an analytic pro-p group. Let us recall the so-called standard description of normal subgroups of GŽ k .. Let T be, as before, the finite set of places of k, defined by T s ¨ finite : GŽ k ¨ . is compact4 . Then, we say that GŽ k . has the standard description of normal subgroups if each non-central normal subgroup of GŽ k . is open in the T-adic topology given by diagonally embedding GŽ k . in Ł ¨ g T GŽ k ¨ .. This is the analogue of the congruence subgroup property for the group GŽ k . itself. This was conjectured by Platonov and Margulis and has been proved for all the groups except for those of type A n where it is known for some n and open for all odd n G 5 Žsee wPR2, Chap. 9x.. It should be noted Žloc.cit.. that for the CSP to hold for S-arithmetic groups, it is necessary that GŽ k . has a standard description of normal subgroups, and S l T s B. THEOREM 2. Let G be an S-arithmetic subgroup of G. Suppose the ˆ ¨ Ł p SL nŽZ p . for some n G 2. Assume that GŽ k . profinite completion G has the standard description of normal subgroups and also that S l T s B. Then, G has the congruence subgroup property. Proof. Look at the S-congruence kernel C. We have exact sequences $ p
1 ª C ª GŽ k . ª GŽ k . s GŽA S . ª 1 $ p
1 ª C ª G Ž OS . ª G Ž OS . s
Ł G Ž O¨ . ª 1,
¨ fS
where, for any subgroup H of GŽ k ., Hˆ and H denote its S-arithmetic and S-congruence completions, respectively. Now, it is well known from the work of Prasad-Raghunathan Žsee wPR2, Theorem 9.15x. that the finiteness of C is equivalent to its centrality in the above sequences. Further, it is easy to see wPR1, Proposition 2x that if C is not central, it is not of type ŽF. Žrecall that a profinite group is of type ŽF. if the number of continuous homomorphisms to any finite group of finite.. Let us suppose, if possible, that C is not of type ŽF.. Then, using this and the congruence sequence above wPR1x, construct an exact sequence of the type 1ª
d
Ł F ª P ª G¨ ª 1. I
ADELIC PROFINITE GROUPS
763
Here F is a finite simple group, I is an infinite set, ¨ is some place of k, G¨ is a finite central extension of GŽ k ¨ ., and, most importantly, the inverse image dy1 ŽW . of any open compact subgroup of G¨ is a quotient ˆ Note also that dy1 ŽW . is a pro-P group, where of an open subgroup of G. P is the finite set consisting of the primes dividing the order of F and the prime q corresponding to ¨ . By our assumption, G is adelic, and by Corollary 3, dy1 ŽW . has bounded generation for any open compact subgroup W of G¨ . But, as proved at the end of Sections 4 and 5 of wPR1x, if W is deep enough, the inverse image dy1 ŽW . does not have bounded generation. This contradiction proves the theorem. COROLLARY 5. With the assumptions as in the theorem, the profinite completion of G has bounded generation.
ACKNOWLEDGMENTS This work was done during the second author’s visit to the University of Waterloo during October]December 1996. He acknowledges its hospitality and excellent work atmosphere. He also thanks A. Lubotzky for some stimulating conversations.
REFERENCES wCKKx J. Cossey, O. H. Kegel, and L. G. Kovacs, Maximal Frattini extensions, Arch. Math. 35 Ž1980., 210]217. wDx J. Dixon, ‘‘The Structure of Linear Groups,’’ Van Nostrand, Princeton, NJ, 1971. wHx B. Huppert, Normalteiler und maximale untergruppen endlicher gruppen. Math. Z. 60 Ž1954., 409]434. wL1x A. Lubotzky, Subgroup growth and congruence subgroups, In¨ ent. Math. 119 Ž1995., 267]295. wL2x A. Lubotzky, Free quotients and the congruence kernel of SL 2 , J. Algebra 77 Ž1982., 411]418. wPR1x V. P. Platonov and A. S. Rapinchuk, Abstract properties of arithmetic groups and the congruence subgroup problem, Russian Acad. Sci., Iz¨ . 56 Ž1992., 483]508. wPR2x V. P. Platonov and A. S. Rapinchuk, Algebraic Groups and Number Theory,’’ Academic Press, San Diego, 1994. wZx E. I. Zelmanov, On the restricted Burnside problem, in ‘‘Proceedings of the International Congress of Mathematicians,’’ Kyoto, 1991,’’ Vol. I, pp. 395]402.