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Finite-Rank Free Metabelian Subgroups of Finitely Presented Metabelian Groups
JOURNAL OF ALGEBRA ARTICLE NO.
180, 323]333 Ž1996.
0069
Finite-Rank Free Metabelian Subgroups of Finitely Presented Metabelian Groups* Ada Peluso† Department of Mathematics and Statistics, Hunter College, New York, New York 10021 Communicated by Walter Feit Received March 13, 1993
A sufficient condition is obtained for the residual torsion-free nilpotence of certain finitely presented metabelian groups that arise from a matrix representation developed by Magnus Ž1939, Ann. of Math. 40, 764]768. for metabelian groups. Using this condition and a construction due to Baumslag Ž1973, J. Austral. Math. Soc. 16, 98]110., we prove that a free metabelian group of finite rank can be embedded in a finitely presented metabelian group that is also residually torsionfree nilpotent. Q 1996 Academic Press, Inc.
1. INTRODUCTION It is known that every finitely generated metabelian group can be embedded in a finitely presented metabelian group. This remarkable result was obtained independently by Baumslag w2x and Remeslennikov w8x in the early 1970s. It is of interest to try to determine whether a finitely generated metabelian group with a specific property can be embedded in a finitely presented metabelian group with the same property. The property of residual nilpotence was considered by the author in w7x, and there it was proved that a finitely generated free metabelian group can be embedded in a finitely presented metabelian group that is also residually nilpotent. Here we are interested in residual torsion-free nilpotence. By definition, a group G is residually P, where P is any group property, if G possesses a family Kl4 of normal subgroups such that GrKl is P for all l, and F l Kl s 1. A finitely generated free metabelian group is residually torsion-free nilpotent. In fact, Magnus w5x proved that a finitely generated * This research was supported Žin part. by a grant from The City University of New York PSC]CUNY Research Award Program. † E-mail: [email protected]. 323 0021-8693r96 $18.00 Copyright Q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
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ADA PELUSO
free metabelian group can be embedded in the wreath product of two finitely generated free abelian groups, and this wreath product is residually torsion-free nilpotent. ŽSee, for example, the discussion on pages 99]102 in Gruenberg and Roseblade w3x.. A matrix representation of metabelian groups due to Magnus w5x will be used here to establish a sufficient condition for the residual torsion-free nilpotence of certain finitely presented metabelian groups. By this means, we shall prove the following main result. THEOREM. The wreath product of two finitely generated free abelian groups can be embedded in a finitely presented metabelian group that is residually torsion-free nilpotent. The above discussion then yields the desired result about free metabelian groups as an immediate corollary. COROLLARY. A free metabelian group of finite rank is a subgroup of a finitely presented metabelian group that is also residually torsion-free nilpotent.
2. A MATRIX REPRESENTATION Consider the Žstandard restricted. wreath product of two free abelian groups U and V of ranks m and n, respectively. Let ZV denote the integral group ring of V. If U is free abelian on u1 , u 2 , . . . , u m 4 , then it is easily seen w7x that the base group T of U X V is a free ZV-module on u1 , u 2 , . . . , u m 4 . That is, m
Ts
[ ZVu .
Ž 1.
i
is1
The group V is a subgroup of the multiplicative group of ZV. And U X V may be regarded as the semidirect product T i V, where the action of V on T is conjugation and is denoted by a¨ for a g T and ¨ g V. We identify the elements a and Ž1, a. for all a g T, and similarly we identify the elements ¨ and Ž ¨ , 0. for all ¨ g V. With these identifications, ¨ y1 a¨ s a¨ . Let us consider the following multiplicative group of 2 = 2 matrices: MŽT , V . s
½ž
¨
a
0 : a g T, ¨ g V . 1
/
5
M ŽT, V . is a finitely generated metabelian group. It follows from the multiplication in a semidirect product and from the fact that V is free
FINITE-RANK FREE METABELIAN SUBGROUPS
325
abelian and T is a free ZV-module that the map
Ž ¨ , a. ¬ ¨
ž
a
0 1
/
gives the isomorphism U X V ( MŽT , V . . We will embed M ŽT, V . in a group which can be shown to be a finitely presented metabelian group that is residually torsion-free nilpotent. Let V be free abelian on t 1 , t 2 , . . . , t n4 . According to a construction due to Baumslag w2x, the embedding of the metabelian group M ŽT, V . in a finitely presented metabelian group makes use of n polynomials f 1 , f 2 , . . . , f n of the form f i s 1 q c i , 1 t i q c i , 2 t i2 q ??? qc i , d iy1 t id iy1 q t id i , where c i, 1 , . . . , c i, d iy1 , d i are integers and d i is positive Ž1 F i F n.. We do not need to impose any further condition on the f i until Section 4. Let S be the multiplicative semigroup in ZV generated by
f 1 , . . . , fn 4 j 14 ,
Ž 2.
V s gp Ž t 1 , t 2 , . . . , t n , f 1 , f 2 , . . . , f n . .
Ž 3.
and let
If R is any commutative ring with unity and S is a multiplicative semigroup Žcontaining 1. in R, we designate by R S the ring of fractions RS s
½
r s
: r g R, s g S .
5
Similarly, for an R-module T, we write TS for the corresponding module of fractions. ŽSee Chapter 3 in Atiyah and Macdonald w1x for the notion of rings of fractions and modules of fractions.. In our case, we let R s ZV, where V is free abelian of rank n. And T Žthe base group of U X V . is a free R-module with finite basis. Hence, with the semigroup S given by Ž2., TS is a free R S -module. ŽSee, for example, Section 3.5 in Northcott w6x.. There is a canonical map
u : T ª TS as well as a natural embedding V ¨ V,
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where V is defined in Ž3.. These maps induce a homomorphism M Ž T , V . ª M Ž TS , V . defined by
ž
¨
a
0 ¨ ¬ 1 au
/ ž
0 , 1
/
for a g T, ¨ g V. Now, M ŽTS , V . is metabelian. It has been proved by Thomson w9x that M ŽTS , V . is finitely presented. Hence, the wreath product of two finitely generated free abelian groups is a subgroup of the finitely presented metabelian group M ŽTS , V .. It remains to verify that M ŽTS , V . is residually torsion-free nilpotent. This will be carried out by using the sufficient condition to be obtained in Lemma 3.3.
3. PRELIMINARY RESULTS We begin by considering matrices over commutative rings and obtain a sufficient condition for residual torsion-free nilpotence. From this follows easily a more general result for matrices over modules. In Lemmas 3.1 and 3.2 the following notation is used. Let A be any commutative ring with unity, denote by Aq the additive group of A, and let U be a subgroup of the multiplicative group of A. The multiplicative matrix group M Ž A, U . is given by M Ž A, U . s
½ž
u a
0 : a g A, u g U . 1
/
5
Let N be the set of positive integers. If I is an ideal in A, we denote by Iˆ the isolator of I in A, by which we mean Iˆs a g A: ' n g N 2 na g I 4 . Iˆ is an ideal in A, and $ ArIˆ is torsion-free abelian. The isolator of k I , k g N, is designated by I k. For the ideal I in A generated by 1 y u: u g U 4 , we will write I s id A Ž 1 y u: u g U . . LEMMA 3.1.
If Aq and U are torsion-free, then M Ž A, U . is torsion-free.
This follows immediately from the assumption about Aq and U.
FINITE-RANK FREE METABELIAN SUBGROUPS
327
LEMMA 3.2. Let I s id AŽ1 y u: u g U ., where U is torsion-free, and assume that the ring A has the property `
$
F I n s 0.
ns1
Then M Ž A, U . is residually torsion-free nilpotent. $
Proof. The canonical map A ª Ar I n, n g N, induces the homomorphism $
M Ž A, U . ª M Ž ArI n , U . defined by
ž
u 0 $ ¬ 1 a qIn
u a
/ ž
0 1
/
Ž 4.
for a g A, u g U. For n g N, let $
k nU M s ker M Ž A, U . ª M Ž ArI n , U . .
ž
/
$
Now, since Ar I n and U are torsion-free, the image Mrk nU M is torsionfree. It will be shown that Mrk nU M is torsion-free nilpotent. We use the result that, in general, HrK is nilpotent if and only if K contains some term of the lower central series of H. It can be verified that for n g N, $
k nU M ( I n . In fact, from Ž4.,
k nU M s
½ž
1 a
$ 0 : a gIn . 1
/
5
Just as above, it is also true that
k n M s ker Ž M Ž A, U . ª M Ž ArI n , U . . ( I n for n g N. And induction on n shows that if gn M denotes the nth term of the lower central series of M, then, for n g N,
gn M 9 k n M.
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To see this, let n s 1. For any two matrices Xs
ž
u a
0 1
/
Ys
and
ž
¨
b
0 1
/
in M Ž A, U ., the commutator is 1
w X, Y x s a ¨ y 1 q b 1 y u Ž . Ž .
ž
0 , 1
/
where a, b g A and u, ¨ g U. Now, I s id AŽ1 y u: u g U .. Hence, aŽ ¨ y 1. q bŽ1 y u. g I. That is,
w X , Y x g k 1 M s ker Ž M Ž A, U . ª M Ž ArI, U . . . And so w M, M x s g 1 M 9 k 1 M. The rest of the induction proof follows as expected. In fact, from the assumption g i M 9 k i M, we have g iq1 M 9 w I i, M x. We then compute a commutator in w I i, M x and use the fact that the ideal I i is generated by Ł irs1Ž1 y u r .: u r g U 4 . $ From the definition of the isolator of an ideal, we have I n 9 I n. Now, $ gn M 9 k n M implies gn M 9 I n 9 I n, and so
gn M 9 k nU M for n g N. The groups Mrk nU M are therefore torsion-free nilpotent. $ U n Furthermore, k n M ( I implies `
F k nU M s 0
ns1
because of the assumption in the lemma. This shows that M Ž A, U . is residually torsion-free nilpotent. We recall that we need to verify that the matrix group M ŽTS , V . is residually torsion-free nilpotent. An analog of Lemma 3.2 for modules is required. Let R be a commutative ring with unity, T a left R-module generated by t 1 , t 2 , . . . , t n4 , and U a torsion-free subgroup of the multiplicative group of R. Analogously to preceding remarks, the matrix group M ŽT, U . is then given by MŽT , U . s
½ž
u r 1 t 1 q r 2 t 2 q ??? q rn t n
0 1 : u g U, ri g R , 1 F i F n .
/
Let J s id R Ž 1 y u: u g U . .
5
FINITE-RANK FREE METABELIAN SUBGROUPS
329
The set JT consists of elements of T of the form a 1 a1 q a 2 a2 q ??? q a k a k with a i g J, a i g T Ž1 F i F k .. Clearly JT is a submodule of T. In agreement with the earlier notion for ideals, the isolator of a submodule K of T is given by Kˆ s a g T : ' n g N 2 na g K 4 . Kˆ is a submodule of T. The analog of Lemma 3.2 that we need for modules is the following lemma. LEMMA 3.3. With the notation preceding this lemma, the group M ŽT, U . is $ ` n residually torsion-free nilpotent if F ns 1 J Ts 0. The proof of Lemma 3.3 uses exactly the same argument as that of $ n Lemma 3.2. This is because the canonical map T ª Tr J T, n g N, induces the homomorphism $
M Ž T , U . ª M Ž TrJ n T, U . defined by
ž
u a
u 0 $ ¬ 1 a q J nT
/ ž
0 1
/
for a g T, u g U.
4. PROOF OF THE THEOREM We apply Lemma 3.3 to our situation, namely, the group M ŽTS , V ., where T, S, and V are given by Ž1., Ž2., and Ž3. in Section 2. Keeping in mind that V is the free abelian group of rank n and that R s ZV, let I s id R Ž 1 y ¨ : ¨ g V . . Then, if V is generated by t 1 , t 2 , . . . , t n4 , we have I s id R Ž 1 y t 1 , 1 y t 2 , . . . , 1 y t n . . This can be verified with the induction technique used by Johnson w4, Sect. 11.2x in his proof about the generating set of an augmentation ideal. Recalling that R S is the ring of fractions determined by R and S, we denote by IS the extension of the ideal I to R S . That is, IS is the ideal in R S generated by the elements of I. A typical element of IS can be expressed in the form a 1 r 1X q a 2 r 2X q ??? qa k r kX , where a i g I and riX g R S ,
330
ADA PELUSO
1 F i F k. It is clear that IS s
½a s
: a g I, s g S ,
5
and that IS has the same generators as I. According to Lemma 3.3, the group M ŽTS , V . is residually torsion-free nilpotent if `
$
F ISj TS s 0.
Ž 5.
js1
Therefore, we need to know when such an intersection Ž5. is equal to 0. The answer is given in Lemmas 4.1 and 4.3. We remark at this point that, for the rest of the presentation, for any $ k k ˆ Ž . ideal J we will write J in place of J . This is possible because in the case under discussion all ideals are generated by 1 y t 1 , . . . , 1 y t n4 . $
LEMMA 4.1. With the notation described abo¨ e, if F `js1ŽIS . j s 0, then $
F `js1 ISj TS s 0.
Proof. From the remark made just prior to Lemma 4.1 about the $ $ $ equality of J k and Ž Jˆ. k , we note that in our situation, ISj TS s ŽIS . j TS . Since R is an integral domain, the ring of fractions R S is a field and therefore a Noetherian ring. TS is a finitely generated free R S -module and therefore a Noetherian R S -module. ŽSee, for example, Chapter 6 in Atiyah and Macdonald w1x.. $ $ ` Ž .j Hence, if we put K s F I T , we have ISK s K, which means that js1 S S $ there exists a g IS such that a x s x for all x g K. These notions appear in the proof of what is frequently referred to as the Intersection Theorem. ŽSee Section 4.6 in Northcott w6x.. Then, a 2 x s a x, or Ž a 2 y a . x s 0 for all x g K. Now, K is a submodule of TS$and since TS is a free R S -module, then a 2 y a s 0, or a 2 s a . But F j ŽIS . j s 0 implies, then, that a s 0, which gives x s 0. If M is a multiplicative semigroup of an integral domain R and J is an ideal in R, we define the M-isolator of J in R by J Ž M . s r g R: ' m g M 2 mr g J 4 . If J s J Ž M ., we say that J is M-isolated in R. LEMMA 4.2.
JM l R s J if and only if J is M-isolated in R.
Proof. We prove this by showing that JM l R s J Ž M .. Now, r g JM l R implies that r s arm Žfor a g J, m g M . and r g R. This in turn implies
FINITE-RANK FREE METABELIAN SUBGROUPS
331
that mr s a , or r g J Ž M .. Hence, JM l R : J Ž M .. The reverse inclusion is immediate: x g J Ž M . implies x g R and mx g J for some m g M, i.e., x g R and x g JM . As earlier, I is the ideal in ZV generated by 1 y t 1 , . . . , 1 y t n4 . In the polynomial ring R 0 s Zw t 1 , . . . , t n x, we let I0 s id R 0Ž 1 y t 1 , 1 y t 2 , . . . , 1 y t n . . In this case, S represents the multiplicative semigroup in R 0 generated by t 1 , . . . , t n , f 1 , . . . , f n4 j 14 . LEMMA 4.3.
$
If I0j is S-isolated in R 0 for all j g N, then F `js1ŽIS . j s 0.
Proof. The result in Lemma 4.2 is applied to the polynomial ring R 0 and the associated multiplicative semigroup S. From the assumption that I0j is S-isolated in R 0 , we obtain
Ž I0j . S l R 0 s I0j . But Ž I0j .S s Ž I0 .Sj s ISj . Also, in our case, R 0rI0 is torsion-free and so I0 s Iˆ0 . We can therefore write $
Ž IS .
j
l R 0 s I0j .
Now, F `js1 I0j s 0 because of the nature of I j , that is, I0j is generated by $ 0 j ` Łms1Ž1 y t i .: 1 F im F n4 . Hence, F js1ŽIS . j s 0 also. m We now choose a set of polynomials f 1 , f 2 , . . . , f n4 that guarantees that I0j is S-isolated in R 0 for all j g N. We know from w2x that the wreath product of two free abelian groups of ranks m and n, respectively, can be embedded in a finitely presented metabelian group by introducing n polynomials from the polynomial ring R 0 s Zw t 1 , . . . , t n x with the form f i s 1 q c i , 1 t i q c i , 2 t i2 q ??? q c i , d iy1 t id iy1 q t id i , where c i, 1 , . . . , c i, d iy1 , d i are integers and d i is positive Ž1 F i F n.. We impose on the f i the additional requirement that none of them vanish when the variables t i Ž1 F i F n. have the value 1, that is, f i Ž 1, 1, . . . , 1 . / 0,
i s 1, 2, . . . , n.
We shall refer to such polynomials as choice polynomials. To verify that I0j is S-isolated for all j g N it is sufficient to verify that I0 is S-isolated. It remains, then, to prove the following lemma.
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LEMMA 4.4. With the choice polynomials f 1 , . . . , f n defined abo¨ e, I0 is S-isolated in R 0 . Proof. We recall that I0 is the ideal in R 0 s Zw t 1 , . . . , t n x generated by 1 y t 1 , . . . , 1 y t n4 . First, we show that g Ž t 1 , t 2 , . . . , t n . g I0 if and only if g Ž1, 1, . . . , 1. s 0. Let us abbreviate Ž t 1 , . . . , t n . and Ž1, 1, . . . , 1. by Ž˜t . and Ž˜1., respectively. If g Ž˜t . g I0 , then g Ž ˜t . s Ž 1 y t 1 . h1 Ž ˜t . q ??? q Ž 1 y t n . h n Ž ˜t . , where h i Ž˜t . g R 0 Ž1 F i F n.. Hence, g Ž˜1. s 0. Conversely, if g Ž˜1. s 0 and g Ž˜t . f I0 , then g Ž˜t . s a0 q g 0 Ž˜t ., where 0 / a0 g Z and g 0 Ž˜t . g I0 . This gives the contradiction: g Ž˜1. s a0 / 0. Therefore, g Ž˜t . g I0 if and only if g Ž˜1. s 0. It can now be shown that I0 is S-isolated in R 0 ; i.e., it can be verified that I0 s I0 Ž S . s g g R 0 : 's g S 2 sg g I0 4 . S is the multiplicative semigroup in R 0 generated by t 1 , . . . , t n , f 1 , . . . , f n4 j 14 . Therefore, s g S has the form t 1m1 t 2m 2 ??? t nm n f 1n 1 f 2n 2 ??? f nn n with m i , n i nonnegative integers Ž1 F i F n.. And thus sŽ˜1. / 0 from the definition of the choice polynomials f 1 , . . . , f n . Certainly, by definition, I0 Ž S . < I0 . Let g g I0 Ž S .. Then, sg s h g I0 for some s g S. From what was just proved, hŽ˜1. s 0. Thus, g Ž˜1. s 0, or g g I0 . To sum up, the wreath product of two finitely generated free abelian groups is a subgroup of a finitely presented metabelian group that depends on a set of polynomials, and the proper selection of these polynomials guarantees that the latter group is residually torsion-free nilpotent. Thus, the theorem stated in Section 1 has been verified. And since a free metabelian group of finite rank can be embedded in the wreath product of two finitely generated free abelian groups, the indicated corollary follows.
ACKNOWLEDGMENTS The author thanks Vladimir Remeslennikov for some very insightful conversations pertaining to the problem treated in this paper. The author also is grateful to the referee for some very helpful comments.
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333
REFERENCES 1. M. F. Atiyah and I. G. Macdonald, ‘‘Introduction to Commutative Algebra,’’ Addison]Wesley, Reading, MA, 1969. 2. G. Baumslag, Subgroups of finitely presented metabelian groups, J. Austral. Math. Soc. 16 Ž1973., 98]110. 3. K. W. Gruenberg and J. E. Roseblade ŽEds.., ‘‘Group Theory: Essays for Philip Hall,’’ Academic Press, London, 1984. 4. D. L. Johnson, ‘‘Presentations of Groups,’’ Cambridge Univ. Press, Cambridge, UK, 1990. 5. W. Magnus, On a theorem of Marshall Hall, Ann. of Math. 40 Ž1939., 764]768. 6. D. G. Northcott, ‘‘Lessons on Rings, Modules and Multiplicities,’’ Cambridge Univ. Press, Cambridge, UK, 1968. 7. A. Peluso, An embedding theorem for finitely generated free metabelian groups, J. Pure Appl. Algebra, 98 Ž1995., 73]81. 8. V. N. Remeslennikov, On finitely presented groups, in ‘‘Proceedings, Fourth All-Union Symposium on the Theory of Groups, Novosibirsk, 1973,’’ pp. 164]169. 9. M. W. Thomson, Subgroups of finitely presented solvable linear groups, Trans. Amer. Math. Soc. 231 Ž1977., 133]142.
180, 323]333 Ž1996.
0069
Finite-Rank Free Metabelian Subgroups of Finitely Presented Metabelian Groups* Ada Peluso† Department of Mathematics and Statistics, Hunter College, New York, New York 10021 Communicated by Walter Feit Received March 13, 1993
A sufficient condition is obtained for the residual torsion-free nilpotence of certain finitely presented metabelian groups that arise from a matrix representation developed by Magnus Ž1939, Ann. of Math. 40, 764]768. for metabelian groups. Using this condition and a construction due to Baumslag Ž1973, J. Austral. Math. Soc. 16, 98]110., we prove that a free metabelian group of finite rank can be embedded in a finitely presented metabelian group that is also residually torsionfree nilpotent. Q 1996 Academic Press, Inc.
1. INTRODUCTION It is known that every finitely generated metabelian group can be embedded in a finitely presented metabelian group. This remarkable result was obtained independently by Baumslag w2x and Remeslennikov w8x in the early 1970s. It is of interest to try to determine whether a finitely generated metabelian group with a specific property can be embedded in a finitely presented metabelian group with the same property. The property of residual nilpotence was considered by the author in w7x, and there it was proved that a finitely generated free metabelian group can be embedded in a finitely presented metabelian group that is also residually nilpotent. Here we are interested in residual torsion-free nilpotence. By definition, a group G is residually P, where P is any group property, if G possesses a family Kl4 of normal subgroups such that GrKl is P for all l, and F l Kl s 1. A finitely generated free metabelian group is residually torsion-free nilpotent. In fact, Magnus w5x proved that a finitely generated * This research was supported Žin part. by a grant from The City University of New York PSC]CUNY Research Award Program. † E-mail: [email protected]. 323 0021-8693r96 $18.00 Copyright Q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
324
ADA PELUSO
free metabelian group can be embedded in the wreath product of two finitely generated free abelian groups, and this wreath product is residually torsion-free nilpotent. ŽSee, for example, the discussion on pages 99]102 in Gruenberg and Roseblade w3x.. A matrix representation of metabelian groups due to Magnus w5x will be used here to establish a sufficient condition for the residual torsion-free nilpotence of certain finitely presented metabelian groups. By this means, we shall prove the following main result. THEOREM. The wreath product of two finitely generated free abelian groups can be embedded in a finitely presented metabelian group that is residually torsion-free nilpotent. The above discussion then yields the desired result about free metabelian groups as an immediate corollary. COROLLARY. A free metabelian group of finite rank is a subgroup of a finitely presented metabelian group that is also residually torsion-free nilpotent.
2. A MATRIX REPRESENTATION Consider the Žstandard restricted. wreath product of two free abelian groups U and V of ranks m and n, respectively. Let ZV denote the integral group ring of V. If U is free abelian on u1 , u 2 , . . . , u m 4 , then it is easily seen w7x that the base group T of U X V is a free ZV-module on u1 , u 2 , . . . , u m 4 . That is, m
Ts
[ ZVu .
Ž 1.
i
is1
The group V is a subgroup of the multiplicative group of ZV. And U X V may be regarded as the semidirect product T i V, where the action of V on T is conjugation and is denoted by a¨ for a g T and ¨ g V. We identify the elements a and Ž1, a. for all a g T, and similarly we identify the elements ¨ and Ž ¨ , 0. for all ¨ g V. With these identifications, ¨ y1 a¨ s a¨ . Let us consider the following multiplicative group of 2 = 2 matrices: MŽT , V . s
½ž
¨
a
0 : a g T, ¨ g V . 1
/
5
M ŽT, V . is a finitely generated metabelian group. It follows from the multiplication in a semidirect product and from the fact that V is free
FINITE-RANK FREE METABELIAN SUBGROUPS
325
abelian and T is a free ZV-module that the map
Ž ¨ , a. ¬ ¨
ž
a
0 1
/
gives the isomorphism U X V ( MŽT , V . . We will embed M ŽT, V . in a group which can be shown to be a finitely presented metabelian group that is residually torsion-free nilpotent. Let V be free abelian on t 1 , t 2 , . . . , t n4 . According to a construction due to Baumslag w2x, the embedding of the metabelian group M ŽT, V . in a finitely presented metabelian group makes use of n polynomials f 1 , f 2 , . . . , f n of the form f i s 1 q c i , 1 t i q c i , 2 t i2 q ??? qc i , d iy1 t id iy1 q t id i , where c i, 1 , . . . , c i, d iy1 , d i are integers and d i is positive Ž1 F i F n.. We do not need to impose any further condition on the f i until Section 4. Let S be the multiplicative semigroup in ZV generated by
f 1 , . . . , fn 4 j 14 ,
Ž 2.
V s gp Ž t 1 , t 2 , . . . , t n , f 1 , f 2 , . . . , f n . .
Ž 3.
and let
If R is any commutative ring with unity and S is a multiplicative semigroup Žcontaining 1. in R, we designate by R S the ring of fractions RS s
½
r s
: r g R, s g S .
5
Similarly, for an R-module T, we write TS for the corresponding module of fractions. ŽSee Chapter 3 in Atiyah and Macdonald w1x for the notion of rings of fractions and modules of fractions.. In our case, we let R s ZV, where V is free abelian of rank n. And T Žthe base group of U X V . is a free R-module with finite basis. Hence, with the semigroup S given by Ž2., TS is a free R S -module. ŽSee, for example, Section 3.5 in Northcott w6x.. There is a canonical map
u : T ª TS as well as a natural embedding V ¨ V,
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ADA PELUSO
where V is defined in Ž3.. These maps induce a homomorphism M Ž T , V . ª M Ž TS , V . defined by
ž
¨
a
0 ¨ ¬ 1 au
/ ž
0 , 1
/
for a g T, ¨ g V. Now, M ŽTS , V . is metabelian. It has been proved by Thomson w9x that M ŽTS , V . is finitely presented. Hence, the wreath product of two finitely generated free abelian groups is a subgroup of the finitely presented metabelian group M ŽTS , V .. It remains to verify that M ŽTS , V . is residually torsion-free nilpotent. This will be carried out by using the sufficient condition to be obtained in Lemma 3.3.
3. PRELIMINARY RESULTS We begin by considering matrices over commutative rings and obtain a sufficient condition for residual torsion-free nilpotence. From this follows easily a more general result for matrices over modules. In Lemmas 3.1 and 3.2 the following notation is used. Let A be any commutative ring with unity, denote by Aq the additive group of A, and let U be a subgroup of the multiplicative group of A. The multiplicative matrix group M Ž A, U . is given by M Ž A, U . s
½ž
u a
0 : a g A, u g U . 1
/
5
Let N be the set of positive integers. If I is an ideal in A, we denote by Iˆ the isolator of I in A, by which we mean Iˆs a g A: ' n g N 2 na g I 4 . Iˆ is an ideal in A, and $ ArIˆ is torsion-free abelian. The isolator of k I , k g N, is designated by I k. For the ideal I in A generated by 1 y u: u g U 4 , we will write I s id A Ž 1 y u: u g U . . LEMMA 3.1.
If Aq and U are torsion-free, then M Ž A, U . is torsion-free.
This follows immediately from the assumption about Aq and U.
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LEMMA 3.2. Let I s id AŽ1 y u: u g U ., where U is torsion-free, and assume that the ring A has the property `
$
F I n s 0.
ns1
Then M Ž A, U . is residually torsion-free nilpotent. $
Proof. The canonical map A ª Ar I n, n g N, induces the homomorphism $
M Ž A, U . ª M Ž ArI n , U . defined by
ž
u 0 $ ¬ 1 a qIn
u a
/ ž
0 1
/
Ž 4.
for a g A, u g U. For n g N, let $
k nU M s ker M Ž A, U . ª M Ž ArI n , U . .
ž
/
$
Now, since Ar I n and U are torsion-free, the image Mrk nU M is torsionfree. It will be shown that Mrk nU M is torsion-free nilpotent. We use the result that, in general, HrK is nilpotent if and only if K contains some term of the lower central series of H. It can be verified that for n g N, $
k nU M ( I n . In fact, from Ž4.,
k nU M s
½ž
1 a
$ 0 : a gIn . 1
/
5
Just as above, it is also true that
k n M s ker Ž M Ž A, U . ª M Ž ArI n , U . . ( I n for n g N. And induction on n shows that if gn M denotes the nth term of the lower central series of M, then, for n g N,
gn M 9 k n M.
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To see this, let n s 1. For any two matrices Xs
ž
u a
0 1
/
Ys
and
ž
¨
b
0 1
/
in M Ž A, U ., the commutator is 1
w X, Y x s a ¨ y 1 q b 1 y u Ž . Ž .
ž
0 , 1
/
where a, b g A and u, ¨ g U. Now, I s id AŽ1 y u: u g U .. Hence, aŽ ¨ y 1. q bŽ1 y u. g I. That is,
w X , Y x g k 1 M s ker Ž M Ž A, U . ª M Ž ArI, U . . . And so w M, M x s g 1 M 9 k 1 M. The rest of the induction proof follows as expected. In fact, from the assumption g i M 9 k i M, we have g iq1 M 9 w I i, M x. We then compute a commutator in w I i, M x and use the fact that the ideal I i is generated by Ł irs1Ž1 y u r .: u r g U 4 . $ From the definition of the isolator of an ideal, we have I n 9 I n. Now, $ gn M 9 k n M implies gn M 9 I n 9 I n, and so
gn M 9 k nU M for n g N. The groups Mrk nU M are therefore torsion-free nilpotent. $ U n Furthermore, k n M ( I implies `
F k nU M s 0
ns1
because of the assumption in the lemma. This shows that M Ž A, U . is residually torsion-free nilpotent. We recall that we need to verify that the matrix group M ŽTS , V . is residually torsion-free nilpotent. An analog of Lemma 3.2 for modules is required. Let R be a commutative ring with unity, T a left R-module generated by t 1 , t 2 , . . . , t n4 , and U a torsion-free subgroup of the multiplicative group of R. Analogously to preceding remarks, the matrix group M ŽT, U . is then given by MŽT , U . s
½ž
u r 1 t 1 q r 2 t 2 q ??? q rn t n
0 1 : u g U, ri g R , 1 F i F n .
/
Let J s id R Ž 1 y u: u g U . .
5
FINITE-RANK FREE METABELIAN SUBGROUPS
329
The set JT consists of elements of T of the form a 1 a1 q a 2 a2 q ??? q a k a k with a i g J, a i g T Ž1 F i F k .. Clearly JT is a submodule of T. In agreement with the earlier notion for ideals, the isolator of a submodule K of T is given by Kˆ s a g T : ' n g N 2 na g K 4 . Kˆ is a submodule of T. The analog of Lemma 3.2 that we need for modules is the following lemma. LEMMA 3.3. With the notation preceding this lemma, the group M ŽT, U . is $ ` n residually torsion-free nilpotent if F ns 1 J Ts 0. The proof of Lemma 3.3 uses exactly the same argument as that of $ n Lemma 3.2. This is because the canonical map T ª Tr J T, n g N, induces the homomorphism $
M Ž T , U . ª M Ž TrJ n T, U . defined by
ž
u a
u 0 $ ¬ 1 a q J nT
/ ž
0 1
/
for a g T, u g U.
4. PROOF OF THE THEOREM We apply Lemma 3.3 to our situation, namely, the group M ŽTS , V ., where T, S, and V are given by Ž1., Ž2., and Ž3. in Section 2. Keeping in mind that V is the free abelian group of rank n and that R s ZV, let I s id R Ž 1 y ¨ : ¨ g V . . Then, if V is generated by t 1 , t 2 , . . . , t n4 , we have I s id R Ž 1 y t 1 , 1 y t 2 , . . . , 1 y t n . . This can be verified with the induction technique used by Johnson w4, Sect. 11.2x in his proof about the generating set of an augmentation ideal. Recalling that R S is the ring of fractions determined by R and S, we denote by IS the extension of the ideal I to R S . That is, IS is the ideal in R S generated by the elements of I. A typical element of IS can be expressed in the form a 1 r 1X q a 2 r 2X q ??? qa k r kX , where a i g I and riX g R S ,
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1 F i F k. It is clear that IS s
½a s
: a g I, s g S ,
5
and that IS has the same generators as I. According to Lemma 3.3, the group M ŽTS , V . is residually torsion-free nilpotent if `
$
F ISj TS s 0.
Ž 5.
js1
Therefore, we need to know when such an intersection Ž5. is equal to 0. The answer is given in Lemmas 4.1 and 4.3. We remark at this point that, for the rest of the presentation, for any $ k k ˆ Ž . ideal J we will write J in place of J . This is possible because in the case under discussion all ideals are generated by 1 y t 1 , . . . , 1 y t n4 . $
LEMMA 4.1. With the notation described abo¨ e, if F `js1ŽIS . j s 0, then $
F `js1 ISj TS s 0.
Proof. From the remark made just prior to Lemma 4.1 about the $ $ $ equality of J k and Ž Jˆ. k , we note that in our situation, ISj TS s ŽIS . j TS . Since R is an integral domain, the ring of fractions R S is a field and therefore a Noetherian ring. TS is a finitely generated free R S -module and therefore a Noetherian R S -module. ŽSee, for example, Chapter 6 in Atiyah and Macdonald w1x.. $ $ ` Ž .j Hence, if we put K s F I T , we have ISK s K, which means that js1 S S $ there exists a g IS such that a x s x for all x g K. These notions appear in the proof of what is frequently referred to as the Intersection Theorem. ŽSee Section 4.6 in Northcott w6x.. Then, a 2 x s a x, or Ž a 2 y a . x s 0 for all x g K. Now, K is a submodule of TS$and since TS is a free R S -module, then a 2 y a s 0, or a 2 s a . But F j ŽIS . j s 0 implies, then, that a s 0, which gives x s 0. If M is a multiplicative semigroup of an integral domain R and J is an ideal in R, we define the M-isolator of J in R by J Ž M . s r g R: ' m g M 2 mr g J 4 . If J s J Ž M ., we say that J is M-isolated in R. LEMMA 4.2.
JM l R s J if and only if J is M-isolated in R.
Proof. We prove this by showing that JM l R s J Ž M .. Now, r g JM l R implies that r s arm Žfor a g J, m g M . and r g R. This in turn implies
FINITE-RANK FREE METABELIAN SUBGROUPS
331
that mr s a , or r g J Ž M .. Hence, JM l R : J Ž M .. The reverse inclusion is immediate: x g J Ž M . implies x g R and mx g J for some m g M, i.e., x g R and x g JM . As earlier, I is the ideal in ZV generated by 1 y t 1 , . . . , 1 y t n4 . In the polynomial ring R 0 s Zw t 1 , . . . , t n x, we let I0 s id R 0Ž 1 y t 1 , 1 y t 2 , . . . , 1 y t n . . In this case, S represents the multiplicative semigroup in R 0 generated by t 1 , . . . , t n , f 1 , . . . , f n4 j 14 . LEMMA 4.3.
$
If I0j is S-isolated in R 0 for all j g N, then F `js1ŽIS . j s 0.
Proof. The result in Lemma 4.2 is applied to the polynomial ring R 0 and the associated multiplicative semigroup S. From the assumption that I0j is S-isolated in R 0 , we obtain
Ž I0j . S l R 0 s I0j . But Ž I0j .S s Ž I0 .Sj s ISj . Also, in our case, R 0rI0 is torsion-free and so I0 s Iˆ0 . We can therefore write $
Ž IS .
j
l R 0 s I0j .
Now, F `js1 I0j s 0 because of the nature of I j , that is, I0j is generated by $ 0 j ` Łms1Ž1 y t i .: 1 F im F n4 . Hence, F js1ŽIS . j s 0 also. m We now choose a set of polynomials f 1 , f 2 , . . . , f n4 that guarantees that I0j is S-isolated in R 0 for all j g N. We know from w2x that the wreath product of two free abelian groups of ranks m and n, respectively, can be embedded in a finitely presented metabelian group by introducing n polynomials from the polynomial ring R 0 s Zw t 1 , . . . , t n x with the form f i s 1 q c i , 1 t i q c i , 2 t i2 q ??? q c i , d iy1 t id iy1 q t id i , where c i, 1 , . . . , c i, d iy1 , d i are integers and d i is positive Ž1 F i F n.. We impose on the f i the additional requirement that none of them vanish when the variables t i Ž1 F i F n. have the value 1, that is, f i Ž 1, 1, . . . , 1 . / 0,
i s 1, 2, . . . , n.
We shall refer to such polynomials as choice polynomials. To verify that I0j is S-isolated for all j g N it is sufficient to verify that I0 is S-isolated. It remains, then, to prove the following lemma.
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LEMMA 4.4. With the choice polynomials f 1 , . . . , f n defined abo¨ e, I0 is S-isolated in R 0 . Proof. We recall that I0 is the ideal in R 0 s Zw t 1 , . . . , t n x generated by 1 y t 1 , . . . , 1 y t n4 . First, we show that g Ž t 1 , t 2 , . . . , t n . g I0 if and only if g Ž1, 1, . . . , 1. s 0. Let us abbreviate Ž t 1 , . . . , t n . and Ž1, 1, . . . , 1. by Ž˜t . and Ž˜1., respectively. If g Ž˜t . g I0 , then g Ž ˜t . s Ž 1 y t 1 . h1 Ž ˜t . q ??? q Ž 1 y t n . h n Ž ˜t . , where h i Ž˜t . g R 0 Ž1 F i F n.. Hence, g Ž˜1. s 0. Conversely, if g Ž˜1. s 0 and g Ž˜t . f I0 , then g Ž˜t . s a0 q g 0 Ž˜t ., where 0 / a0 g Z and g 0 Ž˜t . g I0 . This gives the contradiction: g Ž˜1. s a0 / 0. Therefore, g Ž˜t . g I0 if and only if g Ž˜1. s 0. It can now be shown that I0 is S-isolated in R 0 ; i.e., it can be verified that I0 s I0 Ž S . s g g R 0 : 's g S 2 sg g I0 4 . S is the multiplicative semigroup in R 0 generated by t 1 , . . . , t n , f 1 , . . . , f n4 j 14 . Therefore, s g S has the form t 1m1 t 2m 2 ??? t nm n f 1n 1 f 2n 2 ??? f nn n with m i , n i nonnegative integers Ž1 F i F n.. And thus sŽ˜1. / 0 from the definition of the choice polynomials f 1 , . . . , f n . Certainly, by definition, I0 Ž S . < I0 . Let g g I0 Ž S .. Then, sg s h g I0 for some s g S. From what was just proved, hŽ˜1. s 0. Thus, g Ž˜1. s 0, or g g I0 . To sum up, the wreath product of two finitely generated free abelian groups is a subgroup of a finitely presented metabelian group that depends on a set of polynomials, and the proper selection of these polynomials guarantees that the latter group is residually torsion-free nilpotent. Thus, the theorem stated in Section 1 has been verified. And since a free metabelian group of finite rank can be embedded in the wreath product of two finitely generated free abelian groups, the indicated corollary follows.
ACKNOWLEDGMENTS The author thanks Vladimir Remeslennikov for some very insightful conversations pertaining to the problem treated in this paper. The author also is grateful to the referee for some very helpful comments.
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REFERENCES 1. M. F. Atiyah and I. G. Macdonald, ‘‘Introduction to Commutative Algebra,’’ Addison]Wesley, Reading, MA, 1969. 2. G. Baumslag, Subgroups of finitely presented metabelian groups, J. Austral. Math. Soc. 16 Ž1973., 98]110. 3. K. W. Gruenberg and J. E. Roseblade ŽEds.., ‘‘Group Theory: Essays for Philip Hall,’’ Academic Press, London, 1984. 4. D. L. Johnson, ‘‘Presentations of Groups,’’ Cambridge Univ. Press, Cambridge, UK, 1990. 5. W. Magnus, On a theorem of Marshall Hall, Ann. of Math. 40 Ž1939., 764]768. 6. D. G. Northcott, ‘‘Lessons on Rings, Modules and Multiplicities,’’ Cambridge Univ. Press, Cambridge, UK, 1968. 7. A. Peluso, An embedding theorem for finitely generated free metabelian groups, J. Pure Appl. Algebra, 98 Ž1995., 73]81. 8. V. N. Remeslennikov, On finitely presented groups, in ‘‘Proceedings, Fourth All-Union Symposium on the Theory of Groups, Novosibirsk, 1973,’’ pp. 164]169. 9. M. W. Thomson, Subgroups of finitely presented solvable linear groups, Trans. Amer. Math. Soc. 231 Ž1977., 133]142.