On the logical strength of the automorphism groups of free nilpotent groups
aa r X i v : . [ m a t h . L O ] D ec ON THE LOGICAL STRENGTH OF THEAUTOMORPHISM GROUPS OF FREE NILPOTENTGROUPS
VLADIMIR TOLSTYKH
Abstract.
Considering a particular case of a problem posed by S. She-lah, we prove that the automorphism group of an infinitely generatedfree nilpotent group of cardinality λ first-order interprets the full second-order theory of the set λ in empty language. Introduction
In his paper [8] of 1976, S. Shelah suggested a general program of the studyof the logical strength of first-order theories of the automorphism groups offree algebras. Recently he has again attracted attention to that program inhis survey [9]. Namely, the problem 3.14 from [9] asks for which varieties V of algebras, letting F λ for a free algebra in V with λ > ℵ free generators, canwe syntactically interpret in the first-order theory of Aut( F λ ) the full second-order theory of the set λ in empty language (possibly for sufficiently largecardinals λ ). Recall that a theory T in a logic L is said to be syntacticallyinterpretable in a theory T in a logic L if there is a mapping χ → χ ∗ fromthe set of all L -sentences to the set of all L -sentences such that χ ∈ T ⇐⇒ χ ∗ ∈ T . It should be pointed out that one the main results of [8] states that for any variety of algebras V the first-order theory of the endomorphism semi-group End( F λ ) syntactically interprets Th ( λ ) provided that a cardinal λ is greater than or equal to the power of the language of V . The situationwith the automorphism groups seems to be more difficult and the reasonis obvious: despite being complicated in many cases, the endomorphismsemi-groups of free algebras, as Shelah’s analysis in [8] demonstrates, canbe viewed as combinatorial objects.There exist only a few examples of varieties for which the Shelah’s problemis completely investigated: some varieties have the desired property (forinstance, the variety of all vector spaces over an arbitrary division ring andthe variety of all groups [10, 12]), some do not (the variety of all sets with nostructure [6, 7], the automorphism groups of free algebras are here symmetric
Mathematics Subject Classification.
Primary: 03C60; Secondary: 20F19, 20F28.
Key words and phrases.
Automorphism groups, free groups, nilpotent groups, inter-pretations, first-order theories, high-order theories. groups ). To the best of the author’s knowledge, there are no general resultson the subject (however, Shelah introduces in [9, §
3] a wide class of so-called Aut-decomposable varieties, which are in many ways analogous tothe variety of all sets).The purpose of the present paper is to prove that the automorphismgroups Aut( F λ ) of free groups F λ in all varieties of nilpotent groups N s with s > λ for all infinite λ. We also consider a numberof related questions; it is proved, in particular, that the first-order theory ofthe automorphism group of a finitely generated free nilpotent group of class > Reducing nilpotency class
Suppose that N is a free nilpotent group of class s > K m ( N ) , where m is a naturalnumber, denote the kernel of the homomorphism fromthe group Aut( N ) to the group Aut( N/N m +1 ) induced by the natural ho-momorphism N → N/N m +1 , from N to the free nilpotent group N/N m +1 ofnilpotency class m. In particular, K ( N ) is equal to IA( N ) , to the subgroupof so-called IA- automorphisms of N, and K s ( N ) = { id } . Lemma 2.1.
Suppose that γ is an IA -automorphism. Then γ commuteswith every element of the subgroup K m ( N ) modulo the subgroup K m +1 ( N ) . Proof.
According to [1], the groups K m ( N ) form the lower central series ofthe group K ( N ) = IA( N ); every element of an arbitrary group G commuteswith the elements of the k th term of the lower central series of G modulothe ( k + 1)th term [4, Section 5.3]. (cid:3) Like in our previous papers [11, 13], any automorphism θ of N, whichinverts all elements of some basis of N will be called a symmetry . Lemma 2.2.
Let θ be a symmetry. (a) Suppose that c is an element of N m . Then θ either fixes c modulo N m +1 ( when m is even ) , or inverts c modulo N m +1 ( when m is odd ) ; (b) Suppose that γ is an element of K m ( N ) . Then the conjugate of γ by θ either equals to γ modulo K m +1 ( N ) ( when m is even ) or to the inverse of γ modulo K m +1 ( N ) ( when m is odd ) .Proof. (a) Assume that B is a basis of N such that θ sends each element of B to its inverse. Since the group N m /N m +1 is abelian it suffices to provethat θ acts in a prescribed way on generators [ x i , x i , . . . , x i m ] N m +1 , where UTOMORPHISM GROUPS OF FREE NILPOTENT GROUPS 115 x i , . . . , x i m are elements of B [4, Section 5.3]. We have θ [ x i , [ x i , . . . , x i m ]] ≡ [ x i − , [ x i , . . . , x i m ] ( − m − ] ≡ [ x i , [ x i , . . . , x i m ]] ( − m (mod N m +1 ) . (b) By (a). (cid:3) Suppose that ϕ is an involution from Aut( N ) and ϕ , ϕ , . . . , ϕ m , . . . arearbitrary conjugates of ϕ. For every σ in Aut( N ) let us construct the se-quence { σ m ( ϕ , ϕ , . . . , ϕ m ) : m ∈ N } of automorphisms of N as follows: σ = σ,σ = ϕ σ ϕ σ − ,σ = ϕ σ ϕ σ ,σ = ϕ σ ϕ σ − ,. . . More formally, for every m > σ m +1 = ( ϕ m +1 σ m ϕ m +1 σ m − , if m is even ,ϕ m +1 σ m ϕ m +1 σ m , if m is odd . The following result generalizes the corresponding fact from [13] provedthere for free nilpotent groups of nilpotency class 2 . Proposition 2.3.
Let N be a free nilpotent group of nilpotency class s. Thenan involution θ ∈ Aut( N ) is a symmetry modulo IA( N ) ( that is, coincideswith some symmetry modulo the group IA( N )) if and only if for every σ from Aut( N ) and every tuple θ , θ , . . . , θ s of conjugates of θ the automorphism σ s ( θ , θ , . . . , θ s ) of N is trivial.Proof. Suppose that θ = θ ∗ γ, where θ ∗ is a symmetry and γ is an IA-automorphism. Since θ is an involution, then θ ∗ γ = γ − θ ∗ . Any member θ k of the tuple θ , θ , . . . , θ s , a symmetry modulo IA( N ) , also has the form θ ∗ γ k for a suitable IA-automorphism γ k . Let us prove by induction on m that the automorphism σ m = σ m ( θ , . . . , θ m ) is an element of K m ( N ) . This will follow the necessity part of the Proposi-tion.Indeed, if m = 1 , then σ m is an IA-automorphism, that is a member of K ( N ) . Assume that σ m ∈ K m ( N ) and let m be, for instance, even. Wehave by Lemma 2.2(b) and Lemma 2.1: σ m +1 = θ m +1 σ m θ m +1 σ m − = γ m +1 − θ ∗ σ m θ ∗ γ m +1 σ m − ≡ γ m +1 − σ m γ m +1 σ m − ≡ id(mod K m +1 ( N )) .
16 VLADIMIR TOLSTYKH
Let us prove the converse. It is well-known that every automorphism ofthe abelianization N of N, the free abelian group N/ [ N, N ] , can be lifted upto an automorphism of N (see, for instance, [5, §
4] or [3, Section 3.1, Section4.2]). Then it suffices to prove that for every involution of Aut( N ) , whichis not − id , there exist an infinite sequence of the form (2.1), constructedinside Aut( N ) , which contains no trivial members.It can be seen quite easily that every involution f ∈ Aut( N ), which isnot − id , has two f -invariant direct summands B, C of N with N = B ⊕ C and rank B = 2; moreover, the action of f on B can chosen so that f | B isneither id B , nor − id B ([13, Theorem 1.4], [2, Lemma 1]). This reduces theproblem to the automorphism groups of two-generator free abelian groups;for the sake of simplicity we shall work with the group GL(2 , Z ) . According to the just mentioned result from [2], every involution in GL(2 , Z )is conjugate either to the involution (cid:18) − (cid:19) , or to the involution (cid:18) (cid:19) ;hence the group GL(2 , Z ) has exactly two conjugacy classes of non-centralinvolutions. One readily checks that for every integer m (2.2) (cid:18) m − (cid:19) ∼ (cid:18) − (cid:19) and (cid:18) m − − (cid:19) ∼ (cid:18) (cid:19) , where ∼ denotes the conjugacy relation.Let S be a non-central matrix from GL(2 , Z ) and m an integer number.Suppose that X ( m ) = (cid:18) m − (cid:19) S (cid:18) m − (cid:19) S,Y ( m ) = (cid:18) m − (cid:19) S (cid:18) m − (cid:19) S − . There are no difficulties in the verification of the following fact: some elementof the ‘general’ matrix X (and Y ) depends linearly on m. It follows that fora suitable integer m the matrix X ( m ) ( Y ( m )) is again non-central. Thismeans that, starting with a non-central matrix, we can construct an infinitesequence of the form (2.1) having no central matrices; in particular, therewill be no trivial matrices in this sequence. Exactly the same argument,using matrices of the form (cid:18) m − − (cid:19) , proves the similar result for the second conjugacy class of non-central invo-lutions in GL(2 , Z ) . (cid:3) UTOMORPHISM GROUPS OF FREE NILPOTENT GROUPS 117
Corollary 2.4.
Symmetries modulo
IA( N ) form a definable family in thegroup Aut( N ) . Let T − ( N ) denote the set of all automorphisms { σ } of N such that forevery θ, which is a symmetry modulo IA( N ) , the conjugate of σ by θ isequal to σ − . Similarly T + ( N ) denotes the set of all automorphisms of N, which commute with every symmetry modulo IA( N ) . Proposition 2.5.
Let N be a free nilpotent group of nilpotency class s. Then K s − ( N ) = T + ( N ) ∪ T − ( N ) . Therefore K s − ( N ) , the kernel of a surjectivehomomorphism from Aut( N ) to the automorphism group of a free nilpotentgroup of nilpotency class s − and of the same rank as one of N, is adefinable subgroup of Aut( N ) . Proof.
An arbitrary element σ from T + ( N ) ∪ T − ( N ) must commute withany product of two symmetries: if, for instance, σ ∈ T − ( N ) , θ and θ aretwo symmetries then θ θ σ ( θ θ ) − = θ θ σθ θ = θ σ − θ = σ. On the other hand, one finds among the automorphisms of N, which canbe expressed as a product of two symmetries, conjugations (inner automor-phisms of N ) by primitive elements (that is, members of bases of N ). Thisimplies that σ commutes with every element of Inn( N ) . Hence σ preserveseach element of N modulo the center of N. The center of N is equal to thesubgroup N s [3, Section 3.1], and therefore σ ∈ K s − ( N ) . Let τ be a conjugation by a primitive element x of N. We are going torepresent τ as a product of two symmetries. The element x is a member ofsome basis B of N. Suppose θ is a symmetry, which inverts each element of B . Then if a symmetry θ is defined as follows θ x = x − ,θ y = x − y − x, ∀ y ∈ B \ { x } , the product of θ θ of θ and θ is equal to τ. Conversely, according to Lemma 2.2 (b) every element of K s − ( N ) eitherlies in T + ( N ) , or in T − ( N ) . (cid:3) Interpretations
Theorem 3.1.
Let N be a free nilpotent group of class > . Then theautomorphism group of N first-order interprets the automorphism group ofa free nilpotent group of class and of rank which is the same as one of N ( uniformly in N ). Proof.
By Proposition 2.5. (cid:3)
Until otherwise stated, we shall assume that N is a free nilpotent groupof class A denotes the abelianization N of N. It can be shown that Inn( N ) , the subgroup of all conjugations, is a ∅ -definable subgroup of Aut( N ) [13, Corollary 3.2]. The group Inn( N ) is
18 VLADIMIR TOLSTYKH isomorphic to the free abelian group A. Thus, we can interpret in Aut( N )the free abelian group A and the automorphism group of A with the actionon the elements of A. We can also interpret in Aut( N ) the family D of all direct summands of A with inclusion relation and a binary relation, say R such that R ( B, C ) ←→ A = B ⊕ C. One can prove that an involution f from Aut( A ) is diagonalizable in somebasis of A if only if there are no elements of order three in the set K ( f ) K ( f ) , where K ( f ) denotes the conjugacy class of f (see proof of Proposition 2.4in [13]). Hence the fixed-point subgroups of diagonalizable involutions canbe used to interpret the direct summands. Having the group A interpretedin Aut( N ) , we can easily interpret the inclusion relation and the relation R on the family D . Summing up, we see that the group
Aut( N ) first-order interprets themulti-sorted structure M with the following description: • the sorts of M are the free abelian group A, its automorphism groupAut( A ) and the family D of all direct summands of A ; • all sorts carry their natural relations; the relations of D are theinclusion relation and the relation R ; • M has as one of the basic relations the membership relation on A ∪D ; • there are relations defining the action of Aut( N ) on other sorts. Lemma 3.2.
Let A be of infinite rank. Then the first-order theory of thestructure M syntactically interprets the full second-order theory of the set | A | ( in empty language ) , uniformly in A. Proof.
It follows from the results in Section 4 of [8], that the first-ordertheory of the endomorphism semi-group End( A ) of A syntactically interpretsTh ( | A | ) (for the sake of convenience the reader may refer to [10], where thevery similar case of varieties of vector spaces is considered in some details inthe proof of Proposition 10.1; an analysis of the proof shows that it worksalso for free Z -modules, or, in other words, for free abelian groups).To complete the proof, we could therefore interpret in M the endomor-phism semi-group of A. There is a (folklore) trick by which the endomor-phisms can be interpreted in structures similar to M constructed over mod-ules. This trick can be briefly characterized as follows: three submodules,such that any two of them are direct complements of each other, are usedto interpret the endomorphism semi-group of one of them. A detailed de-scription of the trick for infinite-dimensional vector spaces can be found in[10] (see the proof of Proposition 9.3); the reader is again referred to [10] tosee that everything works for free Z -modules as well. (cid:3) The following result solves the problem posed by S. Shelah (see the In-troduction) for all varieties of nilpotent groups N s , where s > . UTOMORPHISM GROUPS OF FREE NILPOTENT GROUPS 119
Theorem 3.3.
The first-order theory of the automorphism group of any in-finitely generated free nilpotent group N of class > syntactically interpretsthe full second-order theory of the set | N | . The first-order theory of
Aut( N ) is therefore unstable and undecidable.Proof. By Theorem 3.1 and Lemma 3.2. (cid:3)
We have also solved the problem of classification of elementary types ofthe automorphism groups of infinitely generated free groups from varieties N s : Theorem 3.4.
Let N and N be infinitely generated free nilpotent groups ofthe same class > . Then the automorphism groups
Aut( N ) and Aut( N ) are elementarily equivalent if and only if the sets | N | and | N | ( with nostructure ) are equivalent in the full second-order logic.Proof. By Theorem 3.1 and Lemma 3.2. (cid:3)
Let N again denote a free nilpotent group of class 2 (recall that A standsfor the abelianization of N and M is the multi-sorted structure constructedover A ).We are going to estimate the logical strength/complexity of the first-ordertheory of Aut( N ) in the case, when N is finitely generated. Lemma 3.5.
Let A be of rank at least . Then the structure M first-orderinterprets ( with parameters ) the ring of integers Z . Proof.
Let us consider two direct summands
B, C of A such that A = B ⊕ C and rank B = 2 . Write G for the group of all automorphisms of A which preserve B andpoint-wise fix C. Clearly, the structure h G, B i with natural relations (thatis, with all relations on sorts along with the action of G on B ) is isomorphicto the two-sorted structure h GL(2 , Z ) , Z i (taken in the same language asone of h G, B i ). It is a well-known and simple result that the latter two-sortedstructure first-order interprets the ring of integers Z . (cid:3) As an immediate corollary we have the following fact.
Theorem 3.6.
Suppose that N is a finitely generated free nilpotent group ofclass > . Then the first-order theory of the group
Aut( N ) is unstable andundecidable. References [1] S. Andreadakis,
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Department of Mathematics, Istanbul Bilgi University, Kus¸tepe 80310 S¸is¸li-Istanbul, Turkey
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