Aspects of free groups

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Journal of Algebra 263 (2003) 119–125

www.elsevier.com/locate/jalgebr

Aspects of free groups

André Nies1

Department of Mathematics, The University of Chicago, 5734 S. University Av., Chicago, IL 60637, U

Received 23 January 2002

Communicated by Robert Kottwitz

Abstract

If n-tuples g, h in a rank 2 free group satisfy the same existential formulas, then thereautomorphism takingg to h. Fix a primep. ThenFk is, up to isomorphism, the onlyk-generatedgroup having all finitek-generatedp-groups as homomorphic images. The theory of non-Abefree groups has no prime model. 2003 Published by Elsevier Science (USA).

1. Introduction

This works contributes to the understanding of the model theory and the structfree groups. In the first two sections, we study the extent to which the first-order propof ann-tuple g in a free groupF determine its orbit. We show that inF2, if n-tuplesg, hsatisfy the same existential formulas, then there is an automorphism ofF2 takingg to h. Inparticular,F2 is ω-homogeneous (as defined in model theory). We present partial realong these lines for other free groups.

In the third section, we give a characterization ofFk . Fix a primep. ThenFk is the onlyk-generated group having all finitek-generatedp-groups as homomorphic images.

In [1, Question (F14)], it was asked whether any finitely generated (f.g.) residuallygroupG with the same finite homomorphic images asFk is isomorphic toFk . By our result,this is true ifG has rank at mostk.

E-mail address:[email protected] address:http://www.math.uchicago.edu/~nies.

1 Partially supported by NSF grant DMS-9803482

0021-8693/03/$ – see front matter 2003 Published by Elsevier Science (USA).doi:10.1016/S0021-8693(02)00665-8

120 A. Nies / Journal of Algebra 263 (2003) 119–125

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Understanding f.g. groups through their finite images has been a recurrent theminstance, both Wehrfritz and Robinson proved that a f.g. solvable group is nilpotentits finite images are nilpotent (see [11, Theorem 15.5.3]).

We thank the referee for pointing out that the weaker theorem thatFk is the onlyk-generated group having all finitek-generated nilpotent groups as homomorphic imacan be derived by combining results in the literature: Pickel [10] proved a result impthat any two f.g. class-c nilpotent groups with the same finite quotients are isomorpMagnus [7] showed that ak-generated groupG with the same quotients by the membeof the lower central sequence asFk is free. One applies Pickel’s result to those quotien

We recall an important notion from model theory. A structureA is aprime modelof atheoryT if it is the least model ofT , in the sense thatA is elementarily embedded intany other model ofT . For instance,(N,+,×) is a prime model of Th(N,+,×). In thefinal section we observe how the recent result of Kharlampovich and Myasnikov [5all non-Abelian free groups have the same first-order theory implies that this theory hprime model. This contrasts with a result of the author [9, Corollary 5.3]: the free clnilpotent group of rank 2U is a prime model of its theory. (In [9], we investigate f.g. grouwhich can be axiomatized by a single first-order sentence within the class of f.g. gand showU has this property.)

2. Test tuples are ∃-determined

We begin with some definitions. We use the first-order language with the sym{◦, −1,1} for the group operation, inverse and the neutral element. An∃-formula is aformula ∃z1 . . .∃zr ψ , whereψ is a quantifier free formula. Ifψ contains no negatiosign, the formula ispositive. LetG be a countable group. Ifg, h ∈ Gn, we writeg ≡∃ h ifg, h satisfy the same∃-formulas. Ann-orbit of G is an orbit under the action of Aut(G)

onGn.Giveng, if h is in the samen-orbit as(g), thenh satisfies the same first-order formu

asg. If the converse holds,g is calledfirst-order determined. The following is a strongeproperty:g is ∃-determinedif eachh such thath ≡∃ g is in the samen-orbit asg.

ConsiderG = Fk . Next we define setsT (g) and T ∗(g) (for k = 2) which capturehow then-tuple g can be expressed by terms of the group language applied to eleu1, . . . , uk.

Definition 2.1. Let G= F(x1, . . . , xk) be the free rankk group and letg ∈Gn.

(i) T (g)={t1, . . . , tn ∈G: ∃u1, . . . , uk ∈ G

∧1�i�n

gi = ti (u1, . . . , uk)

}.

(ii) If k = 2, let

T ∗(g)={t1, . . . , tn ∈ G: ∃u1, u2 ∈ G

([u1, u2] �= 1 and

∧1�i�n

gi = ti (u1, u2)

)}.

A. Nies / Journal of Algebra 263 (2003) 119–125 121

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Clearly, for fixed termst1, . . . , tn, “ t1, . . . , tn ∈ T (g)” can be expressed by a positi∃-formula aboutg, and “t1, t2 ∈ T ∗(g)” can be expressed by an∃-formula.

Lemma 2.2.

(i) T (g)⊆ T (h) iff there is an endomorphismα of G such thatα(gi)= hi (1 � i � n).(ii) In casek = 2, T ∗(g) ⊆ T ∗(h) iff there is an monomorphismα of G such that

α(gi)= hi (1 � i � n).

Proof. The direction from right to left is trivial for both (i) and (ii). To prove thother direction, choose a baseb1, . . . , bk for Fk . For (i), chooset1, . . . , tn such thatgi = ti(b1, . . . , bk) for eachi. SinceT (g) ⊆ T (h), we may picku1, . . . , uk ∈ G such thathi = ti (u1, . . . , uk). If α is the endomorphism ofG given byα(bi) = ui , thenα(gi) = hi(1� i � n).

For (ii), sinceT ∗(g)⊆ T ∗(h), we can chooseu1, u2 ∈ G as above such that, in additio[u1, u2] �= 1, so thatα is a monomorphism. ✷

The following was studied by Turner [12].

Definition 2.3. A tuple g = g1, . . . , gn of elements of a groupG is called a test tuple if eacendomorphismφ :G → G fixing all elementsgi is an automorphism. The tuple is calla test tuple for monomorphisms if each monomorphismφ :G → G fixing all elementsgi ,1 � i � n, is an automorphism.

For instance, ifg generatesG, or, more generally, a subgroup of finite index ofG, theng is a test tuple.

Proposition 2.4. Supposeg is a test tuple inFk . Theng is ∃-determined.

Proof. If g ≡∃ h, thenT (g) = T (h). By Lemma 2.2, we can choose endomorphismsα,β

of Fk such thatα(gi) = hi andβ(hi) = gi (1 � i � n). Clearlyβ ◦ α fixes g. Thereforeαis an automorphism, showing thatg, h are in the samen-orbit. ✷Corollary 2.5. If g generates a subgroup of finite index ofFk , theng is ∃-determined.

Corollary 2.6. If α :Fk �→ Fk preserves∃-formulas in both directions, thenα is an auto-morphism.

Proof. Choose a baseb1, . . . , bk for Fk . α(b) satisfies the same∃-formulas asb does, soby Corollary 2.5,α(b) must be base. ✷

Let V be a variety. Except for (ii) of Lemma 2.2, the definitions and the proofs ofresults carry over to the relatively free groupFV (k) for the variety. Thus, a base, or mogenerally, a tuple generating a subgroup of finite index is∃-determined inFV (k).


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3. F2 is ∃-homogeneous

A groupG is called∃-homogeneousif for all n � 1, alln-tuples inG are∃-determined.To prove thatF2 is ∃-homogeneous, we need (ii) of the following result. The resultobtained by Turner [12] for 1-tuples, but the proof carries over to the general case.that aretract of a groupG is a subgroup Im(ρ), whereρ :G �→ G is an endomorphismsuch thatρ ◦ ρ = ρ.

Theorem 3.1 (Turner [12]).

(i) g is a test tuple inFk iff, for each proper retractR ofFk , there is ani such thatgi /∈ R.(ii) g is a test tuple for monomorphisms inFk iff, for each proper free factorL ofFk , there

is ani such thatgi /∈ L.

Theorem 3.2. F2 is ∃-homogeneous.

Proof. Supposeg, h aren-tuples inF2 andg ≡∃ h. ThenT ∗(g) = T ∗(h). First assumegis a test tuple for monomorphisms. By (ii) of Lemma 2.2, we can choose monomorpα,β of F2 such thatα(gi) = hi andβ(hi)= gi (1 � i � n). Thenβ ◦ α fixes g, soα is anautomorphism.

Otherwise, by Theorem 3.1, the subgroup generated byg is contained in a proper frefactor of F2. Thus there is a primitive elementc such thatgi = cri (1 � i � n), whereri ∈ Z. We apply the following fact.

Lemma 3.3. Supposec ∈ Fk is a primitive element andα(d) = c for a monomorphismαof Fk . Thend is primitive as well.

Proof. Pick a basec, b2, . . . , bk of F(k). Let U be the range ofα. We claim thatcis primitive in U . Choose a Nielsen reduced baseu1, . . . , uk of U (see Lyndon andSchupp [6]). Sincec ∈ U , a noncancelling product ofm elements of this base or theinverses is a word of length�m [6, Proposition 2.13]. Then necessarilyc = ui or c = u−1

i

for somei. Thusc is primitive inU , and henced = α−1(c) is primitive inFk . ✷Sinceg ≡∃ h, there isd ∈ F2 suchhi = dri for eachi < n. SinceT ∗(g1) = T ∗(h1) and

the extraction of roots is unique in a free group,T ∗(c) = T ∗(d). By (ii) of Lemma 2.2there is a monomorphismα such thatα(d) = c. Then, by Lemma 3.3,d is primitive. Thusg, h are in the samen-orbit. ✷

The question remains whether the free groupsFk , k � 3, are∃-homogeneous as we(or even,ω-homogeneous at all). By Proposition 2.4 and (i) of Theorem 3.1, it remto be shown thatg is first-order determined wheng is contained in a proper retractR.(The following might be useful here: by Bergman [2], retracts in f.g. free groups are cunder intersection, so that there is a unique least retractR containingg.)


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4. Finite homomorphic images

Recall that ifV ⊆ F(x1, x2, . . .) andA is group, thenV (A) is the (fully invariant)subgroup ofA generated by all elementsv(a1, . . . , ar), wherev ∈ V ∩ F(x1, . . . , xr) anda1, . . . , ar ∈ A. For instance, ifV = {[x1, . . . , xr ]}, thenV (A) is therth member of thelower central series, also denoted byγrA. The following lemma is easily verified.

Lemma 4.1.

(i) If A,B are groups andf :A →B is an epimorphism, thenf mapsV (A) ontoV (B).(ii) If N �A, thenV (A/N)= (V (A)N)/N .

Theorem 4.2. Fix a primep. Fk is, up to isomorphism, the onlyk-generated group havinall finite k-generatedp-groups as homomorphic images.

Proof. Let F = Fk . It suffices to show the following: ifa ∈ F − {1} generates the normsubgroupN , then there is ak-generated finitep-groupQ which is not a quotient ofF/N .

SinceF is residually nilpotent [4, Theorem 14.2.2], there is a leastc > 1 such thata /∈ γcF . Let H = F/γcF be the free class-c nilpotent group of rankk, and letN =(NγcF )/γcF . It suffices to find a finite class-c nilpotentp-groupQ of rank � k whichis not a quotient ofF/(γcFN) = H/N . For suppose such aQ is a quotient ofF/N , andchoose an onto homomorphismf :F/N → Q. By (i) of Lemma 4.1, the induced ma(F/N)/γc(F/N) → Q/γcQ =Q is onto. But, by (ii) of the same lemma, the group onleft equalsF/(γcFN).

Let A = γc−1H . Thena = (γcF )a ∈ A, andA is contained in (and in fact, equals) tcenter ofH . ClearlyA is f.g., since f.g. nilpotent groups satisfy the maximum condifor subgroups. Because free nilpotent groups are torsion free [8, 31.62],A is a f.g. freeAbelian group. Choose a basisb1, . . . , bm of A in a way thata = b

q

1 for someq ∈ N, andpick r so thatpr > q .

Claim 4.3. For each primep and eachr � 1, there is class-c nilpotent finitep-groupQ ofrank� k such thatγc−1Q ∼= Z

mpr .

The groupQ given by the claim is as desired, namelyQ is not a quotient ofH/N . Forsuppose so. Then, by (i) of Lemma 4.1,A/N = γc−1(H/N) maps ontoγc−1Q ∼= Z

mpr via

an epimorphismρ. But A/N ∼= Zq × Zm−1. Let q = psw wheres < r and (p,w) = 1.

Let c be a generator forZq in Zq × Zm−1, and letL = ker(ρ)〈c〉. Then we obtain an

epimorphism(Zq × Zm−1)/L → Z

mpr /ρ(L). The group on the right has rankm, since

ρ(L) is contained in the subgroup of elements of order� ps . Contradiction.

Proof of the Claim. We obtainQ as a quotient ofH . As above, letb1, . . . , bm be abasis of the free Abelian groupA = γc−1H , and letR = prA. SinceR is containedin the center,R � H . SinceH/R is a f.g. nilpotent group, it is residually finite (see [Example 17.2.8]). So we may chooseM, R � M �H such thatH/M is finite, and none o


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the finitely many nontrivial linear combinations of theRbi is inM/R. LetQ0 =H/M. By(i) of Lemma 4.1,γc−1Q0 = (γc−1HM)/M is the image ofγc−1(H/R) under the naturamapη :H/R → Q0 (whose kernel isM/R). Then, since the elementsη(Rbi) are linearlyindependent,γc−1Q0 ∼= Z

mpr . Finally, Q0 is the direct product of itsq-Sylow subgroups

Let Q be thep-Sylow subgroup, thenQ is a quotient ofQ0 andγc−1Q= γc−1Q0. ✷

5. Prime models

Using Kharlampovich and Myasnikov [5], we observe that the theory of non-Abfree groups has no prime model. We use the following well-known fact.

Proposition 5.1. Let A be a countable structure. ThenA is a prime model of its theory ifeachn-orbit is first-order definable without parameters inA.

In F(a, b), the orbit of[a, b] is the union of the conjugacy classes of[a, b] and[b, a],while in Fk, k > 2, each nontrivial orbit splits into infinitely many conjugacy classes[6, I.5.1]). This implies

Lemma 5.2. If the1-orbit of [a, b] is definable without parameters inF(a, b), thenF(a, b)

does not satisfy the same sentences asFk for 2< k.

Proof. If the orbit is definable by a formulaφ(x), the sentence saying that there are exatwo conjugacy classes of elements satisfyingφ is true inF(a, b), but in no groupFk fork > 2 (since the set of elements satisfyingφ contains an orbit). ✷Theorem 5.3. Th(F2) has no prime model.

Proof. If there is a prime modelP , it can be elementarily embedded intoF2. ThenP isa free group. IfP has rank> 2, then for suitable wordsti in F2, three elementsg0, g1, g2of a free generating set ofP satisfy the existential formula∃x ∃y ∧

i=0,...,2gi = ti (x, y)

in F2, but not inP . ThusP is a free groupF(a, b) of rank 2. By Proposition 5.1, thorbit of [a, b] is definable without parameters, whenceF(a, b) does not satisfy the samsentences asFk for 2< k, using Lemma 5.2. This contradicts the result of Kharlampovand Myasnikov [5]. ✷

An important question which remains in the model theory of free groups is wheththeory of a non-Abelian free group is stable. Stability would mean that there is no foφ(x, y), wherex, y are tuples of variables of the same length, which determines lorders on (not necessarily definable) arbitrarily large finite sets of tuples. It is knowthe theory is not superstable (see [3, p. 694]).

Acknowledgment

The author thanks V. Remeslennikov and E. Turner for helpful comments.


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References

[1] G. Baumslag, A. Myasnikov, V. Shpilrain, Open problems in combinatorial group theory, in: ConMath., Vol. 250, 1999, pp. 1–27, also see Magnus homepage at http://zebra.sci.ccny.cuny.edu/web.

[2] G. Bergman, Supports of derivations, free factorizations, and ranks of fixed subgroups in free groupsAmer. Math. Soc. 351 (1999) 1531–1550.

[3] W. Hodges, Model Theory, in: Encyclopedia Math., Cambridge Univ. Press, Cambridge, 1993.[4] M. Kargapolov, J. Merzljakov, Fundamentals of the Theory of Groups, Springer-Verlag, 1979.[5] O. Kharlampovich, A. Myasnikov, Tarski’s problem about the elementary theory of free groups has a p

solution, Electron. Res. Announc. Amer. Math. Soc. 4 (1998) 101–108.[6] R.C. Lyndon, P.E. Schupp, Combinatorial Group Theory, 1977.[7] W. Magnus, Über freie Faktorgruppen und freie Untergruppen gegebener Gruppen, Monatsh. Math. P

(1939) 307–313.[8] H. Neumann, Varieties of Groups, in: Ergeb. Math. Grenzgeb., Vol. 37, 1967.[9] A. Nies, Separating classes of groups by first-order formulas, Internat J. Algebra Comput., to appea

[10] F. Pickel, A property of finitely generated residually finite groups, Bull. Amer. Math. Soc. 15 (1976)350.

[11] D. Robinson, A Course in the Theory of Groups, Springer-Verlag, 1988.[12] E. Turner, Test words for automorphisms of free groups, Bull. London Math. Soc. 28 (1996) 255–26

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