We build two non-abelian CSA-groups in which maximal abelian subgroups are
conjugate and divisible

Jaligot, Eric

Neman, Azadeh

English

arXiv

We build two non-abelian CSA-groups in which maximal abelian subgroups are conjugate and divisible

Hal-Diderot

Embeddings and chains of free groups

We build two nonabelian CSA-groups in which maximal abelian sub-groups are conjugate and divisible, as the countable unions of increasing chains of CSA-groups and by keeping the constructions as free as possible in each case. For n ≥ 1, let Fn denote the free group on n generators. We view all groups G as first-order structures 〈G, ·, −1, 1〉, where ·, −1, and 1 denote respectively the multiplication, the inverse, and the identity of the group. The following striking results are proved in a series of papers of Sela culminating in [Sel07]. Fact 1 [Sel05, Sel06a, Sel06b, Sel07] (1) For any 2 ≤ n ≤ m, the natural embedding Fn ≤ Fm is an elementary embedding. (2) For any n ≥ 2, the (common) complete theory Th (Fn) is stable. We refer to [Hod93] for model theory in general, and to [Poi87] and [Wag97] for stability theory and in particular stable groups. Let F denote the free group over countably many generators. Fact 1 has the following corollary. Corollary 2 The natural embeddings F2 ≤  ·  · ·Fn ≤  ·  ·  ·  ≤ F are all elementary. In particular each Fn is an elementary substructure of F, and Th (F) is stable. A CSA-group is a group in which maximal abelian subgroups A are mal-normal, i.e., such that A ∩ Ag 6 = 1 implies that g is in A for any element g of the ambient group. The class of CSA-groups contains all free groups and is studied from various points of view. We refer to [JOH04, JMN08] for a model theoretic approach in combination of questions concerning particular groups [Che79, Jal01, Cor03], and to [KMRS08] for an approach more related to com-putational aspects in limit groups. We prove the following lemma on embeddings of torsion-free CSA-groups in which maximal abelian subgroups are cyclic. 1 ha l-

Eric Jaligot

Azadeh Neman

CiteSeerX

HAL-UJM

Embeddings and chains of free groupsEric Jaligot, Azadeh NemanTo cite this version:Eric Jaligot, Azadeh Neman. Embeddings and chains of free groups. 2008. <hal-00286075v2>HAL Id: hal-00286075https://hal.archives-ouvertes.fr/hal-00286075v2Submitted on 13 Jun 2008HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.Embeddings and chains of free groupsEric Jaligot, Azadeh NemanMay 27, 2008AbstractWe build two nonabelian CSA-groups in which maximal abelian sub-groups are conjugate and divisible, as the countable unions of increasingchains of CSA-groups and by keeping the constructions as free as possiblein each case.For n ≥ 1, let Fn denote the free group on n generators. We view all groupsG as first-order structures 〈G, ·, −1, 1〉, where ·, −1, and 1 denote respectivelythe multiplication, the inverse, and the identity of the group. The followingstriking results are proved in a series of papers of Sela culminating in [Sel07].Fact 1 [Sel05, Sel06a, Sel06b, Sel07](1) For any 2 ≤ n ≤ m, the natural embedding Fn ≤ Fm is an elementaryembedding.(2) For any n ≥ 2, the (common) complete theory Th (Fn) is stable.We refer to [Hod93] for model theory in general, and to [Poi87] and [Wag97]for stability theory and in particular stable groups.Let F denote the free group over countably many generators. Fact 1 has thefollowing corollary.Corollary 2 The natural embeddings F2 ≤ · · ·Fn ≤ · · · ≤ F are all elementary.In particular each Fn is an elementary substructure of F , and Th (F ) is stable.A CSA-group is a group in which maximal abelian subgroups A are mal-normal, i.e., such that A ∩ Ag 6= 1 implies that g is in A for any element g ofthe ambient group. The class of CSA-groups contains all free groups and isstudied from various points of view. We refer to [JOH04, JMN08] for a modeltheoretic approach in combination of questions concerning particular groups[Che79, Jal01, Cor03], and to [KMRS08] for an approach more related to com-putational aspects in limit groups.We prove the following lemma on embeddings of torsion-free CSA-groups inwhich maximal abelian subgroups are cyclic.1Lemma 3 Let k ≥ 2, and let G be a countable torsion-free CSA-group in whichmaximal abelian subgroups are cyclic. Let a0 be any generator of a maximalabelian subgroup of G. Then G embeds into a torsion-free CSA-group H = 〈F, r〉in which maximal abelian subgroups are cyclic, where F is a free-group overcountably many generators and rk = a0, and where maximal abelian subgroupsof G are H-conjugate. In particular any element of G has a k-th root in H.Proof:G has countably many maximal abelian subgroups, and countably manyconjugacy classes of such maximal abelian subgroups, which can be enumeratedby i < ω.For each such conjugacy class, fix a maximal (cyclic) abelian subgroup Ai,and inside Ai fix a generator ai. Notice that we can always take for a0 any givengenerator of a maximal abelian subgroup of G.We define inductively on i an increasing family of supergroups Gi of G asfollows.• G0 = G.• Gi+1 is the HNN -extension 〈Gi, ti | atii = ai+1〉.We note that each Gi+1 is generated by G0 together with the elements t0, ...,ti.Let nowGω =⋃i<ωGi.It is clear that Gω is generated by G0, together with the elements ti’s. Byconstruction, it is also clear that in Gω any two distinct elements ti and tjsatisfy no relation by [LS77, Britton’s Lemma]. In particular they generate afree group on two generators ti and tj .We also see that in Gω any two maximal abelian subgroups of G0 are con-jugate, actually by the subgroup of Gω generated by all the elements ti. Inparticular in Gω one hasG0 ⊆ 〈a0〉〈(ti)i<ω | 〉and Gω is generated by 〈a0〉 and 〈(ti)i<ω | 〉.Consider now an abelian torsion-free cyclic supergroup R of A0 = 〈a0〉 gen-erated by an element r such that rk = a0. As k ≥ 2 by assumption, r does notbelong to the cyclic subgroup 〈a0〉 of R.Now one can form the free product of R and Gω with amalgamated subgroup〈a0〉, sayGω+1 = R ∗〈a0〉 Gω.As rk = a0 and Gω is generated by a0 and the ti’s, one gets that Gω+1 isgenerated by the ti’s together with r. HenceGω+1 = 〈r, (ti)i<ω〉2and the second set of generators freely generate the free group F .Now the natural embeddingG ≃ G0 ≤ Gω+1 ≃ His the desired embedding.We note that the class of CSA-groups is inductive, as pointed out in [JOH04].This follows indeed from the fact that the class is axiomatizable by universalaxioms. As H is a direct limit of CSA-groups, it is also a CSA-group. We notethat maximal abelian subgroups of H are also cyclic by the results of [JOH04].By Lemma 3, one can find an infinite sequence of embeddings1G ≤ 2G ≤ · · · ≤ k−1G ≤ kG ≤ · · ·where 1G is a nonabelian countable free group and such that for each k ≥ 2the embedding k−1G ≤ kG is as in Lemma 3, i.e., such that maximal abeliansubgroups of k−1G are conjugate in kG and each element in k−1G is k-divsible inkG. We note that each group kG is a torsion-free CSA-group in which maximalabelian subgroups are cyclic.Consider now the groupG =⋃k≥1kGWe see that G is a CSA-group, as the class of CSA-groups is inductive. Asmaximal abelian subgroups coincide with centralizers of nontrivial elements inCSA-groups, one sees that maximal abelian subgroups of G are conjugate byconstruction. Again the construction implies that each element g of G is n-divisible for each n, and as maximal abelian subgroups coincide with central-izers of nontrivial elements one concludes that maximal abelian subgroups aredivisible.Of course G is nonabelian since it contains the nonabelian free group 1G,and we note that it is also torsion-free. We obtain thus a non-abelian CSA-group in which maximal abelian subgroups are conjugate and divisible, as acountable union of torsion-freeCSA-groups in which maximal abelian subgroupsare cyclic.As the free group is stable, one may wonder about the stability of the groupG built above. Recall that the stable sets are the sets defined by a formulaϕ(x, y) for which there exists an integer n, called the ladder index of ϕ, boundinguniformly the size of n-ladders of ϕ, that is the sets of tuples(x1, · · · , x1 ; y1, · · · , yn)such that ϕ(xi, xj) is true if and only if i ≤ j. We refer to [Hod93]. We recallalso that by Ramsey’s theorem boolean combinations of stable sets are stable[Wag97, 0.2.10], and that the replacement of variables by parameters obviouslydoes not affect the stability of definable sets.3To check the stability of quantifier-free definable sets, the following lemmamay be relevant for the type of groups considered here.Fact 4 Let ϕ(x, y) be a quantifier-free formula in the langage of groups, thatis a boolean combination of equations into the variables involved in the tuples xand y. If a group G is a union of an increasing family of subgroups Gi, and ifϕ defines a stable set in each Gi with a uniform bound on the ladder indices ofϕ in each Gi, then ϕ defines a stable set in G.Proof:By assumption, there exists a uniform bound on the ladder index of ϕ in Gi,when i varies, and thus we find an integer n such that no subgroup Gi can havea ladder of size n.Now one sees that G cannot have a ladder of size n also, as otherwise allelements of the tuples involved in such a ladder would belong to one of thesubgroups Gi, a contradiction as ϕ is quantifier-free.This proves that ϕ defines a stable set in G. Another construction of a non-abelian CSA-group in which maximal abeliansubgroups are conjugate and divisible and with a temptative to keep certainparts of the stability of the free group is as follows. It now consists of addingall roots simultaneously.Instead of starting from the free group F , let us start with 1G = Q ∗ F .Fix a0 the element corresponding to the element 1 of Q (in additive notation).Passing from 1G to 2G is now done as follows. Enumerate by a1, a2, ... etc,generators of maximal abelian cyclic subgroups of 1G, picking up exactly onemaximal abelian cyclic subgroup in each conjugacy class of such subgroups.Now embed 〈a0〉 in a copy of Q, in such a way that a0 represents 1 in Q (inadditive notation), and form the free product of 1G and this new copy of Q,with amalgamated subgroup 〈a0〉. One gets then a new CSA-group, and one canconjugate a0 to a1 by forming an appropriate HNN -extension, with an elementt0. One repeats this process as in Lemma 3, obtaining CSA-groups at each stepby the results of [JOH04]. Calling 2G the union, one gets a CSA-group withone conjugacy class of divisible maximal abelian subgroups, and generated byone such subgroup and a countable free group (generated by all the ti’s addedwhen forming the successive HNN -extensions).One can then build similarly an infinite sequence of embeddings1G ≤ 2G ≤ · · · ≤ k−1G ≤ kG ≤ · · ·such that for each k ≥ 2 the embedding k−1G ≤ kG is as in the processdescribed above. In particular each kG is generated by a divisible abelian groupisomorphic to Q and a free group F . Again the union of all these groups is aCSA-group in which maximal abelian subgroups are conjugate and divisible.We thank Vincent Guirardel, Abderezak Ould Houcine, Zlil Sela, and AlinaVdovina who pointed out a mistake in the first version of the present work.4References[Che79] G. Cherlin. Groups of small Morley rank. Ann. Math. Logic, 17(1-2):1–28, 1979.[Cor03] L. J. Corredor. (∗)-groups and pseudo-bad groups. Rev. ColombianaMat., 37(2):51–63, 2003.[Hod93] W. Hodges. Model theory, volume 42 of Encyclopedia of Mathematicsand its Applications. Cambridge University Press, Cambridge, 1993.[Jal01] E. Jaligot. Full Frobenius groups of finite Morley rank and theFeit-Thompson theorem. Bull. Symbolic Logic, 7(3):315–328, 2001.[JMN08] E. Jaligot, A. Muranov, and A. Neman. Independence property andhyperbolic groups. Bull. Symbolic Logic, 14(1):88–98, 2008.[JOH04] E. Jaligot and A. Ould Houcine. Existentially closed CSA-groups.J. Algebra, 280(2):772–796, 2004.[KMRS08] O. Kharlampovich, A. Myasnikov, V. Remeslennikov, and D. Serbin.Exponential extensions of groups. J. Group Theory, 11(1):119–140,2008.[LS77] R. C. Lyndon and P. E. Schupp. Combinatorial group theory.Springer-Verlag, Berlin, 1977. Ergebnisse der Mathematik und ihrerGrenzgebiete, Band 89.[Poi87] B. Poizat. Groupes stables. Bruno Poizat, Lyon, 1987. Une ten-tative de conciliation entre la ge´ome´trie alge´brique et la logiquemathe´matique. [An attempt at reconciling algebraic geometry andmathematical logic].[Sel05] Z. Sela. Diophantine geometry over groups. V1. Quantifier elimina-tion. I. Israel J. Math., 150:1–197, 2005.[Sel06a] Z. Sela. Diophantine geometry over groups. V2. Quantifier elimina-tion. II. Geom. Funct. Anal., 16(3):537–706, 2006.[Sel06b] Z. Sela. Diophantine geometry over groups. VI. The elementarytheory of a free group. Geom. Funct. Anal., 16(3):707–730, 2006.[Sel07] Z. Sela. Diophantine geometry over groups VIII: Stability. preprint:http://www.ma.huji.ac.il/∼zlil/, 2007.[Wag97] F. O. Wagner. Stable groups. Cambridge University Press, Cam-bridge, 1997.5

Embeddings and chains of free groups

Abstract

Similar works

Full text

Available Versions

Hal-Diderot

CiteSeerX

HAL-UJM