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Diophantine Geometry Over Groups Vii: the Elementary Theory of a Hyperbolic Group

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Diophantine Geometry Over Groups Vii: the Elementary Theory of a Hyperbolic Group DIOPHANTINE GEOMETRY OVER GROUPS VII: THE ELEMENTARY THEORY OF A HYPERBOLIC GROUP Z. Sela1,2 This paper generalizes our work on the structure of sets of solutions to systems of equations in a free group, projections of such sets, and the structure of elementary sets defined over a free group, to a general torsion-free (Gromov) hyperbolic group. In particular, we show that every definable set over such a group is in the Boolean algebra generated by AE sets, prove that hyperbolicity is a first order invariant of a finitely generated group, and obtain a classification of the elementary equivalence classes of torsion-free hyperbolic groups. Finally, we present an effective procedure to decide if two given torsion-free hyperbolic groups are elementarily equivalent. In [18]-[24] we studied sets of solutions to systems of equations in a free group, and developed basic techniques and objects that are required for the analysis of sentences and elementary sets defined over a free group. The techniques we devel- oped, enabled us to present an iterative procedure for a quantifier elimination for predicates defined over a free group, and answer some of A. Tarski’s problems on the elementary theory of a free group. We were also able to classify those finitely generated groups that are elementarily equivalent to a free group. In this paper we generalize our entire work from free groups to torsion-free hyperbolic groups. In the first section we study limit groups over hyperbolic groups, generalize Guba’s theorem on the equivalence of an infinite system of equations to some finite subsystem [9], and associate a canonical Makanin-Razborov diagram with a given system of equations defined over a torsion-free hyperbolic group. As over free groups, this Makanin-Razborov diagram encodes the entire set of solutions to the system. In the second section we generalize Merzlyakov theorem on the existence of a formal solution associated with a positive sentence [11], and the construction of a formal solution to a general AE sentence which is known to be true over some variety defined over a free group, presented in [19], to AE sentences that are known to be true over some variety defined over a torsion-free hyperbolic group. In the third section, we generalize our study of exceptional solutions of parametric systems of equations defined over a free group [20], to hyperbolic groups. The main result of this section is the existence of a global bound (independent of the parameters specialization) on the number of families of exceptional solutions to parametric systems of equations defined over a torsion-free hyperbolic group. In the fourth section, we generalize the iterative procedure for validation of an AE sentence defined over a free group, to AE sentences defined over a torsion-free hyperbolic group. As over free groups, this terminating iterative procedure is the basis for our analysis of elementary sets defined over a hyperbolic group. 1 In the fifth section we generalize the construction of a core resolution, presented in [22], to resolutions defined over a torsion-free hyperbolic group. In the sixth section, we generalize the analysis of the Boolean algebra of AE sets defined over a free group ([22],[23]), to study the Boolean algebra of AE sets defined over a torsion- free hyperbolic group, and finally prove that every definable set over a torsion-free hyperbolic group is in the Boolean algebra generated by AE sets. In the seventh section we use the quantifier elimination obtained in the sixth section, to study some basic properties of the elementary theory of a torsion-free hyperbolic group. We prove that hyperbolicity is a first order invariant of a f.g. group, i.e., we show that a f.g. group that is elementarily equivalent to a torsion- free hyperbolic group must be hyperbolic as well. We continue by classifying all the f.g. groups that are elementarily equivalent to a given torsion-free hyperbolic group. We obtain this classification, by associating an elementary core with a given torsion-free hyperbolic group, and prove that two torsion-free hyperbolic groups are elementarily equivalent if and only if their elementary cores are isomorphic. We conclude this section by proving that the universal theory of a torsion-free hyperbolic group is decidable, and use it to construct an effective procedure to decide if two given torsion-free hyperbolic groups are elementarily equivalent. Some of the work presented in this paper has been generalized since it first appeared. Emina Alibegovic has studied sets of solutions to systems of equations over limit groups [1]. In a sequence of papers Daniel Groves has studied systems of equations over torsion-free relatively hyperbolic groups with abelian parabolic groups [8]. Francois Dahmani has given a simpler and more elegant solution to the universal theory of a torsion-free hyperbolic group [5]. The work that is presented in this paper, was partially supported by an Israel academy of sciences fellowship, and NSF grant no. DMS9729992 through the IAS. §1. Γ-Limit Groups and their Makanin-Razborov Diagrams Let Γ =<γ1,...,γk > be a torsion-free (Gromov) hyperbolic group, and let X be its Cayley graph. Let G =<g1,...,gm > be a f.g. group. Let {hj} ⊂ Hom(G, Γ) be a set of homomorphisms from G to Γ, and suppose that the homomorphisms {hj} belong to distinct conjugacy classes (i.e., for every j1, j2, 1 ≤ j1 < j2, and −1 every γ ∈ Γ, γhj1 γ 6= hj2 ). For each index j we fix an element γj ∈ Γ having ”minimal displacement” under the action λhj and set µj to be: −1 −1 µj = max dX (id., γjhj(gu)γj ) = min max dX (id., γhj(gu)γ ) 1≤u≤m γ∈Γ 1≤u≤m Since the homomorphisms in the sequence {hj} ⊂ Hom(G, Γ) are non-conjugate, the sequence of stretching factors {µj} does not contain a bounded subsequence. ∞ We set {(Xj,xj)}j=1 to be the pointed metric spaces obtained by rescaling the metric on the Cayley graph of Γ, (X,id.), by µj . (Xj,xj) is endowed with a left isometric action of our f.g. group G via the homomorphisms τγj ◦ hj where τγj is the inner automorphism of Γ defined by γj. This sequence of actions of G on the ∞ metric spaces {(Xj,xj)}j=1 allows us to obtain an action of G on a real tree by passing to a Gromov-Hausdorff limit. ∞ Proposition 1.1 ([13], 2.3). Let {Xj}j=1 be a sequence of δj-hyperbolic spaces with δ∞ = lim δj = 0. Let H be a countable group isometrically acting on Xj. 2 Suppose there exists a base point xj in Xj such that for every finite subset P of H, the sets of geodesics between the images of xj under P form a sequence of totally bounded metric spaces. Then there is a subsequence converging in the Gromov topology to a δ∞-hyperbolic space X∞ endowed with a left isometric action of H. ∞ Our spaces {(Xj,xj)}j=1 endowed with the left isometric action of G, satisfy the assumptions of the proposition and they are all trees, so they are 0-hyperbolic, hence, X∞ is a real tree endowed with an isometric action of G. By construction, ∞ the action of G on the real tree X∞ is non-trivial. Let {jn}n=1 be the subsequence ∞ for which {(Xjn ,xjn )}n=1 converge to the limit real tree X∞, and let (Y,y0) denote this (pointed) limit real tree. For convenience, for the rest of this section we denote −1 the homomorphism γjn hjn γjn : G → Γ, by hn. As we did in section 1 of [18], with the limit tree we obtained by using the Gromov-Hausdorff topology we associate natural algebraic objects, the kernel of the action of G on this (limit) real tree and the quotient of G by this kernel which we call the (strict) Γ-limit group. Definition 1.2. The kernel of the action of the group G on the limit tree Y is defined to be: K∞ = {g ∈ G | ∀y ∈ Y g(y) = y} Having the kernel of the action, we define the (strict) Γ-limit group to be: L∞ = G/K∞ and set η : G → L∞ to be the natural quotient map. The following simple facts on the kernel of the action and the (strict) limit group are important observations, and their proof is identical to the proof of lemma 1.3 of [18]. Lemma 1.3. With notation of definition 1.2: (i) L∞ is a f.g. group. (ii) If Y is isometric to a real line then the (strict) limit group L∞ is f.g. free abelian. (iii) If g ∈ G stabilizes a tripod in Y then for all but at most finitely many n’s g ∈ ker(hn) (recall that a tripod is a finite tree with 3 endpoints). In particular, if g ∈ G stabilizes a tripod then g ∈ K∞. (iv) Let g ∈ G be an element which does not belong to K∞. Then for all but at most finitely many n’s g∈ / ker(hn). (v) L∞ is torsion-free. (vi) Let [y1,y2] ⊂ [y3,y4] be a pair of non-degenerate segments of Y and as- sume the stabilizer of [y3,y4] in L∞, stab([y3,y4]), is non-trivial. Then stab([y3,y4]) is an abelian subgroup of L∞ and: stab([y1,y2]) = stab([y3,y4]) Hence, the action of L∞ on the real tree Y is stable. In a torsion-free hyperbolic group every solvable subgroup is infinite cyclic and every maximal cyclic subgroup is malnormal.
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