aa r X i v : . [ m a t h . G R ] F e b MAKANIN-RAZBOROV DIAGRAMS OVER FREE PRODUCTS
E. Jaligot , and Z. Sela , This paper is the first in a sequence on the first order theory of free products. In thefirst paper we generalize the analysis of systems of equations over free and (torsion-free) hyperbolic groups, and analyze systems of equations over free products. To dothat we introduce limit groups over the class of free products, and show that a finitelypresented group has a canonical (finite) collection of maximal limit quotients. Wefurther extend this finite collection and associate a Makanin-Razborov diagram overfree products with every f.p. group. This MR diagram encodes all the quotients ofa given f.p. group that are free products, all its homomorphisms into free products,and equivalently all the solutions to a given system of equations over a free product.
Sets of solutions to equations defined over a free group have been studied exten-sively. Considerable progress in the study of such sets of solutions was made byG. S. Makanin, who constructed an algorithm that decides if a system of equationsdefined over a free group has a solution [Ma], and showed that the universal andpositive theories of a free group are decidable. A. A. Razborov was able to give adescription of the entire set of solutions to a system of equations defined over a freegroup [Ra2], a description that was further developed by O. Kharlampovich and A.Myasnikov [Kh-My].In [Se1] a geometric approach to the study of sets of solutions to systems ofequations over a free group is presented. This was generalized in [Se3] for systemsof equations over (torsion-free) hyperbolic groups, in [Al] to systems of equationsover limit groups, and in [Gr] to systems of equations over toral relatively hyperbolicgroups.In this paper we generalize part of the techniques and results that were obtainedover free groups to study systems of equations over arbitrary free products. Let Σbe a system of equations which is defined over a free product, A ∗ B : w ( x , . . . , x n ) = 1... w s ( x , . . . , x n ) = 1Following [Ra1] we set the associated f.p. group G (Σ) to be: G (Σ) = < x , . . . , x n | w , . . . , w s > Universite de Lyon - CNRS and Universite Lyon 1. Partially supported by a French ANR JC05-47038. Hebrew University, Jerusalem 91904, Israel. Partially supported by an Israel academy of sciences fellowship. h : G (Σ) → A ∗ B , and every such homomorphism corresponds to a solution of the system Σ.Therefore, the study of sets of solutions to systems of equations over the free product A ∗ B is equivalent to the study of all the homomorphisms from a fixed f.p. group G into A ∗ B .We further generalize our point of view, and instead of the set of homomorphismsfrom a given f.p. group G (Σ) into a particular free product, we study the set of allthe homomorphisms from the f.p. group G (Σ) into all possible free products. ByKurosh subgroup theorem, this is equivalent to the study of all the quotients of agiven f.p. group, G (Σ), that are free products.To analyze the set of free product quotients of a given f.p. group, we generalizethe notion of limit groups (over free groups), and define limit groups over f reeproducts . The definition over free products (definition 1) is a generalization ofthe definition of limit groups over free groups, but with each limit group over freeproducts, L , there is an additional structure, a subset of conjugacy classes in thelimit group L , that are called elliptics , that are forced to be mapped to conjugatesof the factors in any homomorphism from the limit group into a free product.After proving some basic properties of limit groups over free products, we as-sociate with them a canonical virtually abelian JSJ decomposition (theorem 11).Limit groups over free products do not satisfy the d.c.c. that hold for limit groupsover free and hyperbolic groups. Still, in theorem 13 we prove a basic d.c.c. thatholds for such limit groups, and applies to descending chain of limit groups overfree products, in which the maps between successive limit groups are proper epi-morphisms that do not map non-trivial elliptic elements to the identity element.This d.c.c. allows us to associate a resolution with each limit group over freeproducts (theorem 18). We further define a natural partial order on the set oflimit quotients over free products of a given f.p. group, and prove that there arefinitely many (equivalence classes of) maximal limit quotients (over free products)of a f.p. group. Finally we extend each of the maximal limit quotients with finitelymany resolutions and obtain a Makanin-Razborov diagram of a f.p. group over freeproducts.The diagram that we associate with a f.p. group encodes all the quotients ofthe given f.p. group that are free products. Unfortunately, our construction is notcanonical, and we state a natural conjecture that if answered affirmatively willenable one to construct a canonical diagram. Also, the construction uses the finitepresentability of the group in question in an essential way. Hence, encoding the setof free product quotients of a f.g. group is left open.The Makanin-Razborov diagram over free products is the first step towards theanalysis of the first order theory of free products that will appear in the sequel.This study was motivated by a question of the first author on the stability of a freeproduct of stable groups. We expect that some of the notions and constructionsthat appear in this paper (and in the sequel) can be generalized to other classes ofgroups, e.g. acylindrical splittings of f.p. groups, and various classes of relativelyhyperbolic groups. §
1. Limit Groups over Free Products
We start the analysis of systems of equations over free products with the def-inition of a limit group over the set of free products. The definition generalizes2he corresponding ones for free, hyperbolic, and relatively hyperbolic groups, but itassociates with a limit group an additional structure - it’s collection of conjugacyclasses of elliptic elements. Also, note that unlike the case of a free or a hyper-bolic group, we consider limit groups over the entire class of free products, and notnecessarily over a given one.
Definition 1.
Let { A n } and { B n } be two sequences of groups (not necessarilyfinitely generated), and let G be a finitely generated group. We say that a sequenceof homomorphisms, { h n : G → A n ∗ B n } , is a convergent sequence, if the followingconditions hold: (i) for each g ∈ G there exists some index n g > , so that for every n > n g , h n ( g ) = 1 , or for every n > n g , h n ( g ) = 1 . (ii) for each g ∈ G there exists some index n eg > , so that for every n >n eg , h n ( g ) is elliptic in the free product A n ∗ B n (i.e., it is contained in aconjugate of A n or B n ), or for every n > n eg , h n ( g ) is not elliptic in A n ∗ B n .With the convergent sequence we associate its stable kernel that is defined to be: K = { g ∈ G | ∃ n g ∀ n > n g h n ( g ) = 1 } and the associate limit group: L = G/K , which we call a limit group over (thecollection of ) free products, and set η : G → L to be the natural quotient map.With the limit group L we associate an additional structure, its collection ofconjugacy classes that are stably elliptic, i.e.: E L = { ℓ ∈ L | ∃ g ∈ G η ( g ) = ℓ ∃ n g > ∀ n > n g h n ( g ) is elliptic } Note that by definition if η ( g ) = η ( g ) , then g is stably elliptic iff g is stablyelliptic. Also, note that every f.g. group can be a limit group over free products,as given a finitely generated group G , we can look at the free product G ∗ B , forsome non-trivial group B , with the fixed sequence of homomorphisms that map G identically onto G in the free product G ∗ B . Note that in this tautological case, theentire (limit) group G is set to be elliptic. Given a convergent sequence of homomorphisms one can pass to a subsequencethat converges into a (possibly trivial) action of the associated limit group on somereal tree.Let A and B be non-trivial groups (not necessarily finitely generated). With thefree product, A ∗ B , we can naturally associate its Bass-Serre tree. Let G be af.g. group G = < g , . . . , g m > , let { A n , B n } be a sequence of pairs of non-trivialgroups, and let { h n : G → A n ∗ B n } , be a sequence of homomorphisms.With the sequence of free products, { A n ∗ B n } , we naturally associate theirBass-Serre trees that we denote, { T n } , with a base point t n (which is one of thevertices in T n ). Each homomorphism, h n : G → A n ∗ B n , gives rise to an action λ h n of the group G on the Bass-Serre tree T n . For each index n we fix an element γ n ∈ A n ∗ B n , so that the homomorphism γ n h n γ n − has ”minimal displacement”,i.e., the element γ n satisfies the equality:max ≤ u ≤ m d T n ( t n , γ n h n ( g u ) γ n − ( t n )) = min γ ∈ A n ∗ B n max ≤ u ≤ m d T n ( t n , γh n ( g u ) γ − ( t n ))3e further set µ n to be: µ n = max ≤ u ≤ m d T n ( t n , γ n h n ( g u ) γ n − ) . First, suppose that the sequence of integers, { µ n } , is bounded. In that casewe can extract a subsequence of the homomorphisms { h n } (still denoted { h n } ),that converges into a limit group (over free products) L , with an associated setof elliptics E L . Furthermore, the sequence of homomorphisms γ n h n γ − n convergesinto a faithful action of L on some simplicial tree with trivial edge stabilizers, thatwe denote T . In that case either the entire group L is elliptic (i.e. E L = L ), or L isinfinite cyclic, or it is freely decomposable and the stabilizer of each vertex groupin T is elliptic. In this case, the limit group L is a free product of elliptic vertexgroups (in T ) with a (possibly trivial) free group.Suppose that the sequence of integers, { µ n } , does not contain a bounded subse-quence. We set { ( X n , x n ) } ∞ n =1 to be the pointed metric spaces obtained by rescalingthe metric on the Bass-Serre trees ( T n , t n ), by µ n . ( X n , x n ) is endowed with a leftisometric action of our f.g. group G via the homomorphisms γ n h n γ − n . This se-quence of actions of G on the metric spaces { ( X n , x n ) } ∞ n =1 allows us to obtain anaction of G on a real tree by passing to a Gromov-Hausdorff limit. Proposition 2 ([Pa], 2.3).
Let { X n } ∞ n =1 be a sequence of δ n -hyperbolic spaceswith δ ∞ = lim δ n = 0 . Let H be a countable group isometrically acting on X n .Suppose there exists a base point x n in X n such that for every finite subset P of H ,the sets of geodesics between the images of x n under P form a sequence of totallybounded metric spaces. Then there is a subsequence converging in the Gromovtopology to a δ ∞ -hyperbolic space X ∞ endowed with a left isometric action of H . Our spaces { ( X n , x n ) } ∞ n =1 endowed with the left isometric action of G , satisfythe 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 { n j } ∞ j =1 be the subsequencefor which { ( X n j , x n j ) } ∞ j =1 converges to the limit real tree X ∞ , and let ( Y, y ) denotethis (pointed) limit real tree.For convenience, for the rest of this section we (still) denote the homomorphism γ n j h n j γ n j − : G → A n j ∗ B n j , by h n . By passing to a further subsequence wecan assume that the sequence of homomorphisms { h n } converges into a limit group(over free products) that we denote L , with elliptic elements E L , and an associatedquotient map, η : G → L , with kernel K . In the sequel we call a limit group overfree products that is obtained from a sequence of homomorphisms with unboundedstretching factors, a strict limit group over free products.The following simple facts on the kernel of the action, K , (see definition 1.1) andthe (strict) limit group L are important observations, and their proof is similar tothe proof of lemma 1.3 of [Se1]. Lemma 3. (i)
Elements in E L fix points in Y . (ii) L is f.g. (iii) If Y is isometric to a real line then the limit group L has a subgroup ofindex at most 2, which is f.g. free abelian. If g ∈ G stabilizes a tripod in Y then for all but finitely many n ’s, g ∈ ker ( h n ) (recall that a tripod is a finite tree with 3 endpoints). In particular,if g ∈ G stabilizes a tripod then g ∈ K . (v) Let g ∈ G be an element which does not belong to K . Then for all butfinitely many n ’s, g / ∈ ker ( h n ) . (vi) Every torsion element in L is elliptic, i.e., it is in E L . (vii) Let [ y , y ] ⊂ [ y , y ] be a pair of non-degenerate segments of Y and as-sume that the stabilizer of [ y , y ] in L , stab ([ y , y ]) , is non-trivial. Then stab ([ y , y ]) is an abelian subgroup of L and: stab ([ y , y ]) = stab ([ y , y ]) Hence, the action of L on the real tree Y is (super) stable. (viii) Let
H < G be a f.g. subgroup and suppose that η ( H ) ⊂ E L . Then for all butfinitely many n ’s, h n ( H ) is elliptic, i.e., h n ( H ) is contained in a conjugateof A n or B n .Proof: Part (i) follows from the definition of the elliptic elements E L . A limitgroup L is a quotient of a f.g. group, hence, it is finitely generated. If Y is a realline, then L contains a subgroup of index at most 2 that acts on the real line byisometries and preserves its orientation. Hence, this subgroup must be f.g. abelian,and it contains no elliptic elements, so it contains no torsion. Therefore, L containsa f.g. abelian subgroup of index at most 2. (iv), (v), and (vii), follow by the sameargument that is used in the case of free and hyperbolic groups ([Se1],1.3). Atorsion element in L is the image of an element g ∈ G , which is mapped to atorsion element by all the homomorphisms, h n : G → A n ∗ B n , for large n . Hence, h n ( g ) is contained in a conjugate of A n or B n for large n , so g is mapped to anelliptic element in L , and (vi) follows. To prove (viii) let H = < u , . . . , u m > .Since H is contained in E L then there exists some n so that for all n > n , all theelements u , . . . , u m and the products u i u j , i, j = 1 , . . . , m , are mapped to ellipticelements by the homomorphism h n . Therefore, for all n > n , h n ( H ) is elliptic,i.e., contained in a conjugate of A n or B n . (cid:3) Recall that in limit groups over free and torsion-free hyperbolic groups, everynon-trivial abelian subgroup is contained in a unique maximal abelian subgroup,and every maximal abelian subgroup is f.g. and malnormal. This is clearly not thecase in limit groups over free products, as every f.g. group is a limit group over freeproducts. However, for the analysis of strict limit groups over free products, weare really interested only in non-elliptic abelian subgroups, as only these occur asstabilizers of non-degenerate segments in real trees on which the strict limit groupsact faithfully, and so that these real trees are obtained as a limit from a sequenceof homomorphisms into free products. Non-elliptic abelian subgroups have similarproperties as abelian subgroups in limit groups over free and torsion-free hyperbolicgroups. The proof is similar to the proof of lemma 1.4 in [Se1].
Lemma 4.
With the notation of definition 1 let u , u , u be non-trivial elementsof L , and suppose that at least one of the elements, u , u , u , is non-elliptic (i.e.,not in E L ), and [ u , u ] = 1 and [ u , u ] = 1 . Then: (i) u , u , u are non-elliptic and [ u , u ] = 1 . let A < L be a non-elliptic abelian subgroup of L . Then A is contained in aunique maximal abelian subgroup in L , which is its centralizer, C ( A ) . Thecentralizer of A , C ( A ) , intersects the set of elliptic elements, E L , trivially. (iii) let A be a non-elliptic abelian subgroup in L . Then the centralizer of A , C ( A ) , is almost malnormal in L . C ( A ) is of index at most 2 in N ( A ) ,the normalizer of A , and for each element ℓ ∈ L , ℓ / ∈ N ( A ) , ℓC ( A ) ℓ − intersects A trivially. Furthermore, if [ N ( A ) : C ( A )] = 2 then N ( A ) isgenerated by C ( A ) and an elliptic element of order 2 that conjugates eachelement in C ( A ) to its inverse.Proof: Let g , g , g be elements of G that are mapped to u , u , u . Since [ u , u ] =1 and [ u , u ] = 1, and the elements u , u , u are non-trivial, there exists some n ,so that for all n > n , [ h n ( g ) , h n ( g )] = 1 and [ h n ( g ) , h n ( g )] = 1, and the ele-ments, h n ( g ) , h n ( g ) , h n ( g ), are non-trivial. Since for some j , 1 ≤ j ≤ u j is notelliptic, there exists some n j > n , so that for all n > n j , h n ( g j ) is a hyperbolic el-ement. Since for n > n j , h n ( g ) is non-trivial and commutes with h n ( g j ), h n ( g ) isa hyperbolic element, and by the same argument so are h n ( g ) and h n ( g ). Hence,all the 3 elements, h n ( g ) , h n ( g ) , h n ( g ), are hyperbolic and have the same axis,so they all commute and [ u , u ] = 1.By part (i) commutativity is transitive for non-elliptic elements of a limit groupover free products. Hence a non-elliptic abelian subgroup is contained in a uniquemaximal abelian subgroup, which is its centralizer, and the centralizer must benon-elliptic as well.Let A < L be a non-elliptic abelian subgroup. Let u ∈ N ( A ) \ C ( A ), and let g ∈ G be an element that is mapped to u . Given a finite set of non-trivial elements g , . . . , g m that are mapped to C ( A ), there exists some integer n , so that for every n > n , h n ( g j ) are hyperbolic, h n ( g i ) commutes with h n ( g j ), and h n ( g ) does notcommute with h n ( g i ), for all i, j = 1 , . . . , m , and h n ( g ) h n ( g j ) h n ( g ) − commuteswith all the elements h n ( g i ), for i, j = 1 , . . . , m . This imply that the elements h n ( g j ) have the same axis in the Bass-Serre tree that is associated with the freeproduct A n ∗ B n , and the element h n ( g ) preserves this axis setwise, and must bean elliptic element. Hence, h n ( g ) is elliptic, and h n ( g ) h n ( g j ) h n ( g ) − = h n ( g j ) − , j = 1 , . . . , m . Furthermore, h n ( g ) is an elliptic element that preserves the axisof the elements h n ( g j ) pointwise, so h n ( g ) = 1. It follows that ucu − = c − forevery c ∈ C ( A ), and u = 1. By the same argument if u , u ∈ N ( A ) \ C ( A ) then u u ∈ C ( A ), hence, [ N ( A ) : C ( A )] = 2.Let ℓ / ∈ N ( A ), and let t ∈ G be an element that is mapped to ℓ . Then there existssome index n , so that for all n > n , h n ( t ) maps the axis of h n ( g ) , . . . , h n ( g m ) toa different axis that intersects the original axis in a bounded (or empty) set. Hence, ℓC ( A ) ℓ − intersects C ( A ) trivially. (cid:3) Lemma 3 shows that the action of L on the real tree Y is (super) stable. Theoriginal analysis of stable actions of groups on real trees applies to f.p. groups ([Be-Fe1]), and the limit group L is only known to be f.g. by part (i) of lemma 3. Still,given the basic properties of the action of L on the real tree Y that we already know,we are able to apply a generalization of Rips’ work to f.g. groups obtained in [Se5]and [Gu]. In [Se5] and [Gu], the real tree Y is divided into distinct components,where on each component a subgroup of L acts according to one of several canonicaltypes of actions. The theorem from [Se5] we present, that was later corrected in6Gu], is going to be used extensively and its statement uses the notions and basicdefinitions that appear in the appendix of [Ri-Se1]. Hence, we refer a reader whois not yet familiar with these notions to that appendix and to [Be-Fe1] and [Be]. Theorem 5 (([Se5],3.1),[Gu]).
Let G be a f.g. group, let H , . . . , H r be subgroupsof G , and suppose that G can not be presented as a free product in which thesubgroups, H , . . . , H r can be conjugated into the factors. Let G admit a (super)stable isometric action on a real tree Y , so that the subgroups, H , . . . , H r , fixpoints in Y . Assume the stabilizer of each tripod in Y is trivial. There exist canonical orbits of subtrees of Y : Y , . . . Y k with the followingproperties:(i) gY i intersects Y j at most in one point if i = j .(ii) gY i is either identical with Y i or it intersects it at most in one point.(iii) The action of stab ( Y i ) on Y i is either discrete or it is of axial type orIET type. G is the fundamental group of a graph of groups with:(i) Vertices corresponding to orbits of branching points with non-trivialstabilizer in Y .(ii) Vertices corresponding to the orbits of the canonical subtrees Y , . . . , Y k which are of axial or IET type. The groups associated with these ver-tices are conjugates of the stabilizers of these components. To a stabi-lizer of an IET component is an associated 2-orbifold. All boundarycomponents and branching points in this associated 2-orbifold stabi-lize points in Y . For each such stabilizer we add edges that connectthe vertex stabilized by it and the vertices stabilized by its boundarycomponents and branching points.(iii) Edges corresponding to orbits of edges between branching points withnon-trivial stabilizer in the discrete part of Y with edge groups whichare conjugates of the stabilizers of these edges.(iv) Edges corresponding to orbits of points of intersection between the or-bits of Y , . . . , Y k . Before concluding our preliminary study of limit groups over free products thatappear as limits of sequences of homomorphisms with unbounded stretching factors { µ n } , and their actions on the limit real tree, we present the following basic factsthat are necessary in the sequel. Proposition 6.
Suppose that L is a (strict) limit group over free products, that isobtained as a limit of homomorphisms into free products with unbounded stretchingfactors { µ n } . E L is its set of elliptic elements, and the limit real tree on which L acts that is obtained from this sequence of homomorphisms is ( Y, y ) . Supposefurther that L does not admit a non-trivial free decomposition in which all theelements in the set E L can be conjugated into the various factors. Then: (i) Stabilizers of non-degenerate segments which lie in the complement of thediscrete and axial parts of Y are trivial in L . (ii) The (set) stabilizer of an axial component in Y is either a maximal abeliansubgroup in L , or it contains a maximal abelian subgroup in L as a subgroupof index 2. Let A be the maximal abelian subgroup that is contained in the set stabilizerof an axial component in Y . A can be presented as a direct sum A = A + A ,where A is the (possibly trivial) point stabilizer of the axial component, and A is a f.g free abelian group that acts freely on the axial component, and A has rank at least 2.Proof: Since the elements in E L fix points in the limit tree Y (Part (i) in lemma 3),the action of L on the real tree Y satisfies the conclusion of theorem 5. Hence, thestabilizer of a segment in an IET component in Y fixes the entire IET component,and in particular it fixes a tripod. By part (iv) of lemma 3 a stabilizer of a tripod istrivial, hence, so is the stabilizer of a non-degenerate segment in an IET componentin Y .Let Ax be an axis of an axial component in Y , and let stab ( Ax ) be its setstabilizer. Let stab + ( Ax ) be the subgroup of stab ( Ax ) that preserve the orientationof Ax . By the same argument that was used in the proof of lemma 4, stab + ( Ax ),is abelian. Since stab ( Ax ) normalizes stab + ( Ax ), lemma 4 implies that the indexof stab + ( Ax ) in stab ( Ax ) is bounded by 2.Let A = stab + ( Ax ), and let A < A be the point stabilizer of Ax . Then, bytheorem 5 (the structure of an axial component) there exists a short exact sequence:1 → A → A → B →
1, where B is a f.g. free abelian group. Since A is abelianand B is free abelian, the short exact sequence splits, and A = A + A , where A isisomorphic to B , hence, A is f.g. free abelian. (cid:3) By theorem 5 and proposition 6 a non-trivial strict limit group over free products,which is not a cyclic group, admits a non-trivial virtually abelian decomposition(i.e., a graph of groups with virtually abelian edge groups). To further study thealgebraic structure of a strict limit group we need to construct its canonical virtu-ally abelian JSJ decomposition. However, unlike the case of limit groups over freeand hyperbolic groups, in constructing the virtually abelian JSJ decomposition ofa strict limit group over free products, we will not be interested in all the virtuallyabelian decompositions of L , but only in those virtually abelian decompositions inwhich all the elements in E L are elliptic (i.e., can be conjugated into non-virtually-abelian, non-QH vertex groups), and for which the (non-trivial) maximal abeliansubgroups that are contained in the virtually abelian edge groups are not in E L .Note that since a non-trivial strict limit group over free products admits a virtuallyabelian decomposition in which the elements E L can be conjugated into non-QH,non-virtually-abelian vertex groups, and the (non-trivial) maximal abelian sub-groups of edge groups are not in E L , if a strict limit group over free products is notvirtually abelian nor a Fuchsian group, its (virtually) abelian JSJ decomposition isnon-trivial.To construct the virtually abelian JSJ decomposition of a strict limit group overfree products we need to study some basic properties of virtually abelian splittings.We start with the following lemma, which is identical to lemma 2.1 in [Se1] (theproofs are identical as well). Lemma 7.
Let L be a strict limit group over free products with set of elliptics E L , and suppose that L admits no free product in which the elements in E L can beconjugated into the various factors. Let A be a non-elliptic abelian subgroup in L , nd let M be the normalizer of A in L . Suppose that M is abelian. Then: (i) If L = U ∗ A V , and the elements in E L can be conjugated into U and V ,and M is not cyclic, then M can be conjugated into either U or V . (ii) If L = U ∗ A , and the elements in E L can be conjugated into U , and M isnot cyclic, then either M can be conjugated into U , or M can be conjugatedonto M ′ , so that L = U ∗ A M ′ . Unlike limit groups over free and torsion-free hyperbolic groups in which normal-izers of non-trivial abelian subgroups are abelian, by proposition 6 the normalizersof non-elliptic abelian subgroups in L are either abelian or virtually abelian, andif they are not abelian, the abelian centralizers of these (non-elliptic) abelian sub-groups are contained in their normalizers as subgroups of index 2. Lemma 7 dealswith the case in which the normalizer of such an abelian subgroup is abelian. Toconstruct the JSJ decomposition of limit groups over free products, we still need toanalyze splittings over non-elliptic abelian subgroups with virtually abelian, non-abelian normalizers. Lemma 8.
Let L be a limit group over free products, and let A be a non-ellipticabelian subgroup in L . Let E L be the set of elliptics in L , and suppose that L admitsno free product decomposition in which the elements of E L can be conjugated intothe factors. Let C ( A ) be the centralizer of A , let M be the normalizer of A , andsuppose that [ M : C ( A )] = 2 . Then: (i) If L = U ∗ A V , and all the elements in E L can be conjugated into U or V ,then either M can be conjugated into either U or V , or M can be conjugatedonto M ′ , and M ′ inherits an abelian amalgamation: M ′ = U ∗ A V , where U < U , V < V , [ U : A ] = [ V : A ] = 2 , and both U and V aregenerated by A and an element of order 2. In this case, M is the semidirectproduct of A with an infinite dihedral group. In this case we can modify thegiven abelian decomposition, and obtain a virtually abelian decomposition, L = U ∗ U M ′ ∗ V V . (ii) If L = U ∗ A , and the elements in E L can be conjugated into U , then either M can be conjugated into U , or M can be conjugated onto M ′ , and M ′ inheritsan abelian amalgamation: M ′ = U ∗ A V , where U < U , V < tU t − , where t is a Bass-Serre generator that is associated with the splitting, L = U ∗ A . [ U : A ] = [ V : A ] = 2 , and both U and V are generated by A and anelement of order 2. In this case, M is the semidirect product of A withan infinite dihedral group. In the HNN case, L = U ∗ A , we can modify thegiven abelian decomposition, and obtain a virtually abelian decomposition, L = ( U ∗ U M ′ ) ∗ V , where with V there are two associated embeddings, oneinto M ′ and one into tU t − . The graph of groups that is associated withthis decomposition contains two vertices (with vertex groups, U and M ′ ),and two edges with edge groups, U and V .Proof: Let L = U ∗ A V and suppose that M , the normalizer of A , that containsthe centralizer of A as a subgroup of index 2, is not elliptic. Then M preserves(setwise) an axis in the Bass-Serre tree that is associated with the amalgamatedproduct U ∗ A V . Since A preserves this axis pointwise, and M contains an (elliptic)element that acts on the axis as an inversion, M/A acts on the axis as an infinitedihedral group. Hence, it inherits from it a splitting: M = U ∗ A V , where U V contain A as a subgroup of index 2, and they are both obtained from A byadding to it an elliptic element of order 2. If we start with the Bass-Serre tree thatis associated with U ∗ A V , add a vertex in the middle of the edge that is stabilizedby A and connected the vertices that are stabilized by U and V , and then fold thecouple of edges that are stabilized by A and associated with the elements of order2 in U and V , we obtain the splitting: L = U ∗ U M ′ ∗ V V , i.e., a splitting of L with two vertex groups U and M ′ , and two edge groups, U and V .Let L = U ∗ A V . the argument that we use in this case is similar. Supposethat M , the normalizer of A , is not elliptic. In this case, M preserves (setwise)an axis in the Bass-Serre tree that is associated with the HNN extension U ∗ A . A preserves this axis pointwise, and M contains an (elliptic) element that actson the axis as an inversion, hence, M/A acts on the axis as an infinite dihedralgroup. Therefore, as in the amalgamated product case, M inherits from this actiona splitting: M = U ∗ A V , where U and V contain A as a subgroup of index 2,and they are both obtained from A by adding to it an elliptic element of order 2. U < U , and V < tU t − , for an appropriate Bass-Serre generator t . If we startwith the Bass-Serre tree that is associated with U ∗ A , add a vertex in the middleof the edge that is stabilized by A and connects the vertices that are stabilizedby U and tU t − , and then fold the couple of edges that are stabilized by A andassociated with the elements of order 2 in U and V , we obtain the splitting: L = ( U ∗ U M ′ ) ∗ V , where V embeds into V and into tU t − , i.e., the limit group L admits a graph of groups decomposition with two vertex groups, stabilized by U and M ′ , and two edges in between these two vertices, one edge is stabilized by U and the second by V . (cid:3) According to lemma 7 we replace each abelian splitting of L of the form L = U ∗ A ,in which all the elements in E L can be conjugated into U , A is a non-elliptic abeliansubgroup in L , and the centralizer of A which is also its normalizer is denoted by M , and M can not be conjugated into U , by the amalgamated product L = U ∗ A M ′ (where M ′ is a conjugate of M ). According to part (i) of lemma 8 we replace eachabelian splitting of L of the form L = U ∗ A V , in which all the elements in E L can be conjugated into U and V , A is a non-elliptic subgroup in L , and M thenormalizer of A contains the centralizer of A as a subgroup of index 2, and M cannot be conjugated into U nor V , by the amalgamated product L = U ∗ U M ′ ∗ V V ,where M ′ is a conjugate of M . If L = U ∗ A , the elements in E L can be conjugatedinto U , A is non-elliptic in L , the normalizer M of A in L contains the centralizer of A as a subgroup of index 2, and M can not be conjugated into U , then we replacethe given HNN extension by a graph of groups with two vertices and two edgesbetween them, according to part (ii) of lemma 8, L = ( U ∗ U M ′ ) ∗ V .By performing these replacements, we get that every non-elliptic abelian sub-group of L with a non-cyclic centralizer is contained in a vertex group in all thevirtually abelian splittings of L under consideration, i.e., splittings in which edgegroups are non-elliptic abelian, or edge groups that contain non-elliptic abelian sub-groups as subgroups of index 2, and all the elements in E L can be conjugated intothe vertex groups. This will allow us to use acylindrical accessibility in analyzingall the abelian splittings of the limit group over free products L that are obtainedfrom converging sequences of homomorphisms into free products. Definition 9 ([Se5],[De],[We]).
A splitting of a group H is called k - acylindrical f for every element h ∈ H which is not the identity, the fixed set of h when act-ing on the Bass-Serre tree corresponding to the splitting has diameter at most k .Following Delzant [De], and Weidmann [We], we say that a splitting of H is ( k, C ) -acylindrical if the stabilizer of a path of length bigger than k in the Bass-Serre treecorresponding to the splitting has stabilizer of cardinality at most C . If a strict limit group over free products L can be written in the form L = V ∗ A V ∗ A V ∗ A V , where A , A , A are subgroups of a maximal abeliansubgroup M , which is its own normalizer, and M is a subgroup of V , then onecan modify the corresponding graph of groups to a tripod of groups with V inthe center, and V , V , V at the 3 roots. Similarly if A , A , A are subgroups ofa maximal abelian subgroup, which is of index 2 in its normalizer M , then if M is contained in one of the vertex groups V i , i = 1 , . . . ,
4, then one can modify thecorresponding graph of groups to a tripod of groups in the same way. If M is notcontained in one of the limit groups, V i , i = 1 , . . . ,
4, then one can modify the givensplitting to a virtually abelian splitting which is a tree with one root and 4 verticesconnected to it, where M is placed at the root, and the subgroups V i , i = 1 , . . . , E L can beconjugated into non-abelian, non-QH vertex groups. Lemma 10.
Let L be a limit group over free products that does not admit a freesplitting in which all the elements in E L can be conjugated into the various factors.A splitting of L , in which all the edge groups are non-elliptic abelian subgroups in L , and in which all the elements in E L can be conjugated into non-QH, non-abelianvertex groups, can be modified using lemmas 7 and 8 to a virtually abelian splitting(of L ) in which all the normalizers of non-elliptic abelian subgroups with non-cycliccentralizers can be conjugated into non-QH vertex groups, and so that the obtainedvirtually abelian splitting is (2 , -acylindrical.Proof: Let L be a limit group over free products that admits no free factorizationin which the elements of E L can be conjugated into the factors. Let Λ be a graphof groups with fundamental group L with non-elliptic abelian edge groups. If thenormalizer of an abelian edge group in Λ can not be conjugated into a vertex groupin Λ, we perform the modification that appears in part (ii) of lemma 7 in case thenormalizer of an edge group is abelian, and the modification of parts (i) and (ii)in lemma 8 in case the normalizer of an edge group is not abelian, where thesemodifications are applicable. After performing these modifications, and slidingoperations, so that the fixed set of a non-elliptic abelian subgroup is star-like, weobtain a graph of groups Λ ′ , with virtually abelian edge groups, in which everynon-elliptic abelian subgroup with non-cyclic centralizer can be conjugated into anon-QH vertex group. In the Bass-Serre tree that corresponds to Λ ′ the fixed setof every non-elliptic element has diameter bounded by 2. Since the centralizersof non-elliptic abelian subgroups are almost malnormal, the stabilizers of pathsof length 3 in Λ ′ are either trivial, or a cyclic subgroup of order 2. Hence, Λ ′ is(2,2)-acylindrical. 11 Lemma 10 shows that if in all the virtually abelian splittings of L under con-sideration, all the normalizers of non-cyclic abelian subgroups can be conjugatedinto vertex groups, these virtually abelian splittings are (2 , Theorem 11 (cf. ([Se1],2.7)).
Suppose that L is a strict limit group over freeproducts with set of elliptics E L , so that L admits no free decomposition in whichthe elements in E L can be conjugated into the various factors. There exists areduced unfolded splitting of L with virtually abelian edge groups, which we callthe virtually abelian J SJ (Jaco-Shalen-Johannson) decomposition of L , with thefollowing properties: (i) Every canonical maximal QH subgroup (CMQ) of L is conjugate to a vertexgroup in the JSJ decomposition. Every QH subgroup of L , in which allthe elements in E L can be conjugated into vertex groups that are adjacentto the QH subgroup or into torsion elements in the QH subgroup, can beconjugated into one of the CMQ subgroups of L . Every vertex group inthe JSJ decomposition which is not a CMQ subgroup of L is elliptic in anyabelian splitting of L under consideration. (ii) Every CMQ subgroup is a Fuchsian group (in general, with torsion), whereall its torsion elements are elliptic in L . The edge groups that are connectedto a CMQ subgroup, that are all cyclic, may be elliptic. (iii) Every edge group that is not connected to a CMQ vertex group in the JSJdecomposition, or an edge group that is connected to a virtually abelianvertex group, contains an abelian subgroup of index at most 2, and thisabelian subgroup is non-elliptic. (iv)
A one edge abelian splitting L = D ∗ A E or L = D ∗ A , in which A is a non-elliptic abelian subgroup, and all the elements in E L can be conjugated into D or E , which is hyperbolic in another such elementary abelian splitting,is obtained from the virtually abelian JSJ decomposition of L by cutting a2-orbifold corresponding to a CMQ subgroup of L along a weakly essentials.c.c.. (v) Let Θ be a one edge splitting of L along a non-elliptic abelian subgroup, L = D ∗ A E or L = D ∗ A , in which all the elements in E L can be conjugated into D or E . Suppose that the given elementary splitting is elliptic with respectto any other such elementary abelian splitting of L . Then Θ is obtainedfrom the JSJ decomposition of L by a sequence of collapsings, foldings,conjugations, and finally possibly unfoldings that reverse the foldings thatare performed according to part (i) of lemma 7 and parts (i) and (ii) oflemma 8. (vi) If J SJ is another JSJ decomposition of L , then J SJ is obtained from theJSJ decomposition by a sequence of slidings, conjugations and modifyingboundary monomorphisms by conjugations (see section 1 of [Ri-Se2] forthese notions)Proof: By lemma 10 the splittings of the ambient limit group (over free products) L that are considered for the construction of the virtually abelian JSJ decomposi-tion of L , have the property that all the elliptic elements E L in L can be conjugated12nto non-QH, non-abelian vertex groups, and every non-cyclic, non-elliptic abeliansubgroup of L can also be conjugated into a non-QH vertex group. Since L ad-mits no free decompositions in which the elements E L can be conjugated into thefactors, there is no pair of elliptic-hyperbolic splittings along non-elliptic abeliangroups (so that the elements E L can be conjugated into vertex groups), i.e., allthe splittings along non-cyclic, non-elliptic abelian groups under consideration, areelliptic-elliptic with respect to all the splittings along non-elliptic abelian groupsunder consideration.Since the modifications of abelian splittings along non-elliptic abelian subgroupsthat are performed according to lemmas 7 and 8, are performed only in case thecentralizers of (non-elliptic) edge groups are non-cyclic, in case centralizers of non-elliptic edge groups are (infinite) cyclic, we consider only cyclic edge groups (andnot dihedral ones). Hence, the only hyperbolic-hyperbolic splittings under consid-eration are pairs of splittings along infinite cyclic groups. For these we can apply[Ri-Se2] (this part of [Ri-Se2], the construction of the quadratic part (section 5 inthe paper), applies to f.g. groups, and not only to f.p. ones), that produces a finitecollection of CMQ subgroups of L , and a quadratic decomposition of L with theCMQ subgroups as part of the vertex groups, so that every splitting of L alonga non-elliptic cyclic group, in which the elements of E L can be conjugated intovertex groups, and so that this splitting is hyperbolic with respect to another suchsplitting, is obtained from the quadratic decomposition of L by cutting one of theCMQ subgroups along a s.c.c. and possibly collapsing the rest of the splitting.Given the quadratic decomposition of L , to complete the construction of the(virtually abelian) JSJ decomposition of L , we successively refine the quadraticdecomposition using splittings that are elliptic with respect to it (and in whichthe elements of E L can be conjugated into vertex groups). This refinement pro-cess terminates after finitely many steps, since all the obtained splittings are (2,2)acylindrical, and by [We] this implies a bound on the combinatorial complexity ofthe obtained splitting. All the properties of the obtained (virtually) abelian JSJdecomposition of L follow in the same way as in section 7 of [Ri-se]. (cid:3) §
2. A Descending Chain Condition
In section 4 of [Se1] we were able to use the cyclic JSJ decomposition of a ( F k )limit group, in order to show that ( F k ) limit groups are f.p. and that a f.g. group isa limit group if and only if it is ω -residually free. For limit groups over a torsion-freehyperbolic group, we were able to prove similar d.c.c. and a.c.c. as in the case of afree group, even though a limit group over hyperbolic groups need not be finitelypresented.Limit groups over free products do not satisfy the d.c.c. and a.c.c. conditions thatlimit groups over free and hyperbolic groups do satisfy. However, weaker principlesdo hold for these limit groups, and these are enough to construct Makanin-Razborovdiagrams, that encode sets of solutions to systems of equations over free products.As we will see one of the keys to formulate and prove the d.c.c. and a.c.c. principlesthat we present for limit groups over free products, is our consideration of limitgroups over the entire class of free products, and not over a given one.We start with a d.c.c. for limit groups over free products which is a key to ourentire approach. It uses the techniques that were used to prove a general d.c.c. for13imit groups over hyperbolic groups, but it is not as general as in the case of limitgroups over free and hyperbolic groups. Definition 12.
Let G be a f.g. group. On the set of limit groups over free productsthat are quotients of G , together with the quotient maps from G to these limitgroups, we define a partial order. Let L , L be two limit groups over free productsthat are quotients of G , with sets of elliptics, E L , E L , in correspondence, and withprescribed quotient maps η i : G → L i , i = 1 , . We write that ( L , η ) > ( L , η ) ,if there exists an epimorphism: τ : L → L , that maps the elliptics in L into theelliptics of L , τ ( E L ) ⊂ E L , and for which τ : L → L satisfies either: (1) τ has a non-trivial kernel. (2) τ is an isomorphism, and τ ( E L ) is a proper subset of E L .If there exists an isomorphism τ : L → L that maps the elliptics in L ontothe elliptics in L , and for which: η = τ ◦ η , we say that ( L , η ) is in the sameequivalence class as ( L , η ) . Note that the relation that is defined on the limitquotients (over free products) of a f.g. group is a partial order. Theorem 13.
Let G a f.g. group. Every strictly decreasing sequence of limit groupsover free products that are quotients of G : ( L , η ) > ( L , η ) > ( L , η ) > . . . for which: (1) the maps: τ i : L i → L i +1 , that satisfy: η i +1 = τ i ◦ η i , are proper quotientmaps (i.e., have non-trivial kernels). (2) the maps τ i do not map non-trivial elements in E L i to the identity elementin L i +1 .terminates after finitely many steps.Proof: The argument that we use is a modification of the argument that is usedto prove theorem 1.12 in [Se3]. Suppose that there exists a f.g. group G for whichthere exists an infinite decreasing sequence of limit groups over free products thatare quotients of G : L > L > L > . . . that satisfy the conditions of the theorem.W.l.o.g. we may assume that the f.g. group G is a free group F d , for some integer d .We fix F d , where d is the minimal positive integer for which there exists an infinitedescending chain of limit groups over free products so that consecutive quotientmaps, τ i : L i → L i +1 , have non-trivial kernels and do not map non-trivial ellipticelements to the identity element, and fix a free basis for F d , F d = < f , . . . , f d > .We set C to be the Cayley graph of F d with respect to the given generating set,and look at an infinite decreasing sequence constructed in the following way. Weset R to be a limit group over free products, which is a quotient of F d , with thefollowing properties:(1) R is a proper quotient of F d .(2) R can be extended to an infinite decreasing sequence of limit groups overfree groups: R > L > L > . . . , that satisfy the conditions of the theorem.(3) The map η : F d → R maps to the identity the maximal number of elementsin the ball of radius 1 in the Cayley graph C , among all possible maps from F d to a limit group over free products L , that satisfy properties (1) and (2).14e continue iteratively. At step n , given the finite decreasing sequence R > R >. . . > R n − , we choose the limit group over free products, R n , to satisfy:(1) R n is a proper quotient of R n − .(2) The finite decreasing sequence of limit groups over free products: R >R > . . . > R n can be extended to an infinite decreasing sequence thatsatisfies the conditions of the theorem.(3) The map η n : F d → R n (that is obtained as a composition of the map F d → R with the sequence of proper epimorphisms: R i → R i +1 , i = 1 , . . . , n − n in the Cayley graph C , among all the possible maps from F d to a limitgroup over free products, L n , that satisfy the properties (1) and (2).To prove theorem 13, we will show that the last descending sequence we con-structed terminates after finitely many steps. With the decreasing sequence R >R > . . . we associate a sequence of homomorphisms into free products: { h n : F d → A n ∗ B n } . For each index n , R n is a quotient of F d , hence, R n is generatedby d elements that are the image of the fixed generators of F d under the quotientmap η n . R n is a limit group over free products. Hence, R n , with its set of elliptics E R n ,is obtained from a convergent sequence of homomorphisms { u s : G n → C s ∗ D s } ,where G n is a f.g. group. Since R n is generated by the image of the elements f , . . . , f d under the quotient map η n , for large enough s , the images u s ( G n ) are d-generated groups, and furthermore, they are generated by the images of d elementsin the f.g. group G n , that are mapped by the quotient map ν n : G n → R n ontothe elements η n ( f ) , . . . , η n ( f d ). Hence, we may assume that the limit groups overfree products, R n , are obtained as limit groups from a sequence of homomorphisms { v s : F d → C s ∗ D s } , and the image of the fixed generating set of the free group F d , is the set of elements η n ( f ) , . . . , η n ( f d ).For each index n , we pick h n to be a homomorphism h n : F d → A n ∗ B n , sothat h n is a homomorphism v s : F d → C s ∗ D s for some large index s , so that h n satisfies the following two conditions:(i) every element in the ball of radius n of C , the Cayley graph of F d , thatis mapped by the quotient map η n : F d → R n to the trivial element, ismapped by h n to the trivial element in A n ∗ B n . Every such element thatis mapped to a non-trivial element by η n , is mapped by h n to a non-trivialelement in A n ∗ B n .(ii) every element in the ball of radius n of C , the Cayley graph of F d , thatis mapped by the quotient map η n : F d → R n to an elliptic element, ismapped by h n to an elliptic element in A n ∗ B n . Every such element that ismapped to a non-elliptic element by η n , is mapped by h n to a non-ellipticelement in A n ∗ B n .From the sequence { h n } we can extract a subsequence that converges into a limitgroup over free products, that we denote R ∞ . By construction, the limit group R ∞ is the direct limit of the sequence of (proper) epimorphisms: F d → R → R → . . . Let η ∞ : F d → R ∞ be the canonical quotient map. Our approach towardsproving the termination of given descending chains of limit groups over free products15s based on studying the structure of the limit group R ∞ , and its associated quotientmap η ∞ . We start this study by listing some basic properties of them. Lemma 14. (i) R ∞ is not finitely presented. (ii) R ∞ can not be presented as the free product of a f.p. group and freely inde-composable elliptic subgroups. (iii) Let R ∞ = U ∗ . . . ∗ U t ∗ F be the most refined (Grushko) free decompositionof R ∞ in which the elliptic elements in R ∞ , E R ∞ , can be conjugated intothe various factors, and F is a f.g. free group. Then there exists an index j , ≤ j ≤ t , for which:(1) U j is not finitely presented nor elliptic.(2) If B is a f.g. subgroup of F d for which η ∞ ( B ) = U j , then U j is astrict limit group over free products of a subsequence of the restrictedhomomorphisms, h n | B .Proof: To prove part (i), suppose that R ∞ is f.p. i.e.: R ∞ = < g , . . . , g d | r , . . . , r s > . Then for some index n , and every index n > n , h n ( r j ) = 1 for j = 1 , . . . , s . Thisimplies that for some index n > n , and every index n > n , each of the groups R n is a quotient of R ∞ , by a quotient map that send the generating set g , . . . , g d of R ∞ to the elements η n ( f ) , . . . , η n ( f d ), a contradiction.Suppose that R ∞ = V ∗ . . . ∗ V t ∗ M where M is f.p. and each of the factors V j is elliptic. Let B , . . . , B t and D be f.g. subgroups of F d , for which η ∞ ( B j ) = V j for j = 1 , . . . , t , and η ∞ ( D ) = M . W.l.o.g. we may assume that the free group F d is generated by the collection of the subgroups B , . . . , B t , D .Since the factors V j , j = 1 , . . . , t , are elliptic, and since the subgroups, B j , j =1 , . . . , t , are f.g. for every index j , j = 1 , . . . , t , there exists an index n j , so that forevery index n > n j , the image η n ( B j ) is elliptic. Since the maps τ i : R i → R i +1 do not map non-trivial elliptic elements (in E R i ) to the identity element, η n ( B j ) isisomorphic to η ∞ ( B j ) = V j via the map η ∞ ◦ η − n .The factor M is assumed f.p., hence, if D = < d , . . . , d s > , then M = < d , . . . , d s | r , . . . , r u > .There exists an index n , for which for every index n > n , η n ( r i ) = 1, for i = 1 , . . . , u .Let m > n j for j = 0 , . . . , t . By our arguments, from the universality of freeproducts, all the relations that hold in R ∞ hold in η m ( F d ) = R m . Hence, R m isa quotient of R ∞ , where the quotient map maps the prescribed generators of R ∞ to the prescribed generators of R m (i.e., the corresponding images of the givenset of generators F d = < f , . . . , f d > ). Since R m +1 is a proper quotient of R m ,this implies that R m +1 is a proper quotient of R ∞ , again by a map that maps theprescribed set of generators of R ∞ to the prescribed set of generators of R m +1 ,which clearly contradicts our assumptions that the sequence of limit groups { R j } is strictly decreasing with R i +1 being a proper quotient of R i for every index i , andthe limit group R ∞ is the direct limit of this decreasing sequence. This concludesthe proof of part (ii).To prove part (iii) note that (1) in part (iii) follows from part (ii). Every factor U j of the limit group L that is not elliptic is a strict limit group that is obtainedfrom a sequence of homomorphisms of some f.g subgroup of F d , and (2) follows.16 R ∞ is a limit group over free products which is a (proper) quotient of all thelimit groups over free products, { R n } . For each index n , the limit group R n waschosen to maximize the number of elements that are mapped to the identity in theball of radius n of F d by the quotient map η n : F d → R n , among all the properlimit (over free products) quotients of R n − that admit an infinite descending chainof limit groups over free products that satisfy the conditions of theorem 13. If R ∞ admits an infinite descending chain of limit groups over free products: R ∞ → L → L → . . . that satisfy the properties in theorem 13, then the limit group (over free products) L admits an infinite descending chain of limit groups that satisfy the conditions oftheorem 13, and since it is a proper quotient of R ∞ , for large enough index n , thequotient map ν n : F d → L maps to the identity strictly more elements of the ballof radius n in the Cayley graph of F d , than the map η n : F d → R n , a contradiction.Hence, R ∞ does not admit an infinite descending chain of limit groups over freeproducts that satisfy the conditions of theorem 13.To continue the proof of theorem 13, i.e., to contradict the existence of the infinitedescending chain of limit groups over free products that satisfies the conditions ofthe theorem, we need a modification of the shortening procedure that was used in[Se1] for ( F k ) limit groups, and in [Se3] for limit groups over hyperbolic groups.Since the description of the shortening procedure is rather long and involved, weprefer not to repeat it, and refer the interested reader to section 3 of [Se1]. Thesame construction that appears in [Se1] applies to (strict) limit groups over freeproducts.Given a f.g. group G , and a sequence of homomorphisms into free products: { u s : G → A s ∗ B s } , that converges into a (strict) limit group over free products, L , the shortening procedure constructs another (sub) sequence of homomorphismsfrom a free group F d (where the f.g. group G is generated by d elements), { v s n : F d → A s n ∗ B s n } , so that the sequence of homomorphisms v s n converges to a limitgroup over free products SQ , and there exists a natural epimorphism L → SQ , thatmaps the elliptic elements in L , E L , monomorphically into the elliptic elements in SQ , E SQ . Definition 15.
We call the limit group over free products, SQ , that is obtainedby the shortening procedure, a shortening quotient of the limit group (over freeproducts) L . By construction, a shortening quotient of a limit group over free products is,in particular, a quotient of that limit group. In the case of freely indecomposable F k -limit groups, a shortening quotient is always a proper quotient ([Se1],5.3). Ifthe limit group over free products that we start with, L , is strict, non-cyclic andadmits no free decomposition in which the elements of E L can be conjugated intothe factors, a shortening quotient of L is a proper quotient of it. More generallywe have the following. Proposition 16.
Let G be a f.g. group, and let { u s : G → A s ∗ B s } be a sequenceof homomorphisms that converges into an action of a non-cyclic, strict limit group ver free products, L , on some real tree Y , where L admits a (possibly trivial)free decomposition in which the elliptic elements, E L , can be conjugated into thefactors, and so that there exists at least one factor in this free decomposition, whichis strict, non-cyclic, and freely indecomposable relative to its intersection with E L .Then every shortening quotient of L , obtained from the sequence { u s } , is a properquotient of L (in which non-trivial elliptic elements in L are not mapped to non-trivial elliptic elements of the shortening quotient).Proof: Suppose that the f.g. group G is generated by d elements. A shorteningquotient SQ of L is obtained from a sequence of homomorphisms { v s n : F d → A s n ∗ B s n } that converges into SQ . Let L be a factor in a (possibly trivial) freedecomposition of L , in which all the elements E L can be conjugated into the variousfactors, so that the factor L is a non-cyclic strict limit group (over free products),which is freely indecomposable relative to its intersection with E L .Let SQ be the image of L in the shortening quotient SQ . Note that SQ isa shortening quotient of L . By construction, the shortening quotient SQ is aquotient of the non-cyclic, strict limit group over free products L , which is freelyindecomposable relative to its elliptic elements, E L . If the sequence of homomor-phisms { v s n } , restricted to some f.g. preimage of L , has bounded stretching factors,i.e., if the shortening quotient SQ is not strict, SQ can not not be (entirely) el-liptic, hence, it must be freely decomposable or cyclic, so it is a proper quotient of L . If SQ is a strict limit group over free products, then the shortening quotient SQ is a proper quotient of L by the shortening argument that is used in the proofof claim 5.3 in [Se1]. (cid:3) The shortening procedure, and proposition 16, enable us to obtain a resolution of the limit group R ∞ , with which we can associate a completion , into which R ∞ embeds. This completion enables us to present R ∞ as a f.g. group which is finitelypresented over some of its elliptic subgroups. Since theorem 13 assumes that thesuccessive maps along the infinite descending chains under consideration, τ i , do notmap non-trivial elliptic elements to the identity element, it is implied that ellipticsubgroups embed along the sequences under consideration. This implies that forlarge enough n , R n is a proper quotient of R ∞ , which contradicts the fact that R ∞ is a proper quotient of all the limit groups (over free products), { R n } , that appearin the infinite descending chain we constructed. Proposition 17.
Let R ∞ be the direct limit of the sequence of limit groups overfree products that we constructed (in order to prove theorem 13), { R n } . Then thereexists a finite sequence of limit groups over free products: R ∞ → L → L → . . . → L s for which: (i) L is a shortening quotient of R ∞ , and L i +1 is a shortening quotient of L i ,for i = 1 , . . . , s − . (ii) The epimorphisms along the sequence are proper epimorphisms, and non-trivial elliptic elements in L i are mapped to non-trivial elliptic elements in L i +1 . (iii) L s = H ∗ . . . ∗ H r ∗ F t where the factors, H , . . . , H r , are elliptic, and theentire elliptic set, E L s , is the union of the conjugates of H , . . . , H r . F t isa (possibly trivial) free group. The resolution: R ∞ → L → L → . . . → L s is a strict resolution ([Se1],5),i.e., in each level non-QH, non-virtually-abelian vertex groups in the virtu-ally abelian JSJ decomposition are mapped monomorphically into the limitgroup in the next level, and QH vertex groups are mapped into non-virtually-abelian, non-elliptic subgroups. (v) The constructed resolution is well-structured (see definition 1.11 in [Se2]for a well-structured resolution).Proof:
By lemma 14 and proposition 16 a shortening quotient of R ∞ is a properquotient of it. Furthermore, non-trivial elliptic elements in R ∞ are mapped tonon-trivial elliptic elements in the shortening quotient. Hence, we set L to be ashortening quotient of R ∞ . If from the sequence of (shortened) homomorphismsthat was used to construct L , it’s possible to extract a subsequence that satisfythe properties of lemma 14, we continue with this subsequence, and use it to geta shortening quotient L of L , which by proposition 16, is a proper quotient of L . Continuing this process iteratively, and recalling that every descending chain oflimit groups over free products that starts with R ∞ and satisfies the assumptionsof the statement of theorem 13, terminates after finitely many steps, we finally getthe sequence of proper epimorphisms: R ∞ → L → L → . . . → L s . Parts (i) and (ii) follow immediately from the construction of the descendingfinite sequence of shortening quotients, and part (iii) follows, since by lemma 14and proposition 16, the descending sequence of shortening quotients terminates,precisely when the obtained limit group is the free product of elliptic factors anda free group. Part (iv) follows since each shortening quotient in the sequenceis constructed from homomorphisms that converge into the previous limit groupin the sequence that were modified by modular automorphisms. Part(v) followssince like in the case of free and hyperbolic groups, every strict Makanin-Razborovresolution, i.e., a resolution that is obtained from a sequence of shortening quotients,is well-structured (see definition 1.11 in [Se2]). (cid:3)
Proposition 17 constructs from a subsequence of the homomorphisms, { h n : F d → A n ∗ B n } , a well-structured resolution of the limit group over free products, R ∞ , that terminates in a limit group L s which is a free product of elliptic sub-groups and a (possibly trivial) free group. In section 1 of [Se2], a completion isconstructed from a given well-structured resolution (see definition 1.11 in [Se2] fora well-structured resolution). This construction that generalizes in a straightfor-ward way to well-structured resolutions of limit groups over torsion-free hyperbolicgroups in [Se3], generalizes in a straightforward way to well-structured resolutionsof limit groups over free products. For the detailed construction of the completionsee definition 1.12 in [Se2].We denote the completion of the well-structured resolution that is constructedin proposition 17, Comp ( Res ). By definition 1.12 and lemma 1.13 in [Se2], eachof the limit groups, R ∞ , L , . . . , L s is embedded into the completion of the con-structed resolution, Comp ( Res ). All the (virtually abelian) edge groups that arenot connected to boundary elements of QH vertex groups (edge groups that areconnected to QH vertex groups are always cyclic), and all the virtually abelian19ertex groups, contain abelian subgroups as subgroups of index at most 2. Further-more, these abelian groups are non-elliptic subgroups of the associated limit groups, R ∞ , L , . . . , L s . Since the only non-elliptic abelian subgroups of the terminal limitgroup, L s , are infinite cyclic, all the edge groups, and all the vertex groups, thatappear in all the levels of the completion, Comp ( Res ), are finitely generated. Inparticular all the edge groups and all the vertex groups that appear in the virtuallyabelian JSJ decompositions of the limit groups over free products, R ∞ , L , . . . , L s ,are finitely generated.Let ρ : R ∞ → Comp ( Res ) be the embedding of the limit group over free prod-ucts, R ∞ , into the completion of the constructed resolution: R ∞ → L → . . . → L s . ρ ( R ∞ ) being a f.g. subgroup of Comp ( Res ) inherits a (finite) virtually abelian de-composition from the virtually abelian decomposition that is associated with thetop level of
Comp ( Res ). The edge groups in that inherited (finite) virtually abeliandecomposition are subgroups of f.g. virtually abelian groups, hence, f.g. virtuallyabelian groups. The vertex groups in that virtually abelian decomposition can ei-ther be conjugated into subgroups of a lower level of the completion, or they can beconjugated into QH groups or into f.g. virtually abelian groups. f.g. subgroups ofvirtually abelian groups are again f.g. virtually abelian. f.g. subgroups of Fuchsiangroups are free products of f.g. Fuchsian groups and f.g. virtually free groups. Hence R ∞ can be reconstructed from finitely many f.g. groups that can be conjugated intolower level of the completion, Comp ( Res ), and finitely many f.g. Fuchsian groups,f.g. virtually free groups, and f.g. virtually abelian groups, by performing free prod-ucts and free products with amalgamation and HNN extensions along f.g. virtuallyabelian groups.Continuing with this decomposition procedure along the lower levels of the com-pletion,
Comp ( Res ), we get that the subgroup ρ ( R ∞ ) (that is isomorphic to R ∞ )can be reconstructed from finitely many f.g. elliptic subgroups in Comp ( R ∞ ),and finitely many f.g. Fuchsian groups, f.g. virtually free groups, and f.g. virtu-ally abelian groups, by performing finitely many operations of free products andfree products with amalgamation and HNN extensions along f.g. virtually abeliangroups. In particular, R ∞ is obtained from finitely many elliptic subgroups of R ∞ by adding finitely many generators and relations.By construction, the limit group (over free products), R ∞ , is the direct limit ofthe decreasing sequence of limit groups, { R n } , which are all quotients of some freegroup, F d . Every f.g. subgroup of F d that is mapped to an elliptic subgroup in R ∞ ,is mapped to elliptic subgroups in all the limit groups, R n , for all n > n for someindex n . R ∞ is generated by finitely many f.g. elliptic subgroups and finitely manyvirtually abelian, virtually free, and f.g. Fuchsian groups together with finitely manyBass-Serre generators that are added in each of the performed HNN extensions(along f.g. virtually abelian subgroups). Since these last groups are all f.p. andelliptic subgroups in each of the limit groups R n are mapped monomorphicallyinto R ∞ by our assumptions on the decreasing sequence, { R n } , there exists someindex n , so that for all n > n , the limit group R n is generated by finitely manyelements that are mapped to the Bass-Serre elements that are used in constructing R ∞ , finitely many subgroups that are isomorphic to the f.g. virtually abelian, f.g.virtually free, and f.g. Fuchsian groups, and finitely many elliptic subgroups thatare isomorphic to the f.g. elliptic subgroups that altogether generate R ∞ . Since R ∞ is generated by these subgroups and the Bass-Serre elements by imposing finitelymany relations, there exists some index n , so that for every n > n these relations20old in R n , which implies that R n is a quotient of R ∞ using a quotient map thatmaps the fixed generating set of R ∞ (the images of a fixed basis of F d ) to thefixed generating set of R n . This implies that R n +1 is a proper quotient of R ∞ bya quotient map that maps the fixed generating set of R ∞ to the fixed generatingset of R n +1 , which contradicts the construction of R ∞ as the direct limit of thedecreasing sequence of limit groups over free products, { R n } . This finally impliesthe d.c.c. that is stated in theorem 13. (cid:3) Theorem 13 proves a basic d.c.c. that holds for limit groups over free products.This d.c.c. is weaker than the ones proved for limit groups over free and hyperbolicgroups ([Se1],[Se3]). Indeed it is stated only for decreasing sequences of limit groupsover free products for which the successive maps do not map non-trivial ellipticelements to the identity. Still, this d.c.c. is the basis for our analysis of limit groupsover free products, and for the analysis of solutions to systems of equations overfree products.We start with the following theorem, which is a rather immediate corollary of thed.c.c. that is stated in theorem 13, that associates a resolution with a given limitgroup over free products, a resolution that has similar properties to the resolutiondescribed in proposition 17.
Theorem 18.
Let L be a limit group over free products. Then there exists a finitesequence of limit groups over free products: L → L → L → . . . → L s for which: (i) L is a shortening quotient of L , and L i +1 is a shortening quotient of L i ,for i = 1 , . . . , s − . In particular, elliptic elements in L i are mappedmonomorphically to elliptic elements in L i +1 . (ii) The epimorphisms along the sequence are proper epimorphisms. (iii) L s = H ∗ . . . ∗ H r ∗ F t where the factors, H , . . . , H r , are elliptic, and theentire elliptic set, E L s , is the union of the conjugates of H , . . . , H r . F t isa (possibly trivial) free group. (iv) The resolution: L → L → L → . . . → L s is a strict resolution ([Se1],5),i.e., in each level non-QH, non-virtually-abelian vertex groups and edgegroups in the virtually abelian JSJ decomposition are mapped monomor-phically into the limit group in the next level, and QH vertex groups aremapped into non-virtually-abelian, non-elliptic subgroups. (v) The constructed resolution is well-structured (see definition 1.11 in [Se2]for a well-structured resolution). As a corollary, the limit group (over freeproducts) L is embedded into the completion of the well-structured resolu-tion: L → L → L → . . . → L s so that all the elliptic elements in L are mapped into conjugates of theelliptic subgroups, H , . . . , H r , of L s .Proof: Theorem 18 generalizes the resolution that was constructed for the limitgroup (over free products) R ∞ , to general limit groups over free products. To proveproposition 17 we used the d.c.c. for resolutions of R ∞ for which the epimorphisms21hat are associated with them do not map non-trivial elliptic elements to the identityelement, that follows from the construction of R ∞ . Theorem 13 proves that such ad.c.c. holds for resolutions of an arbitrary limit group over free products, for whichthe associated epimorphisms do not map non-trivial elliptic elements to the identityelement. With this general d.c.c. the proof of proposition 17 generalizes to generallimit groups over free products. (cid:3) §
3. Finitely Presented Groups
Theorem 13 proves the basic d.c.c. for limit groups over free products, and the-orem 18 associates a resolution with each such limit group, hence, it embeds eachlimit group over free products into a completion, where this completion is a towerover a limit group which is a free product of a (possibly trivial) free group with a(possibly empty) finite collection of f.g. elliptic subgroups.When considering limit groups over free products we analyzed sequences of ho-momorphisms from a f.g. group into free products. Since our goal is to obtain astructure theory for sets of solutions to systems of equations, and the group thatis associated formally with a finite system of equations is f.p. and not only f.g. wemay assume that the limit groups over free products that we are considering areobtained from sequences of homomorphisms from a f.p. group into free products(and not only from a f.g. one).As we will see in the sequel, if we attempt to construct a Makanin-Razborovdiagram that is associated with a f.p. group, we will need to consider only f.g. limitgroups over free products that are recursively presented, i.e., limit groups that canbe embedded into f.p. groups. A modification or a strengthening of the existenceof such an embedding is a key for obtaining further d.c.c. that will eventually allowthe construction of a Makanin-Razborov diagram over free products for a given f.p.group. We start with the following simple observation.
Proposition 19.
Let G be a f.p. group, and let L be a limit group over free productswhich is a quotient of G . Then there exists a limit group over free products ˆ L withthe following properties: (1) there is a f.p. completion, Comp , which is a tower over a free product offinitely many f.p. elliptic subgroups and a free group, so that ˆ L embeds into Comp , and the elliptic elements in ˆ L are mapped into conjugates of thefinitely many elliptic factors in the free decomposition that is associatedwith the limit group that appears in the terminal level of the completion Comp . (2) either ˆ L = L or ˆ L > L (see definition 12 for the relation > on limit groupsover free products).Proof: By theorem 18, the limit group (over free products) L admits a well-structured resolution: L → L → L → . . . → L s and L s admits a free product decomposition: L s = H ∗ . . . ∗ H r ∗ F t where thefactors, H , . . . , H r , are elliptic, and the entire elliptic set, E L s , is the union of theconjugates of H , . . . , H r . F t is a (possibly trivial) free group.22urthermore, with this resolution it is possible to associate a completion, Comp ,and the limit group L embeds into this completion, so that all the elliptic elementsin L are mapped into conjugates of the elliptic subgroups, H , . . . , H r , of L s (thegroups L , . . . , L s admit natural embeddings into the various levels of the com-pletion, Comp , and the elliptics in each of these limit groups are mapped intoconjugates of H , . . . , H r in the completion Comp ).Since L is embedded into the completion Comp , G is naturally mapped into Comp . By construction, the completion Comp is built as a tower over the termi-nal limit group L s . If Comp is f.p. we obtained the conclusion of the proposition,as we can take ˆ L = L , and ˆ L is embedded into the f.p. completion Comp . Hence,we may assume that Comp is not finitely presented, i.e., at least one of the factors, H , . . . , H r , is not finitely presented. In that case we gradually replace Comp bya f.p. completion into which G is mapped.Each of the factors of L s , H , . . . , H r , is f.g. so it is a quotient of some f.g. freegroup. Let F , . . . , F r be f.g. free groups that H , . . . , H r are quotients of. Westart the construction of a f.p. completion that replaces the completion Comp , witha tower T that has in its base level the free group F ∗ . . . ∗ F r ∗ F t , and the next(upper) levels are connected to the lower levels of the constructed tower, preciselyas they are connected in the completion, Comp , i.e., using the same graphs ofgroups, just that the group that is associated with the lowest level in Comp ,which is L s = H ∗ . . . ∗ H r ∗ F t , is replaced by the free group, F ∗ . . . ∗ F r ∗ F t .Note that T is a tower, but it is not necessarily a completion (see definition 1.12in [Se2]), as in general there are no retractions from a group that is associated witha certain level onto the group that is associated with the level below it. Each of thelevels above the base level in T is constructed using a (finite) graph of groups, inwhich some vertex groups are the groups that are associated with the lower level in T . Hence, the group that is associated with a level above the base level, is obtainedfrom a free product of the group that is associated with the lower level with a f.p.group by imposing finitely many relations. Furthermore, the graphs of groups thatare associated with the different levels in T are similar to the graphs of groups thatare associated with the corresponding levels in the completion Comp , and differfrom Comp only in the groups that are associated with the base level.Since each of the groups that are associated with the upper levels in T is obtainedfrom a free product with a f.p. group by imposing finitely many relations, and sincethe graphs of groups that are associated with the upper levels have similar structureas the corresponding graphs of groups that are associated with the levels of thecompletion Comp , and these graphs of groups differ only in the structure of thegroup that is associated with the base level, it is enough to impose only finitelymany relations from the defining relations of the various factors of the limit groupsthat is associated with the base level of Comp , L s , H , . . . , H r , on the associatedfree groups, F , . . . , F r , so that if we replace the group that is associated with thebase level of T , F ∗ . . . , F r ∗ F t , with the obtained f.p. quotient, V ∗ . . . ∗ V r ∗ F t ,and construct from it a tower, T , by imitating the construction of Comp and T (i.e., with similar graphs of groups in all the upper levels), T will be a completion. T is a completion, but it may be that the f.p. group G is not mapped into it. G is mapped into the completion Comp . Hence, once again, since G is f.p. it isenough to impose only finitely many relations from the defining relations of thevarious factors, H , . . . , H r , on the factors, V , . . . , V r , so that if we replace thegroup that is associated with the base level in T with the obtained f.p. quotient,23 ∗ . . . ∗ M r ∗ F t , and construct from it a tower T by imitating the constructionof the towers Comp , T , and T , T is a f.p. completion, and G maps into it.Furthermore, the map from G into the completion Comp , is a compositionof the maps from G to T , composed with the natural quotient map from T to Comp . Hence, if we denote the image of G in T by ˆ L , then ˆ L is a limit groupover free products, its set of elliptics is precisely the intersection of ˆ L with theset of conjugates of M , . . . , M r , and either ˆ L is isomorphic to L and the naturalisomorphism from ˆ L onto L maps the elliptics in ˆ L monomorphically onto theelliptics in L , or the natural epimorphism from ˆ L onto L has a non-trivial kernel,and this epimorphism maps the elliptics in ˆ L onto the elliptics in L , in which caseˆ L > L . (cid:3) Proposition 19, the d.c.c. proved in theorem 13, and the resolution that is asso-ciated with a limit group over free products in theorem 18, enable us to prove thatthere are maximal elements in the set of all limit groups over free products thatare all quotients of a (fixed) f.p. group G , and that there are only finitely manyequivalence classes of such maximal elements. The existence of maximal elementsin the set of limit quotients is valid even for f.g. groups. Proposition 20.
Let G be a f.g. group. Let R , R , . . . be a sequence of limitgroups over free products that are all quotients of the f.g. group G , and for which: R < R < . . . Then there exists a limit group over free products R that is a quotient of G , so thatfor every index m , R > R m .Proof: Identical to the proof in the free and hyperbolic groups cases (see proposi-tion 1.20 in [Se3]). (cid:3)
Proposition 20 proves that given an ascending chain of limit quotients (over freeproducts) of a f.g. group G , there exists a limit quotient of G that bounds all thelimit groups in the sequence. Hence, we can apply Zorn’s lemma (it is enough toconsider countable ascending chains in case of quotients of a f.g. group), and obtainmaximal limit quotients (over free products) of any given f.g. group, and every limitquotient of a f.g. group is dominated by a maximal limit quotient of that group.Proposition 19 proves that if G is in addition f.p. then if R is a limit quotient of G (over free products), then there exists a limit group over free products L , that iseither isomorphic to R or L > R , so that L embeds in a f.p. completion. Hence, ifwe are interested in maximal limit quotients (over free products) of a f.p. group G ,it is enough to consider limit quotients of G that embed in f.p. completions, andthere are clearly at most countably many such limit quotients.In case a group G is f.p. the existence of maximal limit quotients, and the exis-tence of an embedding of maximal limit quotients of a f.p. group G into f.p. com-pletions, imply the finiteness of the (equivalence classes of) maximal limit quotients(over free products) of a f.p. group. Theorem 21.
Let G be a f.p. group. Then there are only finitely many equivalenceclasses of maximal elements in the set of limit quotients (over free products) of G ,and each of these maximal elements embeds in a f.p. completion. roof: Let G be a f.p. group. Since all its maximal limit quotients over freeproducts can be embedded into f.p. completions, there are at most countably manymaximal limit quotients of G (over free products). Suppose that there are infinitelymany non-equivalent maximal limit quotients of G , and let R , R , . . . be the infinitesequence of (non-equivalent) maximal limit quotients (over free products) of G .Each R i is equipped with a given quotient map η i : G → R i , hence, fixing agenerating set for G , we fix a generating set in each of the R i ’s. i.e., we havemaps ν i : F d → R i (assuming G has rank d ), that factor through the epimorphism F d → G .For each index i we look at the collection of words of length 1 in F d that aremapped to the identity, and those that are mapped to elliptic elements by ν i . Thereis a subsequence of the R i ’s for which this (finite) collection of words is identical.Starting with this subsequence, for each R i (from the subsequence) we look at thecollection of words of length 2 in F d that are mapped to the identity and those thatare mapped to elliptic elements by ν i , and again there is a subsequence for whichthis (finite) collection is identical. We continue with this process for all lengths ℓ of words in F d , and look at the diagonal sequence (that we denote R i , R i , . . . ).We choose homomorphisms h j : F d → A j ∗ B j , that factor through the map F d → G , so that for words w of length at most j , h j ( w ) = 1 iff ν i j ( w ) = 1, and h j ( w ) is elliptic iff ν i j ( w ) is elliptic (we can choose such homomorphisms since R i j is a limit quotient of G ). After passing to a subsequence, the homomorphisms h j converge into a limit group over free products M , which is a limit quotient of G .Note that in the (canonical) map F d → M , the elements of length at most j thatare mapped to the identity, and those that are mapped to be elliptic, are preciselythose that are mapped to the identity and those that are mapped to be elliptic bythe map ν i j : F d → R i j . R , R , . . . form the entire list of maximal limit quotients of G over free products.We construct a new sequence of homomorphisms: f j : F d → C j ∗ D j that factorthrough the quotient map F d → G . First, f j has the same property as h j , i.e., theelements of length at most j that are mapped to the identity by f j are preciselythose that are mapped to the identity by ν i j : F d → R i j , and the elements of lengthat most j that are mapped to be elliptic by f j are precisely those that are mappedto be elliptic by ν i j : F d → R i j . Second, since R i j is maximal and is not equivalentto R , . . . , R i j − , there must exist some elements u , . . . , u i j − ∈ F d so that foreach index s , 1 ≤ s ≤ i j −
1, either u s is mapped to the identity in R s , but u s is mapped to a non-trivial element in R i j by ν i j , or u s is mapped to an ellipticelement in R s , but u s is mapped to a non-elliptic element in R i j by ν i j . If the firstholds, we require that f j ( u s ) = 1, and if the second holds we require that f j ( u s ) isnot elliptic.The sequence of homomorphisms, { f j } , converges into the limit group (overfree products) M . We look at a subsequence of the homo. { f j } , so that thesubsequence and its shortenings converge into a resolution of M that satisfy theproperties that are listed in theorem 18, M → L → L → . . . → L s (we stilldenote the subsequence { f j } ).With the resolution M → L → L → . . . → L s , which is a well-structuredresolution by construction, we can naturally associate a completion. Let Comp bethis completion. Since G is naturally mapped onto the limit group M , there existsa natural map, ρ : G → Comp , that factors through the map G → M .Note that by construction, the completion Comp is obtained from the terminal25imit group, L s , of the given resolution of M , by adding finitely many generatorsand relations. Since the group G is f.p. we can repeat the argument that was usedto prove proposition 19, and replace the terminal limit group L s with a (possiblythe same) f.p. group L s that maps onto L s , and starting with L s construct acompletion, Comp , that has the same structure as the completion Comp , exceptthat the terminal limit group (over free products) of
Comp is L s , whereas theterminal limit group of the completion Comp is L s . Furthermore, the group G mapsinto Comp , and since Comp is finitely presented, there exists a subsequence ofthe sequence of homomorphisms { f j } , that factor through the completion Comp .Let M be the image of G in Comp . M is a limit quotient of G (over freeproducts), so there must exist some maximal limit quotient of G , that we denote R b , so that R b is either equivalent to M or R b > M . Now, there exists a subse-quence of the homomorphisms { f j } that factor through the limit group M , hence,this subsequence of homomorphisms factor through the maximal limit quotient R b .By construction, each of the homomorphisms f j does not factor through any ofthe maximal limit groups, R , . . . , R i j − . Hence, for large enough j , none of thehomomorphisms f j factor through the maximal limit quotient R b , a contradiction.Therefore, G admits only finitely many maximal limit quotients (over free prod-ucts), and by proposition 19, each of the maximal limit quotients of G embeds intoa f.p. completion. (cid:3) Theorem 21 proves the existence of finitely many limit quotients of a given f.p.group. Hence, it gives the first level of a Makanin-Razborov diagram of a f.p. groupover free products, and it proves that the groups that appear in the first level ofthe Makanin-Razborov diagram of a f.p. group over free products are canonical(i.e., they are an invariant of the f.p. group). Still, the construction of maximallimit groups over free products, and the proof that there are only finitely many(equivalence classes of) maximal quotients of a f.p. group (over free products), doesnot generalize in a straightforward way to allow the construction of the next levelsin the Makanin-Razborov diagram.Furthermore, theorem 13 proves the basic d.c.c. that is required for analyzinglimit groups over free products. However, it is not sufficient for the construction ofa Makanin-Razborov diagram as it guarantees the termination of strict resolutions,but not of general resolutions in the diagram (if we try to imitate the constructionover free and hyperbolic groups). Hence, to construct a finite Makanin-Razborovdiagram we will need to construct the next levels in the diagram, and in addition toprove an additional d.c.c. that will guarantee the termination of the constructionafter finitely many steps.Let G be a f.p. group. We start the construction of the Makanin-Razborovdiagram over free products of G with the finite collection of (equivalence classes of)maximal limit quotients of G , according to theorem 21. We continue by studyingthe homomorphisms of each of the maximal limit quotients of G into free products.As in the construction of Makanin-Razborov diagrams over free and hyperbolicgroups, we continue by modifying (shortening) these homomorphisms using themodular groups that are associated with the maximal limit quotients (over freeproducts) of the given f.p. group G .Let L be one of the maximal limit quotients (over free products) of G , and let E L be its set of elliptics. First, we factor L into its most refined free decomposition inwhich the elements in E L are elliptic (i.e., contained in conjugates of the factors),26 = U ∗ . . . ∗ U m ∗ F t , where F t is a (possibly trivial free group, and the elementsin E L can be conjugated into the various factors, U , . . . , U m .( L, E L ) is a (maximal) limit quotient of G (over free products), hence, L isobtained as a limit of a sequence of homomorphisms { h n : G → A n ∗ B n } . G isf.p. and is mapped onto L , and L admits the free decomposition, L = U ∗ . . . ∗ U m ∗ F t , where the elliptic elements in E L can be embedded into the various factors U , . . . , U m . By the argument that is used to prove proposition 19, there existfinitely presented groups M , . . . , M m so that the map G → L factors as: G → M ∗ . . . ∗ M m ∗ F t → U ∗ . . . ∗ U m ∗ F t where the two maps are onto, and for each index i , 1 ≤ i ≤ m , M i is mappedonto U i . Since the sequence of homomorphisms { h n } of G converges into ( L, E L ),and the group M ∗ . . . ∗ M m ∗ F t is f.p. and the map from G to L factors throughit, for large enough n the homomorphisms { h n } factor through the map G → M ∗ . . . ∗ M m ∗ F t . Now, if we apply the proof of proposition 19, it follows thatthere are m f.p. completions (over free products), Comp , . . . , Comp m , so that eachof the factors U i is embedded into the completion Comp i so that the elliptics in U i are mapped into elliptics in Comp i (and only elliptics in U i are mapped intoelliptics in Comp i ), and there exist maps: M ∗ . . . ∗ M m ∗ F t → U ∗ . . . ∗ U m ∗ F t → Comp ∗ . . . ∗ Comp m ∗ F t that extend the embeddings from U i to Comp i , for 1 ≤ i ≤ m .Hence, we may continue with each of the factors U i of L in parallel. U i is amaximal limit quotient (over free products) of the f.p. group M i , and by proposition21 it is embedded into a f.p. completion Comp i .Therefore, we may assume that in the sequel, we are given a f.p. group G , and amaximal limit quotient of it, that we still denote, ( L, E L ), and the limit quotient L is freely indecomposable relative to the elliptic subset E L (i.e., L admits no non-trivial free decomposition in which the elements in E L can be conjugated into thefactors).With (the factor) L and E L we naturally associate its virtually abelian JSJ de-composition over free products (theorem 11). We also associate with ( L, E L ) thecollection of homomorphisms of G into free products that factor through ( L, E L ).Fixing a (finite) generating set of a limit group (over free products) L , and givena homomorphism h : L → A ∗ B , we look at a shortest homomorphism among thosethat are obtained by precomposing h with a modular automorphism of L that iscontained in the modular group of automorphisms of L that is associated withthe virtually abelian JSJ decomposition over free products of L (relative to E L ).A limit group over free products that is the limit of a sequence of such shortesthomomorphisms is called a shortening quotient, and denoted SQ . Note that thisdefinition of a shortening quotient is different than the more restricted one givenin definition 15, as in particular, the natural map from a limit group over freeproducts, L , onto a shortening quotient SQ of L , is not always monomorphic onthe set of elliptic elements in L , E L . Still, like in proposition 16, if a shorteningquotient is not elliptic it is a proper quotient of the limit group L . Lemma 22 (cf. proposition 16).
Let L be a limit group over free products,and let E L be its set of elliptics. Suppose that L admits no free decompositions in hich the elements in E L can be conjugated into the factors. Then every shorteningquotient of L which is not (entirely) elliptic is a proper quotient of it.Proof: Identical to the proof of proposition 16. (cid:3)
Like limit quotients (over free products) of a f.g. group, every ascending sequenceof shortening quotient of a limit group over free products is bounded by a shorteningquotient of that limit group.
Lemma 23.
Let L be a f.g. limit group over free products. Let SQ , SQ , . . . be asequence of shortening quotients of L , for which: SQ < SQ < . . . Then there exists a shortening quotient SQ of L , so that for every index m , SQ >SQ m .Proof: Identical to the proof in the hyperbolic group case (proposition 1.20 in[Se3]). (cid:3)
By Zorn’s lemma and lemma 23 it follows that there are maximal elements inthe set of shortening quotients of a f.g. limit group over free products. We call sucha maximal element, a maximal shortening quotient . By lemma 22, if the limitgroup (over free products) L does not admit a free product in which the ellipticelements in L , E L , can be conjugated into the factors, every maximal shorteningquotient of L that is not entirely elliptic is a proper quotient of L . §
4. Covers of Limit Quotients and their Resolutions
The first level in the Makanin-Razborov diagram over free products of a f.p. group G consists of the finitely many maximal limit quotients of G (theorem 21). Over freeand hyperbolic groups we continued to the next level in the diagram by proving thatthere are only finitely many (equivalence classes of) maximal shortening quotients.Over free product we need to prove a finiteness result for shortening quotients andtheir (strict) resolutions, that will enable us to continue to the next level, and sothat the next levels will be constructed in a way for which a termination can beproved.In order to prove that there are only finitely many maximal limit quotients overfree products of a f.p. group over free products (theorem 21), we first showed thatany maximal limit quotient can be embedded into a f.p. completion (proposition19). For maximal shortening quotients of a f.g. limit group over free products wewere not able to prove such a statement. For the continuation of the diagram,we first prove an observation that holds for all the (proper) limit quotients of agiven limit group over free products, that still allows us to construct the Makanin-Razborov diagram over free products for a f.p. group, although we loose some ofthe canonical properties of the diagrams over free and hyperbolic groups.Given a limit group over free products, L , and a limit quotient M of L , theorem24 associates a cover, CM , with M . CM is a limit quotient of L , if L > M ,then
L > CM and M is a limit quotient of CM . The main property of the cover CM that is used in the sequel (and is not always true for M ) is that CM can be28mbedded into a completion, Comp CM , and Comp CM is f.p. relative to the ellipticsubgroups of the given limit group L , i.e., Comp CM is generated from the ellipticsubgroups in L by adding finitely many generators and relations. In particular, thisimplies that if L is recursively presented so is the cover CM . Theorem 24.
Let L be a f.g. limit group over free products, and let E L be its setof elliptics. Let M be a limit quotient of L (over free products), with set of elliptics, E M , and with a quotient map, η : L → M that maps E L into E M .Suppose that L > M , i.e., that the map η has a non-trivial kernel, or that thereexists a non-elliptic element in L that is mapped to an elliptic element in M by η .Let M → M → . . . → M s be a (well-structured) resolution of M , i.e., a res-olution of M that satisfies the properties of the resolution that is associated witha limit group over free products in theorem 18. Then there exists a f.g. limit quo-tient of L , CM , with a set of elliptics, E CM , and a well-structured resolution of CM , CM → CM → . . . → CM s , that satisfies the properties of the resolutions intheorem 18, and a quotient map: τ : L → CM , that maps E L into E CM , so that: (1) there exists a quotient map: ν : CM → M , that maps E CM onto E M , sothat η = ν ◦ τ . (2) if η : L → M has a non-trivial kernel, then τ : L → CM has a non-trivial kernel. If there exists a non-elliptic element in L that is mapped toan elliptic element in M by η , then there exists a non-elliptic element in L that is mapped to an elliptic element in CM by τ . If M i +1 is a properquotient of M i , then CM i +1 is a proper quotient of CM i . (3) if η maps an elliptic element in L to the identity, then τ maps an ellipticelement in L to the identity. (4) if M is a free product of finitely many elliptic subgroups and a free group,so is CM . More generally, CM j is mapped onto M j , ≤ j ≤ s , whereelliptics in CM j are mapped onto elliptics in M j . (5) all the homomorphisms of the given limit group L that factor through thegiven well-structured resolution of M , factor through the resolution CM → CM → . . . → CM s . (6) with the given well-structured resolution, M → M → . . . → M s , we cannaturally associate a completion, Comp M (see definition 1.12 in [Se2]), andwith the resolution CM → CM → . . . → CM s we can naturally associatea completion, Comp CM . CM is embedded into Comp CM , and the ellipticelements in CM are mapped into the terminal limit group CM s . (7) by theorem 18, the elliptic elements, E L , in the limit group L are conjugatesof finitely many (possibly none) f.g. subgroups, E , . . . , E r in L . Then thecompletion, Comp CM , is obtained from (copies of the subgroups) E , . . . , E r by adding finitely many generators and relations, i.e., Comp CM is f.p. rel-ative to the subgroups E , . . . , E r . (8) if M admits a free decomposition, M = V ∗ . . . ∗ V u ∗ F t , where F t is a freegroup, and this free decomposition is respected by the given resolution of M ,then CM has a similar free decomposition, CM = CV ∗ . . . ∗ CV u ∗ F t , whichis respected by the constructed resolution of CM , where the map ν respectsthis free decomposition, i.e., ν ( CV i ) = V i , i = 1 , . . . , u , and ν ( F t ) = F t . Inparticular, the completion, Comp CM , admits a similar free decomposition, Comp CM = Comp ∗ . . . ∗ Comp u ∗ F t , where CV i embeds into Comp i . roof: Let L be a limit group over free products, with set of elliptics E L . Bytheorem 18 there are finitely many subgroups, E , . . . , E r , in L , so that the set ofelliptic elements in L , E L , is the union of the conjugacy classes of E , . . . , E r . Let M be a limit quotient of L , and let M → M → . . . → M s be a well-structuredresolution of M , where M s is a free product of finitely many elliptic factors and apossibly trivial free group.With the given well-structured resolution of M we associate a completion, Comp M . M is a limit quotient of L , and M is a subgroup of the completion, Comp M , so L is mapped into Comp M . Hence, the elliptic subgroups in L , E , . . . , E r , aremapped into conjugates of the elliptic subgroups, that are factors in the terminallimit group M s , in Comp M . If the terminal limit group M s is f.p. relative to thesubgroups, E , . . . , E r , the theorem follows (by taking the cover CM to be M and Comp CM to be Comp M ). Otherwise we modify the argument that was used toprove proposition 19.Since M is embedded into the completion Comp M , L is naturally mapped into Comp M . Each of the factors of the terminal limit group of Comp M , M s , is f.g.so it is a quotient of some conjugates of (copies of) the elliptic subgroups of L , E , . . . , E r and a f.g. free group. We start the construction of a the completion Comp that covers the completion
Comp M , with a tower T that has in its baselevel the free product of a free group (isomorphic to the free factor in the freedecomposition of M s ), the free products of corresponding conjugates of E , . . . , E r with free groups (so that each of the factors of M s is a quotient of each of thesefree products). The next (upper) levels are connected to the lower levels of theconstructed tower T , precisely as they are connected in the completion, Comp M ,i.e., using the same graphs of groups, just that the group that is associated with thelowest level in Comp M , which is M s , is replaced by the prescribed free products. T is a tower, but it is not necessarily a completion (see definition 1.12 in [Se2]),as in general there are no retractions from a group that is associated with a certainlevel onto the group that is associated with the level below it. Each of the levelsabove the base level in T is constructed using a (finite) graph of groups, in whichsome vertex groups are the groups that are associated with the lower level in T .Hence, the group that is associated with a level above the base level, is obtainedfrom a free product of the group that is associated with the lower level with a f.p.group by imposing finitely many relations. Furthermore, the graphs of groups thatare associated with the different levels in T are similar to the graphs of groupsthat are associated with the corresponding levels in the completion Comp M , anddiffer from Comp M only in the groups that are associated with the base level.Each of the groups that are associated with the upper levels in T is obtainedfrom the groups that appear in the lower level of T by a free product with a f.g. freegroup and further imposing finitely many relations. The graphs of groups that areassociated with the upper levels in T have similar structure as the correspondinggraphs of groups that are associated with the levels of the completion Comp M , i.e.,the graphs of groups differ only in the vertex groups that are associated with lowerlevels. Furthermore, these vertex groups differ only in the groups that are associatedwith the base levels of T and Comp M . Hence, it is enough to impose only finitelymany (additional) relations from the defining relations of the various factors of thelimit groups that is associated with the base level of Comp M , M s , on the subgroupthat is associated with the base level of T . This means imposing finitely many(additional) relations on the associated free products of free groups and conjugates30f (copies of) the subgroups, E , . . . , E r , that form the group which is associatedwith the base level of T , so that if we replace the group that is associated with thebase level of T , with the obtained quotient, and construct from the obtained basesubgroup a tower, T , by imitating the construction of Comp M and T (i.e., withsimilar graphs of groups in all the upper levels), T will be a completion (i.e., it isa tower with retractions between consecutive levels). T is a completion, but it may be that the limit group L is not mapped into it. L is mapped into the completion Comp M , and as a limit group it is finitely presentedrelative to its elliptic subgroups. Hence, once again, it is enough to impose onlyfinitely many relations from the defining relations of the various factors of M s , sothat if we replace the group that is associated with the base level in T with theobtained quotient, and construct from it a tower T by imitating the constructionof the towers Comp M , T , and T , T is a completion, it is f.p. relative to the ellipticsubgroups, E , . . . , E r , and L maps into it.We denote the images of the limit group L into the various levels of the comple-tion T , by CM, CM , . . . , CM s . By imposing finitely many additional relationson the base subgroup of T from the relations of the base subgroup of Comp M , M s , one can further guarantee that if M is a proper quotient of L , then CM isa proper quotient of L , if L > M then
L > CM , and similarly, if M j +1 is aproper quotient of M j then CM j +1 is a proper quotient of CM j , and if M j +1 > M j then CM j +1 > CM j . We denote the obtained completion by Comp CM and itsassociated resolution as the resolution that is associated with CM (the obtainedresolution is a well-structured resolution by construction). All the other propertiesof the limit groups, and the associated resolution and completion, Comp CM , thatare listed in the statement of the theorem follow easily from the construction. (cid:3) Given a f.g. limit group over free products L , and its limit quotient M withan associated well-structured resolution, M → M → . . . → M s , that satisfy theassumptions of theorem 24, and for which L > M , we call a limit quotient CM of L , that satisfies the conclusion of the theorem, a cover of the limit quotient M ,its associated well-structured resolution, CM → CM → . . . → CM s , a cover ofthe given resolution of M , and the associated completion, Comp , into which CM is embedded, that was constructed from the given well-structured resolution of M ,a cover completion .In constructing the Makanin-Razborov diagrams of a f.p. or a f.g. group over afree or a hyperbolic group, we were able to show that the set of shortening quotientsof a limit group over these groups contain finitely many equivalence classes ofmaximal shortening quotients. In studying limit groups over free products we arenot able to prove a similar theorem. Over free products we prove that given alimit group L , and fixing a cover for each pair of a shortening quotient and itsassociated well-structured resolution, there exists a finite subcollection of coverswhich is good for all the shortening quotients of L . As we will see in the sequel,a similar statement on the existence of a finite subcollection of cover completions(with a similar proof) is sufficient for the construction of the Makanin-Razborovdiagram over free products. Theorem 25.
Let L be a f.g. limit group over free products, suppose that L isnot (entirely) elliptic and that L admits no free product decomposition in which theelliptic elements in L , E L , can be conjugated into the factors. ith each pair of a shortening quotient M of L , and a well-structured resolu-tion of M , there is an associated quotient map, η M : L → M , that satisfies theassumptions of theorem 24. Hence, by the conclusion of theorem 24, for each pairof a shortening quotient M of L , and its associated well-structured resolution wecan choose a cover CM ( M ) together with a completion, Comp CM , into which CM embeds.From the entire collection of covers of shortening quotients of L and their as-sociated well-structured resolutions, it is possible to choose a finite subcollection ofcovers, CM , . . . , CM e , so that for every maximal shortening quotient M , thereexists an index i , ≤ i ≤ e , for which the quotient map, η : L → M , is a com-position of the two quotient maps: L → CM i → M (where elliptics are mapped toelliptics in these two maps).Proof: The argument that we use is similar to the proof of the finiteness of thenumber of equivalence classes of maximal shortening quotients (over free products)of a f.p. group. Let L be a f.g. limit group. By theorem 24, given a shorteningquotient of it, M , and a well structured resolution of that shortening quotient, thereexists a cover CM of M , and CM can be embedded into a completion, Comp , thatis obtained from the finitely many (conjugates of) elliptic subgroups of L , by addingfinitely many generators and relations. Therefore, there are at most countably manysuch completions, Comp , and hence, at most countably many such covers, CM .Note that by lemma 22 each shortening quotient M of L is either entirely elliptic,or it is a proper quotient of L . In case the shortening quotient M is not entirelyelliptic, it follows by theorem 24, that the associated cover, CM is a proper quotientof L (like the shortening quotient M ).Suppose that the countable collection of covers does not contain a finite subcover,i.e., there is no finite subcollection of the constructed covers, CM , . . . , CM e , sothat for every shortening quotient M , the quotient map L → M factors as acomposition of quotient maps of limit groups over free products: L → CM i → M ,for some index i , 1 ≤ i ≤ e .To contradict the lack of a finite subcover, we start by ordering the collection ofcovers, CM , CM , . . . . Since there is no finite subcover, there must be a sequenceof indices, i , i , . . . , so that a shortening quotient, M i j , which is covered by CM i j ,is not covered by any of the previous covers, CM , . . . , CM i j − .For each index j , the shortening quotient M i j is a limit of shortest homomorphisms,and it is not covered by any of the covers, CM , . . . , CM i j − . Hence, for each index j , there exists a shortest homomorphism h j : L → A j ∗ B j , that does not factorthrough any of the covers, CM , . . . , CM i j − .We look at the sequence of homomorphisms { h j } . A subsequence of this sequenceconverges into a limit group (over free products) R , which is a quotient of thelimit group L . Unless the limit group R is the (possibly trivial) free product offinitely many elliptic factors and a (possibly trivial) free group, a subsequence ofthe shortenings of these homomorphisms converges into a shortening quotient R of R , where the elliptics in R are mapped monomorphically into the elliptics in R ,and R is a proper quotient of R . By continuing iteratively and applying the d.c.c.for decreasing sequences of limit groups over free products (theorem 13), we obtaina finite (strict) resolution R → R → . . . → R s , where R s is a free product offinitely many f.g. elliptic subgroups and a (possibly trivial) free group. For brevity,we still denote the obtained subsequence of shortest homomorphisms, { h j } .32he pair of the shortening quotient R , and its (strict) resolution, R → R → . . . → R s , is one of the pairs of a shortening quotient of the limit group L ,and its associated strict resolutions, with which we have associated the covers, CM , CM , . . . . Hence, one of these covers, CM r , is a cover that is associated withthis pair. Since a cover completion is finitely presented relative to the elliptic sub-groups of L , for large enough indices j , the homomorphisms { h j } factor throughcover completion and hence factor through the cover CM r . That contradicts thechoice of the homomorphisms { h j } , as for large j , h j is supposed not to factorthrough the covers, CM , . . . , CM j − . (cid:3) Theorem 25 proves that given a f.g. limit group L that admits no free decom-position in which the elliptic elements, E L , can be conjugated into the factors, itis possible to find finitely many limit quotients of L , one which is isomorphic to L and is entirely elliptic, and the rest which are proper quotients of L , that coverall its shortening quotients. This finite collection of covers is not canonical, butin principle it can be taken as the next step in the Makanin-Razborov diagram.Except for the entirely elliptic cover that is isomorphic to L , the other covers thatare associated with L are all proper quotients of it, hence, in principle we can con-tinue with the construction iteratively. However, the d.c.c. that we proved is validonly for sequences of strictly decreasing limit quotients, for which the quotients areproper and are monomorphic when restricted to elliptic elements (theorem 13).Therefore, to complete the construction of the Makanin-Razborov diagram of af.p. group over free products we use a different approach. Instead of constructing afinite cover of all the shortening quotients of a given limit group (over free products),we construct a finite cover for all the (strict) resolutions of the given limit group.With each strict resolution of the given limit group we associate a cover of thatresolution (which is a resolution by itself), and there are only countably manysuch covers, as the completion that is associated with the cover resolution is f.p.relative to the elliptic subgroups of the original limit group. Then we use a similarargument to the one that was used in proving theorem 25 to prove that there existsa finite subcollection of the collection of cover resolutions, i.e., that there exists afinite subcollection so that every homomorphisms of the given limit group into freeproducts factors through at least one of the resolutions from the finite subcollectionof cover resolutions. Theorem 26.
Let L be a f.g. limit group over free products. Then there existsfinitely many well-structured resolutions of quotients of L , so that every homomor-phism from L into a free product factors through at least one of these well-structuredresolutions. Furthermore, with each of these (finitely many) well-structured resolu-tions we can naturally associate a completion, and these completions are f.p. relativeto the (finitely many) elliptic subgroups in the given limit group L .Proof: The proof is similar to the proof of theorem 25. First, we factor the limitgroup over free products L into a maximal free decomposition in which the ellipticelements of L , E L , can be conjugated into the factors. We continue with each ofthe factors separately. Hence, we may assume that the limit group L is freelyindecomposable with respect to its set of elliptics, E L . By theorem 24, givena limit quotient of L , that we denote T , and a well-structured resolution of T , T → T → . . . → T s , that is obtained by taking successive shortening quotients33see theorem 18), there exists a cover of T , that we denote CT , which is a limitquotient of L , and a cover of the resolution that is associated with T , which is awell-structured resolution, with an associated cover completion, Comp CT , that isobtained from the finitely many (conjugates of) elliptic subgroups of L , by addingfinitely many generators and relations. Therefore, there are at most countablymany such triples of a cover of a limit quotient, an associated (well-structured)cover resolution, and the corresponding cover completion.Suppose that the countable collection of cover resolutions does not containa finite subcover, i.e., there is no finite subcollection of the constructed covers, CT , . . . , CT e , with associated cover completions, Comp , . . . , Comp e , so that foreach homomorphism h of L into a free product (that maps the elliptics in L , E L , intoelliptic elements), the homomorphism h factors through at least one of the coverresolutions that is associated with the cover completions, Comp , . . . , Comp e .To obtain a contradiction to the lack of finiteness of covering resolutions, we startby ordering the collection of covering completions and their associated resolutions, Comp , Comp , . . . . Since there is no finite subcover for the entire collection ofhomomorphisms of the given limit group L into free products, for each index i ,there exists a homomorphism, h i : L → A i ∗ B i , that does not factor through theresolutions that are associated with the completions, Comp , . . . , Comp i − .Like in the proof of theorem 25, a subsequence of the sequence of homomor-phisms, { h i } , converges into a limit group (over free products) R , which is a quo-tient of the limit group L . Unless R is a (possibly trivial) free product of ellipticsubgroups and a (possibly trivial) free group, a subsequence of the shortenings ofthese homomorphisms converges into a shortening quotient R of R , where theelliptics in R are mapped monomorphically into the elliptics in R , and R is aproper quotient of R . By continuing iteratively and applying the d.c.c. for decreas-ing sequences of limit groups over free products (theorem 13), we obtain a finitewell-structured resolution R → R → . . . → R s , where R s is a free product offinitely many f.g. elliptic subgroups and a (possibly trivial) free group. For brevity,we still denote the obtained subsequence of shortest homomorphisms, { h i } .The pair of the limit quotient R of the given limit group (over free products) L ,and its (well-structured) resolution, R → R → . . . → R s , is one of the pairs of alimit quotient of L , and its associated well-structured resolutions, with which wehave associated the covers, Comp , Comp , . . . . Hence, one of these completions, Comp r , is a cover that is associated with this pair. Therefore, for large enough index i , the homomorphism { h i } factors through the cover resolution that is associatedwith the completion, Comp r . That contradicts the choice of the homomorphisms { h i } , as for each i , h i is supposed not to factor through the cover resolutions thatare associated with the completions, Comp , . . . , Comp i − . (cid:3) §
5. Makanin-Razborov Diagrams of Finitely Presented Groups
Theorem 21 on the finiteness of the number of equivalence classes of maximallimit quotients (over free products) of a f.p. group, together with theorem 26 onthe existence of finitely many (cover) resolutions of some quotients of a given f.g.limit group over free products, so that every homomorphism of the given f.g. limitgroup into free products factors through at least one of the resolutions, allow us toconstruct a Makanin-Razborov diagram of a f.p. group over free products.34iven a f.p. group G , we start with its (canonical) finite collection of maximallimit quotients over free products (theorem 21). With each maximal limit quo-tient we associate a finite collection of well-structured resolutions of it (accordingto theorem 26), so that each homomorphism of the original maximal limit quotientinto free products, factors through at least one of its associated resolutions. Weconstruct the diagram by mapping the given f.p. group G into the f.g. limit groupthat appears in the top level of each of the (finitely many) well-structured resolu-tions that are associated with its collection of maximal limit quotients (in parallel).Since every homomorphism of G into free products, factors through at least one ofits maximal limit quotients, every homomorphism of G into free products factorsthrough at least one of the resolutions in its Makanin-Razborov diagram over freeproducts. That is for every homomorphism of the f.p. group G , there exists at leastone resolution in the Makanin-Razborov diagram, so that the homomorphism canbe written as a successive composition of the epimorphisms between the groupsthat appear in the various levels of the resolutions, modular automorphisms of thelimit groups that appear in the various levels (that are encoded by the virtuallyabelian decompositions that are associated with these groups), and finally a ho-momorphism from the terminal group of the resolution (which is a free product ofelliptic factors and a free group), that sends every elliptic factor into a conjugateof a factor in the image free product.At this stage we slightly improve the diagram. The virtually abelian decompo-sitions that are associated with each of the limit groups that appear in the variouslevels of the well-structured resolutions in the Makanin-Razborov diagrams, aredecompositions that are inherited from the free and virtually abelian JSJ decom-positions of the limit groups that appear along the well-structured resolutions thatthe resolutions in the Makanin-Razborov diagram cover, according to the construc-tion that appears in theorem 24. However, these may not be the Grushko andvirtually abelian decompositions of the limit groups in the Makanin-Razborov di-agram themselves. To fix that, and make sure that all the decompositions in thelimit groups that appear in the Makanin-Razborov diagram are indeed Grushkoand virtually abelian JSJ decompositions (over free products), we slightly modifythe construction of a cover. Theorem 27.
Let L be a f.g. limit group over free products, let M be a limitquotient of L , and let M → M → . . . → M s , be a well-structured resolution of M ,so that M s is a free product of finitely many elliptic factors and a possibly trivialfree group. Suppose that the free products that are associated with the various limitgroups along the resolution, M, M , . . . , M s , are their Grushko free decompositionswith respect to their elliptic subgroups (i.e., the resolution respects the Grushko freedecompositions of the groups along it), and that the virtually abelian decompositionsthat are associated with the limit groups M, M , . . . , M s are their virtually abelianJSJ decompositions over free products.Then there exists a cover CM of M , with a cover resolution, CM → CM → . . . → CM s , that satisfies the properties of a cover that are listed in theorem 24, andfor which the free decompositions along the cover resolution are the Grushko freedecompositions of the limit groups, CM, CM , . . . , CM s , and the virtually abelianJSJ decompositions of these groups over free products have the same structure as thevirtually abelian decompositions that are associated with them along the resolution,i.e., the same structure as the virtually abelian JSJ decompositions of the limit roups, M, M , . . . , M s .Proof: The proof that we use is a modification of the argument that we used toprove theorem 24. Let L be a limit group over free products, with a set of elliptics E L . Recall that by theorem 18, the set of elliptics E L is the union of conjugates ofsome (elliptic) subgroups, E , . . . , E r , in L . Let M be a limit quotient of L , andlet M → M → . . . → M s be a well-structured resolution of M , where M s is a freeproduct of finitely many elliptic factors and a possibly trivial free group.With the given well-structured resolution of M we associate a completion, Comp M .Given the well-structured resolution of M , and its associated completion, Comp M ,we use the construction that was used in proving theorem 24, and construct acompletion, Comp , which is f.p. relative to the elliptic subgroups, E , . . . , E r , andfor which the images of the limit group L into the various levels of Comp , thatwere denoted,
CM, CM , . . . , CM s , satisfy the list of properties that is presentedin theorem 24.By adding finitely many relations to the base subgroup of Comp from the setof relations that are defined on the base subgroup, M s , of the completion Comp M ,we may assume that the abelian decompositions that are inherited by the sub-groups, CM, CM , . . . , CM s , from the abelian decompositions that are associatedwith the various levels of the completion Comp , are similar to the abelian decompo-sitions that are inherited by the various abelian decompositions of the subgroups, M , . . . , M s from the abelian decompositions that are associated with the variouslevels of Comp M .Suppose that the Grushko free decomposition of the limit group M with respectto its elliptic subgroups is M = M ∗ . . . ∗ M b ∗ F v , and this free decompositiontogether with the virtually abelian JSJ decompositions of the factors, M j , overfree products with respect to the elliptic subgroups of M , give rise to an abeliandecomposition , ∆ M . Note that by our assumptions, the completion, Comp M ,respects the Grushko free decomposition of M , and the abelian decompositionsthat are associated with the various levels of Comp M are the virtually abelian JSJdecompositions over free products of the subgroups, M, M , . . . , M s − .We order the relations that the terminal limit group M s of Comp M satisfy, andsequentially impose them on the terminal limit group of the completion, Comp . Weclaim that after adding finitely many of these relations, the free product decompo-sition, and the virtually abelian JSJ decomposition of the corresponding subgroup CM (after adding the relations) will be similar to those of the subgroup M .The cover CM , which is the image of the limit group L in the completion, Comp ,admits a free decomposition CM = CM ∗ . . . ∗ CM b ∗ F v , in which the ellipticsubgroups in CM can be conjugated into the factors. This free decomposition isinherited from the structure of the completion, Comp , as the completions
Comp and
Comp M have the same structure, and Comp M respects the Grushko decomposition(relative to elliptic subgroups) of the limit quotient M , M = M ∗ . . . ∗ M b ∗ F v .Let CM ( n ) be the image of L in the completion, Comp ( n ), that is obtainedfrom Comp by imposing on the terminal level in
Comp the first n relations in M s , the terminal limit group in Comp M . CM ( n ) inherits a free decompositionfrom Comp ( n ), CM ( n ) = CM ( n ) ∗ . . . ∗ CM ( n ) b ∗ F v , a free decomposition inwhich the elliptic subgroups in CM ( n ) can be conjugated into the factors (notethat the elliptic subgroups in CM ( n ) can be conjugated into the factors of theterminal limit group of Comp ( n )). If this free decomposition is not the Grushko36ree decomposition of CM ( n ) with respect to its elliptic subgroups, then at leastone of the factors admits a further non-trivial free decomposition with respect tothe elliptic subgroups.Suppose that there exists a sequence of indices (still denoted n ) for which thefree decomposition of CM ( n ) that is inherited from Comp ( n ) is not the Grushkofree decomposition of CM ( n ) with respect to the elliptic subgroups in CM ( n ). Bypassing to a subsequence (still denoted n ) we may assume that one of the factors,w.l.o.g. CM ( n ) admits a non-trivial free decomposition CM ( n ) = A n ∗ B n , whereeach of the elliptic subgroups in CM ( n ) can be conjugated into one of the otherfactors in the given free decomposition of CM ( n ), to A n or to B n .In that case we look at the actions of the groups CM ( n ) on the (pointed) Bass-Serre trees, ( T n , t n ), that correspond to the (non-trivial) free products, A n ∗ B n .Note that these actions are faithful actions of the groups, CM ( n ) , that the ellipticsubgroups in CM ( n ) that can be conjugated into CM ( n ) can be conjugated into A n or B n , and that by construction, the direct limit of the groups, CM ( n ) , isthe factor M of the limit group M which is assumed to be freely indecomposablerelative to its elliptic subgroups. CM is f.g. so we fix a generating set for it, < g , . . . , g d > , and since the groups CM ( n ) are (limit) quotients of CM , it gives us a generating set for each of thegroups, CM ( n ) . Given the action of CM ( n ) on the Bass-Serre tree, ( T n , t n ),we precompose this action with a (modular) automorphism φ n of CM ( n ) , i.e., anautomorphism that can be expressed as a composition of an automorphism thatcomes from the virtually abelian decomposition that CM ( n ) inherits from thevirtually abelian decomposition that is associated with the top level in Comp ( n )and an inner automorphism, so that the maximal displacement of the base point t n by the action of the tuple of elements, φ n ( g ) , . . . , φ n ( g d ), is minimal among allsuch (modular) automorphisms φ .Since we modify the actions of the groups, CM ( n ) , by precomposing them with(modular) automorphisms, and since the actions are all faithful, there is a subse-quence of twisted actions that converge into an action of the direct limit of thegroups, CM ( n ) , i.e., the factor M of M , on a real tree. Since the automorphisms φ n were chosen to minimize the displacement of the base points under the corre-sponding twisted actions, and since the virtual abelian JSJ decomposition of thelimit group M has the same structure as the virtually abelian decomposition thatis inherited by CM ( n ) from the virtually abelian decomposition that is associatedwith the top level of the completions, Comp ( n ), the set of displacements of the basepoints under the twisted actions has to be bounded. Hence, the factor M of M inherits a non-trivial free decomposition from the limit action, a free decompositionin which all the elliptic subgroups in M can be conjugated into the factors. Thiscontradicts the assumption that M admits no such non-trivial free decomposition.Therefore, there must exist some index n , so that for all n > n , the limit groups CM ( n ) admit no free decomposition in which the elliptic subgroups of CM ( n ) can be conjugated into the factors.By passing to a subsequence, we may assume that all the factors in the free de-composition of the limit groups, CM ( n ), are freely indecomposable relative to theirelliptic subgroups. Suppose that there exists a sequence of indices (still denoted n ) for which the virtually abelian decomposition that at least one of the factors ofthe the groups, CM ( n ), CM ( n ) , . . . , CM ( n ) b , inherits from the virtually abeliandecomposition that is associated with the top level of the completion, Comp ( n ), is37ot the virtually abelian JSJ decomposition over free products of that factor. Wlogwe may assume that this factor is CM ( n ) .Let ∆( n ) be the virtually abelian decomposition that CM ( n ) inherits fromthe virtually abelian decomposition that is associated with the top level of thecompletion Comp ( n ). Let J SJ ( n ) be the virtually abelian JSJ decomposition overfree products of CM ( n ) , and let ∆ M be the virtually abelian decomposition that M inherits from the virtually abelian decomposition that is associated with thetop level of the completion, Comp M , which by our assumptions is the virtuallyabelian JSJ decomposition of M over free products. Since we assumed that thevirtually abelian decompositions, ∆( n ), are not identical to the virtually abeliandecompositions, J SJ ( n ), the virtually abelian JSJ decompositions, J SJ ( n ), mustbe proper refinements of the virtually abelian decompositions, ∆( n ). Note that thestructure of the virtually abelian decompositions, ∆( n ), is similar to that of theabelian decomposition, ∆ M .For every index n , the virtually abelian JSJ decomposition J SJ ( n ) is a properrefinement of the virtually abelian decomposition ∆( n ). Hence, if needed we cancut some of the QH subgroups in J SJ ( n ) along s.c.c. and obtain a new decom-position, Θ( n ), of CM ( n ) that refines ∆( n ), in which all the edge groups and allthe QH vertex groups in ∆( n ) are elliptic, and at least one of the non- QH non-virtually-abelian vertex groups in ∆( n ) is not elliptic. Hence, at least one of thesevertex groups inherits a non-trivial virtually abelian decomposition from Θ( n ), adecomposition in which all the edge groups that are connected to that vertex groupare elliptic.By passing to a further subsequence (still denoted n ), we may assume that thevertex group that inherits a non-trivial virtually abelian decomposition from Θ( n )is a vertex group V ( n ) in ∆( n ) that is mapped to the same vertex group V in ∆ M ,the virtually abelian JSJ decomposition of the limit group M .We fix a free group F r , where r is the rank of the limit group CM , and anepimorphism, τ : F r → CM . We fix a finite generating set for F r . We may assumethat this generating set contains elements that are mapped to elements that gen-erate the edge groups and the vertex groups in the virtually abelian decompositionof CM that is inherited from the top level of the completion, Comp .For each index n , we look at a homomorphism h n : F r → A n ∗ B n that approxi-mates the limit group CM ( n ). This means that h n maps each element in the ballof radius n in the Cayley graph of F r (with respect to the given set of generators),to an elliptic element or to a trivial element if and only if the element is trivial orelliptic in CM ( n ). It maps the elements from the generating sets that are mappedto the edge groups in ∆( n ) to non-elliptic elements. Furthermore, let S < F r bethe subgroup that is generated by those elements in the fixed generating set of F r whose image generate the vertex group in the virtually abelian decomposition of CM that is mapped to the vertex group V in ∆ M , and the edge groups that areconnected to that vertex group. The vertex group V ( n ) is not elliptic in the virtu-ally abelian decomposition of the factor CM ( n ) , Θ( n ), and the edge groups thatare connected to V ( n ) in ∆( n ) are elliptic in Θ( n ). Hence, we may further modifyeach of the homomorphisms h n , by precomposing each of them with Dehn twistsalong edge groups that lie in the graph of groups that is inherited by V ( n ) fromthe graph of groups Θ( n ). We apply this modification, so that for the obtained ho-momorphism, ˆ h n , when restricted to the subgroup S < F r (which is mapped onto38 ( n )), ˆ h n : S → A n ∗ B n , the minimal displacement of a point in the Bass-Serretree, that is associated with the free product A n ∗ B n , under the action of the fixedset of generators of S , will be at least n times larger than the minimal displacementof a point in that Bass-Serre tree, under the action of the fixed set of generatorsthat are mapped to any given edge group that is connected to V ( n ) in ∆( n ).By construction, the homomorphisms, { ˆ h n : S → A n ∗ B n } , converge into a non-trivial action of the vertex group V in ∆ M on some real tree (where the convergenceis into V as a limit group over free products). All the edge groups that are connectedto V in ∆ M fix points in that real tree and they are all non-elliptic subgroups(i.e. each element in these groups is mapped to non-elliptic element in A n ∗ B n for large n ). With this action it is possible to associate a non-trivial graph ofgroups decomposition of V , with abelian edge groups, in which all the edge groupsthat are connected to V are contained in vertex groups in that graph of groupsdecomposition. Hence, using this graph of groups decomposition it is possible tofurther refine the graph of groups, ∆ M , and this clearly contradict the assumptionthat ∆ M is the virtually abelian JSJ decomposition of the limit group M .Therefore, for large n , the abelian decompositions, ∆( n ), are the virtually abelianJSJ decompositions of the limit groups over free products, CM ( n ). The sameargument implies the same results for the next limit groups in the constructedresolution, CM ( n ) , . . . , CM s − ( n ), and the theorem follows. (cid:3) The Makanin-Razborov diagram of a f.p. group G over free products is uniform,i.e., it encodes all the homomorphisms from G into arbitrary free products. Equiv-alently, it encodes all the quotients of a f.p. group that are free products. As wewill see in the sequel, the Makanin-Razborov diagram that we constructed sufficesin order to modify the results and the techniques that were used to study the firstorder theory of a free or a hyperbolic group, in order to study the first order the-ory of a free product. We also believe that modifications of it can be applied forstudying homomorphisms of a f.p. group into groups with more general splittings(notably k -acylindrical splittings), and probably homomorphisms into (some classesof) relative hyperbolic groups.Unfortunately, the diagram that we constructed is not canonical, as it uses finitecovers (theorems 25 and 26), and these are not unique. To construct a canonicaldiagram, we believe that it’s better to study only maximal homomorphisms intofree products. Definition 28.
Let G be a f.g. group. On the set of homomorphisms of G intofree products, we define a partial order. Let h i : G → A i ∗ B i , i = 1 , , be twohomomorphisms. Note that the images of the homomorphisms h i inherit (possiblytrivial) free products from the free product decompositions A i ∗ B i , i = 1 , . We writethat h > h , if there exists an epimorphism with non-trivial kernel: τ : h ( G ) → h ( G ) , that maps the elliptics in h ( G ) into the elliptics in h ( G ) , so that for every g ∈ G , h ( g ) = τ ( h ( g )) .If τ is an isomorphism and it maps the elliptics in h ( G ) onto the elliptics in h ( G ) , and for every g ∈ G , h ( g ) = τ ( h ( g )) , we say that h is in the sameequivalence class as h .Note that this relation on homomorphisms into free products, which is a partialorder on homomorphisms, is a special case of the partial order that was defined in efinition 12 for limit groups over free products. To construct a canonical Makanin-Razborov diagram, it seems that one needs toprove the existence of maximal homomorphisms with respect to the above partialorder. The existence of maximal homomorphisms allows one to construct a canon-ical (finite) collection of maximal shortening quotients of a f.g. limit group overfree products, and then prove a d.c.c. that allows the termination of the construc-tion of a diagram, using somewhat similar construction to the one used over freeand hyperbolic groups. To prove the existence of maximal homomorphisms (withrespect to the prescribed partial order), one needs to prove the following naturalconjecture:
Conjecture.
Let G be a f.g. group. Let h , h , . . . be a sequence of homomorphismsof G into free products, for which: h < h < . . . Then there exists a homomorphism h from G into a free product, so that for everyindex m , h > h m (one may even assume that the homomorphisms, { h m } , do notfactor through an epimorphism onto a group of the form M ∗ F for some nontrivialfree group F ). Finally, we note that the Makanin-Razborov diagram over free products that weconstructed is associated with a f.p. group. Some of our arguments are not validfor f.g. groups. In particular, although there exist maximal elements in the set oflimit quotients over free products of a f.g. group, it is not clear if there are onlyfinitely many maximal limit quotients. Therefore, the study of the collection ofhomomorphisms from a given f.g. group into free products remains open.
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