Freiheitssatz for amalgamated products of free groups over maximal cyclic subgroups
FFreiheitssatz for amalgamated products of free groups over maximalcyclic subgroups
Carsten FeldkampFebruary 2, 2021
Abstract
In 1930, Wilhelm Magnus introduced the so-called Freiheitssatz: Let F be a free group with basis X and let r be a cyclically reduced element of F which contains a basis element x P X , then every non-trivial element of the normal closure of r in F contains the basis element x . Equivalently, the subgroupfreely generated by X zt x u embeds canonically into the quotient group F {xx r yy F . In this article, we wantto introduce a Freiheitssatz for amalgamated products G “ A ˚ U B of free groups A and B , where U isa maximal cyclic subgroup in A and B : If an element r of G is neither conjugate to an element of A nor B , then the factors A , B embed canonically into G {xx r yy G . For a group G and an element r P G we denote the normal closure of r in G by xx r yy G . We mostlywrite G {xx r yy instead of G {xx r yy G if it is clear from the context, that the normal closure is taken over G .Further, we write r a, b s “ a ´ b ´ ab for the commutator r a, b s of two elements a, b .In 1930, W. Magnus proved the classical Freiheitssatz : If F is a free group with basis X and r a cyclically reduced element containing a basis element x P X , then the subgroup freely generatedby X zt x u embeds canonically into the quotient group F {xx r yy . This result became a cornerstone ofone-relator group theory and led to different kinds of natural generalizations.One way to generalize the Freiheitssatz of W. Magnus is to study so-called one-relator products . A one-relator product of groups A j ( j P J ) for some index set J is a quotient group p ˚ j P J A j q{xx r yy ,where r is an element of ˚ j P J A j which is not conjugate to an element of a single A j ( j P J ). Notethat one-relator groups are special cases of one-relator products where the A j ( j P J ) are free groups.Thus, knowing the Freiheitssatz of W. Magnus, it is natural to ask under which conditions the factors A j ( j P J ) canonically embed into a one-relator product p ˚ j P J A j q{xx r yy . One result in this contextis the Freiheitssatz for locally indicable groups (cf. Theorem 1.4) independently proved by J. Howie(see [How81]), S. Brodskii (see [Bro80], [Bro84]) and H. Short (see [Sho81]). It is not known whether a r X i v : . [ m a t h . G R ] J a n his generalized Freiheitssatz can be further generalized to torsion-free groups without stronger assump-tions. However, several Freiheitss¨atze for one-relator products of torsion-free groups are known underrestrictions to exponent sums of r due to A. Klyachko (see [Kly93]) and under the condition that r hasa syllable length smaller or equal 8 due to M. Edjvet and J. Howie (see [EH20]). There are also furtherFreiheitss¨atze for one-relator products assuming small cancellation conditions on the symmetric closureof the relation r (see [Juh06]).Another way of generalizing the Freiheitssatz of W. Magnus is to consider more than one relation r .Seeing the additional relations as part of the underlying group, these results are called Freiheitss¨atze forone-relator quotients . Generalizations of that kind can be found in [HS09], [HS10] due to J. Howie andM. Saeed. The aim of this article is to prove the following Freiheitssatz for special one-relator quotientswhich generalizes Chapter 4 of the author’s dissertation [Fel20]. After formulating this result we shortlydiscuss its assumptions and its connection to the results of [HS09].
Main Theorem 1.1.
Let A and B be two free groups and let G “ A ˚ U B be an amalgamated product,where U is a maximal cyclic subgroup in both factors. Further, let r be an element of G which is neitherconjugate to an element of A nor B . Then A and B canonically embed into the quotient group G {xx r yy G . The following example shows that the assumption on U to be a maximal cyclic subgroup cannot beomitted. Example 1.2.
Let A : “ x a |y , B : “ x b, c, d |y and r : “ a p bc q ´ . Further, let U be generated by p : “ a in A and by q : “ p bc q d in B . Then we have in p A ˚ U B q{xx r yy : p bc q d “ a ô p bc q d “ p bc q ô d “ x c, d |y does not embed into p A ˚ U B q{xx r yy even though q contains the basis element b of B .Note that Main Theorem 1.1 is in part already contained in [HS09, Theorems 3.1, Theorem 4.2 andTheorem 5.1] by J. Howie and M. Saeed since many amalgamated products of free groups over maximalcyclic subgroups in both factors are limit groups (in the sense of Z. Sela). For example the fundamentalgroups π p S ` g q with g ě π p S ` g q with g ě g are elementary equivalent to free groups and therefore limit groups. Otherexamples of amalgamated products of free groups over maximal cyclic subgroups in both factors thatare limit groups can be found in the class of so-called doubles of free groups F ˚ x w y–x r w y r F , where w is anelement of a free group F and r F , r w are copies of F, w . These doubles of free groups are word-hyperbolicif and only if x w y is a maximal cyclic subgroup in F (see [GW10]).However, there are also many amalgamated products of free groups over maximal cyclic subgroupsin both factors which are not limit groups. One such example is the group G “ x a, b, c, d, z | r a, b sr c, d s “ z y “ x a, b | y ˚ xr a,b sy – x z r c,d s ´ y x c, d, z | y . In [CCE91], J. Comerford, L. Comerford and C. Edmunds have shown that a non-trivial product oftwo commutators in a free group can never be more than a cube. It follows that z is in the kernel of very homomorphism ϕ : G Ñ F n , where F n is the free group of rank n . Therefore, the finitely gen-erated group G cannot be ω -residually free which is equivalent to G not being a limit group (see [CG05]).In order to prove Main Theorem 1.1 we study special products of groups which we call tree-products(see Definition 2.3) and finally introduce a Freiheitssatz for tree-products (see Theorem 7.1). MainTheorem 1.1 follows directly from this Freiheitssatz. As a tool we also prove the following lemmaconcerning one-relator products of locally indicable and free groups. Recall that a group G is indicable if there exists an epimorphism from G to Z . A group G is locally indicable if every non-trivial, finitelygenerated subgroup of G is indicable. Lemma 1.3.
Let H be a locally indicable group, A be a free group with basis A and let p : “ v ´ gv be anelement of A , where v, g use letters from disjunct sets of basis elements from A and where g j is cyclicallyreduced as well as no proper power. Further, let r be an element of A ˚ H which is not conjugate to anelement of A nor H nor x p |y ˚ H . Then x p |y ˚ H embeds canonically into p A ˚ H q{xx r yy . We want to recall some results that are important for the proof of Main Theorem 1.1. In 1981, J.Howie proved the following theorem which is known as the already mentioned
Freiheitssatz for locallyindicable groups . This result was also proved independently by S. Brodskii (see [Bro80], [Bro84, Theo-rem 1]) and by H. Short (see [Sho81]).
Theorem 1.4 ( see [How81, Theorem 4.3 (Freiheitssatz)] ) . Suppose G “ p A ˚ B q{ N , where A and B are locally indicable groups, and N is the normal closure in A ˚ B of a cyclically reduced word R oflength at least . Then the canonical maps A Ñ G , B Ñ G are injective. J. Howie obtains this Freiheitssatz by studying systems of equations over some group G . A finitesystem W of m P N equations w i p x , x , . . . x n q (1 ď i ď m ) in the variables x , x , . . . x n over the group G corresponds to the presentation S : “ x G, x , x , . . . x n | w , w , . . . , w m y . The system W is said to have a solution over G if it has a solution in some group containing G as a subgroup. Remark 1.5 ( cf. [How81, Proposition 2.3] ) . The group G embeds into S if and only if the system W has a solution over G .Let M be the p m ˆ n q matrix whose p i, j q -th entry is the sum of the exponents of x j occurring in theword w i P G ˚ x x y ˚ ¨ ¨ ¨ ˚ x x n y . A system W is called independent if the associated matrix M has rank m . It is conjectured that any independent system of equations over any group G has a solution over G .J. Howie proved for example the following special case of that conjecture. Theorem 1.6 ( cf. [How81, Corollary 4.2] ) . Let G be a locally indicable group. Then every indepen-dent system of equations over G has a solution over G . Aside from Theorem 1.6 we also use the following result about locally indicable groups. heorem 1.7 ( see [How82, Theorem 4.2] ) . Let A and B be locally indicable groups, and let G bethe quotient of A ˚ B by the normal closure of a cyclically reduced word r of length at least . Then thefollowing are equivalent: p i q G is locally indicable; p ii q G is torsion-free; p iii q r is not a proper power in A ˚ B . First, we recall the definition of staggered presentations in the sense of W. Magnus from [LS01, Sec-tion II.5] with some small alteration: Instead of using only cyclically reduced elements we allow someconjugation that leads to the definition of stabilizing generators. This is done in view of further appli-cations and has nearly no influence on the presented proof. The alteration is not necessary for provingthe main theorem of this article.
Definition 2.1 ( staggered presentations over free groups, cf. [LS01, Section II.5] ) . Let G “ x X Y S | P y be a group presentation, I Ď Z an index set and let Y “ Ů i P I Y i be a subset of X . We assume P “ t v ´ j g j v j | j P J u for a totally ordered index set J , v j P x S |y and cyclically reducedelements g j P x X |y such that each g j contains at least one element of Y . For every p j : “ v ´ j g j v j wedenote with α p j resp. ω p j the smallest resp. largest index i such that p j contains an element of Y i . If j ă k implies α p j ă α p k as well as ω p j ă ω p k , we say that G “ x X Y S | P y is a staggered presentation ( over free groups ) and P a staggered set of x X Y S | y . We call the generators from S stabilizing generators and the generators from X non-stabilizing .Next, we consider an example of a staggered set that is typical for the staggered sets arising in theproof of our main theorem. Example 2.2.
We define X “ t a i | i P Z u Y t b i | i P Z u Y t c i | i P Z u Y “ t a u , Y “ t b , c u , Y “ t a u , Y “ t a , b , c u , Y “ t b , c u , Y “ t a u and Y “ t a u , S “ H . Further, let p ´ “ a b c ´ a a ´ , p “ a b c ´ a a ´ and p “ a b c ´ a a ´ . Then P “ t p ´ , p , p u is a staggered set of x X | y “ x X Y S | y , where Y “ Ů i “ Y i Ă X . Definition 2.3 ( tree-products ) . Let T be a tree. We associate a free group F with basis F ‰ H toevery vertex of T . Every edge of T between two vertices with vertex-groups K and L is associated to an dge-relation p “ q , where p P K and q P L . We call p resp. q an edge-word of K resp. L and say that K and L are connected over the edge-relation p “ q . We demand that no edge-word is a proper powerin its vertex-group and that for each vertex-group F the set of all edge-words of F forms a staggered setof F . Let G be the union of the bases of all vertex-groups and let R be the union of all edge-relations.Then we call the group G : “ x G | R y a tree-product (associated to T ) . We refer to a vertex-group thatis associated to a leaf of T as a leaf-group . If T only contains one vertex, we do not consider that vertexas a leaf.We call a tree-product, associated to a subtree of T , a subtree-product of G . A subtree B of T is a branch if we get a tree by deleting every vertex of B in T and every edge of T that is connected to avertex of B . We call a tree-product, associated to a branch of T , a branch-product of G .Let r be an element of G and A a leaf-group of G . We say that r uses the leaf-group A with basis A if every presentation of r , written in the generators from G , uses at least one generator from A .Let the size | G | of G be the number of its vertex-groups. Finally, let P be the set of the edge-wordsof all leaf-groups in G and let P be the set of all cyclical reductions of the elements of P . Then we call σ : “ Σ p P P | p | the boundary-length of G . Figure 1: Illustration for a tree-product of size 9 with 6 leaf-groups
Notation 2.4.
Let G be a tree-product and let S be a subtree-product of G . We denote by G a S thefree product of subtree-products of G arising by deleting all generators and edge-relations of S alongwith the edge-relations of all edges adjacent to S from the presentation of G . Note that G a B is atree-product for every branch-product B of G . Analogously, we define G a M for a free product G oftree products and a set M of subtree-products of the tree-products from G . Lemma 2.5.
Tree-products are locally indicable groups.Proof.
Let G be a tree-product. Our proof is by induction over the size | G | . For the induction base | G | “
1, we note that, according to Definition 2.3, all vertex-groups of G are free and therefore locally ndicable. For the induction step let G be a tree-product with size n ` n P N . Further, let A be a leaf-group of G with edge-relation p “ q , where p P A . We have G “ p G a A q ˚ q “ p A . By theinduction hypothesis, G a A is locally indicable. Therefore, G is locally indicable by Theorem 1.7. In this section, we find for an arbitrary element r of a free product G ˚ H , where G is a tree-productand H a locally indicable group, a uniquely determined minimal subtree-product S of G such that S ˚ H contains at least one conjugate of r . Definition 3.1.
Let G be a tree-product and H a locally indicable group. Further let r be an element of G ˚ H which is not conjugate to an element of X ˚ H for any vertex-group X of G . We call a presentationof a conjugate r r of r contracted conjugate if the following properties hold:The presentation r r contains under all presentations of all conjugates of r only generators from H anda minimal subtree-product T of G . Further, let r r be of the form Π ni “ v p i q fulfilling the following threeproperties.(i) Every v p i q contains either only basis elements of a single vertex-group or is an elements of H zt u .(ii) Cyclically proceeding v p i q ( i P Z n ) contain elements of different vertex-groups or of a vertex-groupand H .(iii) No element v p i q is a power of an edge-word of a leaf-group of T .We call the elements v p i q pieces of the contracted conjugate r r . The number n of pieces of r r is the length of r r and is denoted by || r r || G ˚ H (or || r r || ). Finally, we call the minimal subtree-product T minimaltree-product of r P G ˚ H .The following lemma shows that Definition 3.1 is well-defined. Lemma 3.2.
Let G be a tree-product and H be a locally indicable group. Further, let r be an elementof G ˚ H which is not conjugate to an element of X ˚ H for any vertex-group X of G . Then the minimaltree-product of r is uniquely determined.Proof. Let r r be a contracted conjugate of r that is contained in a minimal tree-product T . For anarbitrary leaf-group A of T we consider the branch-products which arise from G by deleting T a A alongwith all edge-relations of the edges that connect T a A to the rest of the tree-product. We denote thebranch-product containing A by Z A and the free product of all remaining branch-products by R A . Withthe aim of obtaining a contradiction let r ˚ be a contracted conjugate of r that is contained in a minimaltree-product T ˚ different from T . By definition, we have | T | “ | T ˚ | . Since T and T ˚ are different thereis at least one leaf-group A of T such that T ˚ does not contain any vertex-group of Z A . For such aleaf-group A we define X : “ p G a R A q ˚ H, Y : “ p G a Z A q ˚ H and S : “ p G a Z A a R A q ˚ H. ote that G ˚ H “ X ˚ S Y. (3.1)Since X Xp T ˚ ˚ H q is a subgroup of S and the tree-product G a Z A a R A is smaller than T , the contractedconjugate r ˚ is not an element of X . Thus, there exists an element w P G ˚ H with w ´ r rw “ r ˚ R X. Since r r is an element of T Ă X , w is not an element of X . We consider the normal form w “ x y x . . . y m x m ` ( m ě
1) resp. the amalgamated product (3.1), where x , x m ` P p X z S q Y t u , x i P X z S for 2 ď i ď m and y i P Y z S for 1 ď i ď m . We write x ´ m ` y ´ m ¨ ¨ ¨ y ´ x ´ r rx loomoon : “ u y x ¨ ¨ ¨ y m x m ` “ r ˚ P Y. For x “ x ‰ u is a conjugate of r r and can therefore not be an element of S . In this section we introduce some operations for tree-products that will be helpful for the proof of ourmain theorem.
Definition 4.1 ( root-products ) . Let G be a tree-product and G be the union of the bases of allvertex-groups of G . Further let R be the union of all edge-relations of G . We consider a subset M Ă G that only contains basis elements of leaf-groups of G . For every element a P M we choose an element n p a q P Z zt u . Then we say that r G : “ x G Y t r a | a P M u | R , a “ r a n p a q p a P M qy is a root-product of G which is obtained by extracting the roots r a of the elements a P M . Remark 4.2.
If the cyclical reductions of the edge-words of all the leaf-groups of G contain at leasttwo basis elements, every root-product of G is also a tree-product.Before we introduce the next operation for tree-products we define special isomorphisms of tree-products: Definition 4.3 ( leaf-isomorphisms ) . Let G be a tree-product, n P Z zt u and let a, b be two stabilizingor two non-stabilizing generators of the same leaf-group A of G . Then we refer to the transition fromthe old generator b to the new generator ¯ b : “ ba n or ¯ b : “ a n b as a leaf-isomorphism (of the tree-product G ) if the resulting group presentation is still a tree-product. emark 4.4. Note that in the stabilizing case both choices and in the non-stabilizing case at least oneof the choices ¯ b : “ ba n or ¯ b : “ a n b results in building a new tree-product since every edge-word (writtenin the old generating set) is of the form v ´ pv where v only contains stabilizing generators and p iscyclically reduced and no proper power.Root-products and leaf-isomorphisms can help to generate the conditions for applying the homomor-phism described in the following definition (see Remark 4.7). Definition 4.5 ( leaf-homomorphisms ) . Let G be a tree-product, A be a leaf-group of G and H alocally indicable group. Let p “ q ( p P A ) be the edge-relation of the edge connecting A and G a A .Further let a be a basis element of A which is contained in p with exponential sum 0. Then we callthe homomorphism ϕ : G ˚ H Ñ Z which maps a to 1 and all other generators of G ˚ H to 0 a leaf-homomorphism (for a ). If we consider G together with an element r P G ˚ H that has exponential sum0 respectively a , we call ϕ a leaf-homomorphism for r P G ˚ H .In the following, we are mostly interested in the kernels of leaf-homomorphisms. Remark 4.6.
The kernel ker p ϕ q of a leaf-homomorphism ϕ : G ˚ H Ñ Z has the following structure: For H we obtain countably infinitely many free factors H i : “ a ´ Ha of ker p ϕ q . We consider the free productof these factors as one locally indicable free factor r H of ker p ϕ q . For every generator b ‰ a of A we obtaincountably infinitely many generators b i : “ a ´ i ba i ( i P Z ). These generators build a vertex-group r A ofker p ϕ q . For the edge-word p P A we obtain a staggered set P “ t p i | i P Z u of r A , where p i : “ a ´ i pa i .Instead of G a A there are countable finitely many copies X i : “ a ´ i p G a A q a i of G a A . Every copy X i contains a copy q i : “ a ´ i qa i of the edge-word q of the edge-relation p “ q associated to the edgeconnecting A with G a A . We connect X i with r A over the edge with edge-relation p i “ q i and therebyobtain a tree-product K with ker p ϕ q “ K ˚ r H . The edge-words p i P r A ( i P Z ) are strictly shorter thanthe word p P A since all letters a ˘ vanish and the word p i contains for every letter b ‰ a ˘ in p exactlyone letter b j ( j P Z ).It is possible that a tree-product G does not allow the application of a leaf-homomorphism. Thefollowing remark describes a procedure to either construct an isomorphic tree-product with strictlyshorter boundary-length or to move on to a tree-product r G allowing the application of a leaf-productsuch that G can be embedded into r G . Remark 4.7.
Let G be a tree-product, A be a leaf-group of G and H a locally indicable group. Further,let p “ q ( p P A ) be the edge-relation of the edge connecting A with G a A and let a , b be two non-stabilizing basis elements of A which are contained in p with exponential sums p a , p b ‰
0. First, we goover to the root-product r G of G setting a “ r a p b . Afterwards, we consider the leaf-isomorphism ψ of A in r G which is given by r b “ b r a p a or r b “ r a p a b . It follows that p , written using the new generators r a and r b has exponential sum 0 respectively r a . If p does not contain the generator r a , the boundary-length of r G respectively the new generating set is strictly smaller than the boundary-length of G . In the case that p ontains r a there is a leaf-homomorphism ϕ for the generator r a of the leaf-group A in r G . Since the onlypossible chances in the word lengths of p P G and p P r G come from the usage of the generators a / r a andthese generators vanish by building the kernel ker p ϕ q (see Remark 4.6), the length of p i P r A Ă ker p ϕ q is not only strictly shorter than the length of p P A Ă ψ p r G q , but also strictly shorter then the length of p P A Ă G .The following algorithm gives a contracted conjugate r r ˚ in ker p ϕ q for a contracted conjugate r r of anelement r in a tree-product G that is contained in the kernel ker p ϕ q of a leaf-homomorphism ϕ of G . InLemma 4.9, we prove the functionality of that algorithm. Algorithm 4.8.
Let G be a tree-product, H a locally indicable group and r an element of G ˚ H withcontracted conjugate r r . Further let r be an element of the kernel of a leaf-homomorphism ϕ for a basiselement a of the leaf-group A in G , where ker p ϕ q “ K ˚ r H is given with the structure described inRemark 4.6. The following algorithm rewrites the contracted conjugate r r of r P G ˚ H into a contractedconjugate r r ˚ of a ´ (cid:96) ra (cid:96) P ker p ϕ q ( (cid:96) P Z ) with || r r ˚ || ker p ϕ q ď || r r || G ˚ H .For reasons of symmetry, it suffices to only consider contracted conjugates r r ˚ of r “ r P ker p ϕ q : Tofind a contracted conjugate of a ´ (cid:96) ra (cid:96) P ker p ϕ q ( (cid:96) P Z ) we can apply the algorithm to r “ r and replaceall indices i ( i P Z ) in the resulting contracted conjugate r r ˚ by i ` (cid:96) .Let r r be given in the form r r “ Π ni “ v p i q (see Definition 3.1). First, we step by step construct apresentation r r of r r as an element of ker p ϕ q . For the start we set λ “ r r “ P ker p ϕ q and j “ j -th generator e p j q in r r . If e p j q “ a we set λ “ λ ´ e p j q “ a ´ we set λ “ λ `
1. In the case e p j q ‰ a ˘ we add the letter e p j q λ “ a ´ λ e p j q a λ to the word r r . If e p j q is the lastletter in r r , we go to Step 2. Elsewise we set j “ j ` λ , r r and j .The presentation r r given by Step 1 can be written in the form r r “ Π ni “ r v p i q , where the subwords r v p i q for which v p i q does not lie in A can be obtained from v p i q by adding the same index to every letterof v p i q . Subwords r v p i q are trivial if and only if v p i q is a power a k ( k P Z zt u ). In that case the indices ofthe adjacent subwords r v p i ´ q and r v p i ` q differ by k .Now, we delete every trivial subword r v p i q . If r v p i q is trivial and r v p i ´ q , r v p i ` q are elements of r H ,we merge them into one subword. If r v p q is trivial and r v p q , r v p n q are elements of r H , we permute thepresentation r r cyclically by r v p q and merge r v p n q r v p q . We proceed analogously in the case that r v p n q “ r v p q , r v p n ´ q P r H . Let r r “ Π n i “ v i q be the resulting presentation. Then r r satisfies the properties(i) and (ii) of Definition 3.1. Let T be the minimal subtree-product of K such that r r only containsgenerators from T and r H . The following steps rewrite r r into a conjugate r r ˚ of r in ker p ϕ q that alsosatisfies property (iii) of Definition 3.1.Step 2. We replace all subwords v i q ( i P Z n “ t , , . . . , n u ) which are powers of edge-words p of aleaf-group L of T with edge-relation p “ q ( p P L ) by the corresponding power of q . If v i ´ q or v i ` q are elements of the vertex-group containing the edge-word q , we merge v i q with those subwords(after cyclically permutation if necessary). If there is a leaf-group of T such that no generator of T is sed in the new presentation, we delete this leaf-group and the edge-relation of the edge connecting theleaf-group to the rest of T . We repeat Step 2 with the new presentation and subtree-product T untilthere is no subword v i q left which is a power of an edge-word of a leaf-group of T . Let r r “ Π n i “ v i q and T be the resulting presentation and subtree-product.Step 3. We step by step consider all leaf-groups B of T . Let m be the number of all subwords v i q written in the basis of B in the presentation r r “ Π n i “ v i q . Further, let τ : Z m “ t , , . . . , m u Ñ Z n “ t , , . . . , n u be the function which maps j to the position i of the j -th subword written in thebasis of B . We merge the subwords v i q from p T a B q ˚ r H that are placed before the first, after thelast or between two subwords v τ p j q ( j P t , , . . . , m u ) to new subwords w p j q ( j P t , , . . . , m uq . Afterconjugation with w p q as well as merging w p q and w p m q to a new w p m q we may write w. l. o. g. r r “ Π mj “ v τ p j qq w p j q . (4.1)In the case that there are subwords w p i q which correspond as elements of ker p ϕ q to powers of theedge-word p P B , we merge them with their cyclical neighbours v p τ p i qq and v p τ p i ` qq to a new subwordfrom B . After that, we repeatedly delete trivial subwords and merge newly cyclically adjacent w p k q - or v k q -subwords. Finally, we divide the remaining subwords from p T a B q ˚ r H into the original subwords v p i q and return to Step 2 with the resulting presentation along with T as the new T .In the case that no subword w p i q of the presentation (4.1) corresponds as an element of ker p ϕ q to apower of the edge-word p P B , we repeat Step 3 with the next leaf-group of T . When we checked allleaf-groups of T in one run of Step 3 without going back to Step 2, we set r r ˚ : “ r r “ Π n i “ v i q .The following lemma secures the functionality of Algorithm 4.8. Lemma 4.9.
Algorithm 4.8 ends after finitely many iterations. The element r r ˚ given by Algorithm 4.8is a contracted conjugate of a ´ i ra i P ker p ϕ q and the length of r r ˚ is shorter or equal to the length of thecontracted conjugate r r of r P G ˚ H .Proof. The finiteness of the algorithm and the inequality || r r ˚ || ker p ϕ q ď || r r || G ˚ H follow from the fact thatno step increases the number of subwords r v p i q / v i q / v i q in the current presentation and every step ofthe algorithm which redirects to a previous step shortens the number of those subwords.It remains to show that r r ˚ is indeed a contracted conjugate of r P ker p ϕ q . First, we note that r P ker p ϕ q “ K ˚ r H (see Remark 4.6) cannot be conjugate to an element of X ˚ r H for a vertex-group X of K since otherwise r P G ˚ H would be conjugate to an element of Y ˚ H , where Y is a vertex-groupof G and therefore would not have a contracted conjugate (see Definition 3.1).The validity of the properties (i) and (ii) for contracted conjugates (see Definition 3.1) is maintainedby every step. Moreover, the validity of property (iii) is secured by Step 2. Note that the algorithmonly ends if Step 3 leaves the presentation unchanged. So the output r r ˚ satisfies the properties (i)-(iii)of Definition 3.1. We end the proof of this lemma by showing that no conjugate of r in ker p ϕ q can becontained in the free product of r H and a smaller subtree-product than T . For this purpose we consider n arbitrary leaf-group B of T . Similar to the proof of Lemma 3.2 we consider the branch-productsarising from K by deleting all vertex-groups of T a B along with the edge-relations of the adjacentedges. We denote the branch product containing B by Z B and writeker p ϕ q “ K ˚ r H “ Z B ˚ p “ q ` p K a Z B q ˚ r H ˘ . (4.2)Note that in this last run of Step 3 the presentation (4.1) is for every leaf-group B of T a cyclicallyreduced normal from of a conjugate of r “ r P ker p ϕ q respectively the amalgamated product B ˚ p “ q ` p T a B q ˚ r H ˘ with length greater or equal two. This presentation is in particular a cyclically reduced normal formrespectively the amalgamated product from (4.2) with length greater or equal two. Thus, no conjugateof r in ker p ϕ q can be contained in p K a Z B q ˚ r H . So the minimal tree-product of r contains for everyleaf-group B of T at least one vertex-group of Z B . Therefore the minimal tree-product of r containsthe tree-product T . Since r r ˚ P T ˚ r H , the tree-product T is the minimal tree-product of r in ker p ϕ q . Remark 4.10.
Let G be a tree-product, H a locally indicable group and r r “ Π ni “ v p i q a contractedconjugate in G ˚ H . Further, let ϕ be a leaf-isomorphism from G ˚ H for a leaf-group A . Then, bywriting every piece v p i q P A using the new generators from ϕ p A q , we get a contracted conjugate r r ˚ of ϕ p G ˚ H q with the same length.The next lemma will allow us to apply the induction hypotheses in the proofs of our embeddingtheorems. Lemma 4.11.
Using the notation of Remark 4.6 (and Remark 4.7) let r be a contracted conjugate in G ˚ H , let T be the minimal tree-product of an element r P ker p ϕ q and let σ be the boundary-length of T .We denote the contracted conjugate of a ´ i ra i P ker p ϕ q by r i . Further let T i be the minimal tree-productof r i and let σ i be the border length of T i . Then we have p|| r i || ´ | T i | , σ i q ă p|| r || ´ | T | , σ q respectively the lexicographical order (where the first component is weighted higher).Proof. Because of Lemma 4.9, we have || r i || ď || r || . Since every rooted tree is the union of all uniquepaths from the root to the leafs and T i contains for every leaf-group L of T at least one copy of L , wealso have | T i | ě | T | , thus: || r i || ´ | T i | ě || r || ´ | T | ô || r i || ´ | T i | “ || r || ´ | T | ô p|| r i || “ || r || ^ | T i | “ | T |q Let p “ q (where p P A ) be the edge-relation to the edge connecting A with G a A . The equality | T i | “ | T | implicates T i “ p a ´ i p T a A q a i q ˚ q i “ p i r A . As noticed in Remark 4.6 and Remark 4.7 we have | p i | r A ă | p | A . very other edge-word of T i is obtained from the associated edge-word of T by adding indices. So thelengths of the edge-words different from p do not change. Altogether, we get the inequality σ i ă σ inthe case || r i || ´ | T i | “ || r || ´ | T | .We end this section with the following definition. Definition 4.12 ( reduction- and fan-generators ) . Let ϕ : G ˚ H Ñ Z be a leaf-homomorphism for r mapping a generator a of a leaf-group A of G to 1. We use the notation of Remark 4.6 and considerthe element r as the element r in ker p ϕ q . If r is an element of p r A ˚ p i “ q i X i q ˚ r H for an element i P Z ,we call a a reduction-generator of r P G ˚ H . Elsewise we call a a fan-generator of r P G ˚ H . For the proof of the Magnus-Freiheitssatz and the Magnus property of free groups (see [Mag30]), W.Magnus defined staggered presentations over free groups (cf. [LS01, Section II.5], also Definition 2.1). Wegeneralize that definition firstly to the case of locally indicable groups and secondly to the case of specialtree-products. For the connection between the following definition and Definition 2.1 see Remark 5.2.
Definition 5.1 ( staggered presentations over locally indicable groups ) . Let I , J Ă Z be indexsets, let U be an arbitrary locally indicable group, let V i ( i P I ) be non-trivial locally indicable groupsand let r j ( j P J ) be cyclically reduced elements of U ˚ ˚ i P I V i , such that every r j uses at least oneof the free factors V k ( k P I ). Further, let W : “ ˚ i P I V i . We denote the smallest (resp. largest) index (cid:96) P I such that r j uses the free factor V (cid:96) by α r j (resp. ω r j ). We call p U ˚ W q{xx r j | J yy a staggeredpresentation (over locally indicable groups) if the inequalities α r m ă α r n and ω r m ă ω r n hold for all m , n P J with m ă n . Remark 5.2.
Definition 2.1 is apart from the stabilizing generators the special case of Definition 5.1for free groups. This becomes apparent in the following way: Let G “ x X Y S | P y be a staggeredpresentation in the sense of Defintion 2.1 with respect to subsets Y i ( i P I ) of X . By cyclically reducingall elements of P we get a set P “ x r j | j P J y of cyclically reduced relations satisfying x X Y S | P y “ x X Y S | P y “ x X | P y ˚ x S |y . Defining V i : “ x Y i |y ( i P I ) and U : “ xp X Y S qz Y |y , where Y “ Ů i P I Y i , we see that x X Y S | P y is astaggered presentation in the sense of Definition 5.1 for the special case of free groups V i and U .Before we continue examining staggered presentations we recall a definition of aspherical presenta-tions. Definition 5.3 ( cf. [LS01, Chapter III, Proposition 10.1] ) . A presentation G “ x X | R y is aspherical if and only if there are no non-trivial identities in x X |y among the relations R .In the proofs of our embedding theorems we will use the following results. heorem 5.4 ( cf. [LS01, Chapter III, Proposition 11.1] ) . If G “ x X | R y where R consists of asingle relator, or more generally, if the presentation is staggered (in the sense of Definition 5.1, but withfree groups V i and U ), then the presentation is aspherical. Remark 5.5.
Let G “ x X Y S | P y be a staggered presentation in the sense of Defintion 2.1 withrespect to subsets Y i ( i P I ) of X , where P “ t p j | j P J u for an index set J . Then the free group x P |y embeds canonically into x X Y S |y , i. e. there are no non-trivial identities among the elementsof P in x X Y S |y . This can easily be seen by the following observation: From Remark 5.2 we noticethat G {xx S yy “ x X | P y , where P consists of the images of the elements p j P P under the canonicalhomomorphism from G to G {xx S yy is a staggered presentation in the sense of Definition 5.1, but withfree groups V i and U . Combining Definition 5.3 and Theorem 5.4 we see that there are no non-trivialidentities among the elements of P in x X |y . However, such an identity would follow from a non-trivialidentity among the elements of P in x X Y S |y by applying the homomorphism sending the elements of X to themselves and the elements of S to the trivial element. Theorem 5.6 ( see [LS01, Chapter III, Proposition 10.2] ) . If G “ x X | R y is aspherical, and noelement of R is conjugate to another or to its inverse, then the following condition holds: Let p ¨ ¨ ¨ p n “ where each p i “ u i r e i i u ´ i for some u i P F , r i P R , and e i “ ˘ . Then the indices fall into pairs p i, j q such that r i “ r j , e i “ ´ e j , and u i P u j N C i where C i (the centralizer of r i ) is the cyclic group generatedby the root s i of r i “ s m i i . Next, we prove the following corollary of Theorem 1.4 about staggered presentations of locally indi-cable groups.
Corollary 5.7.
Let U be a locally indicable group and p U ˚ W q{xx r j | J yy be a staggered presentationfor W “ ˚ i P I V i with locally indicable groups V i . Further, let w be a non-trivial element of U ˚ W . Let α, ω P I be indices such that w is contained in the normal closure of the elements r j ( j P J ) in U ˚ W and uses only free factors V k with α ď k ď ω . Then w is already contained in the normal closure of theelements r (cid:96) in U ˚ W with α ď α r (cid:96) and ω r (cid:96) ď ω .Proof. Let V ď µ : “ ˚ k ď µ V k , V ě µ : “ ˚ k ě µ V k and V µ,ν : “ ˚ µ ď k ď ν V k for some indices µ, ν P I . We fixan arbitrary element m P J . First, we prove for an n P J with m ď n the isomorphy p U ˚ W q{xx r k | m ď k ď n yy– ` p U ˚ V ď ω rn ´ q{xx r k | m ď k ď n ´ yy ˘ ˚ U ˚ V αrn,ωrn ´ ` p U ˚ V ě α rn q{xx r n yy ˘ (5.1)by induction on n ´ m . For the induction base ( m “ n ) the isomorphy follows directly from Theorem 1.4.Note for the induction step ( n Ñ n `
1) that U ˚ V α rn ` ,ω rn embeds due to Theorem 1.4 canonicallyinto the factor p U ˚ V ě α rn q{xx r n yy which embeds due to the induction hypothesis into p U ˚ W q{xx r k | m ď k ď n yy . Since U ˚ V α rn ` ,ω rn also embeds due to Theorem 1.4 canonically into p U ˚ V ě α rn ` q{xx r n ` yy e derive the desired isomorphy p U ˚ W q{xx r k | m ď k ď n ` yy– ` p U ˚ V ď ω rn q{xx r k | m ď k ď n yy ˘ ˚ U ˚ V αrn ` ,ωrn ` p U ˚ V ě α rn ` q{xx r n ` yy ˘ . This ends the proof of the isomorphy (5.1).For the purpose of a contradiction, let w be an element of xx r j | j P J yy U ˚ W which is also anelement of the free product H ˚ V α,ω for some indices α, ω , but is not contained in the normal closure xx r (cid:96) | α ď α r (cid:96) , ω r (cid:96) ď ω yy U ˚ W . Let us choose two indices m, n P J such that w is an element of thenormal closure xx r (cid:96) | m ď (cid:96) ď n yy U ˚ W and n ´ m is minimal with this property. We only considerthe case ω ă ω r n since the case α r m ă α follows analogously. Consider the amalgamated productfrom isomorphy (5.1). Because of the inequality ω ă ω r n , w is an element of the left factor of thisamalgamated product. Since w is trivial in p U ˚ W q{xx r k | m ď k ď n yy it must already be trivial in p U ˚ V ď ω rn ´ q{xx r k | m ď k ď n ´ yy . This contradicts the minimality of n ´ m . We begin this section by proving Lemma 1.3.
Proof of Lemma 1.3.
W. l. o. g. let r be cyclically reduced. Our proof is by induction over the length | g | of the cyclical reduction g of p . For the base of induction let | g | “
1. Then p is a primitive elementof A . We extend p to a basis A of A . Since r is not an element of x p |y ˚ H , r contains a basis elementfrom A zt p u which is a free basis of A {xx p yy . We have p A ˚ H q{xx r yy “ ` p A {xx p yyq ˚ x p |y ˚ H ˘ {xx r yy . So, by Theorem 1.4, x p |y˚ H embeds canonically into p A ˚ H q{xx r yy . For the induction case we choose twobasis elements a, b P A which are contained in g . We want to construct a basis-element r a with p r a “ p a “
0, we set r a : “ a, r b : “ b , and if p a ‰
0, but p b “
0, we set r a : “ b, r b : “ a . For p a ‰ ‰ p b wedefine r a by a “ r a p b and set r b : “ b r a p a . Note that for p written in the new basis A : “ p A zt a, b uq Y t r a, r b u we have p r a “ r r a “ ϕ : A ˚ H Ñ Z sending r a to 1 and every other basis element from A along with every element of H to 0. It is easy to see that the normal closure of r P A ˚ H correspondsto the normal closure of the elements r i ( i P Z ) in ker p ϕ q , where r i “ r a ´ i r r a i . We haveker p ϕ q “ r A ˚ ˚ i P Z H i , where H i : “ r a ´ i H r a i and r A is the free group with basis r A : “ t x i | x P A zt r a u , i P Z u for x i “ r a ´ i x r a i . We define p i : “ r a ´ i p r a i and g i : “ r a ´ i g r a i ( i P Z ). To show the desired embedding it is sufficient to prove the embedding of x p |y ˚ H into ker p ϕ q{xx r i | i P Z yy . By assumption, we have r P p A ˚ H qz H . Thus, each r i P r A ˚ ˚ i P Z H i ontains at least one piece of at least one free factor H i . It follows that ker p ϕ q{xx r i | i P Z yy is a staggeredpresentation over locally indicable groups (with V i : “ H i , cf. Definition 5.1). Let j P Z such that r j contains an element from H . By Corollary 5.7, it remains to show the embedding of x p |y ˚ H intoker p ϕ q{xx r j yy . Because a free product of locally indicable groups is locally indicable, this embeddingfollows by the induction hypothesis which can be seen in the following way: Comparing g written in thebasis A and g in the basis r A we see that every letter a vanished without replacement and every otherletter x was replaced with a letter x i . Since g contained at least one letter a by assumption, we have | g | ą | g | . This justifies the application of the induction hypothesis.Case 2. r r a ‰ ‰ p r b In this case the matrix ¨˝ p r a p r b r r a r r b ˛‚ has full rank. Let G : “ A ˚ p “ c px c |y ˚ H q . Because of Theorem 1.6 and Remark 1.5 it follows that x c |y ˚ H and therefore x p |y ˚ H embeds canonically into G {xx r yy “ p ` A ˚ H q{xx r yy ˘ ˚ p “ c x c |y . Finally, wededuce that x p |y ˚ H embeds canonically into p A ˚ H q{xx r yy .Case 3. r r a ‰ “ p r b Note that because of p r b “ p a “ p b “ r a and defined r a : “ a, r b : “ b .Thus, if r r b “
0, we can apply Case 2 with reversed roles of r a and r b . If r r b ‰ a by a “ ¯ a r b and set ¯ b : “ b ¯ a r a . Note that for r , g written using the basis elements of ¯ A : “ p A zt a, b uq Y t ¯ a, ¯ b u we have r ¯ a “ p ¯ a “
0. Thus, we can also apply Case 2; this time with ¯ a and ¯ b taking over the roles of r a and r b .Next, we prove an easy lemma which enables us to pass from tree-products to certain root-productsin the proofs of our embedding theorems. Lemma 6.1.
Let H be a locally indicable group, let G be a tree-product and let r G be a root-product of G which is obtained though the relation a “ r a k , where k P Z zt u and a is a generator of a leaf-group A .Further, let r be an element of G ˚ H and let U be a subgroup of G such that U ˚ H embeds canonicallyinto p r G ˚ H q{xx r yy . Then U ˚ H embeds canonically into p G ˚ H q{xx r yy .Proof. If x a y A embeds into p G ˚ H q{xx r yy , we set m “ 8 . Elsewise let m P N be the smallest power suchthat a m is trivial in p G ˚ H q{xx r yy . Then the desired statement follows directly from p r G ˚ H q{xx r yy – p G ˚ H q{xx r yy ˚ a “ r a k x r a | r a km “ y , since an element of U ˚ H which is trivial in the left factor of the amalgamated product must also betrivial in the amalgamated product itself.The following lemmata are further tools that will be used repeatedly in the proofs of our embeddingtheorems. For the proof of the first lemma we use a theorem of A. Karrass, W. Magnus and D. Solitar: heorem 6.2 ( see [KMS60, Theorem 3] ) . Let G be a group with generators a, b, c, . . . and a singledefining relation V k p a, b, c, . . . q , k ą , where V p a, b, c, . . . q is not itself a true power. Then V has order k and the elements of finite order in G are just the powers of V and their conjugates. Lemma 6.3.
Let G be a tree-product, S be a subtree-product of G and H be a locally indicable group.Further, let r be an element of G ˚ H with contracted conjugate r r and minimal tree-product T . Finally,let T contain at least one vertex-group of S . If ` S X T ˘ ˚ H embeds canonically into p T ˚ H q{xx r r yy , then S ˚ H embeds canonically into p G ˚ H q{xx r yy .Proof. First, we note that T Y S and T X S are subtree-products of G since T , S are subtree-productsof G and T contains by assumption at least one vertex-group of S . We may define P p q : “ ` p T ˚ H q{xx r r yy ˘ ˚ p S X T q˚ H ` S ˚ H ˘ , (6.1)where the embedding of the amalgamated subgroup into the left factor follows due to the assumption. Let U p j q ( j P t , , . . . , k u , k P N ) be the branch-products obtained by deleting every vertex-group of T and S along with the edge-relations of the edges adjacent to T or S . Further, let p j “ q j ( j P t , , . . . , k u , k P N )with q j P U p j q be the edge-relations of the edges connecting U p j q to G a U p j q . Let (cid:96) P N Y t8u be thesmallest power such that p (cid:96) is trivial in P p q . We prove the following claim. For all (cid:96) P N , j P t , , . . . , k u the element q j is of order (cid:96) in U p j q {xx q (cid:96)j yy . To prove that claim we consider the quotient group r U p j q of U p j q which we construct by taking thequotient with the normal closure of q (cid:96)j and every edge-word of U p j q apart from q j . This quotient groupis a free product of staggered presentations over free groups in the sense of Definition 2.1. Note thatby cyclically reducing all stabilizing generators we arrive at a staggered presentation of locally indicablegroups in the sense of Definition 5.1 for the special case of free groups (see Remark 5.2). Let Q be thevertex-group of U p j q containing q j . If q λj would be trivial in U p j q {xx q (cid:96)j yy for some λ P N with λ ă (cid:96) , then q λj would also be trivial in r U p j q . Using Corollary 5.7 (for the special case of free groups) we concludethat q λj would be trivial in Q {xx q (cid:96)j yy . This contradicts Theorem 6.2. Therefore, we can define inductivelyfor j P t , , . . . , k u P p j q “ P p j ´ q ˚ p j “ q j p U p j q {xx q (cid:96) j j yyq , where (cid:96) j P N Yt8u is the smallest power such that p (cid:96) j j is trivial in P p j ´ q . Note that P p k q “ p G ˚ H q{xx r yy .Altogether, we constructed p G ˚ H q{xx r yy by iteratively building amalgamated products starting with thegroup S ˚ H (see (6.1)). Thus, S ˚ H embeds canonically into p G ˚ H q{xx r yy . Lemma 6.4.
Let G be a tree-product, S a subtree-product of G and H a locally indicable group. Further,let r be an element of G ˚ H with contracted conjugate r r . Under the condition that the minimal tree-product T of r contains at least one vertex-group of S and assuming that at least one edge-word ofa leaf-group of T that is also part of G a S is a primitive element, S ˚ H embeds canonically into p G ˚ H q{xx r yy . roof. Due to Lemma 6.3 it is sufficient to consider the case that G is the minimal tree-product T of r . Let A be a leaf-group of G a S and G with edge-relation p “ q , where p P A is a primitive element.Such a group exists by assumption. We extend p to a basis of A and replace the old basis with this newbasis. Since the leaf-group A is part of the minimal tree-product T and all generators p ˘ in r r can bereplaced with the help of the edge-relation p “ q we know that r r contains at least one basis element a ‰ p of A . We have p G ˚ H q{xx r yy “ ` p G a A q ˚ q “ p x p y ˚ p A {xx p yyq ˚ H ˘ {xx r yy “ ` p G a A q ˚ H ˚ p A {xx p yyq ˘ {xx r yy . Because of Theorem 1.7 and S Ď G a A , we get the desired embedding.The following proposition is the first step in proving our Freiheitssatz for tree-products (see Theo-rem 7.1). Proposition 6.5.
Let G be a tree-product and S be a subtree-product of G . Further, let H be a non-trivial locally indicable group and r a contracted conjugate in p G ˚ H qz G . Assuming that the minimaltree-product of r contains at least one vertex-group of respectively S and G a S , S ˚ H embeds canonicallyinto p G ˚ H q{xx r yy .Proof. Because of Lemma 6.3, we may assume that G is the minimal tree-product of r . Let σ be theboundary-length of G and let | G a S | be the number of all vertex-groups of G a S . We prove thedesired statement by induction over the tuple p|| r || ´ | G a S | , σ q in lexicographical order (where the firstcomponent is weighted higher).As the base of induction we consider the cases || r || ´ | G a S | P Z , σ “ || r || ´ | G a S | ď σ P N . If | G a S | “
1, let A : “ G a S . Elsewise we choose a leaf-group A of G and G a S . In the case σ “ A is a primitive element of A . Therefore, we derive the desired embedding withLemma 6.4. In the case || r ||´| G a S | ď B of G a S such that r contains no basiselement of B . The vertex-group B cannot be a leaf-group of G since we assumed that G is the minimaltree-product of r . We consider the branch-products of G which arise by deleting the vertex-group B along with the edge-relations of the adjacent edges in G . Since B is no leaf-group, there is at least onebranch-product Y among the resulting branch-products which is completely contained in G a S . Wedenote the free product of the remaining branch-products by R . Let p i “ q i ( i P I ) with p i P B be theedge-relations of the edges adjacent to B . Since, by assumption, r is no element of G , Theorem 1.4 givesus the canonical embedding of Y ˚ R into p Y ˚ R ˚ H q{xx r yy . Let R p q , R p q , . . . , R p k q ( k P N ) be the freefactors of R . Then every q i is contained in different factors of the free product Y ˚ R p q ˚ R p q ˚ ¨ ¨ ¨ ˚ R p k q .Because of Lemma 2.5 all free factors are locally indicable and therefore in particular torsion-free. Thus,the group Q freely generated by t q i | i P I u embeds into Y ˚ R and hence into ( Y ˚ R ˚ H q{xx r yy . Sincethe elements p i ( i P I ) of B form a staggered set and B {xx p i | i P I yy is a staggered presentation over freegroups, the group P freely generated by t p i | i P I u embeds canonically into B as noticed in Remark 5.5. o we may write: p G ˚ H q{xx r yy – B ˚ P – Q ` p Y ˚ R ˚ H q{xx r yy ˘ Finally, note that R ˚ H embeds into the right factor of the amalgamated product by Theorem 1.4.Therefore, S ˚ H embeds into p G ˚ H q{xx r yy .For the induction step we consider the leaf-group A of G a S and G along with the edge-relation p “ q ( p P A ) of the edge adjacent to A . If the cyclical reduction of p contains only one basis element a of A , we have p “ v ´ a ˘ v , where v consists only of stabilizing generators, due to Definition 2.3.Thus, p is a primitive element of A and the desired embedding follows analogously to the induction basefor σ “
2. Let p contain at least two different non-stabilizing basis elements a , b of A . We recall thenotation p b for the exponent sum of p P A respectively b .If p a “ p b “
0, we consider the root-product r G of G given through r a r b “ a , which is a tree-productbecause of p ‰ v ´ a ˘ v . By Lemma 6.1 it is sufficient to prove the embedding of S ˚ H into p r G ˚ H q{xx r yy .We apply the leaf-isomorphism given by r b : “ b r a r a or r b : “ r a r a b (cf. Remark 4.4). In slight abuse ofnotation we denote the image of r G under the leaf-isomorphism and the new contracted conjugate againby r G and r . The new presentation of r P r G ˚ H is also a contracted conjugate with the same length as theold presentation r P G ˚ H . Since p r a “ p is the only edge-word possibly containing r a , the exponentsum r r a is well-defined. We have p r a “ r r a “
0. If p does not contain the generator r a , the tree-product r G has a shorter boundary-length than G and the desired embedding follows by the induction hypothesis.If p contains r a , we go directly to Case 2. Thus, in the following part of the proof up to Case 2 we canassume that at least one exponential sum p a or p b is not 0. W. l. o. g. let p b ‰
0. Similar to the situation p a “ p b “ a “ r a p b the root-product r G : “ G ˚ a “ r a pb x r a |y of G and apply theleaf-isomorphism of r G which is given by r b : “ b r a p a or r b : “ r a p a b (cf. Remark 4.4). We have p r a “
0. If p does not contain the generator r a , the tree-product r G has a shorter boundary-length than G and thedesired embedding follows by the induction hypothesis. Thus, assume that p contains r a .Case 1. Let r r a ‰ p b ‰ p r b ‰
0. So (for some arbitrary fixedpresentation r ) the matrix ¨˝ p r a p r b r r a r r b ˛‚ has full rank and the desired embedding follows from Remark 1.5and Theorem 1.6.Case 2. Let r r a “ ϕ : r G ˚ H Ñ Z with ϕ p r a q “
1. Let A be the basis of the leaf-group A in G . As noticed in Remark 4.6, we have ker p ϕ q “ K ˚ r H , where r H “ ˚ (cid:96) P Z H (cid:96) for H (cid:96) : “ a ´ (cid:96) Ha (cid:96) ( (cid:96) P Z )and K is a tree-product which can be constructed in the following way. We start with a vertex-group r A : “ x r A |y for r A : “ t r a ´ (cid:96) y r a (cid:96) | y P A zt a u , (cid:96) P Z u and connect r A over respectively one edge-relation p (cid:96) “ q (cid:96) ( p (cid:96) P A ) with countably infinitely many copies( r G a A q (cid:96) : “ a ´ (cid:96) p r G a A q a (cid:96) ( (cid:96) P Z ) of r G a A . Note that | p (cid:96) | r A ă | p | A for all (cid:96) P Z (cf. Remark 4.7). y assumption, the contracted conjugate r P G ˚ H uses the factor H . We define r i : “ a ´ i ra i .Then ker p ϕ q{xx r i | i P Z yy is a staggered presentation for V i “ H i (cf. Definition 5.1). Algorithm 4.8gives us a contracted conjugates r ˚ i of r i P ker p ϕ q ( i P Z ) with || r ˚ i || ker p ϕ q ď || r || G ˚ H (see Lemma 4.9).Since all elements of S which are not contained in the kernel of ϕ cannot be elements of the normalclosure of r in r G ˚ H , it suffices to prove the embedding of the copy S ˚ H Ă p r G a A q ˚ H of S ˚ H in ker p ϕ q{xx r ˚ i | i P Z yy . By renaming, if necessary, let r ˚ : “ r ˚ be w. l. o. g. an element r ˚ i ( i P Z )which uses the free factor H of ker p ϕ q . Then, by Corollary 5.7, an element w P S ˚ H is trivial inker p ϕ q{xx r ˚ i | i P Z yy if and only if it is trivial in ker p ϕ q{xx r ˚ yy . The embedding that remains to showdepends on the minimal tree-product T of r ˚ :Case 2.1. T contains at least one vertex-group of S .Applying Lemma 6.3 on the subtree-product S of K we see that it suffices to consider T ˚ r H insteadof ker p ϕ q . So our aim is to prove the embedding of p S X T q ˚ H into p T ˚ r H q{xx r ˚ yy . Note that r a iseither a reduction- or fan-generator (see Definition 4.12). If r a is a reduction-generator of r P r G , we have T “ r A ˚ p “ q p r G a A q . As noticed above we have | p | r A ă | p | A . Therefore, the boundary-length of T isshorter than the boundary-length of G and the desired embedding follows by the induction hypothesis.Thus, assume that r a is a fan-generator of r P r G . We define S : “ S X T . Using this notation our aimis to prove the embedding of S ˚ H into p T ˚ r H q{xx r ˚ yy . We want to apply the induction hypothesis.Because of || r ˚ || ker p ϕ q ď || r || G ˚ H it suffices to show | T a S | ą | G a S | . Note that for every leaf-group of G there is at least one copy of this leaf-group in T since G is the minimal tree-product of r and T is theminimal tree-product of r . Because every rooted tree is the union of all unique paths from the root tothe leafs, T contains at least one copy of every vertex-group of T a S and we get | T a S | ě | G a S | . Byassumption, r a is a fan-generator. Thus, T contains at least two copies C , C µ of the unique vertex-group C of G adjacent to A . If C is a vertex-group of G a S , the vertex-groups C and C µ are vertex-groupsof T a S . If C is a vertex-group of S then C µ is a vertex-group of T a S . In both cases we have anadditional vertex-group of T a S and therefore | T a S | ą | G a S | . By applying the induction hypothesisfor ( || r ˚ || ´ | T a S | , σ ), where σ is the boundary-length of T , we derive the desired embedding.Case 2.2. T contains no vertex-group of S .Let S be the free factor of K a T containing S . Further, let D be the vertex-group of K a S that isconnected by an edge to S and let d “ e with d P D be the edge-relation of that edge. We first want toshow that D ˚ r H embeds canonically into p T ˚ r H q{xx r ˚ yy .If r a is a reduction-generator of r P r G , we have D “ r A , T “ r A ˚ p j “ q j p r G a A q j with j P Z zt u and | p j | r A ă | p | A , so the boundary-length of T is shorter than the boundary length of G (cf. Case 2.1).Because of | T | “ | G | and | S | ě D ˚ r H embeds canonically into p T ˚ r H q{xx r ˚ yy . In the case that r a is a fan-generator we have | T | ě | G | ` D ˚ r H . sing Lemma 6.3 it follows that D ˚ r H embeds into pp K a S q ˚ r H q{xx r ˚ yy . Thus, we may writeker p ϕ q{xx r ˚ yy “ ` pp K a S q ˚ r H q{xx r ˚ yy ˘ ˚ Z ˚ Ă H p S ˚ r H q , where the generator of Z is mapped to d in the left and e in the right factor. Finally, we conclude that S ˚ r H and therefore in particular S ˚ H embeds into ker p ϕ q{xx r ˚ yy . Using Proposition 6.5 we prove the following Freiheitssatz, which in comparison to Proposition 6.5 omitsthe condition that H is non-trivial. Theorem 7.1 ( Freiheitssatz for tree-products ) . Let G be a tree-product, S a subtree-product of G and H a (possibly trivial) locally indicable group. Further, let r be an element of G ˚ H whose minimaltree-product contains at least one vertex-group of respectively S and G a S . Then S ˚ H embeds canonicallyinto p G ˚ H q{xx r yy . In a similar way to the proof of the Magnus-Freiheitssatz (cf. [Mag30]), the proof of Theorem 7.1will be simultaneous to the proof of Proposition 7.4. Before we formulate this proposition, we introduce α - (resp. ω -)branch-limits and staggered presentations for tree-products analogously to Definition 5.1. Notation 7.2.
Let I Ď Z Yt˘8u be an index set, r H be a locally indicable group, let K be a tree-productand Z i ( i P I ) be branch-products of K that share no common vertex-groups. For m, n P Z Y t˘8u wedefine K m,n “ K a t Z (cid:96) | (cid:96) P I ^ p (cid:96) ă m _ n ă (cid:96) qu . Definition 7.3 ( staggered presentations over tree-products and α -/ ω -branch-limits ) . Let I , J Ď Z be index sets, r H be a locally indicable group, let K be a tree-product and Z i ( i P I ) be branchproducts of K that share no common vertex-groups. Further, let r j ( j P J ) be elements of K ˚ r H suchthat every minimal tree-product of these elements contains at least one vertex-group of respectively Z i and K a Z i for at least one i P I . By α r j resp. ω r j we denote the greatest resp. smallest index λ P Z Y t˘8u such that the minimal tree-product of r j is contained in K λ, ˚ r H resp. K ´8 ,λ ˚ r H . Wecall α r j and ω r j the α - (resp. ω -)branch-limit of r j in K . If we have α r m ă α r n and ω r m ă ω r n for all m , n P J with m ă n we say that p K ˚ r H q{xx r j | J yy is a staggered presentation (over the tree-product K with respect to the branch-products Z i ( i P I )) . Proposition 7.4.
Let p K ˚ r H q{xx r j | J yy be a staggered presentation over a tree product K with respectto branch-products Z i ( i P I ). Further, let u be an element of the normal closure of the elements r i ( i P Z ) in K ˚ r H , such that u P K α,ω ˚ r H for some α, ω P Z . Then u is an element of the normal closureof the elements r k with α ď α r k and ω r k ď ω in K ˚ r H . imultaneous proof of Theorem 7.1 and Proposition 7.4. We consider the tree-product G , the locally indicable group H and the element r from Theorem 7.1.Because of Lemma 6.3 it is sufficient to consider the case that G is the minimal tree-product of r andthat r is a contracted conjugate. Let σ be the boundary-length of G . Our proof will be by inductionover the tuple p|| r || ´ | G | , σ q in lexicographical order (where the first component is weighted higher).Let T j ( j P J ) be the minimal tree-products of the elements r j of Proposition 7.4. W. l. o. g. we assumethat the elements r j are contracted conjugates and that J “ Z or J “ t , , . . . , δ u for some δ P N .Let σ j ( j P J ) be the boundary-lengths of the minimal tree-products T j . First, we prove the following:Statement 1. For fixed p t, s q P Z ˆ N the statement of Theorem 7.1 with p|| r || ´ | G | , σ q ď p t, s q implicatesthe statement of Proposition 7.4 with p max j P J p|| r j || ´ | T j |q , σ q ď p t, s q .Let u be a non-trivial element of the normal closure of the elements r j ( j P J ) in K ˚ r H . Further, let r j ( m ď j ď n ) be the elements r k with α ď α r k and ω r k ď ω . To get a contradiction we assume that u is not contained in the normal closure of the elements r i ( m ď i ď n ) in K ˚ r H . We choose indices ζ , ϑ with ζ ď ϑ such that u is contained in the normal closure of the elements r k ( ζ ď k ď ϑ ) in K ˚ r H andsuch that ϑ ´ ζ is minimal with this property. Because of the assumption we have ζ ă m or n ă ϑ . Byinverting all indices, if necessary, we can assume w. l. o. g. n ă ϑ .Our proof is by induction over λ : “ ϑ ´ ζ P N . For the base of induction ( ϑ “ u is an element ofthe normal closure of r ϑ in K ˚ r H . The inequality n ă ϑ implicates the inequality ω ă ω r ϑ . Thus, u isan element of p K ´8 ,ω rϑ ´ q ˚ r H . Because of p|| r || ´ | T | , σ q ď p t, s q , we derive with Theorem 7.1 andLemma 6.3 that p K ´8 ,ω rϑ ´ q ˚ r H embeds into ` p K ´8 ,ω rϑ q ˚ r H ˘ {xx r ϑ yy . So u has to be trivial in K ˚ r H which is a contradiction.For the induction step ( λ Ñ λ `
1) we write p K ˚ r H q{xx r k | ζ ď k ď ϑ yy– ` p K ´8 ,ω rϑ ´ ˚ r H q{xx r k | ζ ď k ď ϑ ´ yy ˘ ˚ K αrϑ,ωrϑ ´ ˚ Ă H ` p K α rϑ , ˚ r H q{xx r ϑ yy ˘ , where the embeddings of the amalgamated subgroup K α rϑ ,ω rϑ ´ ˚ r H follow from the induction hypothesis.Since u is trivial in K ˚ r H {xx r k | ζ ď k ď ϑ yy and is contained in the subgroup K ´8 ,ω rϑ ´ ˚ r H , it isalready trivial in the left factor p K ´8 ,ω rϑ ´ ˚ r H q{xx r k | ζ ď k ď ϑ ´ yy . This contradiction to theminimality of ϑ ´ ζ ends the proof of Statement 1.Since the case r P p G ˚ H qz G is covered by Proposition 6.5, we can further assume r P G for the proofof Theorem 7.1. Thus, we have p G ˚ H q{xx r yy “ p G {xx r yyq ˚ H and it suffices to prove the embedding of S in G {xx r yy . Let A be a leaf-group of G which is not contained in S . Because of Statement 1 we cannotonly use the induction hypothesis for Theorem 7.1, but also for Proposition 7.4. Moreover, by provingTheorem 7.1 we also prove Proposition 7.4.As the induction base we consider the cases || r || ´ | G | ă ´ σ “
2. For σ “ A is a primitive element and the desired embedding follows from Lemma 6.4. In the case | r || ´ | G | ă ´ B of G such that r does not use a basis element of B and B isnot directly connected to A . Note that B is also no leaf-group of G and can in particular not be A since G is the minimal tree-product of r . We consider the branch-products of G which arise by deleting allgenerators of the vertex-group B along with the edge-relations of the edges adjacent to B in G . Let Z A be the branch-product which contains the leaf-group A and let R A be the free product of the remainingbranch-products. Since B is not directly connected to A we have | Z A | ě
2. Note that Z A ˚ R A is atree-product using the fact that R A is locally indicable by Lemma 2.5. We also note that r uses at leastthe vertex-group A of Z A and one vertex-group of R A because G is the minimal tree-product of r . If r P Z A ˚ R A possesses a minimal tree-product in Z A , we get the canonical embedding of p Z A a A q ˚ R A into p Z A ˚ R A q{xx r yy by Proposition 6.5. If r P Z A ˚ R A does not possess a minimal tree-product in Z A , we have r P A ˚ R A because of Definition 2.3. Let p “ q with p P A be the edge-relation of theedge connecting A and Z A a A . By Lemma 1.3 we get the canonical embedding of x p |y ˚ R A into p A ˚ R A q{xx r yy . It follows p Z A ˚ R A q{xx r yy “ ` p Z A a A q ˚ R A ˘ ˚ Z ˚ R A ` p A ˚ R A q{xx r yy ˘ , where the generator of Z is mapped to p in the right and to q in the left factor. Altogether, we have thecanonical embedding of p Z A a A q ˚ R A into p Z A ˚ R A q{xx r yy in every case.Let u (cid:96) “ v (cid:96) with (cid:96) P L for some index set L and u (cid:96) P B be the edge-relations of the edges adjacentto B in G . Considering the free factors of R A as individual factors, every v (cid:96) is contained in a differentfactor of p Z A a A q ˚ R A . Since all factors are locally indicable due to Lemma 2.5 they are in particulartorsions-free. Thus, the free group V with basis t v i | i P I u embeds into p Z A a A q˚ R A . By Definition 2.1,the elements u i ( i P I ) form a staggered-set of B and B {xx u i | i P I yy is a staggered presentation (overfree groups). Because of Remark 5.5 the free group U with basis t u i | i P I u embeds canonically into B .Combining the embeddings proved so far we may write G a A – B ˚ U – V ` p Z A a A q ˚ R A ˘ and G {xx r yy – p G a A q ˚ p Z A a A q˚ R A ` p Z A ˚ R A q{xx r yy ˘ . (7.1)Now, the embedding of G a A and therefore in particular of S into G {xx r yy follows from the amalgamatedproduct (7.1).For the induction step we consider the leaf-group A and the edge-relation p “ q ( p P A ) of the edgeadjacent to A . The following preliminary considerations of the induction step as well as Case 1 are verysimilar to the corresponding part in the proof of Proposition 6.5. In order to avoid unnecessary doublingwe will shorten argumentations if they are already given in the proof of Proposition 6.5. In the casethat the cyclical reduction of p contains only one basis element a of A , p is a primitive element of A andthe desired embedding follows analogously to the induction base for σ “
2. So we can assume that p contains at least two different non-stabilizing basis elements a , b of A .If p a “ p b “
0, we consider the root-product r G of G given through a “ r a r b and apply the leaf-isomorphism given by r b : “ b r a r a or r b : “ r a r a b . Because of Lemma 6.1 it is sufficient to prove the mbedding of S into r G {xx r yy . As usual, we denote in slight abuse of notation the image of r G underthe leaf-isomorphism and the new contracted conjugate again by r G and r . In this situation we have p r a “ r r a “
0. If p does not contain the generator r a , the tree-product r G has shorter boundary-lengththan G and the desired embedding follows by the induction hypothesis. If p contains r a we go directlyto Case 2. Thus, in the following up to Case 2 we can assume that at least one exponential sum p a or p b is not 0. W. l. o. g. let p b ‰
0. We construct the root-product r G : “ G ˚ a “ r a pb x r a y of G . Next, we applythe leaf-isomorphism of r G which is given by r b : “ b r a p a or r b : “ r a p a b (cf. Remark 4.4). It follows p r a “ p contains r a . We consider two cases for r r a .Case 1. Let r r a ‰ p r b ‰
0. Therefore, (for some arbitrary fixed presentation r )the matrix ¨˝ p r a p r b r r a r r b ˛‚ has full rank and the desired embedding follows from Remark 1.5 and Theorem 1.6.Case 2. Let r r a “ ϕ : r G Ñ Z with ϕ p r a q “
1, define K : “ ker p ϕ q and use the notationsof Remark 4.6 and Notation 7.2. To show that S embeds into r G {xx r yy it suffices to show that the copy S of S embeds into K {xx r i | i P Z yy . Because of Lemma 4.9 we can assume w. l. o. g. that all r i ( i P Z )are contracted conjugates. Let Z be a leaf-group of r G different from A and let Z i : “ r a ´ i Z r a i . Since weassumed G and therefore r G to be the minimal tree-product of r we know that each minimal tree-product T i of an element r i P K ( i P Z ) contains at least one leaf-group Z k along with a vertex-group from K a Z k for at least one k P Z . Thus, because of symmetry, K {xx r i | i P Z yy is a staggered presentation withrespect to the branch-products Z i ( i P Z ). Let j P Z be an index such that T j contains the leaf-group Z . Note that S is part of K , (cf. Notation 7.2). Thus, by Lemma 4.11, we can apply the inductionhypothesis for Proposition 7.4. If S does not contain the leaf-group Z , we immediately arrive at thedesired embedding of S into K {xx r i | i P Z yy . Thus, we may assume in the following that Z is part of S . By the induction hypothesis for Proposition 7.4 it is sufficient to show the canonical embedding of S into K {xx r j yy . Because of Lemma 6.3 it even suffices to prove the embedding of S : “ T j X S into T j {xx r j yy .Note that r a has to be a reduction- or fan-generator (see Definition 4.12). If r a is a reduction-generator,we have T j “ r A ˚ p “ q p r G a A q . As noticed in Remark 4.7 we also have | p | r A ă | p | A . So the boundary-length of T j is strictly smaller than the boundary length of r G . This in combination with Lemma 4.9allows us to apply the induction hypothesis for Theorem 7.1. We get the desired embedding of S into T j {xx r j yy . It remains to consider the case that r a is a fan-generator. In this case we have | T j | ą | r G |p“ | G |q since T j contains the vertex-group r A , at least one copy of every vertex-group of r G a A and at least twocopies of the vertex-group B of r G adjacent to A . Combining this with Lemma 4.9 we are able to applythe induction hypothesis for Theorem 7.1 and deduce the embedding of S into T j {xx r j yy . Proof of the Main Theorem
In this section we prove Main Theorem 1.1 as a corollary of Theorem 7.1 (Freiheitssatz for tree-products).Let p respectively q be the generator of the subgroup of A respectively B which is identified withthe amalgamated subgroup U . We choose bases A of A and B of B such that p is cyclically reducedrespectively A and q is cyclically reduced respectively B . Such bases are always available by replacingthe basis elements with suitable conjugates if necessary. With the new bases, G has the form of atree-product of size 2 (cf. Definition 2.3). Since, by assumption, r is neither conjugate to an element of A nor B , G is the minimal tree-product of r (cf. Definition 3.1). Thus, the desired embeddings followdirectly from Theorem 7.1. Acknowledgements.
I wish to thank Professor Jim Howie for his encouragement as well as hishelp to understand the connection between the results of [HS09] and the main theorem of this article.
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