aa r X i v : . [ m a t h . G R ] M a r ON INDEPENDENT FAMILIES OF NORMAL SUBGROUPS IN FREEGROUPS
O.V. KULIKOVA
Abstract.
Consider a presentation P = < x | S ni = r i > . Let R i be the normal closure of theset r i in the free group F with basis x , P i = < x | r i > , N i = Q j = i R j . In the present article,using geometric techniques of pictures, generators for R i ∩ N i [ R i , N i ] , i = 1 , . . . , n , are obtained from aset of generators over {P i | i = 1 , . . . , n } for π ( P ). As a corollary, we get a sufficient conditionfor the family { R , . . . , R n } to be independent. Introduction
Consider a presentation P = < x | r > , where r = S ni = r i . For i = 1 , . . . , n , let R i be thenormal closure of the set r i in the free group F with basis x , P i = < x | r i > , N i = Q j = i R j . Set R = Q ni =1 R i , G = F / R and G i = F / R i . In this paper we study the quotient groups R i ∩ N i [ R i , N i ] , i = 1 , . . . , n , which are a natural measure of the redundancy between R , ..., R n .Geometric techniques of spherical pictures [1, 2] are used to prove the main theorem of thispaper, in which we determine generators for R i ∩ N i [ R i , N i ] , i = 1 , . . . , n , providing that a set of sphericalpictures over P , generating π ( P ) over {P i | i = 1 , . . . , n } , is known. The idea of applying thesetechniques to determine generators was already used in [3] for some presentations.The family { R , . . . , R n } is said to be independent , if R i ∩ N i = [ R i , N i ] for i = 1 , . . . , n .Independence may be considered as ensuring that certain intersections are as small as possi-ble, or as ensuring that certain commutator subgroups are as large as possible. Independenceand related notions have been studied in [4, 5, 9, 10, 11, 13]. For example, it is shown in[9] that { R , . . . , R n } is independent if and only if the inclusions R i → R induce an isomor-phism ⊕ ni =1 ( Z G ⊗ Z G i H R i ) → H R of relation modules. Also it is known [5, 9, 11] that if { R , . . . , R n } is independent, then P is (A) over {P i | i = 1 , . . . , n } , i.e. π ( P ) is generated bythe empty set over {P i | i ∈ I } . It is proved in [5] that the converse holds in case n = 2. Inthe present paper we obtain the converse statement for arbitrary n , as a corollary of the maintheorem.In the first section of this paper we remind the facts from [1] and [2], required to formulateand prove the main results of this paper. The main results themselves are formulated in thesecond section and proved in the third section. The reader may omit the first section and skipto the next ones if the papers [1] and [2] are familiar to him.1. Main definitions
The following definitions, notation and facts from [1] and [2] will be used in the second andthird sections of the present paper.Let P = h x | r i be a presentation for a group G , where x is a set (”generators”) and r is aset of cyclically reduced words on x ∪ x − (”relators”). We assume that no relator of r is freelytrivial, nor is a conjugate of any other relator or its inverse (so called the RH -hypothesis).We let w denote the set of all words on x ∪ x − (reduced or not). Two words u and v in w are identically equal , if they represent the same element in the free semigroup on x ∪ x − .The words u and v are freely equal , if one can be obtained from the other by a finite number of insertions and deletions of inverse pairs x ε x − ε , where x ∈ x , ε = ±
1. The free equivalenceclass of W ∈ w will be denoted by [ W ]. The free group F on x consists of the free equivalenceclasses, where the multiplication is defined by [ W ][ U ] = [ W U ]. We let N denote the normalclosure of { [ R ] | R ∈ r } in F . Thus, G = F / N .If s is a subset of r then we let s w denote the set of all words of the form W S ǫ W − , where W ∈ w , S ∈ s , ǫ = ± r w .Let σ = ( c , ...., c m ), where c i ∈ r w ( i = 1 , ..., m ). We define the inverse σ − of σ to be( c − m , ..., c − ). We define the conjugate W σW − of σ by W ∈ w to be ( W c W − , ..., W c m W − ).We define operations on sequences as follows. Let σ = ( c , ...., c m ), where c i = W i R ǫ i i W − i , W i ∈ w , R i ∈ r , ǫ i = ± , i = 1 , ..., m .(SUB) Substitution.
Replace each W i by a word freely equal to it.(DEL) Deletion.
Delete two consecutive terms if one is identically equal to the inverse of theother.(INS)
Insertion.
The opposite of deletion.(EX)
Exchange.
Replace two consecutive terms c i , c i +1 by either c i +1 , c − i +1 c i c i +1 or by c i c i +1 c − i , c i .Two sequences σ , σ ′ will be said to be (Peiffer) equivalent (denoted σ ∼ σ ′ ) if one can beobtained from the other by a finite number of applications of the operations (SUB), (DEL),(INS), (EX). The equivalence class containing σ will be denoted by < σ > .We can define a binary operation + on the set Σ of all equivalence classes by the rule < σ > + < σ > = < σ σ > . (Here σ σ is the juxtaposition of the two sequences σ , σ .) Under this operation Σ is a group.The identity (zero element) is the equivalence class of the empty sequence, and the inverse(negative) − < σ > of < σ > is < σ − > .We define Π σ to be the product c c ...c m . We say that σ is an identity sequence , if Π σ isfreely equal to 1. We let π denote the (abelian) subgroup of Σ consisting of all elements < σ > ,where σ is an identity sequence. (Occasionally, when we want to emphasize the presentation P , we will write π ( P )). F acts on Σ by the rule [ W ] · < σ > = < W σW − > , where [ W ] ∈ F , < σ > ∈ Σ. The mapping ∂ : Σ → F , where ∂ : < σ > [Π σ ], is a group homomorphism with the image N and the kernel π . The triple (Σ , F , ∂ ) is an example of a (free) crossed module. N acts trivially on π . Itfollows that π is a left G -module with G -action given by [ W ] N · < σ > = [ W ] · < σ > , where[ W ] ∈ F , < σ > ∈ π .Let R ∈ r . Then R = R op ( R ) , where R o is not a proper power and p ( R ) is a positive integer( R o is the root of R , and p ( R ) is the period ). The identity sequences ζ R = ( R, R o R − R o − ) ( R ∈ r )will be called the trivial sequences , and the elements < ζ R > of π will be called the trivial elements . The submodule of π generated by the trivial elementswill be denoted by T . P is called aspherical (respectively, combinatorially aspherical (CA) ) if π = 0 (respectively, π = T ). As is shown in [6], P is aspherical if and only if P is (CA) and no element of r is aproper power.Subpresentations. N INDEPENDENT FAMILIES OF NORMAL SUBGROUPS IN FREE GROUPS 3
Let P = h x | r i be a subpresentation of P , and let (Σ , F , ∂ ) be the associated crossedmodule. There is an obvious mapping of crossed modules( ϕ, ψ ) : (Σ , F , ∂ ) → (Σ , F , ∂ ) , where ϕ ( < σ > ) = < σ > ( < σ > ∈ Σ ); ψ ([ W ] ) = [ W ] ([ W ] ∈ F ) . Restricting ϕ gives a homomorphism j : π ( P ) → π ( P ) . In general j is not injective. It is still unknown whether j is injective for every subpresentation P of aspherical P (Whitehead problem).Consider the presentation P = h x | r i and suppose that r is expressed as a union r = r ∪ r .For λ = 1 ,
2, let P λ = h x | r λ i and j λ : π ( P λ ) → π ( P ) be the homomorphism as discussedabove. Note that Im j λ is a submodule of π ( P ).Let N λ , where λ = 1 ,
2, be the normal closure of { [ R ] | R ∈ r λ } in the free group F . Now F acts on N ∩ N [ N , N ] via conjugation:[ W ] · [ U ][ N , N ] = [ W U W − ][ N , N ] ([ W ] ∈ F , [ U ] ∈ N ∩ N ) . It is easy to show that N (= N N ) acts trivially, and so we get an induced action of G = F / N on N ∩ N [ N , N ] . We can define a G -homomorphism η : π ( P ) → N ∩ N [ N , N ]by the following rule. Let < σ > ∈ π ( P ), and V be the product (taken in order) of theelements of σ which belong to r w1 . Then η ( < σ > ) = [ V ][ N , N ]. It is not hard to show that η is well-defined.The following theorem was proved by Gutierr´z and Ratcliffe [5]. Theorem 1.
Let ξ : Im j ⊕ Im j −→ π ( P ) be induced by the inclusions Im j , Im j −→ π ( P ) .Then the sequence Im j ⊕ Im j −→ ξ π ( P ) −→ η N ∩ N [ N , N ] −→ is exact. If r and r are disjoint then ξ is injective, and so the sequence is short exact. Pictures.Sequences can be studied very successfully using geometric objects called pictures. Pictureswere introduced in [7, 8]. These objects are a very useful tool to solve combinatorial grouptheory problems. See, for example, [1, 2] and the references cited there.Let us remind the definition of pictures (according to [2]).A picture P is a finite collections of pairwise disjoint closed disks ∆ , . . . , ∆ m in a closeddisk D , together with a finite number of disjoint arcs α , . . . , α l properly embedded in D −∪ mi =1 int ∆ i (where ” int ” denotes interior). Loosely speaking, below the disks ∆ , . . . , ∆ m willbe called vertices of P . An arc can be either a simple closed curve having trivial intersectionwith ∂D ∪ ∂ ∆ ∪ . . . ∪ ∂ ∆ m , or a simple non-closed curve which joins two different pointsof ∂D ∪ ∂ ∆ ∪ . . . ∪ ∂ ∆ m . The boundary ∂D of P will be denoted by ∂ P . The corners ofa vertex ∆ of P are the closures of the connected components of ∂ ∆ − ∪ j α j . The regions of P are the closures of the connected components of D − ( ∪ i ∆ i S ∪ j α j ). The components of P are the connected components of ∪ i ∆ i ∪ ∪ j α j . The picture P is connected if it has at most onecomponent. The picture P is spherical if it has at least one vertex and no arc of P meets ∂ P . O.V. KULIKOVA
Assume that a group presentation P = h x | r i and a picture P are given. Fix an orientation ofthe ambient disk D , thereby determining a sense of positive rotation (i.g. clockwise). Assumethat the vertices and arcs of P are labeled by elements of P as follows.(i) Each arc of P is equipped with a normal orientation (indicated by an arrow transverse tothe arc), and is labeled by an element of x ∪ x − .(ii) Each vertex ∆ of P is equipped with a sign ǫ (∆) = ± R (∆) ∈ r .For a corner c of a vertex ∆ of P , W ( c ) denotes the word in x ∪ x − obtained by reading inorder the (signed) labels on the arcs that are encountered in a walk around ∂ ∆ in the positivedirection, beginning and ending at an interior point of c (with the understanding that if wecross an arc, labeled y say, in the direction of its normal orientation then we read y , whereas ifwe cross the arc against the orientation we read y − ). The oriented and labeled picture P is a picture over P if for each corner c of each vertex ∆ of P , W ( c ) is identically equal to a cyclicpermutation of R (∆) ǫ (∆) . We call W ( c ) the label of ∆, and denote it by W (∆).A corner c is a basic corner of ∆ of P if W ( c ) is identically equal to R (∆) ǫ (∆) . The vertex ∆has exactly p basic corners, where p > R (∆).A picture P over P becomes a based picture over P when it is equipped with basepoints asfollows. • Each vertex ∆ has one basepoint , which is a selected point in the interior of a basiccorner of ∆. • P has a global basepoint , which is a selected point in ∂ P that does not lie on any arc of P .The boundary label on a based picture P over P is the word W ( P ) obtained by readingin order the (signed) labels on the arcs of P that are encountered in a walk around ∂ P inthe positive direction, beginning and ending at the global basepoint. Alteration of the globalbasepoint or of the orientation of the ambient disk D changes the boundary label of P onlyup to cyclic permutation and inversion.There is the following pictorial version of the ”van Kampen lemma” (it can be obtained fromthe theorem 1.1 (V) and the lemma 1.2 (V) of [12] and duality). Lemma 1.
A word U in x ∪ x − represents the identity of G defined by P = h x | r i if andonly if there is a based picture P over P with boundary label identically equal to U . The mirror image of a picture P will be denoted by − P . We can form the sum P + P oftwo pictures P and P in the obvious way (Figure 1): ✫✪✬✩r P + ✫✪✬✩r P = ✫✪✬✩r P P Figure 1A transverse path γ in P over P is a path in the closure of D − S i ∆ i which intersects thearcs of P only finitely many times (moreover, if the path intersects an arc then it crossed it,and doesn’t just touch it), no endpoints of γ touches any arc, and whenever γ meets ∂ P or any N INDEPENDENT FAMILIES OF NORMAL SUBGROUPS IN FREE GROUPS 5 ∂ ∆ i , it does so only in its endpoints. Since we will only over consider transverse paths, we willfrom now on drop the use of the adjective ”transverse”.If we travel along a path γ from its initial point to its terminal point we will cross variousarcs, and we can read off the (signed) labels on these arcs, giving a word W ( γ ), the label on γ .Let a simple closed path γ in P encloses a subpicture B of P . This subpicture consists ofthe vertices and (portions of) arcs that are separated from ∂ P by γ . When P is spherical,the compliment of B in P is defined as follows. Delete the interior of B to form an orientedannulus. Identification of ∂ P to a point produces an oriented disk that has the boundary γ ,and which supports a new picture over P . The compliment of B in P is obtained from thisnew picture by a planar reflection. The complement has the same boundary label as B and itsvertices are those of P − B , taken with opposite signs.The subpicture B enclosed by a simple closed path γ will be called a spherical subpicture if γ intersects no arc. A spherical subpicture will be called empty if it neither consists of any vertexnor any portion of any arc.On the connection between sequences and pictures.A spray for a based picture P with n vertices ∆ , ∆ , . . . , ∆ n is a sequence γ = ( γ , γ , . . . , γ n )of simple paths satisfying the following: • for each i = 1 , , . . . , n , γ i starts at the global basepoint of P and ends at a basepointof ∆ ϑ ( i ) , where ϑ is a permutation of { , , . . . , n } (depending on γ ); • for 1 i < j n , γ i and γ j intersect only at the global basepoint; • travelling around the global basepoint clockwise we encounter the paths in the order γ , γ , . . . , γ n .The sequence σ ( γ ) associated with γ is( W ( γ ) W (∆ ϑ (1) ) W ( γ ) − , . . . , W ( γ n ) W (∆ ϑ ( n ) ) W ( γ n ) − ) . A based picture will be said to represent a sequence σ if there is a spray for the picture whoseassociated sequence is σ . Note that if P represents σ then − P represents σ − ; if P , P represents σ , σ respectively then P + P represents σ σ . One can prove (see for example[1]) that every sequence can be represented by a picture, and every identity sequence can berepresented by a spherical picture; conversely, if P is a picture and if γ is a spray for P , then ∂ ( < σ ( γ ) > ) = [ W ( P )] , and if P is a spherical picture and and if γ is a spray for P , then σ ( γ ) is an identity sequence.If γ , γ ′ are two sprays for a picture P , then < σ ( γ ) > − < σ ( γ ′ ) > ∈ T (theorem 1.4, theorem2.4 of [1]).Consider a set X = { P λ | λ ∈ Λ } of based spherical pictures over P . For each λ , let σ λ be theidentity sequence associated with a spray for P λ , and J ( X ) be the submodule of π generatedby { < σ λ > | λ ∈ Λ } . We say that X generates π if π = J ( X ) + T .It follows from Theorem 1 that if π = T , i.e. the presentation is (CA), then N ∩ N [ N , N ] = 0,since η ( T ) = 0. If π = J ( X )+ T then N ∩ N [ N , N ] normally generated by the images of the elements < σ λ > associated with sprays for all pictures P λ ∈ X . Moreover, as noted in [3], the image of < σ λ > under η is [ V λ ][ N , N ], where V λ is the label of a simple closed path in P λ (orientedas ∂ P λ ) separating the vertices with r -labels from the vertices with r -labels.Operations on pictures.Generally below we will not distinguish between pictures which are isotopic.The following operations (”deformations”) can be applied to a based picture P over P ([2]). BRIDGE : (Bridge move) See Figure 2.
O.V. KULIKOVA ✲ ✲ ✛ ✚✙✛✘ ❄✻ x x x x Figure 2
F LOAT : Deletion of a closed arc that separates D into two parts, one of which containsthe global basepoint of P and all remaining arcs and vertices of P (such a closed arc is calleda floating circle ). F LOAT − : (Insertion of a floating circle). The opposite of F LOAT .A folding pair is a connected spherical subpicture that contains exactly two vertices suchthat • these two vertices are labeled by the same relator and have opposite signs; • the basepoints of the vertices lie in the same region; • each arc in the subpicture has an endpoint on each vertex. F OLD : (Deletion of a folding pair). If there is a simple closed path β in D such that thepart of P encircled by β is a folding pair, then remove that part of P encircled by β . F OLD − : (Insertion of a folding pair). The opposite of F OLD .Let X = { P λ | λ ∈ Λ } be a set of based spherical pictures over P . By an X -picture we meaneither a picture P λ from X or its mirror image − P λ . REP LACE ( X ) : Replace a subpicture of P by the complement of that subpicture in an X -picture.Two based spherical pictures are called X -equivalent if one of them can be transformed intothe other one (up to planar isotopy) by a finite sequence of operations BRIDGE , F LOAT ± , F OLD ± , REP LACE ( X ).We introduce two further operations on pictures as follows. DELET E ( X ) : (Deletion of an X -picture). If there is a simple closed path β in D suchthat the part of P enclosed by β is a copy of an X -picture, then delete that part of P enclosedby β . DELET E ( X ) − : (Insertion of an X -picture). The opposite of DELET E ( X ).Note that the operation REP LACE ( X ) includes DELET E ( X ) ± . On the other hand, theresult of the operation REP LACE ( X ) can be obtained by a finite sequence of operations DELET E ( X ) ± , BRIDGE , F LOAT ± , F OLD ± . Thus, in the definition of X -equivalentpictures, REP LACE ( X ) can be replaced by DELET E ( X ) ± .A dipole in a picture over P consists of an arc which meets two corners c , c in distinctvertices such that • the two vertices are labeled by the same relator and have opposite signs; • c and c lie in the same region of the picture; • W ( c ) = W ( c ) − .By a complete dipole over P , we mean a connected based spherical picture over P thatcontains just two vertices, and where each arc of the picture constitutes a dipole. Note thata complete dipole is just a folding pair, in that the vertex basepoints need not lie in the sameregion. If the relator that labels the two vertices of the complete dipole has period one, then a N INDEPENDENT FAMILIES OF NORMAL SUBGROUPS IN FREE GROUPS 7 complete dipole is exactly the same as a folding pair. A complete dipole will be called primite if the relator labeling its vertices has root Q and period p >
1, and there is a path joining thevertex basepoints with label Q f , where gcd ( f, p ) = 1. It follows from Lemma 2.1 [2] that,modulo primitive dipoles, one need not be concerned with choices of vertex basepoints.The following theorem (see Corollary 1 of Theorem 2.6 [1], Theorem 1.6 (2) [2]) will play animportant role in the proof of the main theorem of the present paper. Theorem 2.
Let X be a collection of based spherical pictures. Then X generates π ( P ) ifand only if every spherical picture over P is X -equivalent to the empty picture, where X is theunion of X and the collection of primitive dipoles for all relators of P , which are a properpower. It follows from Theorem 2 (see Corollary 2 of Theorem 2.6 [1]) that P is (CA) (i.e. π = T )if and only if every spherical picture over P is X -equivalent to the empty picture, where X isthe collection of primitive dipoles for all relators of P , which are a proper power.Independent sets.Suppose that {P i | i ∈ I } is a collection of subpresentations of P , and let X i denote thecollection of all based spherical pictures over P i , i ∈ I . We shall say that a set Y of basedspherical pictures over P generates π ( P ) over {P i | i ∈ I } if the G -module π ( P ) is generatedby the homotopy classes [ f P ] ( P ∈ Y ∪ S i ∈ I X i ) ([2]). By the analogue of Theorem 2 (seeTheorem 1.6 (2) [2]), a collection Y of based spherical pictures over P generates π ( P ) over {P i | i ∈ I } if and only if every spherical picture over P is X -equivalent to the empty picture,where X = Y ∪ S i ∈ I X i .This notion is useful when there are certain subpresentations in the presentation which weknow little about, or which in some way are arbitrary. Then, we can often isolate them bydetermining a ”nice” set of generators of π ( P ) relative to these subpresentations. Examplesof sets of generators over subpresentations for π ( P ) can be found in [2].We will say that P is (CA) over {P i | i ∈ I } if π ( P ) is generated over {P i | i ∈ I } byprimitive dipoles; P is (A) over {P i | i ∈ I } if π ( P ) is generated over {P i | i ∈ I } by theempty set.Consider a presentation P = < x | r > , where r = S ni = r i . For i = 1 , . . . , n , let R i be thenormal closure of r i in the free group F with basis x , P i = < x | r i > , N i = Q j = i R j . Set R = Q ni =1 R i , G = F / R G i = F / R i . The family { R , . . . , R n } is said to be independent if R i ∩ N i = [ R i , N i ] for i = 1 , . . . , n . This and related notions have been studied in [4, 5, 9, 10, 11].Note that any primitive dipole over P belongs to some collection X i of all based sphericalpictures over P i ( i = 1 , . . . , n ), since r = S ni = r i . So P is (A) over {P i | i = 1 , . . . , n } if andonly if P is (CA) over {P i | i = 1 , . . . , n } .2. Main results
Theorem 3.
Let F be the free group with basis x , r = S ni = r i be a set of cyclically reducedwords in x ∪ x − , and P = h x | r i be the presentation satisfying the RH -hypothesis. For i = 1 , . . . , n , let R i be the normal closure of the set r i in F , P i = h x | r i i , N i = Q j = i R j , ˆr i = S j = i r j , R = Q ni =1 R i .If π ( P ) is generated over {P i | i = 1 , . . . , n } by a collection Y of based spherical picturesover P , then for i = 1 , . . . , n , the group R i ∩ N i [ R i , N i ] is generated by { [ W RW − ][ R i , N i ] | R ∈ r i ∩ ˆr i , [ W ] ∈ W } ∪ { [ W V Y W − ][ R i , N i ] | Y ∈ Y , [ W ] ∈ W } , where V Y is a label of a simple closed path in a based spherical picture Y , separating the verticeswith r i -labels and the vertices with ( r − r i ) -labels, W ⊆ F is a set of representatives of all thecosets of R in F . O.V. KULIKOVA
It is known [5, 9, 11] that if { R , . . . , R n } is independent, then P is (A) over {P i | i =1 , . . . , n } . The converse statement for n = 2 is shown in [5]. From Theorem 3, we obtain thefollowing generalization (take Y empty). Corollary 1.
Let F be the free group with basis x , r = S ni = r i be a set of cyclically reducedwords in x ∪ x − , where r , . . . , r n are mutually disjoint sets, and P = h x | r i be the presentationsatisfying the RH -hypothesis. For i = 1 , . . . , n , let R i be the normal closure of r i in F , P i = h x | r i i .If P is (A) over {P i | i = 1 , . . . , n } , then { R , . . . , R n } is independent. Thus, for r = F ni = r i , { R , . . . , R n } is independent if and only if P is (A) over {P i | i =1 , . . . , n } . 3. The proof of Theorem 3
Let us consider the case i = 1 (the proof for i = 2 , . . . , n is similar).By N denote the normal closure of { [ R ] | R ∈ r ∩ ˆr } ∪ { [ V Y ] | Y ∈ Y } in F . We needto prove that if π ( P ) is generated over {P j | j = 1 , . . . , n } by Y , then modulo N [ R , N ],an arbitrary element [ U ] ∈ R ∩ N is equal to the identity. This will imply that R ∩ N [ R , N ] isgenerated by { [ W RW − ][ R , N ] | R ∈ r ∩ ˆr , [ W ] ∈ W } ∪ { [ W V Y W − ][ R , N ] | Y ∈ Y , [ W ] ∈ W } , since [ V Y ] ∈ R ∩ N , r ∩ ˆr ⊂ R ∩ N and R = R N .Since [ U ] belongs both to R and to N , by Lemma 1 there are a picture P R over P = h x | r i with boundary label identically equal to U and a picture P N over P ( r − r ) = h x | ( r − r ) i with boundary label identically equal to U − . Suppose that U is identically equal to x x . . . x m ,where x j ∈ x ∪ x − . Then, the arcs α , . . . , α m met ∂ P R have the labels respectively x , . . . , x m ,and the arcs β , . . . , β m met ∂ P N have the labels respectively x m , . . . , x . Paste P R and P N by their boundaries so that for j = 1 , . . . , m , the arc α j extends the arc β m − ( j − and the globalbasepoint O R of P R coincides with the global basepoint O N of P N . In the obtained two-sphere, choose a small closed disk D in the interior of any region of P R and cut D − ∂D out itto get a spherical picture P over P = h x | r i on the disk D with boundary ∂ P (= ∂D ) = ∂D .The pasted boundaries ∂ P R and ∂ P N give a path Equ on D . We will call Equ the equator .The pasted points O R and O N give a point p ∈ Equ . Fix an orientation of
Equ so that thelabel of
Equ − { p } is equal identically to U . Below the label of Equ − { p } under this orientationwill be called the equatorial label . The part of P corresponding to P R (resp., P N ) will becalled the r -hemisphere (resp., the ( r − r ) -hemisphere ).Below we will transform P by planar isotopy, BRIDGE , F LOAT ± , F OLD ± , DELET E ( X ) ± under the following conditions: the equatorial label is not changed modulo N [ R , N ], all ver-tices of P with r -labels remain only in the r -hemisphere, all vertices with ( r − r )-labelsremain only in the ( r − r )-hemisphere. Deformations (operations), satisfied these conditions,will be called admissible . A picture, in which the equator divides the vertices under theseconditions, will be called a picture with equator . The aim of these operations is to reduce thepicture P with equator to a picture with equator with boundary label equal to the identity inthe free group, that will imply that the initial equatorial label, i.e. [ U ], belongs to N [ R , N ].To proof the existence of the desired operations, we will need two auxiliary statements: Lemma2 and Lemma 3. In these lemmas, we use the notation of Theorem 3, we let X denote the union Y ∪ S ni =1 X i , where X i is the collection of all based spherical picture over P i , i = 1 , . . . , n ; byan Z W -picture we mean a spherical picture, containing only one Z -picture and, possibly, closedarcs encircling this Z -picture, where Z is a given collection of spherical pictures. N INDEPENDENT FAMILIES OF NORMAL SUBGROUPS IN FREE GROUPS 9
Lemma 2. If π ( P ) is generated over {P j | j = 1 , . . . , n } by Y , then the picture P with equator Equ can be reduced by a finite number of admissible operations to a picture e P with equator,being a finite sum of X W -pictures and, possibly, also containing some closed arcs encircling thepoint p . Proof of Lemma 2.
Since π ( P ) is generated over {P j | j = 1 , . . . , n } by Y , then by Theorem 2, the basedspherical picture P over P is X -equivalent to the empty picture, i.e., there are a finite se-quence of based spherical pictures P , P , . . . , P s and a finite sequence f , . . . , f s of operations BRIDGE , F LOAT ± , F OLD ± , DELET E ( X ) ± , which transforms P (up to planar isotopy)to the empty picture so that f j : P j − P j , j = 1 , . . . , s , P = P , P s is the empty picture.Note that any folding pair is a spherical picture over some presentation P i , i = 1 , . . . , n , and,hence, it belongs to X i . In this case F OLD ± is a special case of DELET E ( X ) ± , therefore,below F OLD ± will not be considered separately.Since f , . . . , f s are not necessarily admissible, using the sequences P , P , . . . , P s and f , . . . , f s ,we will construct two new sequences Z , Z , . . . , Z s and g , . . . , g s , g s +1 , where, for i = 1 , . . . , s , Z i is a collection of disjoint spherical subpictures in P i , not containing the whole of the equator,and g i is an admissible operation. In addition, the sequence g , . . . , g s , g s +1 will transform P ,as a picture with equator, to e P so that g j : e P j − e P j , where e P = P , e P j = P j ∪ Z j , e P s +1 = e P are pictures with equator, j = 1 , . . . , s . This will prove Lemma 2.We let Z j , j = 1 , . . . , s , denote a finite collection of disjoint disks in the interior of D ,containing the empty spherical subpictures obtained from subpictures of Z j by deletion of allarcs and all vertices.In the case of a planar isotopy, we may assume that this isotopy reduces P j − to P j = F ( P j − ), where F t : D × [0 , → D × [0 ,
1] is a continuous isotopy of the disk D , so that(i) F t leaves all vertices fixed, i.e., for any t ∈ [0 ,
1] and each vertex ∆, F t (∆) = ∆;(ii) for any t ∈ [0 ,
1] and each arc α , the intersection of the arc F t ( α ) and the equator Equ consists of a finite number of points; moreover, if
Equ intersects F ( α ), then it crossesit, and doesn’t just touch it;(iii) for any arc α , the arc F ( α ) does not intersect any disk of Z j − .If for any t ∈ (0 ,
1) and each arc α , the arc F t ( α ) does not intersect any disk of Z j − , thenthis isotopy is called admissible , otherwise it is called inadmissible . An admissible isotopy doesnot change the equatorial label, as an element of the free group. Since an admissible isotopy isan admissible operation, below we will assume operations to be equal if they are equal up toadmissible isotopy. Include operations of inadmissible isotopy in the list of operations f , . . . , f s .We will construct g j , j = 1 , . . . , s , so that g j transforms ( P j − − Z j ) to ( P j − Z j ) in just thesame way (up to admissible isotopy of ( P j − Z j )) as f j transforms ( P j − − Z j ) to ( P j − Z j ). Inaddition we can always assume that the arcs of e P j and the boundary of any subpicture from Z j intersect the equator not more than finitely many times.As Z , take a set of a single spherical subpicture in P = P such that this subpicture isempty and contains the point p ∈ Equ and a connected part of the equator.Let us define Z j and g j , j = 1 , . . . , s , by induction on j .Assume that Z , Z , . . . , Z j − and g , . . . , g j − have been already defined. Construct Z j and g j by f j : P j − P j as follows. The operation f j is one of the operations: an inadmissibleisotopy, BRIDGE , F LOAT ± , DELET E ( X ) ± .Case 1. Inadmissible isotopy.There are an arc α (labeled by x ∈ x ∪ x − ) in P j − and t ∈ (0 ,
1) so that F t ( α ) intersectssome empty spherical subpicture P ξ from Z j − . Denote by P ξ the spherical subpicture from Z j − , which P ξ is obtained from. To obtain Z j from Z j − , (for each arc α , each such subpicture P ξ and each passing of F t ( α ) through P ξ ) add to P ξ a closed arc (labeled by x ), encirclingall objects (that may be arcs, vertices, the point p ) being already in P ξ . The action of g j on( P j − − Z j ) coincides with the action of f j on ( P j − − Z j ). The action of g j on Z j correspondsto the operation BRIDGE performed on the arc α in the interior of the disk of P ξ . So g j doesnot change the equatorial label as an element of the free group. See Figure 3. P ξ P ξ ✲ f j ✲ x ✲ x P ξ P ξ ✉✉✉ (cid:0)(cid:0)❅❅ ✫✪✬✩✉✉✉ (cid:0)(cid:0)❅❅ ✲ g j ✲ x ✛ x ✲ x Figure 3Case 2.
F LOAT , DELET E ( X ).In P j − , there is a spherical subpicture P η containing only a floating circle (resp., only an X -picture). The operation f j deletes P η from P j − .Applying an admissible isotopy to ∂ P η and f j if necessary, we may assume that P η does notcontain the whole of the equator, and ∂ P η does not intersect any disk from Z j − . Note thatthe arcs and the vertices of P η do not intersect Z j − as well.Let P ξ , ..., P ξ m be all spherical subpictures from Z j − in the interior of the disk of P η , andlet P ξ , ..., P ξ m be the corresponding subpictures from Z j − . The collection Z j is obtained from Z j − by deleting P ξ , ..., P ξ m and adding the spherical subpicture from e P j − encircled by thepath ∂ P η (this subpicture contains the disjoint union of P η and P ξ , ..., P ξ m ). The operation g j acts identically (up to admissible isotopy).Case 3. F LOAT − , DELET E ( X ) − .Let f j be an operation F LOAT − (resp., DELET E ( X ) − ). The operation f j inserts aspherical subpicture P η in ( P j − − Z j − ) such that P η contains only a floating circle (resp.,only an X -picture).Applying an admissible isotopy if necessary, we can obtain the following. If f j is F LOAT − ,the subpicture P η does not intersect the equator. If f j is DELET E ( X ) − and P η containsthe vertices only with r -labels or only with ( r − r )-labels, then the whole of P η is in thehemisphere corresponding to the labels of the vertices in P η . If f j is DELET E ( X ) − and P η contains the vertices both with r -labels and with ( r − r )-labels (we may assume that P η ∈ Y ),then P η is disposed so that ˆ Equ = P η ∩ Equ is connected and the equator divides the verticesof P η into two parts: the vertices with r -labels (in the r -hemisphere) and the vertices with N INDEPENDENT FAMILIES OF NORMAL SUBGROUPS IN FREE GROUPS 11 ( r − r )-labels (in the ( r − r )-hemisphere), in addition the label U η of the path ˆ Equ is suchthat [ U η ] ∈ N .Put Z j = Z j − . The action of g j on ( P j − − Z j ) coincides with the action of f j on ( P j − − Z j ),and the action of g j on Z j is identical. The operation g j does not change the equatorial labelmodulo N [ R , N ].Case 4. BRIDGE .We may assume that f j acts only on ( P j − − Z j − ).Put Z j = Z j − . The action of g j on ( P j − − Z j ) coincides with the action of f j on ( P j − − Z j ),and the action of g j on Z j is identical. The operation g j does not change the equatorial labelas an element in the free group.To complete the proof of Lemma 2, it remains to define g s +1 : e P s e P s +1 , where e P s = P s ∪ Z s , e P s +1 = e P . Since P s is the empty picture, the arcs and the vertices of e P s belong to sphericalsubpictures from Z s . The operation g s +1 transforms each spherical picture P µ from Z s asfollows.By construction of Z s , P µ is the union of embedded one in other spherical pictures each ofwhich either is an X W -picture, or contains only closed arcs and, possibly, the point p . Theoperation g s +1 decomposes P µ as a sum of X W -pictures and spherical pictures, containing onlyclosed arcs and, possibly, p , by applying admissibly isotopy and BRIDGE (see an exampleon Figure 4) so that the vertices remain in their own hemispheres, no summand contains thewhole of the equator, and the boundary of each summand intersects the equator not more thanfinitely many times. After that g s +1 deletes all floating circles not encircling p by applying F LOAT .The operation g s +1 does not change the equatorial label as an element of the free group. ✲ ✉ ✉✉ ✉✉ ✉✉ ✉✫ ✪❡ ❡❡ ❡❡ ❡ ❆❆ ❆❆✁✁ ✁✁ ✻ ★✧ ✥✦ ✻ ❄ xx x P ′ P ′ P ′′ P ′′ Figure 4. Partition of subpictures P ′ and P ′′ (Not to complicate the figure, the orientation and the label are indicated only for the arc which is transformed by BRIDGE .) Lemma 3.
If a based spherical picture ˜ P with the equator Equ is a finite sum of X W -picturesand, possibly, contains some closed arcs around the point p , then by a finite number of admissibleoperations, ˜ P can be reduced to a picture with equator with the equatorial label equal to theidentity in the free group. Proof of Lemma 3.
Below we will use the operation
COM M U T E depicted on Figure 5.This operation changes the equatorial label on an element from [ R , N ] (see details in [14]).It can be realized as a planar isotopy and a finite number of DELET E ( X ) ± , BRIDGE , F LOAT ± . ✲ Equ Equ ❡ ✉ ✉ ❡
Figure 5At first, applying
F LOAT and
DELET E ( X ) to e P , we delete all X W -pictures, not intersect-ing Equ . This does not change the equatorial label. Further, by means of planar isotopy and
COM M U T E , we obtain that for each X W -picture P η , the intersection Equ ∩ P η is connected,i.e., Equ divides P η into two parts: a subpicture over P = h x | r i in the r -hemisphere anda subpicture over P ( r − r ) = h x | ( r − r ) i in the ( r − r )-hemisphere. If at least one of theseparts does not contain vertices, the label of Equ ∩ P η is equal to the identity in the free group.Otherwise either P η is an Y W -picture, or P η contains vertices with labels from r ∩ ˆr . In thefirst case the label of Equ ∩ P η is equal to [ W V Y W − ] in the free group, where V Y is the label ofa simple closed path in a based spherical picture Y ∈ Y , separating the vertices with r -labelsand the vertices with ( r − r )-labels. In the second case the label of Equ ∩ P η is equal to theproduct of elements of the form [ W R ± W − ], where R ∈ r ∩ ˆr . In each of these cases thelabel of Equ ∩ P η belongs to N [ R , N ]. Now we delete all X W -pictures from e P . It followsfrom the above arguments that this operation is admissible. After such operation only severalclosed arcs encircling p may remain in e P . So the equatorial label is equal to the identity in thefree group. This completes the proof of Lemma 3.Let us continue the proof of Theorem 3. By Lemma 2, the picture P with the equator canbe reduced by a finite number of admissible operations to a picture e P with equator, being afinite sum of X W -pictures and, possibly, also containing some closed arcs around p . By Lemma3, ˜ P can be reduced by a finite number of admissible operations to a picture with equator withthe equatorial label equal to the identity in the free group. So the equatorial label [ U ] of theinitial picture P is equal to the identity modulo N [ R , N ]. Keywords:
Presentations, subpresentations, asphericity, independent families of normal sub-groups, intersection of normal subgroups, mutual commutant of normal subgroups, sphericalpictures, identity sequences.
N INDEPENDENT FAMILIES OF NORMAL SUBGROUPS IN FREE GROUPS 13
References [1]
S.J. Pride,
Identities among relations of group presentations, in: E. Ghys etal., eds., Proceedings of the Workshop on Group Theory from a GeometricalViewpoint (World Scientific Publishing, Singapore, 1991) 687-717.[2]
W.A. Bogley and S.J. Pride,
Calculating Generators of π , Two-dimensional Homotopy Theory and Combinatorial Group Theory, LondonMath. Soc. Lec. Notes Ser., vol. ( ser. II ), 1993.[3]
W.A. Bogley and M.A.Gutierrez , Mayer-Vietoris sequences in homo-topy of 2-complexes and in homology of groups , J.Pure Appl.Algebra (1992), 39-65.[4] W.A. Bogley,
An embedding for π of a subcomplex of a finite contractibletwo-complex, Glasgow Math. J., (1991) 365-371.[5] M.A. Guti´errez, J.G. Ratcliffe,
On the second homotopy group , Quart.J. Math. Oxford (2) , 1981, 45-55.[6] I.M. Chiswell, D.J. Collins, J. Huebschmann,
Aspherical group pre-sentations , Math. Z. , 1981, 1-36.[7]
K. Igusa,
The generalized Grassmann invariant , Brandeis University,Waltham (Mass), 1979, preprint.[8]
C.P. Rourke,
Presentations and the trivial group , Topology of low dimen-sional manifolds (ed. R.Fenn), Lecture Notes in Mathematics 722 (Springer,Berlin, 1979), pp. 134-143.[9]
A.J. Duncan, G.J. Ellis, N.D. Gilbert,
A Mayer-Vietoris sequence ingroup homology and the decomposition of relation modules , Glasgow Math.J., (1995) 159-171.[10] N.D. Gilbert,
Identities between sets of relations,
Journal of Pure andApplied Algebra 83 (1993), pp. 263-276.[11]
J. Huebschmann,
Aspherical 2-complexes and an unsettled problem ofJ.H.C.Whitehead,
Math. Ann.
R.S. Lyndon , P.E. Schupp,
Combinatorial group theory,
Springer-Verlag, Berlin - Heidelberg - NewYork, 1977.[13]
R.S. Lyndon,
Dependence and independence in free groups,
J. ReineAngew. Math. , 1962, 148-174.[14]
O.V. Kulikova,
On intersections of normal subgroups in free groups , Al-gebra and discrete mathematics, Number 1,36-67, 2003.
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