A Subgroup of a Direct Product of Free Groups whose Dehn Function has a Cubic Lower Bound
aa r X i v : . [ m a t h . G R ] J a n A SUBGROUP OF A DIRECT PRODUCT OF FREE GROUPSWHOSE DEHN FUNCTION HAS A CUBIC LOWER BOUND
WILL DISON
Abstract.
We establish a cubic lower bound on the Dehn function of a certainfinitely presented subgroup of a direct product of 3 free groups.Department of MathematicsUniversity WalkBristol, BS8 1TWUnited Kingdom [email protected] Introduction
The collection S of subgroups of direct products of free groups is surprisingly richand has been studied by many authors. In the early 1960s Stallings [13] exhibiteda subgroup of ( F ) , where F is the rank 2 free group, as the first known exampleof a finitely presented group whose third integral homology group is not finitelygenerated. Bieri [2] demonstrated that Stallings’ group belongs to a sequence ofgroups SB n ≤ ( F ) n , the Stallings-Bieri groups, with SB n being of type F n − butnot of type FP n . (See [6] for definitions and background concerning finitenessproperties of groups.)In the realm of decision theory, Miha˘ılova [9] and Miller [10] exhibited a finitelygenerated subgroup of ( F ) with undecidable conjugacy and membership problems.It is thus seen that even in the 2-factor case fairly wild behaviour is encounteredamongst the subgroups of direct products of free groups.In contrast to this, Baumslag and Roseblade [1] showed that in the 2-factor casethis wildness only manifests itself amongst the subgroups which are not finitelypresented. They proved that if G is a finitely presented subgroup of F (1) × F (2) ,where F (1) and F (2) are free groups, then G is itself virtually a direct product of atmost 2 free groups. This work was extended by Bridson, Howie, Miller and Short[5] to an arbitrary number of factors. They proved that if G is a subgroup of adirect product of n free groups and if G enjoys the finiteness property FP n , then G is virtually a direct product of at most n free groups. Further information onthe finiteness properties of the groups in class S was provided by Meinert [8] whocalculated the BNS invariants of direct products of free groups. These invariantsdetermine the finiteness properties of all subgroups lying above the commutatorsubgroup.Several authors have investigated the isoperimetric behaviour of the finitely pre-sented groups in S . Elder, Riley, Young and this author [7] have shown that the Mathematics Subject Classification.
Dehn function of Stallings’ group SB is quadratic. The method espoused by Brid-son in [3] proves that the function n is an upper bound on the Dehn functions ofeach of the Stallings-Bieri groups. In contrast, there have been no results whichgive non-trivial lower bounds on the Dehn functions of any groups in S . In otherwords, until now there has been no finitely presented subgroup of a direct productof free groups whose Dehn function was known to be greater than that of the am-bient group. The purpose of this paper is to construct such a subgroup: we exhibita finitely presented subgroup of ( F ) whose Dehn function has the function n asa lower bound.Let K be the kernel of a homomorphism θ : ( F ) → Z whose restriction toeach factor F is surjective. By Lemma 3.1 below the isomorphism class of K isindependent of the homomorphism chosen. Results in [8] show that K is finitelypresented. Theorem 1.1.
The Dehn function δ of K satisfies δ ( n ) (cid:23) n . Note that Theorem 1.1 makes no reference to a specific presentation of K since,as is well known, the Dehn function of a group is independent (up to ≃ -equivalence)of the presentation chosen. We refer the reader to Section 2 for background on Dehnfunctions, including definitions of the symbols (cid:23) and ≃ .The organisation of this paper is as follows. Section 2 gives basic definitionsconcerning Dehn functions, van Kampen diagrams and Cayley complexes. Weexpect that the reader will already be familiar with these concepts; the purposeof the exposition is principally to introduce notation. In Section 3 we define aclass of subgroups of direct products of free groups of which K is a member. Thesubsequent section gives finite generating sets for those groups in the class whichare finitely generated and Section 5 proves a splitting theorem which shows howcertain groups in the class decompose as amalgamated products. We then provein Section 6 how, in certain circumstances, the distortion of the subgroup H inan amalgamated product Γ = G ∗ H G gives rise to a lower bound on the Dehnfunction of Γ. Theorem 1.1 follows as a corollary of this result when applied to thesplitting of K given in Section 5.2. Dehn Functions
In this section we recall the basic definitions concerning Dehn functions of finitelypresented groups. All of this material is standard. For further background and amore thorough exposition, including proofs, see, for example, [4] or [11].Given a set A , write A − for the set of formal inverses of the elements of A andwrite A ± for the set A ∪ A − . Denote by A ± ∗ the free monoid on the set A ± . Werefer to elements of A ± ∗ as words in the letters A ± and write | w | for the lengthof such a word w . Given words w , w ∈ A ± ∗ we write w = w if w and w arefreely equal and w ≡ w if w and w are equal as elements of A ± ∗ . Definition 2.1.
Let P = hA |Ri be a finite presentation of a group G . A word w ∈ A ± ∗ is said to be null-homotopic over P if it represents the identity in G . A null- P -expression for such a word is a sequence ( x i , r i ) mi =1 in A ± ∗ × R ± such that w free = m Y i =1 x i r i x − i . SUBGROUP OF A DIRECT PRODUCT OF FREE GROUPS 3
Define the area of a null- P -expression to be the integer m and define the P -area of w , written Area P ( w ), to be the minimal area taken over all null- P -expressions for w . The Dehn function of the presentation P , written δ P , is defined to be the function N → N given by δ P ( n ) = max { Area P ( w ) : w ∈ A ± ∗ , w null-homotopic, | w | ≤ n } . Although the Dehn functions of different finite presentations of a fixed groupmay differ, their asymptotic behaviour will be the same. This is made precise inthe following way.
Definition 2.2.
Given functions f, g : N → N , write f (cid:22) g if there exists a constant C > f ( n ) ≤ Cg ( Cn + C ) + Cn + C for all n . Write f ≃ g if f (cid:22) g and g (cid:22) f . Lemma 2.3. If P and P are finite presentations of the same group then δ P ≃ δ P . For a proof of this standard result see, for example, [4, Proposition 1.3.3].A useful tool for the study of Dehn functions is a class of objects known asvan Kampen diagrams. Roughly speaking, these are planar CW-complexes whichportray diagrammatically schemes for reducing null-homotopic words to the iden-tity. Such diagrams, whose definition is recalled below, allow the application oftopological methods to the calculation of Dehn functions. For background and fur-ther details see, for example, [4, Section 4]. For the definition of a combinatorialCW-complex see, for example, [4, Appendix A].
Definition 2.4. A singular disc diagram ∆ is a finite, planar, contractible combi-natorial CW-complex with a specified base vertex ⋆ in its boundary. The area of∆, written Area(∆), is defined to be the number of 2-cells of which ∆ is composed.The boundary cycle of ∆ is the edge loop in ∆ which starts at ⋆ and traverses ∂ ∆in the anticlockwise direction. The interior of ∆ consists of a number of disjointopen 2-discs, the closures of which are called the disc components of ∆.Each 1-cell of ∆ has associated to it two directed edges ǫ and ǫ , with ǫ − = ǫ .Let DEdge(∆) be the set of directed edges of ∆. A labelling of ∆ over a set A isa map λ : DEdge(∆) → A ± such that λ ( ǫ − ) = λ ( ǫ ) − . This induces a map fromthe set of edge paths in ∆ to A ± ∗ . The boundary label of ∆ is the word in A ± ∗ associated to the boundary cycle.Let P = hA | Ri be a finite presentation. A P -van Kampen diagram for a word w ∈ A ± ∗ is a singular disc diagram ∆ labelled over A with boundary label w andsuch that for each 2-cell c of ∆ the anticlockwise edge loop given by the attachingmap of c , starting at some vertex in ∂c , is labelled by a word in R ± . Lemma 2.5 (Van Kampen’s Lemma) . A word w ∈ A ± ∗ is null-homotopic over P if and only if there exists a P -van Kampen diagram for w . In this case the P -areaof w is the minimal area over all P -van Kampen diagrams for w . For a proof of this result see, for example, [4, Theorem 4.2.2].Associated to a presentation P = hA | Ri of a group G there is a standardcombinatorial 2-complex K P with π ( K P ) ∼ = G . The complex K P is constructedby taking a wedge of copies of S indexed by the letters in A and attaching 2-cellsindexed by the relations in R . The 2-cell corresponding to a relation r ∈ R has | r | WILL DISON edges and is attached by identifying its boundary circuit with the edge path in K P along which one reads the word r .The Cayley 2-complex associated to P , denoted by Cay ( P ), is defined to bethe universal cover of K P . If one chooses a base vertex of Cay ( P ) to represent theidentity element of G then the 1-skeleton of this complex is canonically identifiedwith the Cayley graph of P . Given a P -van Kampen diagram ∆ there is a uniquelabel-preserving combinatorial map from ∆ to Cay ( P ) which maps the base vertexof ∆ to the vertex of Cay ( P ) representing the identity.3. A Class of Subgroups of Direct Products of Free Groups
In this section we introduce a class of subgroups of direct products of free groupsof which the group K defined in the introduction will be a member. We first fixsome notation which will be used throughout the paper. Given integers i, m ∈ N ,let F ( i ) m be the rank m free group with basis e ( i )1 , . . . , e ( i ) m . Given an integer r ∈ N ,let Z r be the rank r free abelian group with basis t , . . . , t r .Given positive integers n, m ≥ r ≤ m , we wish to define a group K nm ( r )to be the kernel of a homomorphism θ : F (1) m × . . . × F ( n ) m → Z r whose restrictionto each factor F ( i ) m is surjective. For fixed n , m and r , the isomorphism class ofthe group K nm ( r ) is, up to an automorphism of the factors of the ambient group F (1) m × . . . × F ( n ) m , independent of the homomorphism θ . This is proved by thefollowing lemma. Lemma 3.1.
Let F be a rank m free group. Given a surjective homomorphism φ : F → Z r , there exists a basis e , . . . , e m of F so that φ ( e i ) = ( t i if ≤ i ≤ r , if r + 1 ≤ i ≤ m .Proof. The homomorphism φ factors through the abelianisation homomorphismAb : F → A , where A is the rank m free abelian group F/ [ F, F ], as φ = ¯ φ ◦ Ab forsome homomorphism ¯ φ : A → Z r . Since ¯ φ is surjective, A splits as A ⊕ A where ¯ φ is an isomorphism on the first factor and 0 on the second factor. There thus existsa basis s , . . . , s m for A so as¯ φ ( s i ) = ( t i if 1 ≤ i ≤ r ,0 if r + 1 ≤ i ≤ m .We claim that the s i lift under Ab to a basis for F . To see this let f , . . . , f m beany basis for F and let ¯ f , . . . , ¯ f m be its image under Ab, a basis for A . Let ρ ∈ Aut( A ) be the change of basis isomorphism from ¯ f , . . . , ¯ f m to s , . . . , s m . It sufficesto show that this lifts under Ab to an automorphism of F . But this is certainly thecase since Aut( A ) ∼ = GL m ( Z ) is generated by the elementary transformations andeach of these obviously lifts to an automorphism. (cid:3) Definition 3.2.
For integers n, m ≥ r ≤ m , define K nm ( r ) to be the kernelof the homomorphism θ : F (1) m × . . . × F ( n ) m → Z r given by θ ( e ( i ) j ) = ( t j if 1 ≤ j ≤ r ,0 if r + 1 ≤ j ≤ m . SUBGROUP OF A DIRECT PRODUCT OF FREE GROUPS 5
Note that K n (1) is the n th Stallings-Bieri group SB n .By a result in Section 1.6 of [8], if r ≥ m ≥ K nm ( r ) is of type F n − but not of type FP n . In particular the group K ∼ = K (2) defined in theintroduction is finitely presented.4. Generating Sets
We give finite generating sets for those groups K nm ( r ) which are finitely gener-ated. We make use of the following notational shorthand: given formal symbols x and y , write [ x, y ] for xyx − y − and x y for yxy − . Proposition 4.1. If n ≥ then K nm ( r ) is generated by S ∪ S ∪ S where S = { e (1) i (cid:0) e ( j ) i (cid:1) − : 1 ≤ i ≤ r, ≤ j ≤ n } ,S = { e ( j ) i : r + 1 ≤ i ≤ m, ≤ j ≤ n } ,S = { (cid:2) e (1) i , e (1) j (cid:3) : 1 ≤ i < j ≤ r } . Proof.
Partition S as S ′ ∪ S ′′ where S ′ = { e ( j ) i : r + 1 ≤ i ≤ m, ≤ j ≤ n } and S ′′ = { e (1) i : r + 1 ≤ i ≤ m } . Project K nm ( r ) ≤ F (1) m × . . . × F ( n ) m onto the last n − → K m ( r ) → K nm ( r ) → F (2) m × . . . × F ( n ) m →
1. Note that S ∪ S ′ projects to a set of generators for F (2) m × . . . × F ( n ) m and that K m ( r ) is the normalclosure in F (1) m of S ′′ ∪ S . If ζ ∈ F (1) m and w ≡ w ( e (1)1 , . . . , e (1) m ) is a word in thegenerators of F (1) m then ζ w = ζ w “ e (1)1 ( e (2)1 ) − ,...,e (1) m ( e (2) m ) − ” . Thus S ∪ S ′ ∪ S ′′ ∪ S generates K nm ( r ). (cid:3) A Splitting Theorem
The following result gives an amalgamated product decomposition of the groups K nm ( r ) in the case that r = m . Note that with slightly more work one could provea more general result without this restriction. We introduce the following notation:given a collection of groups M, L , . . . , L k with M ≤ L i for each i , we denote by ∗ ki =1 ( L i ; M ) the amalgamated product L ∗ M . . . ∗ M L k . Theorem 5.1. If n ≥ and m ≥ then K nm ( m ) ∼ = ∗ mk =1 ( L k ; M ) , where M = K n − m ( m ) and, for each k = 1 , . . . , m , the group L k ∼ = K n − m ( m − is the kernel ofthe homomorphism θ k : F (1) m × . . . × F ( n − m → Z m − given by θ k ( e ( i ) j ) = t j if ≤ j ≤ k − , if j = k , t j − if k + 1 ≤ j ≤ m . WILL DISON
Proof.
Projecting K nm ( m ) onto the factor F ( n ) m gives the short exact sequence 1 → K n − m ( m ) → K nm ( m ) → F ( n ) m →
1. This splits to show that K nm ( m ) has thestructure of an internal semidirect product M ⋊ ˆ F ( n ) m where ˆ F ( n ) m ∼ = F ( n ) m is thesubgroup of F ( n − m × F ( n ) m generated by e ( n − (cid:0) e ( n )1 (cid:1) − , . . . , e ( n − m (cid:0) e ( n ) m (cid:1) − . Since the action by conjugation of e ( n − k (cid:0) e ( n ) k (cid:1) − on M is the same as the actionof e ( n − k , we have that K nm ( m ) = M ⋊ ˆ F ( n ) m ∼ = ∗ mk =1 (cid:16) M ⋊ D e ( n − k (cid:0) e ( n ) k (cid:1) − E ; M (cid:17) ∼ = ∗ mk =1 (cid:16) M ⋊ D e ( n − k E ; M (cid:17) . Define a homomorphism p k : F (1) m × . . . × F ( n − m → Z by p k (cid:0) e ( i ) j (cid:1) = ( j = k ,0 otherwise,and note that L k ∩ ker p k is the kernel K n − m ( m ) of the standard homomorphism θ : F (1) m × . . . × F ( n − m → Z m given in Definition 3.2. Considering the restrictionof p k to L k gives the short exact sequence 1 → K n − m ( m ) → L k → Z → L k = K n − m ( m ) ⋊ h e ( n − k i . (cid:3) Dehn Functions of Amalgamated Products
In this section we will be concerned with finitely presented amalgamated prod-ucts Γ = G ∗ H G where H , G and G are finitely generated groups and H isproper in each G i . Suppose each G i is presented by hA i | R i i , with A i finite. Notethat we are at liberty to choose the A i so as each a ∈ A i represents an element of G i r H . Indeed, since H is proper in G i there exists some a ′ ∈ A i representing anelement of G i r H and we can replace each other element a ∈ A i by a ′ a if necessary.Let B be a finite generating set for H and for each b ∈ B choose words u b ∈ A ± ∗ and v b ∈ A ± ∗ which equal b in Γ. Define E ⊂ ( A ∪ A ∪ B ) ± ∗ to be the finitecollection of words { bu − b , bv − b : b ∈ B} . Then, since Γ is finitely presented, thereexist finite subsets R ′ ⊆ R and R ′ ⊆ R such that Γ is finitely presented by P = hA , A , B | R ′ , R ′ , Ei . Theorem 6.1.
Let w ∈ A ± ∗ be a word representing an element h ∈ H and let u ∈ A ± ∗ and v ∈ A ± ∗ be words representing elements α ∈ G r H and β ∈ G r H respectively. If [ α, h ] = [ β, h ] = 1 then Area P ([ w, ( uv ) n ]) ≥ n d B (1 , h ) where d B is the word metric on H associated to the generating set B .Proof. Let ∆ be a P -van Kampen diagram for the null-homotopic word [ w, ( uv ) n ](see Diagram 1). For each i = 1 , , . . . , n , define p i to be the vertex in ∂ ∆ suchthat the anticlockwise path in ∂ ∆ from the basepoint around to p i is labelled bythe word w ( uv ) i − u . Similarly define q i to be the vertex in ∂ ∆ such that the SUBGROUP OF A DIRECT PRODUCT OF FREE GROUPS 7
PSfrag replacements uu u uuu vv v vvvp p p n q q q n ww Figure 1.
The van Kampen diagram ∆anticlockwise path in ∂ ∆ from the basepoint around to q i is labelled by the word w ( uv ) n w − ( uv ) i − n v − . We will show that for each i there is a B -path (i.e. an edgepath in ∆ labelled by a word in the letters B ) from p i to q i .We assume that the reader is familiar with Bass-Serre theory, as exposited in[12]. Let T be the Bass-Serre tree associated to the splitting G ∗ H G . Thisconsists of an edge gH for each coset Γ /H and a vertex gG i for each coset Γ /G i .The edge gH has initial vertex gG and terminal vertex gG . We will construct acontinuous (but non-combinatorial) map ∆ → T as the composition of the naturalmap ∆ → Cay ( P ) with the map f : Cay ( P ) → T defined below.There is a natural left action of Γ on each of Cay ( P ) and T and we construct f to be equivariant with respect to this as follows. Let m be the midpoint of theedge H of T and define f to map the vertex g ∈ Cay ( P ) to the point g · m , themidpoint of the edge gH . Define f to map the edge of Cay ( P ) labelled a ∈ A i joining vertices g and ga to the geodesic segment joining g · m to ga · m . Since a H this segment is an embedded arc of length 1 whose midpoint is the vertex gG i . Define f to collapse the edge in Cay ( P ) labelled b ∈ B joining vertices g and gb to the point g · m = gb · m . This is well defined since gH = gbH . This completesthe definition of f on the 1-skeleton of ∆; we now extend f over the 2-skeleton.Let c be a 2-cell in Cay ( P ) and let g be some vertex in its boundary. Assumethat c is metrised so as to be convex and let l be some point in its interior. The formof the relations in P ensures that the boundary label of c is a word in the letters A i ∪ B for some i and so every vertex in ∂c is labelled gg ′ for some g ′ ∈ G i . Thus f as so far defined maps ∂c into the ball of radius 1 / gG i ;we extend f to the interior of c by defining it to map the geodesic segment [ l, p ],where p ∈ ∂c , to the geodesic segment [ gG i , f ( p )]. This is independent of the vertex g ∈ ∂c chosen and makes f continuous since geodesics in a tree vary continuouslywith their endpoints. We now define ¯ f : ∆ → T to be the map given by composing f with the label-preserving map ∆ → Cay ( P ) which sends the basepoint of ∆ tothe vertex 1 ∈ Cay ( P ).Since w commutes with u and v we have that ¯ f ( p i ) = w ( uv ) i − u · m = ( uv ) i − u · m =¯ f ( q i ); define S to be the preimage under ¯ f of this point. By construction, the im-age of the interior of each 2-cell in ∆ and the image of the interior of each A i -edgeis disjoint from ¯ f ( p i ). Thus S consists of vertices and B -edges and so finding a B -path from p i to q i reduces to finding a path in S connecting these vertices. Let s i and t i be the vertices of ∂ ∆ immediately preceding and succeeding p i in the WILL DISON boundary cycle. Unless h = 1, in which case the theorem is trivial, the form ofthe word [ w, ( uv ) n ], together with the normal form theorem for amalgamated prod-ucts, implies that all the vertices p i , s i and t i lie in the boundary of the same disccomponent D of ∆. Furthermore, since u and v are words in the letters A and A respectively, the points f ( s i ) and f ( t i ) are separated in T by f ( p i ). Thus s i and t i are separated in D by S and so there exists an edge path γ i in S from p i to someother vertex r i ∈ ∂D . Since γ i is a B -path it follows that the word labelling thesub-arc of the boundary cycle of ∆ from p i to r i represents an element of H , and, byconsidering subwords of [ w, ( uv ) n ], we see that the only possibility is that r i = q i .Thus, for each i = 1 , . . . , n , the path γ i gives the required B -path connecting p i to q i . We choose each γ i to contain no repeated edges.For i = j , the two paths γ i and γ j are disjoint, since if they intersected therewould be a B -path joining p i to p j and thus the word labelling the subarc of theboundary cycle from p i to p j would represent an element of H . Observe that notwo edges in any of the paths γ , . . . , γ n lie in the boundary of the same 2-cell in ∆since each relation in P contains at most one occurrence of a letter in B . Becausethe word labelling ∂ ∆ contains no occurrences of a letter in B the interior of eachedge of a path γ i lies in the interior of ∆ and thus in the boundary of two distinct 2-cells. Since each path γ i contains no repeated edges we therefore obtain the boundArea(∆) ≥ P ni =1 | γ i | . But the word labelling each γ i is equal to h in Γ and so thelength of γ i is at least d B (1 , h ) whence we obtain the required inequality. (cid:3) We are now in a position to prove the main theorem.
Proof of Theorem 1.1.
To avoid excessive superscripts we change notation and write x i , y i for the generators of F ( i )2 , i = 1 , K ∼ = K (2) ∼ = L ∗ M L where,as subgroups of F (1)2 × F (2)2 , L = K (1) is generated by A = { x x − , y , y } , L ∼ = K (1) is generated by A = { x , x , y y − } and M = K (2) is generatedby B = { x x − , y y − , [ x , y ] } . To obtain the generating set for L we have hereimplicitly used the automorphism of F (1)2 × F (2)2 which interchanges x i with y i andrealises the isomorphism between L and K (1).For n ∈ N , define h n to be the element [ x n , y n ] ∈ K (2) and define w n to bethe word [( x x − ) n , y n ] ∈ A ± ∗ representing h n . Note that h n commutes with both y ∈ A and x ∈ A so by Theorem 6.1 the word [ w n , ( y x ) n ], which has length16 n , has area at least 2 n d B (1 , h n ). We claim that d B (1 , h n ) ≥ n .Suppose that in F (1)2 × F (2)2 the element h n is represented by a word w ≡ w ( x x − , y y − , [ x , y ]) in the generators B . Let k be the number of occurrencesof the third variable in the word w . We will show that k ≥ n .Observe that, as a group element, the word w ( x x − , y y − , [ x , y ]) is equal tothe word w ( x , y , [ x , y ]) w ( x − , y − , . Thus we have that [ x n , y n ] is freely equalto w ( x , y , [ x , y ]) and that w ( x − , y − , w ( x , y , x n , y n ] can be converted to the empty word by freeexpansions, free contractions and deletion of k subwords [ x , y ]. Hence [ x n , y n ] isa null-homotopic word over the presentation P = h x , y | [ x , y ] i with P -area atmost k . But P presents the rank 2 free abelian group, and basic results on Dehnfunctions give that [ x n , y n ] has area n over this presentation. Thus k ≥ n . (cid:3) SUBGROUP OF A DIRECT PRODUCT OF FREE GROUPS 9
Note that the above proof also shows that K (2) has at least quadratic distortionin each of K (1) and F (1)2 × F (2)2 . In fact it can be shown that the distortion isprecisely quadratic. Acknowledgements.
I would like to thank my thesis advisor, Martin Bridson, forhis many helpful comments made during the preparation of this article.
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Will Dison, Department of Mathematics, University Walk, Bristol, BS8 1TW, UnitedKingdom
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