J. Group Theory 12 (2009), 783–793 Journal of Group Theory DOI 10.1515/JGT.2009.012 ( de Gruyter 2009
A subgroup of a direct product of free groups whose Dehn function has a cubic lower bound
Will Dison (Communicated by J. Howie)
Abstract. We establish a cubic lower bound on the Dehn function of a certain finitely presented subgroup of a direct product of three free groups.
1 Introduction The collection S of subgroups of direct products of free groups is surprisingly rich and has been studied by many authors. In the early 1960s Stallings [13] exhibited a 3 subgroup of ðF2Þ , where F2 is the rank 2 free group, as the first known example of a finitely presented group whose third integral homology group is not finitely gener- ated. Bieri [2] demonstrated that Stallings’ group belongs to a sequence of groups n SBn c ðF2Þ , the Stallings–Bieri groups, with SBn being of type Fn 1 but not of type FPn. (See [6] for definitions and background concerning finiteness properties of groups.) In the realm of decision theory, Mihaı˘lova [9] and Miller [10] exhibited a finitely 2 generated subgroup of ðF2Þ with undecidable conjugacy and membership problems. It is thus seen that even in the 2-factor case fairly wild behaviour is encountered amongst the subgroups of direct products of free groups. In contrast to this, Baumslag and Roseblade [1] showed that in the 2-factor case this wildness only manifests itself amongst the subgroups which are not finitely pre- sented. They proved that if G is a finitely presented subgroup of F ð1Þ F ð2Þ, where F ð1Þ and F ð2Þ are free groups, then G is itself virtually a direct product of at most two 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 a direct product of n free groups and if G enjoys the finiteness property FPn, then G is virtu- ally a direct product of at most n free groups. Further information on the finiteness properties of the groups in class S was provided by Meinert [8] who calculated the BNS invariants of direct products of free groups. These invariants determine the fi- niteness properties of all subgroups lying above the commutator subgroup. 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 784 W. Dison
Dehn function of Stallings’ group SB3 is quadratic. The method espoused by Bridson in [3] proves that the function n3 is an upper bound on the Dehn functions of each of the Stallings–Bieri groups. In contrast, there have been no results which give non- trivial lower bounds on the Dehn functions of any groups in S. In other words, until now there has been no finitely presented subgroup of a direct product of free groups whose Dehn function was known to be greater than that of the ambient group. The purpose of this paper is to construct such a subgroup: we exhibit a finitely 3 3 presented subgroup of ðF2Þ whose Dehn function has the function n as a lower bound. 3 2 Let K be the kernel of a homomorphism y : ðF2Þ ! Z whose restriction to each factor F2 is surjective. By Lemma 3.1 below the isomorphism class of K is indepen- dent of the homomorphism chosen. Results in [8] show that K is finitely presented.
Theorem 1.1. The Dehn function d of K satisfies dðnÞ n3.
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 F-equivalence) of the presentation chosen. We refer the reader to Section 2 for background on Dehn functions, including definitions of the symbols and F. The organization of this paper is as follows. Section 2 gives basic definitions con- cerning Dehn functions, van Kampen diagrams and Cayley complexes. We expect that the reader will already be familiar with these concepts; the purpose of the expo- sition is principally to introduce notation. In Section 3 we define a class of subgroups of direct products of free groups of which K is a member. The subsequent section gives finite generating sets for those groups in the class which are finitely generated and Section 5 proves a splitting theorem which shows how certain groups in the class decompose as amalgamated products. We then prove in Section 6 how, in cer- tain circumstances, the distortion of the subgroup H in an amalgamated product G ¼ G1 H G2 gives rise to a lower bound on the Dehn function of G. Theorem 1.1 follows as a corollary of this result when applied to the splitting of K given in Sec- tion 5.
2 Dehn functions In this section we recall the basic definitions concerning Dehn functions of finitely presented groups. All of this material is standard. For further background and a more thorough exposition, including proofs, see, for example, [4] or [11]. Given a set A, write A 1 for the set of formal inverses of the elements of A and G G G write A 1 for the set A U A 1. Denote by A the free monoid on the set A 1.We G G refer to elements of A as words in the letters A 1 and write jwj for the length of G free such a word w. Given words w1; w2 A A we write w1 ¼ w2 if w1 and w2 are freely G equal and w1 1 w2 if w1 and w2 are equal as elements of A .
Definition 2.1. Let P ¼ hAjRi be a finite presentation of a group G. A word G w A A is said to be null-homotopic over P if it represents the identity in G. A subgroup of a direct product of free groups 785
m G G1 A null-P-expression for such a word is a sequence ðxi; riÞi¼1 in A R such that
Ym free 1 w ¼ xirixi : i¼1
Define the area of a null-P-expression to be the integer m and define the P-area of w, written AreaPðwÞ, to be the minimal area taken over all null-P-expressions for w. The Dehn function of the presentation P, written dP, is defined to be the function N ! N given by
A G dPðnÞ¼maxfAreaPðwÞ : w A ; w null-homotopic; jwj c ng:
Although the Dehn functions of di¤erent finite presentations of a fixed group may di¤er, their asymptotic behaviour will be the same. This is made precise in the follow- ing way.
Definition 2.2. Given functions f ; g : N ! N, write f g if there exists a constant C > 0 such that f ðnÞ c CgðCn þ CÞþCn þ C for all n. Write f F g if f g and g f .
F Lemma 2.3. If P1 and P2 are finite presentations of the same group then dP1 dP2 .
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 as van Kampen diagrams. Roughly speaking, these are planar CW-complexes which por- tray diagrammatically schemes for reducing null-homotopic words to the identity. Such diagrams, whose definition is recalled below, allow the application of topo- logical methods to the calculation of Dehn functions. For background and further details see, for example, [4, Section 4]. For the definition of a combinatorial CW- complex see, for example, [4, Appendix A].
Definition 2.4. A singular disc diagram D is a finite, planar, contractible combinatorial CW-complex with a specified base vertex ? in its boundary. The area of D, written AreaðDÞ, is defined to be the number of 2-cells of which D is composed. The boundary cycle of D is the edge loop in D which starts at ? and traverses qD in the anticlockwise direction. The interior of D consists of a number of disjoint open 2-discs, the closures of which are called the disc components of D. 1 Each 1-cell of D has associated to it two directed edges e1 and e2, with e1 ¼ e2. Let DEdgeðDÞ be the set of directed edges of D.Alabelling of D over a set A is a map G l : DEdgeðDÞ!A 1 such that lðe 1Þ¼lðeÞ 1. This induces a map from the set of G G edge paths in D to A . The boundary label of D is the word in A associated to the boundary cycle. 786 W. Dison
Let P ¼ hAjRi be a finite presentation. A P-van Kampen diagram for a word G w A A is a singular disc diagram D labelled over A with boundary label w and such that for each 2-cell c of D the anticlockwise edge loop given by the attaching G map of c, starting at some vertex in qc, is labelled by a word in R 1.
G Lemma 2.5 (Van Kampen’s lemma). A word w A A is null-homotopic over P if and only if there exists a P-van Kampen diagram for w. In this case the P-area of 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 ¼ hAjRi of a group G there is a standard combi- natorial 2-complex KP with p1ðKPÞ G G. The complex KP is constructed by taking a wedge of copies of S1 indexed by the letters in A and attaching 2-cells indexed by the relations in R. The 2-cell corresponding to a relation r A R has jrj edges and is at- 1 tached by identifying its boundary circuit with the edge path in KP along which one reads the word r. The Cayley 2-complex associated to P, denoted by Cay2ðPÞ, is defined to be the 2 universal cover of KP. If one chooses a base vertex of Cay ðPÞ to represent the iden- tity element of G then the 1-skeleton of this complex is canonically identified with the Cayley graph of P. Given a P-van Kampen diagram D there is a unique label- preserving combinatorial map from D to Cay2ðPÞ which maps the base vertex of D to the vertex of Cay2ðPÞ 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 groups of which the group K defined in the introduction will be a member. We first fix some ðiÞ notation which will be used throughout the paper. Given integers i; m A N, let Fm ðiÞ ðiÞ A N Zr be the rank m free group with basis e1 ; ...; em . Given an integer r , let be the rank r free abelian group with basis t1; ...; tr. n Given positive integers n; m d 1 and r c m, we wish to define a group KmðrÞ to ð1Þ ðnÞ r be the kernel of a homomorphism y : Fm Fm ! Z whose restriction to ðiÞ each factor Fm is surjective. For fixed n, m and r, the isomorphism class of the n group KmðrÞ is, up to an automorphism of the factors of the ambient group ð1Þ ðnÞ Fm Fm , independent of the homomorphism y. This is proved by the follow- ing lemma.
Lemma 3.1. Let F be a rank m free group. Given a surjective homomorphism r f : F ! Z , there exists a basis e1; ...; em of F so that ti if 1 c i c r; fðeiÞ¼ 0 if r þ 1 c i c m:
Proof. The homomorphism f factors through the abelianization homomorphism Ab : F ! A, where A is the rank m free abelian group F=½F; F ,asf ¼ f Ab for A subgroup of a direct product of free groups 787
r some homomorphism f : A ! Z . Since f is surjective, A splits as A1 l A2 where f is an isomorphism on the first factor and 0 on the second factor. There thus exists a basis s1; ...; sm for A such that ti if 1 c i c r; fðsiÞ¼ 0ifr þ 1 c i c m:
We claim that the elements si lift under Ab to a basis for F. To see this let f1; ...; fm be any basis for F and let f1; ...; fm be its image under Ab, a basis for A. A Let r AutðAÞ be the change of basis isomorphism from f1; ...; fm to s1; ...; sm.It su‰ces to show that this lifts under Ab to an automorphism of F. But this is certainly the case since AutðAÞ G GLmðZÞ is generated by the elementary transformations and each of these obviously lifts to an automorphism. r
n Definition 3.2. For integers n; m d 1 and r c m, define KmðrÞ to be the kernel of the ð1Þ ðnÞ r homomorphism y : Fm Fm ! Z given by t if 1 c j c r; yðeðiÞÞ¼ j j 0ifr þ 1 c j c m:
n Note that K2 ð1Þ is the nth Stallings–Bieri group SBn. n By a result in [8, Section 1.6], if r d 1 and m d 2 then KmðrÞ is of type Fn 1 but not 3 of type FPn. In particular the group K G K2 ð2Þ defined in the introduction is finitely presented.
4 Generating sets
n We give finite generating sets for those groups KmðrÞ which are finitely generated. We make use of the following notational shorthand: given formal symbols x and y, write ½x; y for xyx 1y 1 and xy for yxy 1.
n U U Proposition 4.1. If n d 2 then KmðrÞ is generated by S1 S2 S3 where
ð1Þ ð jÞ 1 S1 ¼fei ðei Þ : 1 c i c r; 2 c j c ng; ð jÞ S2 ¼fei : r þ 1 c i c m; 1 c j c ng; ð1Þ ð1Þ S3 ¼f½ei ; ej : 1 c i < j c rg:
0 U 00 Proof. Partition S2 as S2 S2 where
0 ð jÞ S2 ¼fei : r þ 1 c i c m; 2 c j c ng 788 W. Dison and
00 ð1Þ S2 ¼fei : r þ 1 c i c mg:
n ð1Þ ðnÞ Project KmðrÞ c Fm Fm onto the last n 1 factors to give the short exact sequence
1 n ð2Þ ðnÞ 1 ! KmðrÞ!KmðrÞ!Fm Fm ! 1:
U 0 ð2Þ ðnÞ 1 Note that S1 S2 projects to a set of generators for Fm Fm and that KmðrÞ is ð1Þ 00 U A ð1Þ 1 ð1Þ ð1Þ the normal closure in Fm of S2 S3.Ifz Fm and w wðe1 ; ...; em Þ is a word in ð1Þ the generators of Fm then
w wðeð1Þðeð2ÞÞ 1;...; eð1Þðeð2ÞÞ 1;eð1Þ ;...;eð1ÞÞ z ¼ z 1 1 r r rþ1 m :
U 0 U 00 U n r Thus S1 S2 S2 S3 generates KmðrÞ.
5 A splitting theorem The following result gives an amalgamated product decomposition of the groups n KmðrÞ in the case that r ¼ m. With slightly more work one could prove a more gen- eral result without this restriction. We introduce the following notation: given a col- lection of groups M; L ; ...; L with M c L for each i, we denote by k ðL ; MÞ 1 k i i¼1 i the amalgamated product L1 M ... M Lk.
If n d and m d then K n m G m L M where M K n 1 m Theorem 5.1. 2 1 mð Þ k¼1ð k; Þ, ¼ m ð Þ n 1 and, for k ¼ 1; ...; m, the group Lk G Km ðm 1Þ is the kernel of the homomorphism
ð1Þ ðn 1Þ Zm 1 yk : Fm Fm ! given by 8 n ðnÞ Proof. Projecting KmðmÞ onto the factor Fm gives the short exact sequence n 1 n ðnÞ 1 ! Km ðmÞ!KmðmÞ!Fm ! 1: n This splits to show that KmðmÞ has the structure of an internal semidirect product ^ðnÞ ^ðnÞ ðnÞ ðn 1Þ ðnÞ M z Fm where Fm G Fm is the subgroup of Fm Fm generated by ðn 1Þ ðnÞ 1 ðn 1Þ ðnÞ 1 e1 ðe1 Þ ; ...; em ðem Þ : A subgroup of a direct product of free groups 789 ðn 1Þ ðnÞ 1 Since the action by conjugation of ek ðek Þ on M is the same as the action of ðn 1Þ ek , we have n ^ðnÞ KmðmÞ¼M z Fm G m ðM z heðn 1ÞðeðnÞÞ 1i; MÞ k¼1 k k G m ðM z heðn 1Þi; MÞ: k¼1 k ð1Þ ðn 1Þ Define a homomorphism pk : Fm Fm ! Z by 1ifj ¼ k; p ðeðiÞÞ¼ k j 0 otherwise; V n 1 and note that Lk ker pk is the kernel Km ðmÞ of the standard homomorphism ð1Þ ðn 1Þ m y : Fm Fm ! Z given in Definition 3.2. Considering the restriction of pk to Lk gives the short exact sequence n 1 Z 1 ! Km ðmÞ!Lk ! ! 1 n 1 h ðn 1Þi r which demonstrates that Lk ¼ Km ðmÞ z ek . 6 Dehn functions of amalgamated products In this section we will be concerned with finitely presented amalgamated products G ¼ G1 H G2 where H, G1 and G2 are finitely generated groups and H is proper in each Gi. Suppose that each Gi is presented by hAijRii, with Ai finite. Note that we are at liberty to choose the Ai in such a way that each a A Ai represents an element of 0 GinH. Indeed, since H is proper in Gi there exists some a A Ai representing an ele- 0 ment of GinH and we can replace each other element a A Ai by a a if necessary. A A G Let B be a finite generating set for H and for each b B choose words ub A1 A G U U G and vb A2 which equal b in G. Define E H ðA1 A2 BÞ to be the finite col- 1 1 A lection of words fbub ; bvb : b Bg. Then, since G is finitely presented, there exist 0 0 finite subsets R1 J R1 and R2 J R2 such that G is finitely presented by h 0 0 i P ¼ A1; A2; B j R1; R2; E : A G A A G Theorem 6.1. Let w A1 be a word representing an element h H and let u A1 A G A A and v A2 be words representing elements a G1nH and b G2nH respectively. If ½a; h ¼½b; h ¼1 then n AreaPð½w; ðuvÞ Þ d 2n dBð1; hÞ; where dB is the word metric on H associated to the generating set B. 790 W. Dison Figure 1. The van Kampen diagram D Proof. Let D be a P-van Kampen diagram for the null-homotopic word ½w; ðuvÞn (see Diagram 1). For each i ¼ 1; 2; ...; n, define pi to be the vertex in qD such that the anticlockwise path in qD from the basepoint around to pi is labelled by the word i 1 wðuvÞ u. Similarly define qi to be the vertex in qD such that the anticlockwise path n 1 i n 1 in qD from the basepoint around to qi is labelled by the word wðuvÞ w ðuvÞ v . We will show that for each i there is a B-path (i.e. an edge path in D labelled by a word in the letters B) from pi to qi. We assume that the reader is familiar with Bass–Serre theory, as described in [12]. Let T be the Bass–Serre tree associated to the splitting G1 H G2. This consists of an edge gH for each coset G=H and a vertex gGi for each coset G=Gi. The edge gH has initial vertex gG1 and terminal vertex gG2. We will construct a continuous (but non- combinatorial) map D ! T as the composition of the natural map D ! Cay2ðPÞ with the map f : Cay2ðPÞ!T defined below. There is a natural left action of G on each of Cay2ðPÞ and T and we construct f to be equivariant with respect to this as follows. Let m be the midpoint of the edge H of T and define f to map the vertex g A Cay2ðPÞ to the point g m, the midpoint of the 2 edge gH. Define f to map the edge of Cay ðPÞ labelled a A Ai joining vertices g and ga to the geodesic segment joining g m to ga m. Since a B H this segment is an em- bedded arc of length 1 whose midpoint is the vertex gGi. Define f to collapse the edge in Cay2ðPÞ labelled b A B joining vertices g and gb to the point g m ¼ gb m. This is well defined since gH ¼ gbH. This completes the definition of f on the 1-skeleton of D; we now extend f over the 2-skeleton. Let c be a 2-cell in Cay2ðPÞ and let g be some vertex in its boundary. Assume that c is metrized so as to be convex and let l be some point in its interior. The form of the relations in P ensures that the boundary label of c is a word in the 0 0 letters Ai U B for some i and so every vertex in qc is labelled gg for some g A Gi. Thus f as so far defined maps qc into the ball of radius 1=2 centred on the vertex gGi; we extend f to the interior of c by defining it to map the geodesic segment ½l; p , where p A qc, to the geodesic segment ½gGi; f ðpÞ . This is independent of the vertex g A qc chosen and makes f continuous since geodesics in a tree vary continuously with their endpoints. We now define f : D ! T to be the map given by composing f with the label-preserving map D ! Cay2ðPÞ which sends the basepoint of D to the vertex 1 A Cay2ðPÞ. A subgroup of a direct product of free groups 791 Since w commutes with u and v we have i 1 i 1 f ðpiÞ¼wðuvÞ u m ¼ðuvÞ u m ¼ f ðqiÞ; define S to be the preimage under f of this point. By construction, the image of the interior of each 2-cell in D and the image of the interior of each Ai-edge is disjoint from f ðpiÞ. Thus S consists of vertices and B-edges, and so finding a B-path from pi to qi reduces to finding a path in S connecting these vertices. Let si and ti be the vertices of qD immediately preceding and succeeding pi in the boundary cycle. Unless h ¼ 1, in which case the theorem is trivial, the form of the word ½w; ðuvÞn , together with the normal form theorem for amalgamated products, implies that all of the ver- tices pi, si and ti lie in the boundary of the same disc component D of D. Further- more, since u and v are words in the letters A1 and A2 respectively, the points f ðsiÞ and f ðtiÞ are separated in T by f ðpiÞ. Thus si and ti are separated in D by S and so A there exists an edge path gi in S from pi to some other vertex ri qD. Since gi is a B-path it follows that the word labelling the sub-arc of the boundary cycle of D n from pi to ri represents an element of H, and, by considering subwords of ½w; ðuvÞ , we see that the only possibility is that ri ¼ qi. Thus, for i ¼ 1; ...; n, the path gi gives the required B-path connecting pi to qi. We choose each gi to contain no repeated edges. 0 For i j, the two paths gi and gj are disjoint, since if they intersected there would be a B-path joining pi to pj and thus the word labelling the subarc of the boundary cycle from pi to pj would represent an element of H. Observe that no two edges in any of the paths g1; ...; gn lie in the boundary of the same 2-cell in D since each relation in P contains at most one occurrence of a letter in B. Be- cause the word labelling qD contains no occurrences of a letter in B the interior of each edge of a path gi lies in the interior of D and thus in the boundary of two distinct 2-cells. Since eachP path gi contains no repeated edges we therefore ob- n tain the bound AreaðDÞ d i¼1 2jgij. But the word labelling each gi is equal to h in G and so the length of gi is at least dBð1; hÞ whence we obtain the required inequality. r 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 ðiÞ xi, yi for the generators of F2 ,fori ¼ 1; 2. 3 By Proposition 4.1 and Theorem 5.1 we have K G K2 ð2Þ G L1 M L2 where, ð1Þ ð2Þ 2 1 as subgroups of F2 F2 , L1 ¼ K2 ð1Þ is generated by A1 ¼fx1x2 ; y1; y2g, 2 1 2 L2 G K2 ð1Þ is generated by A2 ¼fx1; x2; y1y2 g and M ¼ K2 ð2Þ is generated by 1 1 B ¼fx1x2 ; y1y2 ; ½x1; y1 g. To obtain the generating set for L2 we have here implic- ð1Þ ð2Þ itly used the automorphism of F2 F2 which interchanges xi with yi and realizes 2 the isomorphism between L2 and K2 ð1Þ. A N n n A 2 For n , define hn to be the element ½x1 ; y1 K2 ð2Þ and define wn to be the word 1 n n A G A ½ðx1x2 Þ ; y1 A1 representing hn. Note that hn commutes with both y2 A1 and 792 W. Dison n x2 A A2, so by Theorem 6.1 the word ½wn; ðy2x2Þ , which has length 16n, has area at 2 least 2ndBð1; hnÞ. We claim that dBð1; hnÞ d n . ð1Þ ð2Þ Suppose that in F2 F2 the element hn is represented by a word 1 1 1 w wðx1x2 ; y1y2 ; ½x1; y1 Þ in the generators B. Let k be the number of occurrences of the third variable in the word w. We will show that k d n2. 1 1 Observe that, as a group element, the word wðx1x2 ; y1y2 ; ½x1; y1 Þ is equal 1 1 n n to the word wðx1; y1; ½x1; y1 Þ wðx2 ; y2 ; 1Þ. Thus ½x1 ; y1 is freely equal to 1 1 wðx1; y1; ½x1; y1 Þ and wðx2 ; y2 ; 1Þ is freely equal to the empty word. Hence n n wðx1; y1; 1Þ is also freely equal to the empty word. It follows that ½x1 ; y1 can be con- verted to the empty word by free expansions, free contractions and deletion of k n n subwords ½x1; y1 . Hence ½x1 ; y1 is a null-homotopic word over the presentation P ¼ hx1; y1 j½x1; y1 i with P-area at most k. But P presents the rank 2 free abelian n n 2 group, and basic results on Dehn functions give that ½x1 ; y1 has area n over this presentation. Thus k d n2. r 2 Note that the above proof also shows that K2 ð2Þ has at least quadratic distortion 2 ð1Þ ð2Þ in each of K2 ð1Þ and F2 F2 . In fact it can be shown that the distortion is pre- cisely quadratic. Acknowledgement. The author would like to thank his thesis advisor, Martin Brid- son, for his many helpful comments made during the preparation of this article. References [1] G. Baumslag and J. E. Roseblade. Subgroups of direct products of free groups. J. London Math. Soc. (2) 30 (1984), 44–52. [2] R. Bieri. Homological dimension of discrete groups. Queen Mary College Mathematics Notes (Mathematics Department, Queen Mary College, London, 1976). [3] M. R. Bridson. Doubles, finiteness properties of groups, and quadratic isoperimetric in- equalities. J. Algebra 214 (1999), 652–667. [4] M. R. Bridson. The geometry of the word problem. In Invitations to geometry and topol- ogy, Oxford Grad. Texts in Math. 7 (Oxford University Press, 2002), pp. 29–91. [5] M. R. Bridson, J. Howie, C. 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Trees (Springer-Verlag, 1980). [13] J. Stallings. A finitely presented group whose 3-dimensional integral homology is not fi- nitely generated. Amer. J. Math. 85 (1963), 541–543. Received 6 August, 2008; revised 11 November, 2008 Will Dison, Department of Mathematics, University Walk, Bristol, BS8 1TW, United Kingdom E-mail: [email protected]