aa r X i v : . [ m a t h . G R ] D ec AN ISOPERIMETRIC FUNCTION FOR BESTVINA-BRADYGROUPS
WILL DISON
Abstract.
Given a right-angled Artin group A , the associated Bestvina-Brady group is defined to be the kernel of the homomorphism A → Z thatmaps each generator in the standard presentation of A to a fixed generator of Z . We prove that the Dehn function of an arbitrary finitely presented Bestvina-Brady group is bounded above by n . This is the best possible universal upperbound. Introduction
Dehn functions and right-angled Artin groups are some of the most studiedobjects in contemporary geometric group theory. Among the most striking worksconcerning right-angled Artin groups is the combinatorial Morse theory introducedby Bestvina and Brady in [1] to solve the long-standing question of whether thereexist groups of type FP(2) which are not finitely presented. The central objects ofstudy in their theory are the Bestvina-Brady groups, which arise as the kernels ofhomomorphisms from right-angled Artin groups to the integers.A finite flag simplicial complex ∆ with vertices v , . . . , v k defines a right-angledArtin group A with presentation P A = h a , . . . , a k | [ a i , a j ] whenever v i and v j are joined by an edge in ∆ i . The
Bestvina-Brady group H ∆ associated to ∆ is defined to be the kernel of thehomomorphism A → Z = h t i which maps each a i t . In [1] the authors prove thatthe group H ∆ is finitely presented if and only if ∆ is simply connected. The purposeof this article is to estimate the complexity of the word problem in Bestvina-Bradygroups by establishing a universal upper bound on their Dehn functions. Theorem 1.1. If ∆ is simply connected then the Dehn function δ of H ∆ satisfies δ ( n ) (cid:22) n . This result is sharp: there exist finitely presented Bestvina-Brady groups whoseDehn functions are ≃ n (see [2]). Theorem 1.1 provides an obstruction to themethod suggested in [2] for producing Bestvina-Brady groups whose Dehn functionsare similar to n k for arbitrary integers k .A significant component of the proof of Theorem 1.1 is a method for producingan isoperimetric function f for a finitely presented group K from an isoperimetricfunction for a cyclic extension of K . A priori , the function f will be an isoperimetricfunction for a presentation of K with infinitely many relators. We introduce thenotion of area-penetration pairs to deal with such non-finite presentations and showhow they can be used to derive an isoperimetric function for a finite presentationof a group from an isoperimetric function for a presentation of the group withinfinitely many relators. The organisation of this paper is as follows. Section 2 begins with the defini-tions of various filling invariants for finitely generated groups; namely, isoperimetricand Dehn functions and area-radius pairs. We then introduce the new notions ofarea-penetration pairs and relative area functions, which we will use to deal withpresentations with infinitely many relators. In Section 3 we prove a general resultconcerning the isoperimetric functions of cyclic extensions. Theorem 1.1 is provedas a corollary of this in Section 4. Finally, in Section 5 we briefly recount the con-struction due to Brady, Forester and Shankar of a finitely presented Bestvina-Bradygroup with Dehn function ≃ n .2. Filling Functions
In this section we define various filling invariants of groups and give some of theirbasic properties. Throughout P = hA | Ri will be a presentation with A finite.2.1. Area Functions.
We recall the basic definitions concerning isoperimetricfunctions for finitely generated groups. For further background and a more thor-ough exposition see, for example, [3] or [8]. Note that the definitions given here arestandard, but we do not make the usual assumption that the presentations involvedhave a finite number of relators.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 F ( A ) the free group on the set A andby A ± ∗ the free monoid on the set A ± . We refer to elements of A ± ∗ as words inthe letters A ± and write ∅ for the empty word. The length of a word w ∈ A ± ∗ iswritten | w | . Given words w , w ∈ A ± ∗ we write w = w if w and w are equalas elements of F ( A ) and w ≡ w if w and w are equal as elements of A ± ∗ . Definition 2.1.
A word w ∈ A ± ∗ is said to be null-homotopic over P if it repre-sents the identity in the group presented by P . A null- P -expression for such a wordis 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 . Define the area of a null- P -expression Σ , written Area Σ , to be the integer m .Define the P -Area of w , written Area P ( w ) , to be the minimal area taken over allnull- 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 > such that f ( n ) ≤ Cg ( Cn + C ) + Cn + C for all n . Write f ≃ g if f (cid:22) g and g (cid:22) f . If P and P are finite presentations of the same group then δ P ≃ δ P (see, e.g.,[3]). N ISOPERIMETRIC FUNCTION FOR BESTVINA-BRADY GROUPS 3
Definition 2.3.
A function f : N → N is an isoperimetric function for a group G if δ Q (cid:22) f for some (and hence any) finite presentation Q of G . Definition 2.4. A null- P -scheme for a null-homotopic word w ∈ A ± ∗ is a sequence w ≡ w w . . . w n ≡ ∅ of words in A ± ∗ such that each w i w − i +1 is null-homotopic. The P -Cost of each transition w i w i +1 is the P -Area of the word w i w − i +1 . Note that the sum of the costs of the transitions in a null- P -scheme gives anupper bound on the area of the word w .2.2. Area-Radius pairs.Definition 2.5.
Define the radius of a null- P -expression Σ = ( x i , r i ) mi =1 , written Rad Σ , to be max mi =1 | x i | . A pair ( α, ρ ) of functions α, ρ : N → N is said to be an area-radius pair for the presentation P if for all null-homotopic words w ∈ A ± ∗ with | w | ≤ n there exists a null- P -expression Σ with Area Σ ≤ α ( n ) and Rad Σ ≤ ρ ( n ) . The following result shows how area-radius pairs transform under change ofpresentation.
Proposition 2.6.
Let P and Q be finite presentations of the same group. If ( α, ρ ) is an area-radius pair for P then there exists an area-radius pair ( α ′ , ρ ′ ) for Q with α ≃ α ′ and ρ ≃ ρ ′ .Proof. Since P can be converted to Q by a finite sequence of Tietze transformations,it suffices to prove the proposition in the situation that P and Q are related by asingle such transformation. There are four cases to consider. Case 1.
Suppose that P = hA | Ri and Q = hA | R , s i where s ∈ A ± ∗ isnull-homotopic over P . A null- P -expression for a word w ∈ A ± ∗ is also a null- Q -expression for w , so ( α, ρ ) is itself an area-radius pair for Q . Case 2.
Suppose that P = hA | R , s i and Q = hA | Ri where s ∈ A ± ∗ is null-homotopic over Q . Let ( x i , r i ) be a null- Q -expression for s with area M and radius K . If w ∈ A ± ∗ is a null-homotopic word of length at most n then there exists a null- P -expression Σ = ( y i , z i ) Li =1 for w with area L ≤ α ( n ) and radius at most ρ ( n ).Substituting Q Mi =1 x i r i x − i for each occurrence of s in the product Q Li =1 y i z i y − i gives a product which is freely equal to w in F ( A ). The corresponding null- Q -expression has area at most M L and radius at most ρ ( n ) + K . Thus ( Lα ( n ) , ρ ( n ) + K ) is an area-radius pair for Q . Case 3.
Suppose that P = hA |Ri and Q = hA , b | R , bu − b i where u b ∈ A ± ∗ and bu − b is null-homotopic over P . Define K = | u b | . Suppose w ∈ ( A ∪ { b } ) ± ∗ is a null-homotopic word of length at most n ; say w ≡ v b ǫ v . . . b ǫ L v L for some v i ∈ A ± ∗ and ǫ i ∈ {± } . Insert cancelling pairs u − b u b into w to obtain the word w ′ ≡ v ( bu − b u b ) ǫ v . . . ( bu − b u b ) ǫ L v L with w ′ free = w . Define v ′ , . . . , v ′ L to be thewords in A ± ∗ such that w ′ ≡ v ′ ( bu − b ) ǫ v ′ . . . ( bu − b ) ǫ L v ′ L and note that P Li =1 | v ′ i | ≤ K | w | ≤ Kn . For each i ∈ { , . . . , L } define τ i ≡ v ′ i v ′ i +1 . . . v ′ L . Then w ′ free = τ L Y i =1 τ − i ( bu b ) ǫ i τ i and | τ i | ≤ P Li =1 | v ′ i | ≤ Kn . The word τ is null-homotopic over Q and hence over P and so there exists a null- P -expression ( x i , r i ) Mi =1 for τ with area at most α ( Kn ) WILL DISON and radius at most ρ ( Kn ). Thus w free = M Y i =1 x i r i x − i L Y i =1 τ − i ( bu − b ) ǫ i τ i and so we obtain a null- Q -expression for w with area at most M + L ≤ α ( Kn ) + n and radius at most max { max i | x i | , max i | v ′ i |} ≤ max { ρ ( Kn ) , Kn } ≤ ρ ( Kn ) + Kn .Thus ( α ( Kn ) + n, ρ ( Kn ) + Kn ) is an area-radius pair for Q . Case 4.
Suppose that P = hA , b | R , bu − b i and Q = hA |Ri where u b ∈ A ± ∗ and bu − b is null-homotopic over Q . Define K = | u b | . Consider the retraction π : ( A ∪ { b } ) ± ∗ → A ± ∗ which is the identity on A and maps b ± u ± b . Note that π induces a retraction F ( A ∪ { b } ) → F ( A ). Suppose w ∈ A ± ∗ is a null-homotopicword of length at most n and let ( x i , z i ) Mi =1 be a null- P -expression for w with area atmost α ( n ) and radius at most ρ ( n ). Let S be the subset of { , . . . , m } consisting ofthose i for which z i ∈ R ± . Then ( π ( x i ) , π ( z i )) i ∈ S is a null- Q -expression for w witharea at most M and radius at most Kρ ( n ). Thus ( α ( n ) , Kρ ( n )) is an area-radiuspair for Q . (cid:3) Changing Between Infinite Presentations.
Up to this point, all the def-initions of this section have been standard; we now introduce something new. Wesaw above that the Dehn functions of all finite presentations of a fixed group havethe same asymptotic behaviour. This is not true, however, for presentations withan infinite number of relators, where the behaviour of the Dehn function may varymarkedly. Indeed, for any group if we take the set of relators to be the set of allnull-homotopic words then we obtain a presentations whose Dehn function is con-stant. In order to regain some control over how the Dehn function changes whenchanging between (possibly non-finite) presentations, we introduce the followingnotions.
Definition 2.7. An index on a set X is a function k · k : X → N . This is extendedto an index on the set X ± by setting k x − k = k x k . An indexed presentation is apair ( P , k · k ) where P = hA | Ri is a presentation and k · k is an index on R .Let ( P , k · k ) be an indexed presentation whose set of generators A is finite. Apair ( α, π ) of functions α, π : N → N is said to be an area-penetration pair for ( P , k · k ) if for all null-homotopic words w ∈ A ± ∗ with | w | ≤ n there exists a null- P -expression ( x i , r i ) mi =1 for w with area m ≤ α ( n ) and with k r i k ≤ π ( n ) for each i . Given X ⊆ A ± ∗ write hhX ii for the normal closure of the image of X in F ( A ) .Let S ⊆ A ± ∗ be a set of words with hhSii = hhRii . Then Q = hA | Si presents thesame group as P . The relational area function of ( P , k · k ) over Q is defined to bethe function N → N ∪ {∞} given by RArea( n ) = max { Area Q ( r ) : r ∈ R , k r k ≤ n } . Proposition 2.8.
Let ( P , k · k ) and Q be as in Definition 2.7. Let ( α, π ) be anarea-penetration pair for ( P , k · k ) and let RArea be the relational area function of ( P , k · k ) over Q . Then the Dehn function δ Q of the presentation Q satisfies δ Q ( n ) ≤ α ( n ) RArea( π ( n )) . Since the proof of this result is straightforward we omit it.
N ISOPERIMETRIC FUNCTION FOR BESTVINA-BRADY GROUPS 5 Isoperimetric Functions for Cyclic Extensions
Let K ⊳ Γ be a pair of finitely presented groups with Γ /K ∼ = Z . In this sectionwe show how a presentation P Γ of Γ gives rise to an infinite presentation P ∞ K for K . The relators of P ∞ K come equipped with an index k · k and we prove that anarea-radius pair for P Γ is actually an area-penetration pair for ( P ∞ K , k · k ).Let A be a finite generating set for K . Choose an element t of Γ whose imagegenerates Γ /K ∼ = Z and let θ be the automorphism of K induced by conjugation by t . Let P K = hA | Ri be a presentation for K and for each a ∈ A let w a be a wordin A ± ∗ representing θ ( a ). Define S to be the set of words { tat − w − a : a ∈ A} andlet P Γ be the presentation hA , t | R , Si of Γ.For each k ∈ Z , let Φ k : A ± ∗ → A ± ∗ be an endomorphism lifting θ k : K → K which commutes with the inversion involution of A ± ∗ . We take Φ to be theidentity. Define the following collections of words in A ± ∗ : R = { Φ k ( r ) : r ∈ R , k ∈ Z }S = { Φ k +1 ( a )Φ k ( w a ) − : a ∈ A , k ∈ Z } . Note that each word in
R ∪ S represents the identity in K . Since R ⊆ R , thepresentation P ∞ K = hA | R , Si presents K . Define an index k · k on R ∪ S by setting k ω k to be the minimal value of | k | such that either ω ≡ Φ k ( r ) for some r ∈ R or ω ≡ Φ k +1 ( a )Φ k ( w a ) − for some a ∈ A .The following theorem is the principal result of this section. The reader may findit instructive to translate the given proof into the language of either van Kampendiagrams (see, e.g., [3]) or pictures (see, e.g., [7]) where the ideas involved areperhaps more intuitive. Theorem 3.1. If ( α, ρ ) is an area-radius pair for P Γ then it is also an area-penetration pair for the indexed presentation ( P ∞ K , k · k ) .Proof. Let w ∈ A ± ∗ be a null-homotopic word of length at most n and let ( x i , z i ) mi =1 be a null- P Γ -expression for w with m ≤ α ( n ) and with | x i | ≤ ρ ( n ) for each i .We write h ( u ) for the exponent sum in the letter t of a word u ∈ ( A ∪ { t } ) ± ∗ anddefine e N to be the submonoid of ( A ∪ { t } ) ± ∗ consisting of all those words u with h ( u ) = 0. Define X to be the set of words { t k at − k : a ∈ A , k ∈ Z } ≤ ( A ∪ { t } ) ± ∗ .Let L be the submonoid of e N generated by X ± and note that L is free on thisbasis. If u ∈ e N write Λ( u ) for the unique word in L which is freely equal to u in F ( A ∪ { t } ) and freely reduced as an element of F ( X ). For each i ∈ { , . . . , m } ,define x i ≡ Λ( x i t − h ( x i ) ) and z i = Λ( t h ( x i ) z i t − h ( x i ) ). Define σ ≡ Q mi =1 x i z i x − i andnote that w free = σ in F ( A ∪ { t } ).Define a homomorphism Ψ : L → A ± ∗ , which commutes with the inversioninvolution of L , by mapping t k at − k Φ k ( a ). Let N be the kernel of the homomor-phism F ( A ∪ { t } ) → Z defined by mapping t to 1 and each a ∈ A to 0, and notethat N is free with basis the image of X . Thus Ψ descends to a homomorphism N → F ( A ) and since w free = σ in N we have Ψ( w ) free = Ψ( σ ) in F ( A ). Observe thatΨ( σ ) ≡ Q mi =1 Ψ( x i )Ψ( z i )Ψ( x i ) − and Ψ( w ) ≡ w since w contains no occurrence ofthe letter t .If z i ≡ a . . . a l ∈ R then z i ≡ t k a t − k . . . t k a l t − k for some k ∈ Z with | k | = | h ( x i ) | ≤ | x i | . Thus Ψ( z i ) ≡ Φ k ( z i ) where | k | ≤ ρ ( n ). If z i ≡ tat − a . . . a l ∈ S then z i ≡ t k +1 at − k − t k a t − k . . . t k a l t − k for some k ∈ Z with | k | = | h ( x i ) | ≤ | x i | . Thus WILL DISON Ψ( z i ) ≡ Φ k +1 ( a )Φ k ( w a ) − where | k | ≤ ρ ( n ). In either case we have Ψ( z i ) ∈ R ∪ S and k Ψ( z i ) k ≤ ρ ( n ). Thus (Ψ( x i ) , Ψ( z i )) mi =1 is a null- P ∞ K -expression for w and,since w was arbitrary, we see that ( α, ρ ) is an area-penetration pair for P ∞ K . (cid:3) Proof of Theorem 1.1
Recall from the introduction that ∆ is a finite, flag simplicial complex defininga right-angled Artin group A with standard presentation P A . The Bestvina-Bradysubgroup of A is defined to be the kernel H ∆ of the homomorphism A → Z = h t i which maps each of the generators of P A to t . The group H ∆ is finitely presentedif and only if ∆ is simply connected [1]; we now describe such a presentation.Let Edge(∆) be the set of directed edges of ∆ (so the cardinality of Edge(∆)is twice the number of 1-simplices in ∆). We write ιe and τ e respectively for theinitial and terminal vertices of e and we write e for the edge e with the oppositeorientation. We say that the directed edges e , . . . , e n form a combinatorial pathin ∆, written e · . . . · e n , if τ e i = ιe i +1 for all i . If furthermore τ e n = ιe then wesay that e · . . . · e n is a combinatorial 1-cycle.In [6] Dicks and Leary show that if ∆ is simply connected then H ∆ is finitelypresented by P H = h Edge(∆) | R H i where R H consists of all words ee for e ∈ Edge(∆) and all words ef g and e − f − g − where e · f · g is a combinatorial 1-cyclein ∆. If we identify the vertices of ∆ with the generators of A , then the embedding H ∆ ֒ → A is given by mapping e ιe ( τ e ) − for each edge e ∈ Edge(∆). In thissection we will prove Theorem 1.1 by demonstrating that the Dehn function δ ofthe presentation P H satisfies δ ( n ) (cid:22) n .Choose a base vertex q and a spanning tree T in the 1-skeleton of ∆. Given n ∈ Z and vertices u and v of ∆ we write p n ( u, v ) for the element e n . . . e nl of Edge(∆) ± ∗ where e · . . . · e l is the unique geodesic combinatorial path in T from u to v . Wewrite p ( u, v ) as shorthand for p ( u, v ). Note that as group elements(1) p n ( u, v ) − = ( e n . . . e nl ) − = e − nl . . . e − n = e ln . . . e n = p n ( v, u )in H ∆ . For each e ∈ Edge(∆) define w e to be the word p ( q, ιe ) ep ( ιe, q ) of Edge(∆) ± ∗ .In [6] it is proved that mapping e w e defines an automorphism θ of H ∆ and that H ∆ ⋊ θ Z is isomorphic to A with e ∈ Edge(∆) corresponding to ιe ( τ e ) − and thegenerator t of Z corresponding to q ∈ A . It is also shown that if e · . . . · e l isa combinatorial 1-cycle then e n . . . e nl is null-homotopic in H ∆ . Define S H to bethe set of words { tet − w e : e ∈ Edge(∆) } in (Edge(∆) ∪ { t } ) ± ∗ so A is finitelypresented by P ′ A = h Edge(∆) , t | R H , S H i .It is proved in [5] that A is CAT(0) (see [4] for the definition of a CAT(0)group) so by [4, Proposition III.Γ.1.6] there exists a finite presentation for A andan area-radius pair ( α, ρ ) for this presentation with α ( n ) ≃ n and ρ ( n ) ≃ n .By Proposition 2.6 it follows that there is an area-radius pair ( α ′ , ρ ′ ) for P ′ A with α ′ ( n ) ≃ n and ρ ′ ( n ) ≃ n .The following lemma details some properties of the automorphism θ of H ∆ .Of these we will only need (vii), but this property is most easily proved via thepreceding sequence of assertions. N ISOPERIMETRIC FUNCTION FOR BESTVINA-BRADY GROUPS 7
Lemma 4.1.
For all e ∈ Edge(∆) and n ∈ Z the following equalities hold in H ∆ : (i) θ ( e ) = p ( q, ιe ) ep ( q, ιe ) − = p ( q, ιe ) e p ( τ e, q ) = p ( q, ιe ) e p ( q, τ e ) − . (ii) θ ( e n ) = p ( q, ιe ) e n p ( ιe, q ) = p ( q, ιe ) e n +1 p ( τ e, q ) = p ( q, ιe ) e n +1 p ( q, τ e ) − . (iii) If e · . . . · e l is a combinatorial path then θ ( e n . . . e nl ) = p ( q, ιe ) e n +11 . . . e n +1 l p ( τ e, q ) . (iv) θ − ( e ) = p − ( q, ιe ) p − ( τ e, q ) = p − ( q, ιe ) ep − ( ιe, q ) = p − ( q, ιe ) ep − ( q, ιe ) − . (v) θ − ( e n ) = p − ( q, ιe ) e n p − ( ιe, q ) = p − ( q, ιe ) e n − p − ( τ e, q ) = p − ( q, ιe ) e n − p − ( q, τ e ) − . (vi) If e · . . . · e l is a combinatorial path then θ − ( e n . . . e nl ) = p − ( q, ιe ) e n − . . . e n − l p − ( τ e, q ) . (vii) θ k ( e ) = p k ( q, ιe ) e k +1 p k ( τ e, q ) .Proof. (i) Follows from equation (1) and the fact that p ( q, ιe ) ep ( τ e, q ) is null-homotopic.(ii) Follows from (i) on telescoping.(iii) Follows from (ii) on telescoping.(iv) Follows from the calculation θ ( p − ( q, ιe ) p − ( τ e, q )) = p ( q, q ) p ( q, ιe ) p ( ιe, q ) p ( q, τ e ) p ( τ e, q ) p ( q, q )= p ( ιe, q ) p ( q, τ e )= e. (v) Follows from (iv) on telescoping.(vi) Follows from (v) on telescoping.(vii) Follows from (iii) and (vi) by induction on | k | . (cid:3) For each n ∈ Z define a homomorphism Φ n : Edge(∆) ± ∗ → Edge(∆) ± ∗ whichcommutes with the inversion involution and is a lift of θ n by mapping e p n ( q, ιe ) e n +1 p n ( τ e, q ).Define the collections of words R H = { Φ n ( r ) : r ∈ R H , n ∈ Z }S H = { Φ n +1 ( e )Φ n ( w e ) − : e ∈ Edge(∆) , n ∈ Z } in Edge(∆) ± ∗ , and consider the presentation P ∞ H = h Edge(∆) | R H , S H i of H ∆ .Define an index k · k on R H ∪ S H by setting k ω k to be the minimum value of | n | such that either ω ≡ Φ n ( r ) for some r ∈ R H or ω ≡ Φ n +1 ( a )Φ n ( w a ) − for some e ∈ Edge(∆).By Theorem 3.1, ( α ′ , ρ ′ ) is an area-penetration pair for the indexed presentation( P ∞ H , k · k ). Thus, to complete the proof of Theorem 1.1 it suffices, by Proposi-tion 2.8, to show that the relational area function RArea of ( P ∞ H , k · k ) over P H satisfies RArea( n ) ≃ n . We devote the remainder of the section to this task.Let Dist be the length metric on the 1-skeleton of T given by setting the lengthof each edge to 1. Define L = max { Dist( u, v ) : u, v ∈ Vert(∆) } . Lemma 4.2.
Area P H (cid:0) Φ n ( ee ) (cid:1) ≤ (2 L + 1) | n | + 1 for all e ∈ Edge(∆) . WILL DISON
Proof.
The calculation (1) shows that p n ( q, v ) − can be converted to p n ( v, q ) at a P H -cost of at most L | n | for all v ∈ Vert(∆). The following is a null- P H -scheme forthe word Φ n ( ee ):Φ n ( ee ) ≡ p n ( q, ιe ) e n +1 p n ( τ e, q ) p n ( q, τ e ) e n +1 p n ( ιe, q ) p n ( q, ιe ) e n +1 e n +1 p n ( ιe, q ) Cost ≤ L | n | p n ( q, ιe ) p n ( ιe, q ) Cost ≤ | n | + 1 ∅ Cost ≤ L | n | Total cost ≤ (2 L + 1) | n | + 1. (cid:3) Lemma 4.3.
Let e · f · g be a combinatorial -cycle in ∆ . Then Area P H ( e n f n g n ) ≤ | n | .Proof. Note that the relators ef g and e − f − g − imply that ef = g − = f e , so[ e, f ] is null-homotopic with P H -Area 2. The following is a null- P H -scheme for theword e n f n g n : e n f n g n e n f n ( f − e − ) n Cost ≤ | n | e n f n f − n e − n Cost ≤ | n | = ∅ Total cost ≤ | n | + | n | ≤ | n | . (cid:3) Lemma 4.4.
Let e · f · g be a combinatorial -cycle in ∆ . Then Area P H (cid:0) Φ n ( ef g ) (cid:1) ≤ | n | + (3 L + 6) | n | + 3 .Proof. The following is a null- P H -scheme for the word Φ n ( ef g ):Φ n ( ef g ) ≡ p n ( q, ιe ) e n +1 p n ( τ e, q ) p n ( q, ιf ) f n +1 p n ( τ f, q ) . . .. . . p n ( q, ιg ) g n +1 p n ( τ g, q ) p n ( q, ιe ) e n +1 f n +1 g n +1 p n ( τ g, q ) Cost ≤ L | n | p n ( q, ιe ) p n ( τ g, q ) Cost ≤ | n + 1 | ∅ Cost ≤ L | n | . Total cost ≤ | n | + (3 L + 6) | n | + 3. (cid:3) Definition 4.5.
Given a combinatorial -cycle C in ∆ , a sequence ( C i ) mi =0 ofcombinatorial -cycles is said to be combinatorial null-homotopy for C if C = C , C m = ∅ and each C i +1 is obtained from C i by one of the following moves: (i) 1-cell expansion : C i = e · . . . · e l C i +1 = e · . . . · e k · e · e · e k +1 · . . . · e l for some k , where e ∈ Edge(∆) ; (ii) 1-cell collapse : Reverse of a -cell expansion; (iii) 2-cell expansion : C i = e · . . . · e l C i +1 = e · . . . · e k · e · f · g · e k +1 · . . . · e l for some k , where e · f · g is a combinatorial -cycle; (iv) 2-cell collapse : Reverse of a -cell expansion. Lemma 4.6. If ( C i ) mi =0 is a combinatorial null-homotopy for the -cycle e · . . . · e l then the word e n . . . e nl has P H -Area ≤ m | n | . N ISOPERIMETRIC FUNCTION FOR BESTVINA-BRADY GROUPS 9
Proof.
Given a combinatorial 1-cycle C = e · . . . · e l , write W n ( C ) for the word e n . . . e nl ∈ Edge(∆) ± ∗ . If the 1-cycle C i is obtained from C i − by a 1-cell expansionor collapse then, by repeated application of a relator ee , the word W n ( C i − ) canbe converted to the word W n ( C i ) at a P H -cost of at most | n | . If the 1-cycle C i isobtained from C i − by a 2-cell expansion or collapse then, by Lemma 4.3, the word W n ( C i − ) can be converted to the word W n ( C i ) at a P H -cost of at most 3 | n | .Define m to be the number of i for which C i is obtained from C i − by a 1-cellexpansion or collapse. Define m to be the number of i for which C i is obtained from C i − by a 2-cell expansion or collapse. Then the P H -Area of e n . . . e nl = W n ( C ) isat most m | n | + 3 m | n | ≤ m + m ) | n | = 3 m | n | . (cid:3) Lemma 4.7.
There exists a constant K such that Area P H (cid:0) p n ( q, ιe ) e n p n ( τ e, q ) (cid:1) ≤ K | n | for all e ∈ Edge(∆) .Proof.
Given e ∈ Edge(∆) write γ ι ( e ) and γ τ ( e ) respectively for the unique com-binatorial geodesic paths in T from q to ιe and from τ e to q . Then γ ι ( e ) · e · γ τ ( e )is a combinatorial 1-cycle for which there exists a combinatorial null-homotopy (cid:0) C i ( e ) (cid:1) m ( e ) i =0 since ∆ is simply-connected. By Lemma 4.6 Area P H (cid:0) p n ( q, ιe ) e n p n ( τ e, q ) (cid:1) ≤ m ( e ) | n | , so we can take K = 3 max { m ( e ) : e ∈ Edge(∆) } . (cid:3) Lemma 4.8.
Let e · f · g be a combinatorial -cycle in ∆ . Then Area P H (cid:0) Φ n ( e − f − g − ) (cid:1) ≤ (3 K + 4) | n | + (6 L + 6) | n | + 5 , where K is the constant from Lemma 4.7.Proof. The following is a null- P H -scheme for the word Φ n ( e − f − g − ):Φ n ( e − f − g − ) ≡ p n ( τ e, q ) − e − n − p n ( q, ιe ) − p n ( τ f, q ) − f − n − . . .. . . p n ( q, ιf ) − p n ( τ g, q ) − g − n − p n ( q, ιg ) − p n ( q, τ e ) e − n − p n ( ιe, q ) p n ( q, τ f ) f − n − p n ( ιf, q ) . . .. . . p n ( q, τ g ) g − n − p n ( ιg, q ) Cost ≤ L | n | free = p n ( q, ιf ) e − n − p n ( τ g, q ) p n ( q, ιg ) f − n − p n ( τ e, q ) . . . p n ( q, ιe ) g − n − p n ( τ f, q ) p n ( q, ιf ) p n ( q, ιf ) − p n ( q, ιf ) e − n − g − n f − n − e − n g − n − f − n p n ( q, ιf ) − Cost ≤ K | n | p n ( q, ιf ) e − n − ( ef ) n f − n − e − n ( ef ) n +1 f − n p n ( q, ιf ) − Cost ≤ | n | + 1 p n ( q, ιf ) e − n − e n f n f − n − e − n e n +1 f n +1 f − n p n ( q, ιf ) − Cost ≤ | n | + 2 | n + 1 | = p n ( q, ιf ) e − f − ef p n ( q, ιf ) − p n ( q, ιf ) gg − p n ( q, ιf ) − Cost ≤ free = ∅ . Total cost ≤ (3 K + 4) | n | + (6 L + 6) | n | + 5. (cid:3) Lemma 4.9.
Area P H (cid:0) Φ n +1 ( e )Φ n ( w e ) − (cid:1) ≤ K | n | + (3 L + 2 L + 2 K ) | n | + L + K for all e ∈ Edge(∆) , where K is the constant from Lemma 4.7. Proof.
Note that if e · . . . · e l is a combinatorial edge-path in ∆ then Φ n ( e . . . e l ) = Q li =1 p n ( q, ιe i ) e n +1 i p n ( τ e i , q ) can be converted to l Y i =1 p n ( q, ιe i ) e n +1 i p n ( q, τ e i ) − = p n ( q, ιe ) e n +11 . . . e n +1 l p n ( q, τ e l ) − at a P H -cost of at most lL | n | . It follows that for all u, v ∈ Vert(∆) the wordΦ n (cid:0) p ( u, v ) (cid:1) can be converted to the word p n ( q, u ) p n +1 ( u, v ) p n ( q, v ) − at a P H -costof at most L | n | .The following is a null- P H -scheme for the word Φ n +1 ( e )Φ n ( w e ) − :Φ n +1 ( e )Φ n ( w e ) − ≡ p n +1 ( q, ιe ) e n +2 p n +1 ( τ e, q ) (cid:2) Φ n (cid:0) p ( q, ιe ) ep ( ιe, q ) (cid:1)(cid:3) − p n +1 ( q, ιe ) e n +2 p n +1 ( τ e, q ) (cid:2) p n +1 ( q, ιe ) p n ( q, ιe ) − . . .. . . p n ( q, ιe ) e n +1 p n ( τ e, q ) p n ( q, ιe ) p n +1 ( ιe, q ) (cid:3) − Cost ≤ L | n | free = p n +1 ( q, ιe ) e n +2 p n +1 ( τ e, q ) p n +1 ( ιe, q ) − . . .. . . p n ( q, ιe ) − p n ( τ e, q ) − e − n − p n +1 ( q, ιe ) − p n +1 ( q, ιe ) e n +2 p n +1 ( τ e, q ) p n +1 ( q, ιe ) . . .. . . p n ( q, ιe ) − p n ( τ e, q ) − e − n − p n +1 ( q, ιe ) − Cost ≤ L | n + 1 | p n +1 ( q, ιe ) e n +2 e − n − e n e − n − p n +1 ( q, ιe ) − Cost ≤ K | n + 1 | + K | n | = ∅ . Total cost ≤ K | n | + (2 L + L + 2 K ) | n | + L + K . (cid:3) Combining Lemmas 4.2, 4.4, 4.8 and 4.9 we see thatRArea( n ) ≤ (3 K + 4) n + (6 L + 2 K + 6) n + L + K + 5 . This completes the proof of Theorem 1.1.5.
A Bestvina-Brady Group with Quartic Dehn Function
In Section 2.5.2 of [2] Brady gives a sequence ( K m ) m ∈ N of finite, flag simplicialcomplexes and suggests that the Bestvina-Brady group associated to K m will haveDehn function δ ( n ) ≃ n m +2 . Theorem 1.1 shows that this cannot be the case.However, the construction does work in the cases m = 1 and m = 2 and the example K thus shows that the bound obtained in Theorem 1.1 cannot be improved ingeneral. We briefly recount that example here.The complex K is the triangulation of the disc shown in Figure 1. Let H K and A K be the Bestvina-Brady and right-angled Artin groups respectively associated to K . Choose an orientation of the edges of K so as the four edges labelled in the fig-ure are orientated as indicated. Let Edge( K ) ≤ Edge( K ) be the index 2 subgroupconsisting of the positively orientated edges. Let P H be the Dicks-Leary presen-tation for H K with generating set Edge( K ), as described in Section 4. Derivefrom P H the presentation Q H for H with generating set Edge( K ) by using Tietzetransformations to remove all the superfluous generators Edge( K ) r Edge( K ) andall the superfluous relators { ee : e ∈ Edge( K ) } . N ISOPERIMETRIC FUNCTION FOR BESTVINA-BRADY GROUPS 11
PSfrag replacements a bc d Figure 1.
For each k ∈ N define w k ∈ Edge ( K ) ± ∗ to be the null-homotopic word( da ) k ( b − c − ) k ( ad ) k ( c − b − ) k , where a , b , c and d are the orientated edges labelledin the figure. In [2] Brady describes how to construct a van Kampen diagramΩ k over the presentation Q H with boundary label w k and Area(Ω k ) ≃ k . It isshown that the presentation 2-complex Σ associated to Q H is aspherical and thatthe diagram Ω k embeds in the universal cover of Σ. It follows that Area( w k ) =Area(Ω k ) and hence that the Dehn function of Q H is ≃ n . Acknowledgements.
I would like to thank my thesis advisor, Martin Bridson, forhis many helpful comments made during the preparation of this article.
References [1] M. Bestvina and N. Brady. Morse theory and finiteness properties of groups.
Invent. Math. ,129(3):445–470, 1997.[2] N. Brady. Dehn functions and non-positive curvature. In
The Geometry of the Word Problemfor Finitely Generated Groups , Advanced Courses in Mathematics. CRM Barcelona, pages1–79. Birkh¨auser Verlag, Basel, 2007.[3] M. R. Bridson. The geometry of the word problem. In
Invitations to geometry and topology ,volume 7 of
Oxf. Grad. Texts Math. , pages 29–91. Oxford Univ. Press, Oxford, 2002.[4] M. R. Bridson and A. Haefliger.
Metric spaces of non-positive curvature , volume 319 of
Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of MathematicalSciences] . Springer-Verlag, Berlin, 1999.[5] R. Charney and M. W. Davis. Finite K ( π, Prospects in topology(Princeton, NJ, 1994) , volume 138 of
Ann. of Math. Stud. , pages 110–124. Princeton Univ.Press, Princeton, NJ, 1995.[6] W. Dicks and I. J. Leary. Presentations for subgroups of Artin groups.
Proc. Amer. Math.Soc. , 127(2):343–348, 1999.[7] R. A. Fenn.
Techniques of geometric topology , volume 57 of
London Mathematical SocietyLecture Note Series . Cambridge University Press, Cambridge, 1983.[8] T. Riley. Filling functions. In