Superexponential Dehn functions inside CAT(0) groups
aa r X i v : . [ m a t h . G R ] F e b SUPEREXPONENTIAL DEHN FUNCTIONS INSIDE
CAT(0)
GROUPS
NOEL BRADY AND HUNG CONG TRAN
Abstract.
We construct 4–dimensional CAT(0) groups containing finitely presented sub-groups whose Dehn functions are exp ( n ) ( x m ) for integers n, m ≥ ( n ) ( x α ) for inte-gers n ≥ α dense in [1 , ∞ ). This significantly expands the known geometric behavior ofsubgroups of CAT(0) groups. Introduction
CAT(0) geometry plays a fundamental role in geometric group theory and low dimensionaltopology. For example, in the proof of the Virtual Fibering Theorem [Ago13], CAT(0) cubecomplexes are used to encode immersed incompressible surfaces in a 3–manifold and this CAT(0)cubical structure is used to understand finite covers of the manifold. In another direction, theCAT(0) framework and Morse theory perspective introduced in [BB97] to produce groups oftype FP which are not finitely presented has resulted in a range of applications including[LN03], [KLS20] and [KV18].CAT(0) groups are fundamental groups of compact, path metric spaces whose universalcovers satisfy the CAT(0) triangle inequality. See [BH99] for definitions and background onCAT(0) spaces and groups. CAT(0) groups have strongly controlled topology and algebra,including having finite Eilenberg-Maclane spaces, solvable word and conjugacy problems, tightlycontrolled solvable subgroups, and quadratic upper bounds for their Dehn functions. They havebeen intensively studied over the years.Much less is known about the geometry and structure of finitely presented subgroups ofCAT(0) groups. In this paper we focus on Dehn functions. In [BBMS97] there are examples ofautomatic groups containing finitely presented subgroups with exponential or polynomial Dehnfunctions of arbitrary degree. The ambient groups in [BBMS97] are CAT(0). In [BF17] thereare examples of CAT(0) groups containing finitely presented subgroups whose Dehn functionsare x α for a dense set of exponents α ∈ [2 , ∞ ). Throughout this time there were no examples ofCAT(0) groups containing finitely presented subgroups with superexponential Dehn functions.Question 10.7 of [Bri07] asks if there is a universal upper bound on the Dehn functions of finitelypresented subgroups of CAT(0) groups.The goal of the present paper is to shed light on the question above. Such a universal upperbound would have to be greater than any finite iteration of exponential functions. Main Theorem (paraphrased) . Let n be a positive integer and define exp ( n ) ( x ) inductively by exp (1) ( x ) = exp( x ) and exp ( k +1) ( x ) = exp(exp ( k ) ( x )) for k ≥ . Then(1) There are –dimensional CAT(0) groups containing finitely presented subgroups whoseDehn functions are equivalent to exp ( n ) ( x m ) for integers m ≥ .(2) There are 6–dimensional CAT(0) groups containing finitely presented subgroups whoseDehn functions are equivalent to exp ( n ) ( x α ) for α dense in [1 , ∞ ) . The CAT(0) groups in the Main Theorem are obtained from two constructions of non-positively curved spaces; the ultra-convex chaining procedure and the factor-diagonal chaining procedure. These are described in detail in Section 5. The ultra-convex chaining procedure builds a non-positively curved (that is, locally CAT(0)) 2–complex from a sequence of non-positively curved 2–complexes by identifying a non-convex rose (bouquet of circles) in one 2–complex in the sequence with an ultra-convex rose in the next 2–complex. Theorem 5.2 providesa key ingredient for this procedure; namely, a CAT(0) F ℓ ⋊ F k group with F k ultra-convex. Thefactor-diagonal chaining process builds non-positively curved spaces from a sequence of metricproduct spaces by identifying a factor space in one product with a diagonal subspace in thenext.The subgroups with large Dehn functions are constructed using a standard technique; namely,by doubling groups over highly distorted subgroups. The subgroup distortion functions containiterated exponentials. These are the result of chaining together a sequence of the hyperbolicfree-by-free groups constructed in Theorem 5.2.The tricky part is to embed these groups into CAT(0) groups. This is achieved by thefactor-diagonal chaining construction and a group embedding result. The group embeddingpart of the main theorem is provided by Proposition 3.6 which uses a result of Bass to provethat a particular morphism of graphs of groups induces an inclusion of fundamental groups.The structure of the group embedding proposition requires that the factor-diagonal chainingparallels (and has as many terms as) the ultra-convex chaining.The various power functions inside the iterated exponential are obtained by choosing suitableCAT(0) ingredients to start the construction. For conclusion (1) of the Main Theorem we useCAT(0) groups of the form ( F k ⋊ Z ) × Z and for conclusion (2) we use the ambient CAT(0)groups which contain the snowflake groups of [BF17].This paper is organized as follows. Section 2 provides background on Dehn functions, dis-tortion functions of subgroups, and CAT(0) spaces. Section 3 establishes the main groupembedding result. Section 4 concerns the computation of the Dehn functions of the finitely pre-sented subgroups. In Section 5 the ambient CAT(0) groups are constructed. Section 6 containsa statement and proof of the Main Theorem and a list of open questions which are related tothis research. Acknowledgements.
We thank Eduardo Mart´ınez-Pedroza for helpful conversations aboutthe questions in Section 6.The first author was supported by Simons Foundation collaboration grant
Background
The main theorem of this paper concerns Dehn functions of finitely presented subgroupsof CAT(0) groups. The large Dehn functions arise from taking doubles of groups over highlydistorted subgroups. The first subsection provides background on equivalence of functions,Dehn functions of groups, and distortion functions of subgroups. The second subsection givesbackground on non-positively curved, piecewise euclidean 2–complexes and on constructionsrelated to products of non-positively curved spaces.2.1.
Dehn functions and subgroup distortion.
The Dehn functions and distortion func-tions defined below are all computed up to an equivalence which we describe.
Convention 2.1.
Let M be the collection of all functions from [1 , ∞ ) to [1 , ∞ ). Let f and g be arbitrary elements of M . We say that f is dominated by g , denoted f (cid:22) g , if there is aconstant C ≥ f ( x ) ≤ Cg ( Cx ) + Cx for all x ≥
1. We say that f is equivalent to g ,denoted f ∼ g , if f (cid:22) g and g (cid:22) f . This defines an equivalence relation on M .We now recall the definitions of Dehn function and subgroup distortion. UPEREXPONENTIAL DEHN FUNCTIONS INSIDE CAT(0) GROUPS 3
Definition 2.2 (Dehn function and isoperimetric inequality) . Let G = h S | R i be a finitelypresented group and w a word in the generating set S representing the trivial element of G .We define the area of w to beArea( w ) = min { N ∈ N | ∃ equality w = N Y i =1 u i r i u − i freely, where r i ∈ R ± } The
Dehn function δ ( x ) of the finite presentation G = h S | R i is given by δ ( x ) = max { Area( w ) | w ∈ ker( F ( S ) → G ) , | w | ≤ x } where | w | denotes the length of the word w .Using Convention 2.1 it is not difficult to show that two finite presentations of the samegroup define equivalent Dehn functions, we therefore speak of “the” Dehn function of G , whichis well-defined up to equivalence.If the Dehn function of the finitely presented group G is dominated by a polynomial functionwe say that G satisfies a polynomial isoperimetric inequality. Definition 2.3 (subgroup distortion) . Let H ≤ G be a pair of finitely generated groups, andlet d H and d G be the word metrics associated to a choice of finite generating set for each. The distortion of H in G is the functionDist GH ( x ) = max { d H (1 , h ) | h ∈ H with d G (1 , h ) ≤ x } . Up to the equivalence relation in Convention 2.1, this function is independent of the choice ofword metrics d G and d H .The following theorem relates subgroup distortion to the Dehn functions of doubles. Theorem 2.4 ([BH99], III.Γ.6.20) . Let G be a finitely presented group with Dehn function δ G and let H ≤ G be a finitely presented subgroup. Let Dist GH be the distortion of H in G withrespect to some choice of word metrics. The Dehn function δ Γ of the double group Γ = G ∗ H G satisfies Dist GH ( x ) (cid:22) δ Γ ( x ) (cid:22) x (cid:0) δ G ◦ Dist GH (cid:1) ( x ) . geometry. The reader is referred to [BH99] for definitions of and background onCAT(0) spaces and groups. The phrase non-positively curved means the same thing as locally
CAT(0). The universal covering space of a non-positively curved space is a CAT(0) space. ACAT(0) group is one which acts properly, cocompactly by isometries on a CAT(0) metric space.In particular, the fundamental group of a compact, non-positively curved space is a CAT(0)group.This paper makes extensive use of non-positively curved piecewise euclidean 2–complexesand metric products of these complexes. We include pertinent definitions and results.A convex polygon in E is a compact subspace of E which is a finite intersection of halfspaces. Definition 2.5 (PE 2–complex) . A piecewise euclidean 2–complex X is a 2–dimensional cellcomplex with the following additional structure.(1) The 0–skeleton X (0) is a discrete set.(2) Each 1–cell e of X consists of a segment C e ⊆ R ⊆ R = E and an attaching map ϕ e : ∂C e → X (0) . An admissible characteristic map of a 1–cell e is the standardcharacteristic map χ e : C e → X (1) precomposed with an isometry of E .(3) Each 2–cell f of X consists of a convex polygon C f ⊆ E and an attaching map ϕ f : ∂C f → X (1) satisfying the following glueing property. The restriction of ϕ f to eachedge of C f is an admissible characteristic map of a 1-cell of X . UPEREXPONENTIAL DEHN FUNCTIONS INSIDE CAT(0) GROUPS 4
Definition 2.6 (link) . Assume that the length of the shortest 1–cell of X is 1. Given v ∈ X (0) ,one can define the link of v in X to be Lk ( v, X ) = ∂B ( v, X ) . This is the boundary of a metric ball of radius 1/4 centered on v . A piecewise spherical metric on Lk ( v, X ) is defined by taking the induced path metric on Lk ( v, X ) defined above and rescalingby 4. In this rescaled metric, the length of an edge of Lk ( v, X ) is equal to the angle at thevertex w in the polygon C f where χ f ( w ) = v and χ f ( C f ∩ B ( w, C f )) is the given edge.The link condition of Theorem 2.7 below will be used multiple times in Section 5 of thispaper. See [BH99], II.5.5 and II.5.6 for a proof. Theorem 2.7 (link condition for non-positively curved 2–complexes) . A finite piecewise eu-clidean 2–dimensional complex K is a non-positively curved space if and only if for each vertex v ∈ K every injective loop in the piecewise spherical metric on Lk ( v, K ) has length at least π . The factor-diagonal chaining construction in Section 5 of this paper uses properties of metricproducts of non-positively curved spaces. Here are relevant definitions and background results.Let (
X, d ) and ( Y, d ) be two metric spaces. The product metric on X × Y is defined by d (( x , y ) , ( x , y )) = p ( d ( x , x )) + ( d ( y , y )) . Let A and X be non-positively curved metric spaces. A map f : A → X is said to be a locallyisometric embedding if every point in A has a neighborhood N such that f | N : N → X is anisometric embedding. A subspace B ⊂ X of a non-positively curved space X is said to be locallyconvex if every point of B has a neighborhood N such that the geodesics connecting every pairof points of B ∩ N are contained in B . Note that if f : A → X is a locally isometric embedding,then f ( A ) is locally convex in X . Also, if B ⊂ X is locally convex, then the inclusion B ֒ → X is a locally isometric embedding. Lemma 2.8 (product spaces and diagonals) . Let X and Y be non-positively curved spaces.Then(1) The metric space X × Y (equipped with the product metric) is non-positively curved.(2) For any choice of y ∈ Y the map f : X → X × Y, x ( x, y ) is a locally isometricembedding. Moreover, the map f induces a monomorphism f ∗ : π ( X ) → π ( X × Y ) with image π ( X ) × .(3) Assume that Y ⊂ X and the inclusion of Y into X is a locally isometric embedding.Then the map g : Y → X × Y, y ( y, y ) induces a locally isometric embedding ofthe scaled metric space √ Y into X × Y and a group monomorphism g ∗ : π ( Y ) → π ( X × Y ) , h ( h, h ) with image ∆ π ( Y ) ≤ π ( X ) × π ( Y ) .Proof. Statement (1) can be seen from Example (3) on page 167 in [BH99]. The proof ofstatement (2) is straightforward from the definition of product metric. The algebra portion ofstatement (3) is standard. We show that g induces a locally isometric embedding of the scaledmetric space √ Y into X × Y as follows: d (cid:0) g ( y ) , g ( y ) (cid:1) = d (cid:0) ( y , y ) , ( y , y ) (cid:1) = (cid:16)(cid:0) d ( y , y ) (cid:1) + (cid:0) d ( y , y ) (cid:1) (cid:17) / = √ d ( y , y ) . (cid:3) Proposition 2.9 ([BH99], II.11.6 (2)) . Let X and X be two non-positively curved metricspaces and let A ⊂ X and A ⊂ X be closed subspaces that are locally convex and complete.If j : A → A is a bijective local isometry, then the quotient of the disjoint union X = X ⊔ X by the equivalence relation generated by [ a ∼ j ( a ) , ∀ a ∈ A ] is also a non-positively curvedmetric space. UPEREXPONENTIAL DEHN FUNCTIONS INSIDE CAT(0) GROUPS 5 The group embedding result
The main result in this section is Proposition 3.6, which states that the fundamental groupof a particular graph of groups based on a segment of length (2 n + 1) (so it is an iteratedamalgamation of (2 n + 2) vertex groups) embeds into the fundamental group of a second suchgraph of groups based on an isomorphic graph. This proposition is used in Section 6 to embedfinitely presented groups with superexponential Dehn functions into CAT(0) groups.Proposition 3.6 is proved using a criterion of Bass (Proposition 3.2) for when a morphismbetween graphs of groups gives rise to an inclusion between their fundamental groups. In orderto apply this criterion we need the preliminary results in Lemma 3.4 and Lemma 3.5.The underlying geometry of the construction in Section 5 requires that the vertex groupsin the lower graph of groups in Figure 1 are expressed as direct products of the fundamentalgroups of graphs of groups related to certain subgraphs of the first graph of groups. Therefore,we start this section with a lemma which will be used to establish the inclusions between thecorresponding vertex groups in the graphs of groups in Proposition 3.6. Lemma 3.1 (embeddings of vertex groups) . Let ( A ⋊ B ) be a semidirect product and θ : B → C be a monomorphism which identifies B with the subgroup θ ( B ) of C . Define H to be the amalgam H = ( A ⋊ B ) ∗ ( B ≡ θ ( B )) C. Then the map ϕ : ( A ⋊ B ) → H × C : ab ( a, b, θ ( b )) defines an embedding of ( A ⋊ B ) into H × C .Proof. Given g ∈ ( A ⋊ B ), by the normal form in semidirect products there exists unique a ∈ A and b ∈ B such that g = ab . This ensures that ϕ is well-defined as a set map on all of ( A ⋊ B )by ϕ ( g ) = ϕ ( ab ) = ( a, b, θ ( b )) . Next, we show that ϕ is a homomorphism. Given g , g ∈ ( A ⋊ B ), there exist unique elements a , a ∈ A and b , b ∈ B such that g i = a i b i for i = 1 ,
2. Therefore, ϕ ( g ) ϕ ( g ) = ϕ ( a b ) ϕ ( a b )= ( a b , θ ( b ))( a b , θ ( b ))= ( a b a b , θ ( b ) θ ( b ))= (( a b a b − ) b b , θ ( b b ))= ϕ (( a b a b − )( b b ))= ϕ ( g g )and so ϕ is a group homomorphism. The last equality holds because ( a b a b − )( b b ) is theunique normal form representative of the element g g in ( A ⋊ B ).Let π : H × C → H : ( h, c ) h be the homomorphism which projects onto the first factor.Given ab ∈ ( A ⋊ B ) we have ( π ◦ ϕ )( ab ) = π ( ϕ ( ab )) = π ( ab, θ ( b )) = ab ∈ H. Thus, π ◦ ϕ is the inclusion map of ( A ⋊ B ) into H = ( A ⋊ B ) ∗ B C . Therefore, ϕ is injective.Note that ϕ ( A ) = { ( a, | a ∈ A } is the subgroup A × H × C . Also, ϕ ( B ) = { ( b, θ ( b )) | b ∈ B } = { ( θ ( b ) , θ ( b )) | b ∈ B } where the last equality is true because the relation θ ( b ) = b holds in the amalgam H which isthe first factor of the direct product. Therefore, ϕ ( B ) is the diagonal subgroup∆ θ ( B ) ≤ C × C ≤ H × C. UPEREXPONENTIAL DEHN FUNCTIONS INSIDE CAT(0) GROUPS 6
In summary, ϕ is a monomorphism which takes ( A ⋊ B ) isomorphically to the subgroup gener-ated by A × θ ( B ) , h A × , ∆ θ ( B ) i ⊆ H × C. (cid:3) The following result tells when a morphism of graphs of groups induces an monomorphismof their fundamental groups. It is Lemma 5.1 from [BF17] and is a reformulation of a basicresult of [Bas93].
Proposition 3.2 (injectivity for graphs of groups) . Suppose A and B are graphs of groupssuch that the underlying graph Γ A of A is a subgraph of the underlying graph of B . Let A and B be their respective fundamental groups. Suppose that there are injective homomorphisms ψ e : A e → B e and ψ v : A v → B v between edge and vertex groups, for all edges e and vertices v in Γ A , which are compatible with the edge-inclusion maps. That is, whenever e has initialvertex v , the diagram A e A v B e B vi e ψ e ψ v j e commutes.If j e ( ψ e ( A e )) = ψ v ( A v ) ∩ j e ( B e ) whenever e has initial vertex v , then the induced homomor-phism ψ : A → B is injective. The previous result motivates the following definition.
Definition 3.3 (Bass conditions) . Let
A, B, C, D be groups and
A BC D α i βj be a diagram of monomorphisms. This diagram is said to satisfy the Bass conditions if(1) the diagram is commutative, and(2) j ( α ( A )) = j ( C ) ∩ β ( B ).We call the second condition the Bass intersection condition .Lemma 3.4 and Lemma 3.5 below describe two situations in which the Bass conditions hold.These will be used repeatedly in the proof of the main embedding result in Proposition 3.6.
Lemma 3.4 (Bass conditions - factor embedding) . Let ( A ⋊ B ) be a semidirect product and θ : B → C be a monomorphism which identifies B with the subgroup θ ( B ) of C . Define H tobe the amalgam H = ( A ⋊ B ) ∗ ( B ≡ θ ( B )) C and let ϕ : ( A ⋊ B ) → H × C be the monomorphism of Lemma 3.1.Then the diagram of monomorphisms and inclusions ( A ⋊ B ) AH × C H ϕ H × satisfies the Bass conditions.Proof. Note that the left vertical arrow in the diagram is the inclusion map A → ( A ⋊ B ) → ( A ⋊ B ) ∗ ( B ∼ = θ ( B )) C = H . Therefore, a ∈ A is mapped to a ∈ H via the left arrow andthis is mapped to ( a, ∈ H × C via the lower arrow. On the other hand a is mapped to a = a. ∈ ( A ⋊ B ) by the top arrow and ϕ maps this to ( a. , θ (1)) = ( a, ∈ H × C , and sothe diagram commutes.The image of A in H × C is A × A × ⊆ ( H × ∩ ϕ ( A ⋊ B ) . To see the reverse inclusion, let ( g , g ) ∈ ( H × ∩ ϕ ( A ⋊ B ). In particular, ( g , g ) ∈ H × g = 1. Now ( g , ∈ ϕ ( A ⋊ B ) implies that ( g ,
1) = ( ab, θ ( b )) for some ab ∈ ( A ⋊ B ).But then θ ( b ) = 1 and so b = 1 since θ is a monomorphism. Therefore, ( g , g ) = ( a, ∈ A × (cid:3) Lemma 3.5 (Bass conditions - diagonal embedding) . Let ( A ⋊ B ) be a semidirect product and θ : B → C be a monomorphism which identifies B with the subgroup θ ( B ) of C . Define H tobe the amalgam H = ( A ⋊ B ) ∗ ( B ≡ θ ( B )) C and let ϕ : ( A ⋊ B ) → H × C be the monomorphism of Lemma 3.1. Let β denote the inverseof the isomorphism θ : B → θ ( B ) . The composition θ ( B ) β → B ֒ → A ⋊ B is denoted by β too.Then the diagram of monomorphisms and inclusions θ ( B ) ( A ⋊ B ) C H × C β ϕ ∆ C satisfies the Bass conditions.Proof. An element c ∈ θ ( B ) is mapped to ( c, c ) via the composition of the right and lowermaps. It is sent to ϕ ( β ( c )) = ϕ (1 .β ( c )) = (1 .β ( c ) , θ ( β ( c ))) = ( β ( c ) , c ) in H × C via thecomposition of the top and left maps. However, the relation θ ( b ) = b holds in the group H andso ( β ( c ) , c ) = ( θ ( β ( c )) , c ) = ( c, c ) and the diagram commutes.The image of θ ( B ) in H × C is the subgroup ∆ θ ( B ) and from the previous paragraph we have∆ θ ( B ) ⊆ ϕ ( A ⋊ B ) ∩ ∆ C . To see the reverse inclusion, let ( g , g ) ∈ ϕ ( A ⋊ B ) ∩ ∆ C . In particular ( g , g ) ∈ ∆ C and so g = g . Now ( g , g ) ∈ ϕ ( A ⋊ B ) implies that ( g , g ) = ( ab, θ ( b )) for some ab ∈ ( A ⋊ B ). Notingthat b = θ ( b ) in the first factor, this gives ( g , g ) = ( aθ ( b ) , θ ( b )) which implies aθ ( b ) = θ ( b )and so a = 1. Therefore, ( g , g ) = (1 .θ ( b ) , θ ( b )) = ( θ ( b ) , θ ( b )) is in the image ∆ θ ( B ) of θ ( B ) in H × C and so the Bass intersection condition is verified. (cid:3) Here is the main group embedding result of the paper.
Proposition 3.6 (graph of groups embedding) . Suppose we are given
UPEREXPONENTIAL DEHN FUNCTIONS INSIDE CAT(0) GROUPS 8 (1) a diagram of groups and inclusions A SH T ε satisfying the Bass conditions,(2) a sequence of semidirect products ( A i ⋊ B i ) for ≤ i ≤ n , and(3) isomorphisms θ i : B i → A i − for ≤ i ≤ n .Define sequences of groups H i and G i for ≤ i ≤ n and L i for ≤ i ≤ n inductively as follows:(1) H is the group in the diagram above which contains A as a subgroup and H i = ( A i ⋊ B i ) ∗ ( B i ≡ θi A i − ) H i − for ≤ i ≤ n ,(2) G = ( H × H ) ∗ (∆ H ≡ H ) T and G i = ( H i × H i − ) ∗ (∆ Hi − ≡ H i − × G i − for ≤ i ≤ n , and(3) L = S contains the subgroup A and L i = ( A i ⋊ B i ) ∗ ( B i ≡ θi A i − ) L i − for ≤ i ≤ n .Then the double of L n over A n embeds into the double of G n over H n where the group H n includes into G n as H n × .Proof. Note that each double can be expressed as the fundamental group of a graph of groupswhose underlying graph is a segment of length (2 n + 1). The top line segment of Figure 1 isthe underlying graph for the graph of groups description of L n ∗ A n L n and the bottom linesegment underlies the graph of groups description of G n ∗ H n G n . The vertex and edge groupsare indicated in Figure 1. S ( A ⋊ B ) ( A n ⋊ B n ) ( A n ⋊ B n ) ( A ⋊ B ) SL n L n G n G n A A A n − A n A n − A A T ( H × H ) ( H n × H n − ) ( H n × H n − ) ( H × H ) TH H H n − H n H n − H H Figure 1.
The morphism between the graphs of groups descriptions of L n ∗ A n L n and G n ∗ H n G n .There is an isomorphism of the underlying graphs in Figure 1 and there are inclusion maps A i → H i = ( A i ⋊ B i ) ∗ ( B i ≡ θi A i − ) H i − between the corresponding edge groups for 1 ≤ i ≤ n .The inclusions ε : A → H are given by hypothesis. Lemma 3.1 establishes embeddings ϕ i : ( A i ⋊ B i ) → H i × H i − between the corresponding vertex groups for 1 ≤ i ≤ n . Theinclusions S → T are given by hypothesis.Next, we verify that the Bass conditions hold for each square of edge group-vertex groupinclusions. There are 2(2 n + 1) such squares, two per segment of the diagram in Figure 1. To UPEREXPONENTIAL DEHN FUNCTIONS INSIDE CAT(0) GROUPS 9 this end, we label the various maps that are used. The first set of maps are the vertex to edgeinclusion maps in the top graph of groups in Figure 1. • Let γ i : A i → ( A i ⋊ B i ) : a a denote the inclusion of A i into the semidirect product. • Let β i : A i → B i +1 : a β i ( a ) denote the inverse of the isomorphism θ i +1 : B i +1 → A i given in the hypothesis. Note that β i composed with inclusion embeds A i into( A i +1 ⋊ B i +1 ) as the B i +1 subgroup; we denote this composition by β i too. • There is an inclusion A → S at each end which is left unlabelled.The next set of maps are the vertex to edge inclusion maps in the bottom graph of groups inFigure 1. • Let δ i : H i → H i +1 × H i : h ( h, h ) denote the diagonal embedding of H i . • Let α i : H i → H i × H i − : h ( h,
1) denote the inclusion map to the first factor. • There is an inclusion H → T at each end which is left unlabelled.The final set of maps are the vertical edge to edge and vertex to vertex monomorphisms betweenthe two graphs of groups. • Let ε i : A i → H i : a a denote the inclusion of A i into ( A i ⋊ B i ) ⊆ H i for 1 ≤ i ≤ n .The inclusion ε : A → H is given by hypothesis. • Lemma 3.1 established that the maps ϕ i : ( A i ⋊ B i ) → H i × H i − : ab ( ab, θ i ( b )) areembeddings for 1 ≤ i ≤ n . • The inclusion S → T at each end is left unlabelled. Segment labelling.
The diagram in Figure 1 contains (2 n + 1) segments, corresponding to the(2 n + 1) edges of each of the underlying graphs. We label these segments in order from left toright by the integers 0 , , . . . , n . With this labelling, the reflection symmetry about the middleedge sends the k th segment to the (2 n − k )th segment (sending the middle or n th segment toitself). Bass conditions for the middle segment.
Consider the following commutative diagram whichrepresents the middle (or n th) segment of the diagram in Figure 1.( A n ⋊ B n ) A n ( A n ⋊ B n )( H n × H n − ) H n ( H n × H n − ) ϕ n γ n ε n γ n ϕ n α n α n By Lemma 3.4 both the left and right half of this diagram satisfy the two Bass conditions(commuting diagram and intersection condition). To see this, set A = A n , ( A ⋊ B ) = ( A n ⋊ B n ), H = H n , C = H n − , and ϕ = ϕ n . Bass conditions for the k th segment for ≤ k ≤ n − . Consider the following commutativediagram which represents the k th segment (and, by reflection symmetry, the (2 n − k )th segment)of the diagram in Figure 1.( A k ⋊ B k ) A k ( A k +1 ⋊ B k +1 )( H k × H k − ) H k ( H k +1 × H k ) ϕ k γ k ε k β k ϕ k +1 α k δ k Lemma 3.4 implies that the left square of the diagram satisfies the Bass conditions. To seethis, take A = A k , ( A ⋊ B ) = ( A k ⋊ B k ), H = H k , C = H k − , and ϕ = ϕ k . UPEREXPONENTIAL DEHN FUNCTIONS INSIDE CAT(0) GROUPS 10
Lemma 3.5 implies that the right square of the diagram satisfies the Bass conditions. Here weneed to identify B with B k +1 and then θ ( B ) with A k , and take ( A ⋊ B ) = ( A k +1 ⋊ B k +1 ), H = H k +1 , C = H k , and ϕ = ϕ k +1 . With this identification the inclusion θ ( B ) → C becomes theinclusion ε k : A k → H k and the monomorphism β : θ ( B ) → ( A ⋊ B ) becomes the monomorphism β k : A k → ( A k +1 ⋊ B k +1 ), and Lemma 3.5 applies. Bass conditions for the terminal segments.
By symmetry, we only need to consider the 0thsegment.
S A ( A ⋊ B ) T H ( H × H ) ε β ϕ δ The proof that the right square satisfies the Bass conditions follows from Lemma 3.5 exactlyas the proof for the right square in the k th segment above. The left square satisfies the Bassconditions by hypothesis. Conclusion.
Finally, Proposition 3.2 implies that the fundamental group of the top graph ofgroups injects into the fundamental group of the bottom graph of groups; that is, the double L n ∗ A n L n embeds into the double G n ∗ H n G n . (cid:3) Remark 3.7 (applications) . There are two main applications of Proposition 3.6. For the firstapplication, A = F is a subgroup of a snowflake group S from [BF17]. The snowflake groupis a subgroup of a 6–dimensional CAT(0) group T and H = ( F ⋊ Z ) is a convex subgroup of T . The fact that the diagram of group inclusions involving A , H , S , and T satisfies the Bassconditions is established in Lemma 5.3 of [BF17].For the second application, H = ( F k ⋊ Z ) is a CAT(0) free-by-cyclic group and A = F k .The groups S and T are defined to be S = ( F k ⋊ Z ) and T = ( F k ⋊ Z ) × Z . Note that T is a 3–dimensional CAT(0) group. The embedding ϕ : S → T has image the subgroup of T generated by ( F k × ∪ ∆ Z and the embedding H → T has image the subgroup H × A , H , S , and T satisfies the Bassconditions follows directly from Lemma 3.4. Remark 3.8 (normality) . The proofs of Lemma 3.4 and Proposition 3.6 rely on the fact thateach A i is normal in ( A i ⋊ B i ). This appears in the use of normal forms for semidirect productsin the proofs of the embedding results. It is interesting to see what goes wrong when one triesto emulate these embedding results using, for example, ascending HNN extensions instead ofsemidirect products.Consider one of the simplest ascending HNN extensions, the Baumslag-Solitar group BS (1 ,
2) = h a, t | tat − = a i . There is an embedded copy of BS (1 ,
2) in the direct product BS (1 , × h t i which is analogousto the ϕ ( A ⋊ B ) subgroup of Lemma 3.1; namely, the group generated by h ( a, , ( t, t ) i . As in Lemma 3.1, this is the group generated by the subgroup h a i× h ( t, t ) i .However, this embedding does not give rise to an embedding of the double of BS (1 ,
2) over h a i into the double of BS (1 , × Z over BS (1 , ×
1. This is because the group BS (1 , × BS (1 , × Z and so the intersection( BS (1 , × ∩ h ( a, , ( t, t ) i UPEREXPONENTIAL DEHN FUNCTIONS INSIDE CAT(0) GROUPS 11 is the normal closure of the cyclic subgroup h ( a, i in h ( a, , ( t, t ) i = BS (1 , Z [1 /
2] and is not finitely generated. For this example, the analogue of Lemma 3.4 doesnot hold and the proof of Proposition 3.6 breaks down.In this case, the double is isomorphic to BS (1 , ∗ Z [1 / BS (1 ,
2) which is not finitely pre-sented. It is a quotient of the double of BS (1 ,
2) over h a i .One may wish to explore this example by using new letters (say u and v ) for the copies of Z . With this notation, the double of BS (1 , × Z over BS (1 , × h a, t, u, v | tat − = a , [ u, a ] = [ u, t ] = [ v, a ] = [ v, t ] = 1 i . The copies of BS (1 ,
2) in each BS (1 , × Z “side” are h a, tu i and h a, tv i . The group h a, tu, tv i is not isomorphic to the double of BS (1 ,
2) over h a i . Instead of an embedding, one obtains aquotient map from BS (1 , ∗ h a i BS (1 ,
2) = h a, s , s | s as − = a = s as − i to h a, tu, tv i = BS (1 , ∗ Z [1 / BS (1 , ≤ ( BS (1 , × h u i ) ∗ BS (1 , × ( BS (1 , × h v i )sending a a , s tu , s tv . For example, [( tv )( tu ) − , a ] = 1 in this group, but thecorresponding word [ s s − , a ] in the double h a, s , s | s as − = a , s as − = a i is not trivial. Remark 3.9 (no short cuts) . In Proposition 3.6, one might be tempted to embed the doubleof H n over A n into a shorter amalgam than is given.One reason for this is the fact that H n embeds into H n × H n − with image the subgroup h A n × , ∆ H n − i . We see this as follows. Given the setup of Lemma 3.1 one can consider the diagram( A ⋊ B ) B C ( A ⋊ B ) × C B × C C × C ϕ ( b,θ ( b )) θ ∆ C ( θ ( b ) ,c ) The right diagram commutes since b ( b, θ ( b )) ( θ ( b ) , θ ( b )) gives the same result as b θ ( b ) ( θ ( b ) , θ ( b )). Also { ( c, c ) | c ∈ C } ∩ { ( θ ( b ) , c ) | b ∈ B, c ∈ C } = { ( θ ( b ) , θ ( b )) | b ∈ B } is the image of B in C × C and so the Bass intersection condition holds.The left diagram commutes since b ( b, θ ( b )) ( b, θ ( b )) gives the same result as b b =1 .b (1 .b, θ ( b )). Furthermore, B × C ∩ { ( ab, θ ( b )) | ab ∈ ( A ⋊ B ) } = { .b, θ ( b )) | b ∈ B } is the image of B in ( A ⋊ B ) × C and so the Bass intersection condition holds.By Proposition 3.2 and using the notation of Lemma 3.1 H = ( A ⋊ B ) ∗ ( B ∼ = θ ( B )) C embedsinto H × C with image the subgroup generated by A × ∪ ∆ C .As a special application, we have that H n = ( A n ⋊ B n ) ∗ ( B n ∼ = θ n ( B n )) H n − embeds into H n × H n − with image the subgroup h A n × , ∆ H n − i . UPEREXPONENTIAL DEHN FUNCTIONS INSIDE CAT(0) GROUPS 12
Now, one might try to embed H n ∗ A n H n directly into the double ( H n × H n − ) ∗ H n ( H n × H n − ).Verifying the Bass intersection condition involves checking if the following equality holds A n × H n × ∩ h A n × , ∆ H n − i . However, ( H n ×
1) is normal in ( H n × H n − ) and so the intersection ( H n × ∩ h A n × , ∆ H n − i is the normal closure of A n × h A n × , ∆ H n − i . For n > A n ×
1) and so the induced map is not an embedding.The point is that A n is normal in ( A n ⋊ B n ), but it is not normal in H n for n >
1. Thisleads to a failure of the Bass intersection condition of Lemma 3.4 and hence to a failure ofinjectivity of the fundamental groups of the graphs of groups as was the case with the examplein Remark 3.8.4.
Dehn functions of the finitely presented subgroups
The purpose of this section is compute the Dehn functions of the subgroups L n ∗ A n L n ofProposition 3.6 for various choices of the terminal vertex groups S . In order to do this we needto first compute the distortion of A n in L n . A key ingredient is the fact that the definition of L n involves an iterated amalgamation of hyperbolic free-by-free groups.The first lemma below will be used inductively in Proposition 4.4 to establish the upperbound on the distortion of A n = F k n +1 in L n . Recall that a function f : [0 , ∞ ) → [0 , ∞ ) issuper-additive if f ( x + y ) ≥ f ( x ) + f ( y ) for all x, y ∈ [0 , ∞ ). Lemma 4.1 (amalgam distortion) . Let G = ( F ⋊ K ) ∗ K H be a group amalgamation where thegroups F , K , and H are all finitely generated. Assume that the distortion function Dist HK isdominated by a non-decreasing, super-additive function f . Then the distortion function Dist GF is dominated by the composite Dist F ⋊ KF ◦ f .Proof. Let G = F ⋊ K and let S F , S K , and S H be finite generating sets of F , K , and H respectively. Assume that S K is a subset of S H . Then S G = S F ∪ S K and S G = S F ∪ S H arefinite generating sets of G and G respectively and S G is a subset of S G . We denote d F , d K , d H , d G , and d G the word metrics on the corresponding groups with respect to the given choiceof generating sets.We observe that the homomorphism G → H taking all generators of S F to the identity andeach generator of S H to itself shows that H is a retract of G and so are isometrically embeddedsubgroups. Therefore, Dist GK = Dist HK (cid:22) f . This implies that there are positive integers C and D = C such that for each x ≥ GK ( x ) ≤ Cf ( Cx ) ≤ f ( Dx ) . We note that the second inequality above comes from the super-additive property of f and D = C . We also increase C such that f ( Dx ) ≥ x for each x ≥ . We now prove that Dist GG ( x ) ≤ f ( Dx ) for all x ≥
1. Indeed, let b be an arbitrary groupelements in G such that d G (1 , b ) ≤ x . Therefore, there is a word w = u v u v · · · u n v n in S G with the length at most x such that(1) Each u i is a word in S G and u is possibly empty.(2) Each v i is not necessarily a word in S K but it represents a group element k i in K and v n is possibly empty.By the construction we note that d G (1 , k i ) ≤ ℓ ( v i ). Therefore, d K (1 , k i ) ≤ Dist GK (cid:0) ℓ ( v i ) (cid:1) ≤ f (cid:0) Dℓ ( v i ) (cid:1) . UPEREXPONENTIAL DEHN FUNCTIONS INSIDE CAT(0) GROUPS 13
This implies that there is a word v ′ i in S K (therefore also in S G ) representing k i such that ℓ ( v ′ i ) = d K (1 , k i ) ≤ f (cid:0) Dℓ ( v i ) (cid:1) . Therefore, w ′ = u v ′ u v ′ · · · u n v ′ n is a word in S G and represents the group element b . Thus, d G (1 , b ) ≤ ℓ ( w ′ ) ≤ ℓ ( u ) + ℓ ( v ′ ) + ℓ ( u ) + ℓ ( v ′ ) + · · · + ℓ ( u n ) + ℓ ( v ′ n ) ≤ f (cid:0) Dℓ ( u ) (cid:1) + f (cid:0) Dℓ ( v ) (cid:1) + f (cid:0) Dℓ ( u ) (cid:1) + f (cid:0) Dℓ ( v ) (cid:1) + · · · + f (cid:0) Dℓ ( u n ) (cid:1) + f (cid:0) Dℓ ( v n ) (cid:1) ≤ f (cid:0) D ( ℓ ( u ) + ℓ ( v ) + ℓ ( u ) + ℓ ( v ) + · · · + ℓ ( u n ) + ℓ ( v n )) (cid:1) ≤ f ( Dx ) . This implies that Dist GG ( x ) ≤ f ( Dx ) for all x ≥
1. Therefore,Dist GF ( x ) ≤ (Dist G F ◦ Dist GG )( x ) ≤ (Dist G F ◦ f )( Dx ) for each x ≥ GF (cid:22) Dist G F ◦ f . (cid:3) The next two lemmas will be used in Proposition 4.4 to establish lower bounds for thedistortion of A n = F k n +1 in L n . Lemma 4.2 (Lemma 11.64 in [DK18]) . Let X be a hyperbolic space. Then there is a constant α ∈ (0 , depending on the hyperbolicity constant of X such that the following hold. If [ x, y ] is a geodesic of length r and m is its midpoint, then every path joining x , y outside the ball B ( m, r ) has length at least α ( r − . Lemma 4.3 (distortion in hyperbolic free-by-free) . Let F ℓ ⋊ F k be a hyperbolic free-by-freegroup. Let d F ℓ and d F k be the word metrics with respect to finite generating sets S ℓ and S k of F ℓ and F k respectively. Given = b ∈ F ℓ there exists a constant A > such that if g ∈ F k with d F k (1 , g ) sufficiently large, then d F ℓ (1 , gbg − ) ≥ A d Fk (1 ,g ) . Proof.
Let G = F ℓ ⋊ F k . Then S G = S ℓ ∪ S k be a generating set of G . We denote d G theword metric on G with respect to the finite generating set S G and assume that ( G, d G ) is a δ –hyperbolic space for some δ >
1. The group monomorphism sending all element in S ℓ to theidentity and sending each element in S k shows that F k is isometric embedded into G with thegiven word metrics.We claim that bF K b − ∩ F k = { } . Assume to the contrary that bF K b − ∩ F k is a non-trivialgroup. Then there are nontrivial two group elements a and a in F k such that ba b − = a .This implies that a − a = ( a − ba ) b − is a group element in the trivial group F k ∩ F ℓ . Therefore a = a which commutes to b ∈ F ℓ . This implies that a belong to the centralizer C ( b ) of b which is virtually cyclic (see [GdlH90]). Therefore, there are some non-zero integers p and q such that a p = b q ∈ F k ∩ F ℓ = { } . Since G is torsion free, the group element a must be trivialwhich is a contradiction. Thus, bF K b − ∩ F k = { } . By Proposition 9.4 in [Hru10] there is a constant
D > (cid:0) N δ ( F k ) ∩ N δ ( bF k ) (cid:1) ≤ D .
Therefore, diam (cid:0) N δ ( F k ) ∩ N δ (( gb ) F k ) (cid:1) = diam g (cid:0) N δ ( F k ) ∩ N δ ( bF k ) (cid:1) ≤ D .
Let n = d F k (1 , g ). Let β (resp. β ) be the geodesic in the Cayley graph Γ( G, S G ) connecting1 and g (resp. connecting gb and gbg − ) with edges labeled by elements in S K . Then verticesof β (resp. β ) are group elements in F k (resp. ( gb ) F k ). Let e be an edge label by b ∈ S F ℓ UPEREXPONENTIAL DEHN FUNCTIONS INSIDE CAT(0) GROUPS 14 connecting g and gb . Let β be a geodesic in Γ( G, S G ) connecting 1 and gbg − . Since ( G, d G )is δ –hyperbolic, there are vertices m in β , x ∈ β , and y ∈ ( e ∪ β ) such that d G ( m, x ) ≤ δ and d G ( m, y ) ≤ δ . This implies that d G ( x, ( gb ) F K ) ≤ d G ( x, m ) + d ( m, y ) + d ( y, ( gb ) F K ) ≤ δ + 2 δ + 1 < δ . Hence x ∈ N δ ( F k ) ∩ N δ (( gb ) F k ). Also, g ∈ N δ ( F k ) ∩ N δ (( gb ) F k ). This implies that d G ( x, g ) ≤ diam (cid:0) N δ ( F k ) ∩ N δ (( gb ) F k ) (cid:1) ≤ D .
Therefore, d G ( g, m ) ≤ d G ( g, x ) + d G ( x, m ) ≤ D + 2 δ. . Let γ be the path in the Cayley graph Γ( G, S ) which connects 1 and gbg − and traces theshortest word in S ℓ representing the element gbg − . Then ℓ ( γ ) = d F ℓ (1 , gbg − ). Moreover, eachvertex of v of γ is an element a ∈ F ℓ . The group monomorphism sending all element in S ℓ tothe identity and sending each element in S k shows that d G ( g, v ) = d F K ( g,
1) = n . This impliesthat γ lies outside the open ball B ( g, n ). Also d G ( g, m ) ≤ D + 2 δ . The path γ lies outside theopen ball B ( m, n − D − δ ). Here we assume that n > D + 2 δ .Let z (resp. z ) be the point in the geodesic segment [1 , m ] (resp. [ gbg − , m ]) of β suchthat the length of [ z , m ] (resp. [ z , m ]) is exactly n − D − δ . Let γ (resp. γ ) be thegeodesic segment of [1 , m ] (resp. the geodesic segment of [ gbg − , m ]) connecting 1 and z (resp.connecting gbg − and z ). Therefore, ℓ ( γ ) = d G (1 , z )= d G (1 , m ) − d G ( z , m ) ≤ (cid:0) d G (1 , g ) + d G ( g, m ) (cid:1) − ( n − D − δ ) ≤ ( n + D + 2 δ ) − ( n − D − δ ) ≤ D + 4 δ . Similarly, we also have ℓ ( γ ) = d G ( gbg − , z )= d G ( gbg − , m ) − d G ( z , m ) ≤ (cid:0) d G ( gbg − , gb ) + d G ( gb, g ) + d G ( g, m ) (cid:1) − ( n − D − δ ) ≤ (cid:0) d G ( g − ,
1) + 1 + d G ( g, m ) (cid:1) − ( n − D − δ ) ≤ ( n + 1 + D + 2 δ ) − ( n − D − δ ) ≤ D + 4 δ + 1 . The subsegment [ z , z ] of β is a geodesic of length 2 (cid:0) n − D − δ (cid:1) and m is its midpoint.By Lemma 4.2 the path γ = γ ∪ γ ∪ γ joining z , z outside the open ball B ( m, n − D − δ )has length at least 2 α ( n − D − δ ) where the constant α > G, d G ). Therefore, d F ℓ (1 , gbg − ) = ℓ ( γ )= ℓ ( γ ) − ℓ ( γ ) − ℓ ( γ ) ≥ α ( n − D − δ ) − (2 D + 4 δ ) − (2 D + 4 δ + 1) ≥ ( √ αn for n sufficiently large. Therefore, A = ( √ α > (cid:3) UPEREXPONENTIAL DEHN FUNCTIONS INSIDE CAT(0) GROUPS 15
The following proposition establishes the Dehn function of the subgroups L n ∗ A n L n of Propo-sition 3.6. Proposition 4.4 (distortion and Dehn functions in amalgams) . Let n be a positive integer andlet F k i +1 ⋊ F k i be a hyperbolic free-by-free group for ≤ i ≤ n . Let S be a finitely generated groupcontaining a free subgroup F k such that Dist SF k is equivalent to a non-decreasing, super-additivefunction f . As in Proposition 3.6 define groups L i for ≤ i ≤ n inductively by L = S and L i = ( F k i +1 ⋊ F k i ) ∗ F ki L i − . Then the distortion of F k n +1 in L n is equivalent to exp ( n ) ( f ( x )) .Moreover, if the Dehn function of L n is dominated by a polynomial function, then the Dehnfunction of the double L n ∗ F kn +1 L n is equivalent to exp ( n ) ( f ( x )) .Proof. We first prove the upper bound for the distortion Dist L n F kn +1 . First of all, note that foreach i ∈ { , , · · · , n } the distortion of F k i +1 in F k i +1 ⋊ F k i is equivalent to e x . The exponentiallower bound for Dist F ki +1 ⋊ F ki F ki +1 follows from Lemma 4.3 and the exponential upper bound canbe seen from Exercise 6.19 in part III.Γ in [BH99]. Next, apply Lemma 4.1 inductively toobtain the upper bound of exp ( n ) ( f ( x )) for Dist L n F kn +1 . The induction works because f is non-decreasing and super-additive by hypothesis and each exp ( i ) ( f ( x )) is also non-decreasing andsuper-additive.We now prove the lower bound of the distortion. For 1 ≤ i ≤ n + 1 let S i be a generatingset of F k i . These choices of generating sets induces the word metrics on F k , F k , · · · , F k n +1 andwe denote them d F k , d F k · · · , d F kn +1 respectively. We fix a finite generating set T of S thatcontains the finite generating set S of F k . Then S L n = (cid:0)S S i (cid:1) S T generates L n and we let d L n be the word metric on L n induced by S L n . We also denote d S the word metric on the group S induced by the generating set T .For each i ∈ { , , · · · , n + 1 } choose b i be an element in the generating set S i of F k i . ByLemma 4.3 there is a constant A > i ∈ { , , · · · , n } if g i is a group elementin F k i with d F ki (1 , g i ) sufficiently large then d F ki +1 (1 , g i b i +1 g − i ) ≥ A d Fki (1 ,g i ) . Define the function F ( x ) = A x for each x ≥
1. Then we observe that two functions F ( n ) ( f ( x ))and exp ( n ) ( f ( x )) are equivalent. Therefore, it is sufficient to show that Dist L n F kn +1 dominatesthe function F ( n ) ( f ( x )).Since Dist SF k is equivalent to f ( x ), there is a positive integer D > f ( x ) ≤ D Dist SF k ( Dx ) for each x ≥ . Let C = D >
1. Then we observe thatDist SF k ( Cx ) = Dist SF k ( D x ) ≥ (1 /D ) f ( Dx ) ≥ f ( x ) for each x ≥ . We note that the last inequality holds due to the fact f ( x ) is super-additive.Let x ≥ g be a group element in F k such that d S (1 , g ) ≤ Cx and d F k (1 , g ) = Dist SF k ( Cx ) ≥ f ( x ) . For each i ∈ { , , · · · , n + 1 } we define g i = g i − b i g − i − by induction on i . Then each g i is agroup element in F k i . By using an argument on i we can prove that for each i ∈ { , , , · · · , n } we have d F ki +1 (1 , g i +1 ) ≥ F ( i ) ( f ( x )) for x sufficiently large . UPEREXPONENTIAL DEHN FUNCTIONS INSIDE CAT(0) GROUPS 16
In particular, d F kn +1 (1 , g n +1 ) ≥ F ( n ) ( f ( x )) for x sufficiently large . By using an argument on i again we can also prove for each i ∈ { , , , · · · , n } we have d L n (1 , g i +1 ) ≤ (3 i C ) x . In particular, d L n (1 , g n +1 ) ≤ (3 n C ) x . Therefore,Dist L n F kn +1 (cid:0) (3 n C ) x (cid:1) ≥ d F kn +1 (1 , g n +1 ) ≥ F ( n ) ( f ( x )) for x sufficiently large . This implies that Dist L n F kn +1 dominates the function F ( n ) ( f ( x )). Therefore, the distortion of F k n +1 in L n is equivalent to exp ( n ) ( f ( x )).We now prove that the Dehn function of L n ∗ F kn +1 L n is exp ( n ) ( f ( x )). Let D n be the doubleof L n over F k n +1 and denote the Dehn function of D n by δ D n . By hypothesis, the Dehn function δ L n of L n is dominated by a polynomial function x d . By Theorem 2.4 we haveDist L n F kn +1 ( x ) (cid:22) δ D n ( x ) (cid:22) x (cid:0) δ L n ◦ Dist L n F kn +1 (cid:1) ( x ) , which implies thatexp ( n ) ( f ( x )) (cid:22) δ D n ( x ) (cid:22) x (cid:0) exp ( n ) ( f ( x )) (cid:1) d ≤ (cid:0) exp ( n ) ( f ( x )) (cid:1) d +1 . Note that the two functions exp ( n ) ( f ( x )) and (cid:0) exp ( n ) ( f ( x )) (cid:1) d +1 are equivalent (the exponent d + 1 is absorbed into the super-additive function exp ( n − ( f ( x ))). Then the Dehn function ofthe double of L n over F k n +1 is equivalent to exp ( n ) ( f ( x )). (cid:3) As we shall see in Section 5, the hypothesis in the previous proposition that L n has apolynomial isoperimetric inequality holds automatically in the case that the terminal vertexgroup S is a CAT(0) F k ⋊ Z group. The next proposition shows that L n has a polynomialisoperimetric inequality in the case where the terminal vertex group S is a snowflake group.The proof of the next proposition uses the machinery of singular disk diagrams and combi-natorial 2–complexes. The reader may wish to refer to Section 2 of [BKS20] for backgroundon combinatorial complexes and singular disk fillings of combinatorial loops; specifically, Defi-nitions 2.4, 2.9, 2.10, and 2.11. Proposition 4.5 (Dehn function of amalgams) . Let G = A ∗ C B be a group amalgamationwhere A and B are finitely presented groups and C is a finitely generated group whose distortion Dist GC ( x ) is equivalent to a super additive function. Then δ G ( x ) (cid:22) x max { δ A (cid:0) x Dist GC ( x ) (cid:1) , δ B (cid:0) x Dist GC ( x ) (cid:1) } , where δ A , δ B , and δ G are Dehn functions of A , B , and G respectively.Proof. Pick finite generating sets S A , S B , and S C for the groups A , B , and C respectivelywith the property that the image of S C in A is contained in S A and the image of S C in B iscontained in S B . There are finite presentations h S A | R A i of A and h S B | R B i of B and a possiblyinfinite presentation h S C | R C i of C . Let P A , P B , P C denote the corresponding (combinatorial)presentation 2–complexes. There are combinatorial maps f A : P C → P A inducing the inclusion C → A and f B : P C → P B inducing the inclusion C → B .The space [ P A ⊔ ( P C × [0 , ⊔ P B ] / ∼ where ∼ is generated by ( x, ∼ f A ( x ) and ( x, ∼ f B ( x ) for all x ∈ P C is a 3–dimensional cellcomplex with fundamental group A ∗ C B . This complex has finite 2–skeleton. The 1–skeleton isthe union of a wedge of | S A | circles and a wedge of | S B | circles whose base vertices are connectedby an edge (the image of the base vertex of P C times [0 , e and UPEREXPONENTIAL DEHN FUNCTIONS INSIDE CAT(0) GROUPS 17 orient it from 0 to 1. The 2–cells are indexed by the disjoint union of the sets R A , R B , and S C ;the last set indexing square 2–cells with boundary cece for c ∈ S C . Therefore, the 2–skeletonof the universal cover is a geometric model for A ∗ C B . Denote this 2–skeleton by X .We need to bound the area of an edgepath loop in X as a function of its length. Let γ be acombinatorial edgepath loop in X . A positively oriented edge e in γ corresponds to the pathleaving a copy of e P A in X and entering a copy of e P B . The path γ will travel in the 1–skeletonof this copy of e P B and then exit along a (possibly different) coset of C via an edge labelled e . Similarly, an edge e in γ is followed by an edge e . Therefore, the orientations of the edgeslabelled e alternate along γ . We match the e and e edges of γ in pairs as follows.One can think of γ as a combinatorial map γ : S → X for some cell structure on S ; that is,a cellular map which sends open 1–cells of S to open 1–cells of X . Pull back the edge labelsin X to give a labelling of the edges of the domain S of γ . Since X is simply-connected, thereis a continuous map g : D → X such that g | ∂D = γ . Homotope g rel boundary so that it istransverse to each of the spaces e P (1) C × { } in X . The preimage of the union of these spacesis a 1–submanifold of D ; that is, a collection of embedded circles and embedded arcs meeting ∂D = S transversely in the middle of the edges with labels e or e . These arcs provide thedesired matching of e and e edges of S .We use the disk D with boundary S subdivided into | γ | X together with the collection of arcs connecting the midpoints of e and e edges in pairsas a template for building a singular disk diagram (van Kampen diagram) for the loop γ . Weconstruct (or fill in) the singular disk diagram in two steps as follows.First, each arc connecting an e – e pair is mapped into a space e P (1) C × { } in X , the endpointsof the arc sent to vertices of this space. Choose a geodesic edgepath λ in e P (1) C × { } connectingthese vertices. The labels on the edges of the geodesic λ are from S C . Now connect the e – e pair by the strip of 2–cells λ × e . This forms an e –corridor connecting the pair of edges e – e of S .Second, note that the union of the e –corridors divides the disk D into a number of compli-mentary regions. The number of such regions is one more than the number of e –corridors inthe first step. This is bounded above by | γ | . Each region R i has boundary ∂R i which maps asa combinatorial edgepath loop into one of the lifts of the spaces P A or P B in X . Fill this loop ∂R i with a singular disk diagram over the corresponding presentation for A or B . Glue eachsuch singular disk diagram to the union of ∂D and the e –corridors by identifying its boundarywith ∂R i . The result is a singular disk filling ∆ for γ .Now we estimate the area of ∆ as a function of the length | γ | = x . All estimates andbounds in the following argument will be up to equivalence. First note that the length of eachof the e –corridors is bounded above by Dist GC ( x ). Therefore, the length of the boundary of acomplimentary region is bounded above by x + x Dist GC ( x ) ≤ x Dist GC ( x ) (cid:22) x Dist GC (2 x )where the last inequality holds by the fact that Dist GC is equivalent to a super additive function.Therefore, the area of this region is bounded above bymax { δ A ( x Dist GC (2 x )) , δ B ( x Dist GC (2 x )) } . Finally, there are at most x regions in ∆ and therefore the total area of ∆ is bounded above bythe sum of the areas of the e –corridors and the areas of the complimentary regions x Dist GC ( x ) + x max { δ A ( x Dist GC (2 x )) , δ B ( x Dist GC (2 x )) } . This is bounded above by 2 x max { δ A ( x Dist GC (2 x )) , δ B ( x Dist GC (2 x )) } and so the propositionfollows from the definition of (cid:22) . (cid:3) UPEREXPONENTIAL DEHN FUNCTIONS INSIDE CAT(0) GROUPS 18
Remark 4.6 (application) . Our primary application of Proposition 4.5 involves a situationwhere C is a retract of A . In this case B is a retract of A ∗ C B and so B is undistorted in A ∗ C B . Therefore, the distortion of C in A ∗ C B is equivalent to the distortion of C in B .In our application we know that the distortion of C in B , the Dehn function of A , and theDehn function of B are all bounded above by polynomial functions. Proposition 4.5 impliesthat the Dehn function of A ∗ C B is also bounded above by a polynomial function.5. The ambient
CAT(0) groups
The purpose of this section is to build a CAT(0) structure for the ambient group G n ∗ H n G n ofProposition 3.6. This structure is built in stages; first chaining together a sequence of CAT(0)free-by-free groups, then performing a factor-diagonal chaining process (defined in subsection 5.3below), and finally taking a double. The first 2 subsections provide some of the basic buildingblocks which are used in this construction.The goal of subsection 5.1 is to construct 2–dimensional CAT(0) groups which are isomorphicto F ℓ ⋊ F k for k ≥ A i ⋊ B i ) in Proposition 3.6. Thisis the content of Theorem 5.2. The third condition in Theorem 5.2 ensures that the group F k is “ultra-convex” (see Definition 5.1 below) in F ℓ ⋊ F k . This allows us to glue the complex Y k for F k ⋊ F k to the complex Y k for F k ⋊ F k by isometrically identifying the roses R k in eachspace and obtain a non-positively curved result.In subsection 5.2 we give examples of particular free-by-cyclic groups which play the role ofthe group H in Proposition 3.6 in various applications.In subsection 5.3 we use a factor-diagonal chaining construction to combine the spaces ob-tained in subsections 5.1 and 5.2 together with a non-positively curved spaces correspondingto the terminal vertex group T in Figure 1 in order to build non-positively curved spacescorresponding to the groups G n and the double G n ∗ H n G n of Proposition 3.6.5.1. Building 2–dimensional, hyperbolic,
CAT(0) F ℓ ⋊ F k groups. We start with a defi-nition.
Definition 5.1.
Let X be a non-positively curved, piecewise euclidean 2–complex. A 1–dimensional subcomplex Y ⊆ X is ultra-convex if for each 0–cell v ∈ Y the set of pointsof Lk ( v, Y ) are mutually at least 2 π apart in Lk ( v, X ).In particular, the free group π ( Y ) injects into π ( X ). We say that this subgroup is ultra-convex in π ( X )Here is the main result of this subsection. Theorem 5.2 (CAT(0) hyperbolic ( F ℓ ⋊ F k ) with ultra-convex F k ) . Let k ≥ be an integer.There exists an integer ℓ > k and a connected, non-positively curved, 2–dimensional, piecewiseeuclidean cell complex Y k whose fundamental group is ( F ℓ ⋊ F k ) with the following properties.(1) Y k has one 0–cell, v , and each loop in the link, Lk ( v, Y k ) , has length strictly greaterthan π .(2) The subgroup F ℓ , resp. F k , is the fundamental group of a rose R ℓ , resp. R k , in Y (1) k based at v . In the case k = 1 , the rose R is a circle and F = Z .(3) The rose R k ⊆ Y k is ultra-convex in Y k . The proof (which occupies the rest of this subsection) follows the general outline of [BM02]where one takes a suitable branched cover of the total space of a graph of spaces. However,for the purposes of the present paper, one needs to take special care with the initial choiceof the graph of spaces. Our choice in Step 1 below facilitates the proof of the ultra-convexitycondition and also ensures that the branched cover can be achieved via a single cyclic coveringof the complement of the branch points.
UPEREXPONENTIAL DEHN FUNCTIONS INSIDE CAT(0) GROUPS 19
The proof has a number of technical steps. We provide an overview of the these steps forthe reader’s reference. The numbers in the list below correspond to the numbers of the stepsin the proof.(1) We define X k as the total space of a graph of 1–dimensional spaces whose underlyinggraph, Θ k , is a rose of k circles. We define a rose–valued Morse function X k → Θ k . Thespace X k is given a piecewise euclidean 2–complex structure.(2) The complex X k has a single 0–cell, v , and we describe the link Lk ( v, X k ).(3) The space X k − { v } retracts onto a spine which is a graph, D k . This retraction inducesa graph immersion from the barycentric subdivision of Lk ( X k , v ) to D k .(4) List all the short loops in Lk ( v, X k ) and express these as nontrivial elements in H ( D k ).(5) Construct a finite cyclic covering space of D k in which each of the nontrivial elementsin H ( D k ) listed above has one preimage. The piecewise euclidean metric on X k − { v } lifts to the corresponding finite, cyclic cover of X k − { v } . The completion of this metricyields a branched cover c X k → X k . There is a k –leaved rose R k which is ultra–convexin c X k and which maps isomorphically to Θ k via the Morse function. Also, π ( c X k , v ) ishyperbolic.(6) There is a subcomplex Y k ⊆ c X k which has CAT(0), hyperbolic fundamental group ofthe form F ℓ ⋊ F k . The free-by-free structure is established using the Morse functionrestricted to Y k . Step 1. The total space X k of a graph of spaces. In this step we describe X k as the totalspace of a graph of spaces, give X k the structure of a piecewise euclidean complex, and definea graph-valued Morse function X k → Θ k . The underlying graph.
Let k ≥ k denote a graph with one 0–cell, w ,and directed edges labelled by x i for 1 ≤ i ≤ k . The graph of spaces.
Next we define a graph of spaces based on Θ k . The vertex space X w is abouquet R of 8 circles. Denote the 0–cell of R by v and the directed edges by a i for 1 ≤ i ≤ X x j is defined to be a graph O consisting of a circuit of 8 bigons as shown inFigure 2. The edges are oriented as shown and labelled by α i and β i for 1 ≤ i ≤
8. The verticesare labelled n i and p i for 1 ≤ i ≤ n (negative)vertices towards the p (positive) vertices. n p n p n p n p β α β α β α β α β α β α β α β α Figure 2.
The edge space X x j = O for each 1 ≤ j ≤ k .The map f : X x i → X w at the terminal end of each edge x i is given by collapsing the β –circuit to v and mapping each α i to a i for 1 ≤ i ≤ UPEREXPONENTIAL DEHN FUNCTIONS INSIDE CAT(0) GROUPS 20
The map f : X x i → X w at the initial end of each edge x i is given by collapsing the α –circuitto v and mapping each β i to a i +4 for 1 ≤ i ≤ The total space X k . Recall from (see Scott-Wall [SW79]) that the total space X k of this graphof spaces is defined as the quotient X k = ( X w ⊔ G ≤ j ≤ k X x j × [0 , / ∼ where the equivalence relation is generated by ( x, ∼ f ( x ) and ( x, ∼ f ( x ) for all x ∈ X x j .We now describe the cell structure of X k .Since X w = R has one vertex, v , and the underlying graph Θ k has one vertex, the complex X k has only one 0–cell. We denote this 0–cell by v .Each vertex n i (resp. p i ) of O determines an oriented edge n i × [0 ,
1] (resp. p i × [0 , X x j × [0 ,
1] and hence in X k . We label these edges by n i,j (resp. p i,j ). There are 4 k ofeach of these types of edges. The oriented edges a i for 1 ≤ i ≤ X w alsocontribute to the 1-skeleton of X k . Therefore, X (1) k is a bouquet of 8 k + 8 oriented edges at thevertex v ; namely, p i,j and n i , j for 1 ≤ i ≤ ≤ j ≤ k and the edges a , . . . , a .Each edge α i of O determines a 2–cell ( α i × [0 , X x j × [0 , f collapses α i to v ∈ X w , this 2–cell factors through a 2–simplex ( α i × [0 , / ( x, ∼ ( x ′ ,
0) in X k for each1 ≤ j ≤ k . We denote these 2–simplices by α i,j . Likewise, each edge β i of O determines k X k denoted by β i,j for 1 ≤ j ≤ k . Figure 3 shows how these simplices gluetogether in cycles of length 8. a a a a a a a a a a a a a a a a n p n p n p n p n p n p n p n p α α α α α α α α β β β β β β β β Figure 3.
A template for the 16 k X k . The actual 2–simplicesare labelled by α i → α i,j and β i → β i,j , and the corresponding 1–simplices arelabelled by n i → n i,j , p i → p i,j , and a i → a i for 1 ≤ j ≤ k . The piecewise euclidean structure of X k . We give X k a piecewise euclidean structure in whicheach 2–cell is a regular euclidean triangle of side length 1. The Morse function X k → Θ k . As in [BM02] we define a rose-valued Morse function X k → Θ k which sends the bouquet of 8 edges labelled a i to w , sends each n i,j and each p i,j to x j for1 ≤ j ≤ k , and which extends linearly over the 2–simplices. In the case k = 1 this is a standardcircle-valued Morse function. Step 2. The link Lk ( X k , v ) . One can use the piecewise euclidean metric on X k and define Lk ( v, X k ) to be the boundary of the closed metric ball of radius 1 / v in X k . Each 1-cell e of X k contributes two vertices e + and e − to Lk ( v, X k ); the edge e is oriented from e − towards e + . A pair of adjacent edges in a 2–cell of X k contributes an edge to Lk ( v, X k ). UPEREXPONENTIAL DEHN FUNCTIONS INSIDE CAT(0) GROUPS 21 n +4 n +3 n +2 n +1 p +4 p +3 p +2 p +1 n − n − n − n − p − p − p − p − a − a − a − a − a − a − a − a − a +1 a +2 a +3 a +4 a +5 a +6 a +7 a +8 Figure 4.
Template for the link Lk ( v, X k ).The graph in Figure 4 is a template for Lk ( v, X k ). To obtain the link Lk ( v, X k ) first take k copies of the template shown. Next, relabel the vertices in the j th copy as follows: n ± i → n ± i,j , p ± i → p ± i,j and don’t change the labelling of the a ± i vertices. Finally, glue these k graphs togetheralong the a ± i vertices.The template graph has 32 vertices and 48 edges. Therefore, Lk ( v, X k ) obtained from the k copies as above has 16 k + 16 vertices and 48 k edges. This count agrees with the fact that X k has 8 k + 8 1–cells and 16 k LK ( v, X k ) andeach triangular 2–cell contributes 3 edges to Lk ( v, X k ).Using the terminology and notation of Definition 2.4 and Definition 2.5 in [BM02], the 1–cells a i (for 1 ≤ i ≤
8) are horizontal with respect to the Θ k –valued Morse function. For eachoriented edge x j ∈ Θ k the x j –link of v in X k is given by Lk x j ( v, X k ) = the circle spanned by n − i,j and p − i,j for 1 ≤ i ≤ x j –link of v ∈ X k is given by Lk x j ( v, X k ) = the circle spanned by n + i,j and p + i,j for 1 ≤ i ≤ Lk x j ( v, X k ) as the ascending (or negative) link of v in the x j –direction and of Lk x j ( v, X k ) as the descending (or positive) link of v in the x j –direction.Note that the links Lk x j ( a i , X k ) and Lk x j ( a i , X k ) of the horizontal 1–cells a i are all singletons. Step 3. The spine D k of X k − { v } and the immersion Lk ( v, X k ) ′ → D k . The spine of X k − { v } is a graph which we denote by D k and which is defined as follows. The vertices of D k are thebarycenters b α of 2-cells α of X k and the barycenters b e of 1–cells e of X k . An b α vertex of D k isadjacent to an b e vertex if and only if the 1–cell e is a face of the 2–cell α in X k .As is shown in Figure 5, the spine graph D k embeds into X k and there is a deformationretraction of X k − { v } onto D k . Each open edge deformation retracts onto its barycenter(midpoint). These deformation retractions extend to a deformation retraction of a 2–cell α (minus its vertices) onto a tripod graph as indicated in Figure 5. These deformation retractionsglue together over open edges to yield a deformation retraction X k − { v } → D k . UPEREXPONENTIAL DEHN FUNCTIONS INSIDE CAT(0) GROUPS 22 b α b e b e b e Figure 5.
The retraction X k − { v } → D k inside of a 2–cell α with edges e , e ,and e . The dashed lines indicate the retraction; the solid lines are in the spine.One way to visualize the spine D k is to embed portions of it into the diagrams in Figure 3. Atemplate for the spine is described in Figure 6. The actual template is obtained from the graphin Figure 6 by identifying vertices in pairs which have the same labels (the b a i vertices). In orderto obtain the spine D k take k copies of the resulting template graph, indexed by 1 ≤ j ≤ k , andrelabel the vertices in the j th copy as follows: b α i → b α i,j , b β i → b β i,j , and b a i → b a i for 1 ≤ i ≤ b n i → b n i,j , b p i → b p i,j for 1 ≤ i ≤
4. Finally, D k is obtained by glueing these k copies togetherby identifying along the vertices b a i .Each edge in Lk ( v, X k ) lies in (the corner of) a 2–cell and is sent by the retraction (seeFigure 5) to an edgepath of length 2 in the tripod retract of the 2–cell. In this way theretraction induces a graph map ι : Lk ( v, X k ) ′ → D k from the first barycentric subdivision of the link to the spine. One can see that this map isan immersion of graphs as follows. It is clearly an immersion at the barycenter points. To seethat it is an immersion at other vertices, let e and e be two edges of Lk ( v, X k ) based at avertex of Lk ( v, X k ). These edges belong to distinct 2–cells of X k and so their barycenters aremapped to distinct points in D k . Therefore, the images of these edges are distinct in D k , andso the map is an immersion at all vertices.Since ι : Lk ( v, X k ) ′ → D k is a graph immersion, it induces an injection π ( Lk ( v, X k )) → π ( D k ). Step 4. Computing distance and determining short loops in Lk ( v, X k ) . The link Lk ( v, X k )inherits a piecewise spherical structure in which all edges have length π/ X k . In order to obtain a non-positively curved complex, the link needsto be large; that is, all embedded loops in the link have length at least 2 π . Combinatorially,this means that all circuits in the link graph have to be of combinatorial length 6 or more. Ifall loops in the link have combinatorial lengths strictly greater than 6, then the universal coverof X k is a CAT(0) space with a cocompact group action by isometries which does not admitany embedded flat planes. By Flat Plane Theorem (see [BH99] III.H.1.5) π ( X k ) is hyperbolic.The goal is to obtain a branched cover complex of X k , branched over the vertex v , with one0–cell whose link has no loops of combinatorial length 6 or less. This branched cover is obtainedby taking a finite covering of X k − { v } , lifting the locally euclidean metric to this covering spaceand taking the metric completion.There is an embedded copy of Lk ( v, X k ) in X k − { v } ; namely, the sphere of radius 1 / v . In a finite covering space e X → X k − { v } the preimage of Lk ( v, X k ) is a covering space e L → Lk ( v, X k ) . UPEREXPONENTIAL DEHN FUNCTIONS INSIDE CAT(0) GROUPS 23 b a b α b β b a b α b β b a b α b β b a b α b β b a b α b β b a b α b β b a b α b β b a b α b β b p b p b p b p b n b n b n b n b a b a b a b a b a b a b a b a Figure 6.
A template for the spine D k of X k − { v } . The template is obtainedfrom the graph in this figure by identifying vertices with the same labels (the b a i vertices).We want the covering space e X → X k − { v } and the induced covering space e L → Lk ( v, X k ) tosatisfy 4 conditions.(1) The preimage e L is connected.This property ensures that the metric completion of the covering space of X k − { v } has just one 0–cell.(2) The preimage of each of the combinatorial length 8 circuits n +1 ,j p +1 ,j n +2 ,j p +2 ,j n +3 ,j p +3 ,j n +4 ,j p +4 ,j and each of the combinatorial length 8 circuits n − ,j p − ,j n − ,j p − ,j n − ,j p − ,j n − ,j p − ,j is connectedfor 1 ≤ j ≤ k .This significance of this will become apparent later on in the final step of the con-struction of the free-by-free groups.(3) The preimage e L does not contain embedded loops of combinatorial length 6 or less.This property ensures that the metric completion of the covering space e X of X k − { v } is non-positively curved and that its fundamental group is hyperbolic.(4) There are lifts of each of the open edges n ,j for 1 ≤ j ≤ k to the covering space of X v −{ v } which determine a collection of 2 k vertices in the covering space e L of Lk ( v, X k )whose mutual combinatorial distances are all at least 6.This property will be used to establish conclusion (3) of Theorem 5.2.Now we describe all loops in Lk ( v, X k ) of combinatorial length 6 or less and show that theymap to homologically nontrivial 1–cycles in D k . First of all, note that Lk ( v, X k ) is a bipartitegraph on the disjoint union of vertices { a − , . . . , a − , p + i,j , p − i,j | ≤ i ≤ , ≤ j ≤ k } and { a +1 , . . . , a +8 , n + i,j , n − i,j | ≤ i ≤ , ≤ j ≤ k } . UPEREXPONENTIAL DEHN FUNCTIONS INSIDE CAT(0) GROUPS 24
This is clear from the template graph in Figure 4 and the earlier description of how to ob-tain Lk ( v, X k ) from k copies of the template. This means that there are no cycles of oddcombinatorial length in Lk ( v, X k ); in particular, there are no m –cycles for m = 1 , , Lk ( v, X k ).Next we show that each of the 4–cycles and 6–cycles in Lk ( v, X k ) has image in D k which ishomologically nontrivial. To do this, we establish a simple criterion which guarantees that aloop in Lk ( v, X k ) has homologically nontrivial image in D k . First, we need some definitions. Definition 5.3 (type) . Say that a vertex of D k is of type 1 (resp. type 2 ) if it is the barycenterof a 1–cell (resp. 2–cell) of X k .Consider a loop γ in Lk ( v, X k ). By construction/definition, the immersion ι : Lk ( v, X k ) ′ → D k takes vertices of γ to type 1 vertices of D k and barycenters of edges of γ to type 2 verticesin D k . Definition 5.4 (HNT) . A loop γ in Lk ( v, X k ) satisfies condition (HNT) if there is a type 1vertex in ι ( γ ′ ) whose preimage in γ is a single vertex. Lemma 5.5.
Let γ ⊆ Lk ( v, X k ) be a loop which satisfies the (HNT) condition. Then ι ( γ ′ ) isdetermines a nontrivial element of H ( D k ) .Proof. We argue by contraction. Suppose that γ is a 1–cycle in Lk ( v, X k ) which satisfiescondition (HNT) and ι ( γ ′ ) is homologically trivial in D k .Since γ ′ is a 1–cycle in Lk ( v, X k ) ′ the image ι ( γ ′ ) is a 1–cycle in D k . By assumption, thereis a type 1 vertex u ∈ ι ( γ ′ ) whose preimage is just one vertex u ∈ γ . Because ι is a graphimmersion, the two edges in γ ′ which are adjacent to u get sent to distinct edges e and e of ι ( γ ′ ) adjacent to u .Since ι ( γ ′ ) is homologically trivial in D k , the 1–cells in its image need to be crossed at leasttwice (with total coefficient sum being zero). In particular, the distinct edges e and e at u are crossed at least twice. This contradicts the fact that the preimage of u is a single vertex u . (cid:3) The following terminology will be used in determining the structure of 4–cycles and 6–cyclesin Lk ( v, X k ). Definition 5.6.
Edges of Lk ( v, X k ) which contain the a + i vertices (resp. the a − i vertices) arecalled right edges (resp. left edges ). The remaining edges (which connect the n ± i,j vertices tothe p ± i ′ ,j vertices) are called middle edges . Lemma 5.7.
The image under the map ι : Lk ( v, X k ) ′ → D k of each 4–cycle of Lk ( v, X k ) ishomologically nontrivial in D k .Proof. First of all, we argue that there are no 4–cycles which involve the middle edges. Since themiddle edges form disjoint circuits of combinatorial length 8, there are no 4–cycles composedentirely of middle edges. A path of 3 consecutive middle edges has one endpoint of the form n ± i,j and the other endpoint of the form p ± i ′ ,j ′ and so cannot be completed in Lk ( v, X k ) to acircuit of combinatorial length 4. If a path of two consecutive middle edges starts at (withoutloss of generality) some p + i,j it must end at p + i ′ ,j . If we connect one of these endpoints to an a + vertex by an edge, the other edges from that a + vertex connect to either p − vertices or to p + a,b vertices for b = j and so cannot form a 4–cycle. Finally, a circuit which contains a segmentof combinatorial length 3 consisting of a single middle edge with a left edge on one end and aright edge on the other has combinatorial length at least 6. UPEREXPONENTIAL DEHN FUNCTIONS INSIDE CAT(0) GROUPS 25
Therefore, 4–cycles in Lk ( v, X k ) are composed entirely of left edges or of right edges. Tounderstand their structure we remove all of the open middle edges from Lk ( v, X k ). The resultinggraph is the disjoint union of 8 joins; 4 composed of right edges and 4 of left edges. Here aretwo such joins (the first composed of right edges and the second composed of left edges): { a +1 , a +2 } ∗ { p +1 ,j , p − ,j | ≤ j ≤ k } and { a − , a − } ∗ { n +1 ,j , n − ,j | ≤ j ≤ k } . The others have a similar structure. In each case, the 4–cycles are obtained by taking the joinof the 2 element set with a 2 element subset of the 2 k element set. This describes all of thenontrivial 4–cycles in Lk ( v, X k ).Consider a nontrivial 4–cycle γ in Lk ( v, X k ). It is the join of some set { a ± i , a ± i ′ } where i = i ′ with another set involving p ± or n ± vertices. In particular, the images b a i and b a i ′ have singletonpreimages in γ ′ and so the image ι ( γ ′ ) is homologically nontrivial by the (HNT) condition. (cid:3) Lemma 5.8.
The image under the map ι : Lk ( v, X k ) ′ → D k of each 6–cycle of Lk ( v, X k ) ishomologically nontrivial in D k .Proof. We first observe that each 6–cycle must contain a middle edge. As we observed inLemma 5.7 the complement of the union of the open middle edges in Lk ( v, X k ) is the disjointunion of complete bipartite graphs, each of which is a join a 2 vertex set with a 2 k vertex set.Every immersed segment of combinatorial length 2 which starts and ends in the 2 vertex hasdistinct endpoints. Therefore, the combinatorial lengths of closed, immersed paths in thesebipartite graphs must be 4 m for positive integers m . In particular, there are no 6–cycles.Let γ be an arbitrary 6–cycle in Lk ( v, X k ). By the previous paragraph γ contains at leastone middle edge, and so must contain a vertex u of the form p ± i,j . We show that { u } is thefull preimage of { ι ( u ) } . Assume to the contrary that the preimage of ι ( u ) contains two distinctvertices in γ . Then γ must contain the pair of distinct vertices { p + i,j , p − i,j } . Since two vertices p + i,j and p − i,j have the same color in the bipartite graph Lk ( v, X k ), there is no segment with oddcombinatorial length connecting them. Therefore, there is a subsegment of γ of combinatoriallength 2 connecting p + i,j and p − i,j . In particular, p + i,j and p − i,j are both adjacent to some vertex in Lk ( v, X k ) and we observe that it can not happen. Therefore, the preimage of ι ( u ) consists ofa single vertex. This implies that γ satisfies (HNT) condition and ι ( γ ′ ) determines a nontrivialelement in H ( D k ). (cid:3) The following lemma will be used in establishing the ultra-convexity result (part (3)) ofTheorem 5.2.
Lemma 5.9.
The combinatorial distance in Lk ( v, X k ) between n +1 ,j and n − ,j is at least 6.Proof. Note that the map from the link graph to the template graph defined by mapping a ± i to a ± i , n ± i,j to n ± i , p ± i,j to p ± i , and extending over the 1–skeleton is distance non-increasing.Therefore, it is sufficient to compute distances between n +1 and n − in the template graph.More precisely, we will show that each immersed segment γ connecting n +1 and n − must havecombinatorial length at least 6.Since n +1 and n − have the same color in the bipartite structure on the template graph, thecombinatorial length of γ must be even. Therefore, we only need to show the combinatoriallength of γ is greater than 4. We will argue by starting at n +1 and considering all possibilitiesfor the first edges e of γ .There are only four possibilities for e : one of two left edges or one of two middle edges. If e is one of the two left edges adjacent to n +1 , the initial segment of γ of combinatorial length2 must connect n +1 to n − . Since n − and n − are not both adjacent to some vertex, the distancebetween them must be greater than 2. This implies that the combinatorial length γ must begreater than 4. UPEREXPONENTIAL DEHN FUNCTIONS INSIDE CAT(0) GROUPS 26
We now assume that e is a middle edge adjacent to n +1 . Then the other endpoint of e iseither p +1 or p +4 . Therefore, the initial segment of combinatorial length 2 of γ must connect n +1 to one of the vertices in { a +1 , a +2 , n +2 , a +7 , a +8 , n +4 } . Using the same argument in the previousparagraph we see that the distance between each of these points and n − must be greater than2. Therefore, the combinatorial length of γ must be greater than 4. (cid:3) Step 5. The cyclic branched cover b X k → X k . We define a finite, cyclic branched cover b X k → X k which is branched over the vertex v by first taking a finite, cyclic covering e X → X k − { v } , liftingthe piecewise euclidean metric to e X and then considering the metric completion. Since D k is adeformation retract of X k − { v } we can model this covering by a cyclic cover of D k .The non-positive curvature of Y k and conclusion (1) of Theorem 5.2 will be established bykeeping careful control of links as we form the branched cover b X k of X k . In particular, weneed to ensure that each of the 4–cycles and 6–cycles in Lk ( v, X k ) are unwound nontrivially inthe covering space e L . Also, we need to ensure that each of the 2 k loops which are 8–cycles ofmiddle edges has a single preimage in e L .To this end, define L to be the collection of all the nontrivial 4–cycles and 6–cycles and the2 k ι : Lk ( v, X k ) ′ → D k is the graph immersion inducedby the retraction map X v − { v } → D k . Define C = { [ ι ( γ ′ )] ∈ H ( D k ) | γ ∈ L } . By Lemma 5.7 and Lemma 5.8 the images of the 4–cycles and 6–cycles are homologicallynontrivial in D k . Each of the 8–cycles of middle edges satisfies the (HNT) condition and sohas homologically nontrivial image in D k . The set C is the collection of all of these nontrivialelements of H ( D k ).The finite cyclic cover of D k corresponds to a homomorphism π ( D k ) → Z N for some integer N >
1. This homomorphism is expressed as the composition π ( D k ) → H ( D k ) → Z → Z N where the middle homomorphism ℓ : H ( D k ) → Z is given by inner product with a suitableelement of H ( D k ) chosen so that the image under ℓ of each of the elements of C above isnonzero in Z . See Lemma 3.1 in [BM02] for one way of doing this.We now describe how to choose N , the degree of the cover. Recall that Lk ( v, X k ) is asubspace of X k − { v } and that the preimage of Lk ( v, X k ) in the cyclic cover e X → X k − { v } isa covering space e L → Lk ( v, X k ). There are two things to consider in choosing 1 < N ∈ Z .(1) The first consideration concerns non-positive curvature and hyperbolicity. If N ∈ Z ischosen to be relatively prime to each of the integers ℓ ( c ) for c ∈ C , then each loop in L has preimage a single N –fold cover loop in e L .In particular, e L is connected. This means that the metric completion of the liftedpiecewise euclidean metric on e X just adds a single vertex ˆ v . Furthermore, Lk (ˆ v, b X v ) = e L has no loops of combinatorial length less than 8 (because it is bipartite and, byconstruction, has no loops of combinatorial length 6 or less). Therefore, b X v is a non-positively curved space and π ( b X v ) is hyperbolic.(2) The second consideration concerns conclusion (3) of Theorem 5.2.Fix a maximal tree T D ⊆ D k in the spine graph. Define P to be the collection of alledgepaths in Lk ( v, X k ) of combinatorial length at most 6. Given γ ∈ P the path ι ( γ ′ )is an edgepath in D k . There exists a unique geodesic edgepath δ γ in T D whose initialpoint is the terminal point of ι ( γ ′ ) and whose terminal point is the initial point of ι ( γ ′ ).The concatenation ι ( γ ′ ) · δ γ defines a 1–cycle in D k . UPEREXPONENTIAL DEHN FUNCTIONS INSIDE CAT(0) GROUPS 27
Some of the 1–cycles ι ( γ ′ ) · δ γ are homologically nontrivial, and some of these homologyclasses are sent by the map ℓ : H ( D k ) → Z to nontrivial integers. Define an integer M by M = max (cid:8) | ℓ ([ ι ( γ ′ ) · δ γ ]) | (cid:12)(cid:12) γ ∈ P (cid:9) . One can understand the cyclic covering space f D k → D k by taking the maximal tree T D ⊂ D k chosen above and considering N lifts of it labelled T (1) D , . . . , T ( N ) D arrangedalong a line. For each 1 ≤ i ≤ N , the vertex b n ( i )1 ,j ∈ T ( i ) D is the barycenter (midpoint) ofan open 1–cell n ( i )1 ,j ⊂ e X . This 1–cell corresponds to vertices n ( i ) ± ,j in the lift e L of thelink.If we choose N to be larger than ( M + 1) k and relatively prime to the integers ℓ ( c )for c ∈ C as above (we could take N to be a prime), then the mutual combinatorialdistances between the following 2 k vertices (cid:26) (cid:16) n (( M +1) j )1 ,j (cid:17) + , (cid:16) n (( M +1) j )1 ,j (cid:17) − (cid:12)(cid:12)(cid:12)(cid:12) ≤ j ≤ k (cid:27) in e L are at least 6. Indeed, the combinatorial distance between (cid:16) n (( M +1) j )1 ,j (cid:17) + and (cid:16) n (( M +1) j ′ )1 ,j ′ (cid:17) − for j = j ′ is greater than 6 by our choices of M and N and constructionof the cyclic covering. In the case j = j ′ it is at least 6 because the covering map e L → Lk ( v, X k ) is distance non-increasing and we computed the combinatorial distancebetween n +1 ,j and n − ,j in Lk ( v, X k ) to be at least 6 in Lemma 5.9.In conclusion, the metric completion b X k of e X is a piecewise euclidean 2–complex whose 2-cells are euclidean equilateral triangles. It has one 0–cell b v and Lk ( b v, b X k ) = e L has no loops ofcombinatorial length less than 8. This means that b X k is non-positively curved and that π ( b X k )is a hyperbolic group.The complex b X k has 8 N a ( p ) i for 1 ≤ i ≤ ≤ p ≤ N . It has 4 kN n ( p ) i,j and 4 kN p ( p ) i,j for 1 ≤ p ≤ N , 1 ≤ i ≤
4, and 1 ≤ j ≤ k . Ithas 16 kN triangular 2–cells.The branched covering map b X k → X k composes with the rose-valued Morse function X k → Θ k to give a rose-valued Morse function b X k → Θ k . By construction, this function satisfies thecondition ( C ) but not ( HC ) of Definition 2.6 in [BM02] and by Corollary 2.9 of that paperthe kernel of the induced map π ( b X k , b v ) → F k is finitely generated but not finitely presented.The connectivity of Lk x j ( b v, b X k ) and of Lk x j ( b v, b X k ) follows from the fact that the preimage ofeach of the 2 k loops (of combinatorial length 8) composed of middle edges in Lk ( v, X k ) is asingle circle of combinatorial length 8 N in e L = Lk ( b v, b X x ).The 8 N a ( p ) i are horizontal with respect to the Morse function above and thecorresponding groups elements, also denoted by a ( p ) i , generate the kernel of π ( b X k , b v ) → F k .The rose R k of k edges with labels n (( M +1) j )1 ,j for 1 ≤ j ≤ k is ultra-convex in b X k anddetermines a free subgroup of π ( b X k ) which maps isomorphically to the group F k under theepimorphism induced by the Morse function. Step 6. The subcomplex Y k ⊆ b X k . One obtains a subcomplex Y k ⊆ b X k by removing (any)one of the open horizontal 1–cells a ( p ) i together with the interior of each of the 2 k b X k , the 2–complex Y k ispiecewise euclidean, locally CAT (0) and π ( Y k ) is hyperbolic. Furthermore, the rose R k is UPEREXPONENTIAL DEHN FUNCTIONS INSIDE CAT(0) GROUPS 28 ultra-convex in Y k and determines a free subgroup π ( Y k ) which maps isomorphically to F k under the epimorphism induced by the Morse function.The effect of this operation on the Morse function links is to remove a single edge from eachof Lk x j ( b v, b X k ) and Lk x j ( b v, b X k ), turning them from circles of combinatorial length 8 N into linesegments of combinatorial length 8 N −
1. Therefore, the Morse function restricted to Y k satisfiesthe condition C of Definition 2.6 in [BM02] and Theorem 2.7(2) of that paper implies that thekernel of the induced homomorphism is free1 → F N − → π ( Y k ) → F k → . Note that kernel F N − is the fundamental group of the rose of the 8 N − Y k . Also, the rose R k ⊂ Y k has fundamental group F k which maps isomorphically tothe quotient F k in the short exact sequence above. This means that π ( Y k ) ∼ = ( F N − ⋊ F k ) andthe distance in the piecewise spherical metric on Lk ( b v, Y k ) between any two points of Lk ( b v, R k )is at least 6( π/
3) = 2 π . Therefore, the rose R k and the subgroup F k are ultra-convex. Thisconcludes the proof of Theorem 5.2.5.2. Non-positively curved complexes for F p ⋊ Z groups. In this subsection we describevarious CAT(0) free-by-cyclic groups which play the role of the group H in Proposition 3.6.There are many examples in the literature of CAT(0) free-by-cyclic groups whose kernelsare exponentially or polynomially distorted of arbitrary degree. We give explicit examples ofsuch groups here for completeness and also to show that they can be amalgamated with thefree-by-free groups of subsection 5.1. In particular, we show that the geometric model of the F k ⋊ Z group contains an embedded rose R k corresponding to the kernel F k . The kernel F k plays the role of the group A in Proposition 3.6. For each class of examples, we specify thecorresponding vertex group S and record the distortion of A = F k in S . Example (1).
Free-by-cyclic groups with polynomially distorted kernels.
Consider the groupsΓ k defined by Γ k = h a , . . . , a k , t | a ta − = t, and a i a i − a − i = t for 2 ≤ i ≤ k i . The corresponding presentation 2–complex for Γ k has one 0–cell, ( k + 1) 1–cells labelled a , . . . , a k , t , and k square 2–cells. When given a piecewise euclidean structure in which each2–cell is a euclidean square of side length 1, this complex is non-positively curved (there are nocircuits of combinatorial length less than 4 in the link) and so Γ k is a CAT(0) group.There is a circle-valued Morse function on the presentation 2–complex for Γ k obtained bymapping each edge homeomorphically to a circle and extending linearly over the 2–cells. Thisinduces a map Γ k → Z sending each of the generators a , . . . , a k , t to a generator of Z . Theascending and descending links of this Morse function are trees (the ascending link is a coneon the discrete set a − , . . . , a − k with cone point t − and the descending link is a line segment ofcombinatorial length k with vertices labelled (in order) t + , a +1 , . . . , a + k ) and so Γ k is an F k -by- Z group.A set of generators of the kernel, { x , . . . , x k } , is defined by x = ta − and x i = a i − a − i for2 ≤ i ≤ k . These are represented by the diagonals of the square 2–cells of the presentation2–complex; these form a rose, R k , of k petals based at the vertex of the presentation 2–complex.A representative of the monodromy ϕ is given by conjugation by t ; namely ϕ ( x i ) = tx i t − .One verifies that ϕ ( x ) = x , ϕ ( x ) = x x and in general that ϕ ( x i ) = x x . . . x i for 3 ≤ i ≤ k .One proves by induction that ϕ n ( x ) has length 1 and that ϕ n ( x i ) has length equivalent to n i − for 2 ≤ i ≤ k . The induction step uses the fact that ϕ ( x i ) = x . . . x i = ϕ ( x i − ) x i and so ϕ n ( x i ) = ϕ n ( x i − ) ϕ n − ( x i − ) . . . ϕ ( x i − ) x i and that these are all positive words. Alower bound of x k for the distortion of the kernel F k in G k is given by the element t n x nk t − n = UPEREXPONENTIAL DEHN FUNCTIONS INSIDE CAT(0) GROUPS 29 ( ϕ n ( x k )) n . An upper bound is established by considering a word w ( x i , t ) of length x which rep-resents an element of F k and using Britton’s Lemma to successively remove innermost t ± . . . t ∓ pinch pairs. Therefore, the distortion of the F k kernel in G k is equivalent to x k . In the applica-tion, the terminal vertex group S in the top graph of groups in Figure 1 is taken to be F k ⋊ ϕ Z and so the distortion of F k in S is equivalent to x k .These presentation 2–complexes are examples of the the building block complex K that isused in the next subsection. In this case, K has one 0–cell. Example (2).
CAT(0) hyperbolic free-by-cyclic groups.
There are many ways of producingCAT(0) hyperbolic free-by-cyclic groups. For example, one can use the previous subsection with k = 1 to obtain a non-positively curved piecewise euclidean 2–complex Y with fundamentalgroup F ℓ ⋊ Z . By construction, the kernel F ℓ is the fundamental group of a rose, R ℓ , in Y . Thedistortion of F ℓ in F ℓ ⋊ Z is exponential because the semidirect product is hyperbolic. In theapplication, the terminal vertex group S in the top graph of groups in Figure 1 is taken to be F ℓ ⋊ Z and so the distortion of F ℓ in S is equivalent to exp( x ).The complex Y is another example of the building block complex K that is used in the nextsubsection. In this case, K has one 0–cell. Example (3).
The
CAT(0) F ⋊ Z group of [BF17] . A key ingredient in the construction in[BF17] of 6–dimensional CAT(0) groups which contain finitely presented snowflake subgroups isa particular CAT(0) group of the form F ⋊ ϕ Z with palindromic monodromy ϕ of exponentialgrowth.The group F ⋊ ϕ Z is the fundamental group of a 2–dimensional non-positively curved piece-wise euclidean cell complex, labelled Z in section 4 of [BF17]. This complex has 2 vertices.We subdivide this complex by introducing horizontal edges labelled x and y at one of the twovertices (these are drawn and labelled in Figure 2 of [BF17]). This subdivision introduces anembedded rose R in the 1–skeleton which represents the kernel F .In [BF17] there are CAT(0) groups and snowflake subgroups built from the m –fold cycliccovers of Z corresponding to F ⋊ ϕ m Z for each integer m ≥
1. The preimage of the rose R in the m –fold cyclic cover consists of m disjoint roses, each isomorphic to R . Although thedistortion of F in F ⋊ ϕ m Z is exponential, in the application the terminal vertex group S inthe top graph of groups in Figure 1 is the snowflake group of [BF17]. The distortion of F inthe snowflake group S is x α .The m –fold cyclic covers of the subdivided Z (for m ≥
1) are examples of the building blockcomplex K used in the next subsection. In this case, K may have more than one 0–cell (the m –fold covering of the subdivided Z complex has 2 m Non-positively curved complexes for G n and G n ∗ H n G n . The CAT(0) structures for G n and G n ∗ H n G n are built from the blocks in subsections 5.1 and 5.2 together with a CAT(0)space corresponding to the terminal vertex group T in the lower graph of groups. The buildingprocess uses the following three constructions in order:(1) Ultra-convex chaining;(2) Factor-diagonal chaining;(3) Doubling.The chaining operations in (1) and (2) are iterated adjunctions of spaces. The reader could see[Hat02] on page 12 and page 13 for the definition of adjunction (where it is called attachingspaces ) and notation. We describe these two chaining operations in detail. Ultra-convex chaining.
Start with a 2–dimensional, non-positively curved complex, K , withfundamental group F k ⋊ Z and a subcomplex R k ⊆ K which is a rose based at a vertex v ∈ K and which represents the F k kernel in π ( K , v ). Three different types of suchcomplexes K are described in subsection 5.2 above. UPEREXPONENTIAL DEHN FUNCTIONS INSIDE CAT(0) GROUPS 30
By repeatedly applying Theorem 5.2, one obtains a sequence of of 2–dimensional, non-positively curved complexes K , Y k , Y k , . . . , Y k n with fundamental groups π ( Y k i ) = F k i +1 ⋊ F k i , together with roses R k ⊆ K , R k ⊆ Y k , . . . , R k n ⊆ Y k n − and locally isometric embeddings f i : R k i ֒ → Y k i with ultra-convex images, for 1 ≤ i ≤ n . Here, π ( R k i +1 ) is the kernel F k i +1 and π ( f i ( R k i )) is the quotient F k i of π ( Y k i ).Define iterated adjunction spaces, K n , inductively as follows K i = Y k i ⊔ f i K i − for 1 ≤ i ≤ n .The next lemma describes the geometry of K n and relates its fundamental group to groups H i , defined as in Proposition 3.6. Lemma 5.10 (ultra-convex chaining) . Let K , Y k , . . . , Y k n and K , . . . , K n be as defined above.Define the groups H i for ≤ i ≤ n by H = π ( K ) and H i = ( F k i +1 ⋊ F k i ) ∗ F ki H i − for ≤ i ≤ n as in Proposition 3.6. Then the following hold for ≤ i ≤ n (1) H i = π ( K i ) ;(2) Each K i is a 2–dimensional, non-positively curved complex;(3) Each inclusion K i − ֒ → K i is a locally isometric embedding.Moreover, if all loops in the vertex links of K are strictly greater than π (so that H is ahyperbolic group), then each group H i is hyperbolic for i ≥ .Proof. Property (1) follows by induction and van Kampen’s theorem.By hypothesis K is a 2–complex, and each Y k i is a 2–complex by Theorem 5.2. Therefore,the K i are 2–complexes.The complexes K i are shown to be non-positively curved by induction. Since the K i are2–dimensional piecewise euclidean complexes, it is sufficient to verify that every loop in the linkof each vertex of K i has length at least 2 π . This is true for K by hypothesis. For each i ≥ v ∈ K i is obtained by gluing together the links of base vertices in K i − and Y k i , along a set of 2 k i points (namely, the link of the base vertex in the rose R k i )which is 2 π –separated in the latter link Lk ( v , X k ) = Lk ( v , X k − ) ∪ Lk ( v ,R ki ) Lk ( v , Y k i ) . The links of other vertices in K i − (if any) are unaffected by the adjunction of Y k i .By induction, all loops in the the link of each vertex in K i − have length at least 2 π . ByTheorem 5.2 all loops in Lk ( v , Y k i ) have length strictly greater than 2 π . So we only need toconsider the loops created when the link of the base vertex X i − is glued to the link of thevertex in Y k along the set Lk ( v , R k i ). By ultra-convexity, the mutual distances between these2 k i points on the Y k i side are at least 2 π . The distances between these points on the K i − side are all non-zero. Therefore, the new circuits created in the glueing all have length strictlygreater than 2 π and so each K i is non-positively curved (see Theorem 2.7).Next, we show that each inclusion K i − ֒ → K i is a locally isometric embedding. To do this,we need to verify that local geodesic paths in K i − remain local geodesics in K i . It suffices tocheck that this is the case at the base vertex v ∈ K i − . A local geodesic through v determinesa pair of points in the link Lk ( v , K i − ) which are at least π apart. The only way for this to failto be a local geodesic in K i is if these two points are less than π apart in the link Lk ( v , K i ). Butthis is impossible by the structure of Lk ( v , X k ) described above and since paths in Lk ( v , Y k i )which connect points of Lk ( v , K i ) are at least 2 π in length. UPEREXPONENTIAL DEHN FUNCTIONS INSIDE CAT(0) GROUPS 31
Finally, suppose that all loops in the the link of each vertex in K have length strictly greaterthan 2 π . By an analogous induction argument as above, all loops in the the link of each vertexin K i have length strictly greater than 2 π . Therefore, the universal cover of each complex K i is a CAT(0) space with a cocompact group action by isometries which does not admit anyembedded flat planes. By Flat Plane Theorem (see [BH99] III.H.1.5) each group H i = π ( K i )is hyperbolic. (cid:3) Remark 5.11.
The idea of glueing two non-positively curved 2–complexes by identifying anultra-convex graph in one complex with a (possibly non-convex) copy of the graph in the othercomplex to obtain a non-positively curved result was used in [BB06].
Factor-diagonal chaining.
It may be helpful if the reader refers to the schematic diagram inFigure 7 for this discussion. Each right triangle in that figure corresponds to a metric productof non-positively curved spaces whose labels are given on the edges adjacent to the right angle.The hypothenuse corresponds to a diagonally embedded subspace in this product space.The factor-diagonal chaining construction uses two ingredients.(1) The output of the ultra-convex chaining construction. That is, the 2–dimensional, non-positively curved complex K n together with the nested sequence of locally isometricallyembedded subcomplexes K ⊆ K ⊆ · · · ⊆ K n of Lemma 5.10.(2) A non-positively curved space Z with fundamental group T and a locally isometricembedding K ֒ → Z , identifying π ( K ) with the subgroup H ≤ T of Proposition 3.6.The idea behind the factor-diagonal chaining construction is that K i × K i − contains a locallyconvex diagonal copy of K i − which can be glued to the first factor of K i − × K i − . However,as we saw in Lemma 2.8, the diagonal copy of K i − is isometric to the scaled space √ K i − sowe need to keep scaling by √ ≤ i ≤ n , define Z i to be the space ( √ n − i K i × ( √ n − i K i − with theproduct metric. It is non-positively curved. Since K i − locally isometrically embeds into K i , the space ( √ n − i K i − × ( √ n − i K i − locally isometrically embeds into ( √ n − i K i × ( √ n − i K i − . Therefore, the diagonal∆ ( √ n − i K i − ⊆ ( √ n − i K i − × ( √ n − i K i − locally isometrically embeds into ( √ n − i K i × ( √ n − i K i − = Z i . In the product metric,the subspace ∆ ( √ n − i K i − is isomorphic to the scaled space( √ n − i +1 K i − = ( √ n − ( i − K i − . (2) For each integer 1 ≤ i ≤ n , there is a locally isometric embedding of ∆ ( √ n − i K i − ⊆ Z i into Z i − with image the first factor ( √ n − ( i − K i − . Furthermore, there is a locallyisometric embedding of ∆ ( √ n − K ⊆ Z into ( √ n Z with image ( √ n K .(3) Define the space Z to be the iterated adjunction of the spaces ( √ n Z , Z , . . . , Z n usingthese subspaces and locally isometric embeddings. The space Z is represented by onehalf of the diagram in Figure 7.The next lemma describes the geometry of the space Z (resp. the double of Z over thesubspace K n ) and shows that its fundamental group is the group G n (resp. G n ∗ H n G n ) ofProposition 3.6. Lemma 5.12 (factor-diagonal chaining) . Let n be a positive integer and let K ⊆ K ⊆ · · · ⊆ K n UPEREXPONENTIAL DEHN FUNCTIONS INSIDE CAT(0) GROUPS 32 be a sequence of non-positively curved spaces as in Lemma 5.10. Suppose that K locally isomet-rically embeds into a non-positively curved space Z whose fundamental group is T , identifying π ( K ) with the subgroup H ≤ T of Proposition 3.6. • Define spaces Z i for ≤ i ≤ n and Z by the factor-diagonal chaining constructionabove. • Let G = ( H × H ) ∗ (∆ H ≡ H ) T and G i = ( H i × H i − ) ∗ (∆ Hi − ≡ H i − × G i − for ≤ i ≤ n be as defined in Proposition 3.6.Then Z is a non-positively curved space satisfying the following properties:(1) Z has dimension max { , dim Z } ;(2) π ( Z ) = G n ;(3) the space K n locally isometrically embeds as the first factor into K n × K n − ⊆ Z inducingthe inclusion of H n as the first factor H n × ≤ H n × H n − ≤ G n .Moreover, the double of Z over K n in (3) above is a non-positively curved space of dimensionequal to max { , dim Z } with fundamental group G n ∗ H n G n where H n includes as the first factorin G n .Proof. The space Z is defined by the factor-diagonal chaining above. Since each K i is 2-dimensional, the products ( √ n − i ( K i × K i − ) are 4–dimensional and so Z has dimension equalto max { , dim Z } .For each 1 ≤ i ≤ n the inclusion ( √ n − i +1 K i − → ( √ n − i ( K i × K i − ) is a locally isometricembedding with image ∆ ( √ n − i K i − and induces the diagonal embedding H i − → ∆ H i − ≤ H i × H i − . Also, for each 1 ≤ i ≤ n inclusion ( √ n − i K i → ( √ n − i ( K i × K i − ) is a locallyisometric embedding with image the first factor space and induces the embedding H i → H i × ≤ H i × H i − . Finally, by hypothesis there is a locally isometric embedding ( √ n K → ( √ n Z inducing the group embedding H → T . Using Lemma 2.8, Proposition 2.9, van Kampen’sTheorem, and working by induction on n we conclude that Z is a non-positively curved spaceand has fundamental group G n . Z n Z n Z n − Z n − Z Z ( √ n Z ( √ n Z Z i = ( √ n − i ( K i × K i − ),for 1 ≤ i ≤ n . K n K n K n − ∆ K n − √ K n − √ K n − ( √ ) K n − ∆ √ K n − ∆ ( √ n − K ( √ n − K ( √ ) n K ∆ ( √ ) n − K ( √ ) n − K K n − √ K n − ∆ K n − √ K n − ( √ ) K n − ∆ √ K n − ∆ ( √ n − K ( √ n − K ( √ ) n K ∆ ( √ ) n − K ( √ ) n − K Figure 7.
The heart of the CAT(0) construction. The space Z corresponds toone half of this diagram. UPEREXPONENTIAL DEHN FUNCTIONS INSIDE CAT(0) GROUPS 33
There is a locally isometric embedding K n → K n × K n − ⊆ Z with image the first factor of K n × K n − and which induces the inclusion H n → H n × ≤ H n × H n − ≤ G n .Since the inclusion K n → Z above is a locally isometric embedding, the double of Z over K n is non-positively curved by Lemma 2.8 and Proposition 2.9. It has dimension equal to thedimension of Z and has fundamental group G n ∗ H n G n . (cid:3) The Main Theorem and open questions
The main theorem of this paper follows by combining the CAT(0) constructions of Section 5for various choices of the space Z , the group embedding result of Section 3, and the Dehnfunction computations of Section 4. We recall the construction here. Construction.
The construction takes as input the following collection of spaces and groups.(i) A non-positively curved space Z with fundamental group T .(ii) A locally isometrically embedded subspace K ⊆ Z whose fundamental group is F k ⋊ Z which is identified with a subgroup H ≤ T .(iii) A rose R k ⊆ K whose fundamental group is the kernel F k ≤ F k ⋊ Z = H .(iv) A subgroup S ≤ T such that S ∩ H = F k is the free kernel of H and such that Dist SF k is equivalent to a non-decreasing, super-additive function f ( x ).For 1 ≤ i ≤ n let F k i +1 ⋊ F k i be an inductively defined sequence of hyperbolic, CAT(0) groupswhich are constructed as in Theorem 5.2. As in Proposition 3.6, define sequences of groups H i and G i for 1 ≤ i ≤ n and L i for 0 ≤ i ≤ n inductively as follows:(1) H is the group in (2) above and H i = ( F k i +1 ⋊ F k i ) ∗ F ki H i − for 1 ≤ i ≤ n ,(2) G = ( H × H ) ∗ (∆ H ≡ H ) T where T is given in (1) above and G i = ( H i × H i − ) ∗ (∆ Hi − ≡ H i − × G i − for 2 ≤ i ≤ n , and(3) L = S where S is the group given in (4) above and L i = ( F k i +1 ⋊ F k i ) ∗ F ki L i − for 1 ≤ i ≤ n .Let C n be the double of G n over H n and let D n be the double of H n over F k n +1 . Proposition 6.1. C n is a CAT(0) group with geometric dimension at most max { , dim Z } and C n contains a copy of D n as a subgroup. Moreover, if the Dehn function of group L n is domi-nated by some polynomial function, the the Dehn function of D n is equivalent to exp ( n ) ( f ( x )) .Proof. The group C n is CAT(0) of dimension at most max { , dim Z } by Lemma 5.12. Thegroup embedding D n ≤ C n is established in Proposition 3.6. The Dehn function computationis given in Proposition 4.4. (cid:3) Theorem 6.2 (main theorem) . Let n be a positive integer and define exp ( n ) ( x ) inductively by exp (1) ( x ) = exp( x ) and exp ( k +1) ( x ) = exp(exp ( k ) ( x )) for k ≥ . Then(1) There are CAT(0) groups of geometric dimension 4 containing finitely presented sub-groups whose Dehn functions are exp ( n ) ( x m ) for integers m ≥ . The maximal rank offree abelian subgroups in these groups is 4.(2) There are CAT(0) groups of geometric dimension 4 containing finitely presented sub-groups whose Dehn functions are exp ( n ) ( x ) for integers n ≥ . The maximal rank offree abelian subgroups in these groups is 2. UPEREXPONENTIAL DEHN FUNCTIONS INSIDE CAT(0) GROUPS 34 (3) There are
CAT(0) groups of geometric dimension 6 containing finitely presented sub-groups whose Dehn functions are exp ( n ) ( x α ) for α dense in [1 , ∞ ) . The maximal rankof free abelian subgroups in these groups is 6.Proof. We prove the theorem by choosing suitable input data for the construction above.For Statement (1) we let H = F k ⋊ ϕ h t i be one of the free-by-cyclic groups from Example (1)of Subsection 5.2. In that subsection we proved that Dist F k ⋊ ϕ h t i F k is equivalent to x m and F k ⋊ ϕ h t i is the fundamental group of a complex K satisfying condition (iii) and half ofcondition (ii) of the construction above. Let Z = K × S with the product metric. Then Z is a non-positively curved space of dimension 3 and the inclusion K ⊂ Z into the first factoris a locally isometric embedding. This gives condition (i) and the remainder of condition (ii).Let T = π ( Z ) = ( F k ⋊ ϕ h t i ) × h u i and H = π ( K ) = F k ⋊ ϕ h t i . Let S be the subgroupof T generated by F k and tu . Then S is the free-by-cyclic group F k ⋊ ϕ h tu i , S ∩ H = F k ,and the distortion function Dist SF k is also equivalent to x m . This gives condition (iv) of theconstruction.In this case, the two groups L n and H n are isomorphic and therefore they are both CAT(0)groups by Lemma 5.10. This implies that the Dehn function of L n is at most quadratic.Therefore, the CAT(0) group C n constructed above contains a subgroup whose Dehn functionis exp ( n ) ( x m ) by Proposition 6.1. Also, the group C n has geometric dimension at most 4 byProposition 6.1 and at least 4 by the existence of Z subgroups. Therefore, the geometricdimension of C n is exactly 4 and the maximal rank of free abelian subgroups in C n is also 4.The proof of Statement (2) is almost identical to the proof of Statement (1) except we let H = F k ⋊ ϕ h t i be a CAT(0) hyperbolic free-by-cyclic group as in Example (2) of Section 5.2and we consider the CAT(0) group C n − instead of the CAT(0) group C n . As we proved forStatement (1), the CAT(0) group C n − in this case has geometric dimension at most 4 and itcontains a subgroup D n − whose Dehn function is exp ( n ) ( x ) by Proposition 6.1. Moreover, thesubgroup H × H of C n has dimension exactly 4 (by [Dra19] and Chapter VIII Corollary (7.2)in [Bro94]). Therefore, the geometric dimension of C n − is also exactly 4.The group C n − contains a Z subgroup since (for example) it contains the subgroup T =( F k ⋊ Z ) × Z . We argue that C n − does not contain Z by contradiction. Assume that C n − contains Z . Note that each group H j ≤ C n − in this case is hyperbolic (see Lemma 5.10) andso C n − = G n − ∗ H n − G n − is the fundamental group of a graph of groups whose vertex groupsare direct products of two hyperbolic groups and edge groups are hyperbolic. Therefore, theintersection of Z with each edge group conjugate is either Z or trivial. Since Z does not splitnontrivially over Z or 1, we conclude that Z is conjugate into some vertex group. This is acontradiction since each vertex group is a direct product of two hyperbolic groups and does notcontain Z . This implies that the maximal rank of free abelian subgroups in C n − is 2.For Statement (3) we let H = F × Z be the CAT(0) group from Example (3) of Subsec-tion 5.2. In this case, S is the snowflake group and T the ambient CAT(0) group from [BF17].The precise connection with [BF17] is described in Table 1. Note that the first two rows of thetable give condition (i) of the construction. Rows 3 and 4 of the table give condition (ii) ofthe construction. The rose R of condition (iii) is described in Example (3) of Subsection 5.2.Finally, condition (iv) is assured by the last two rows of the table. In particular, the Bassintersection property is a key result in [BF17].The distortion Dist SF k = Dist SF is computed implicitly in [BF17] and is stated explicitly inLemma 5.6 in [BT]. It is equivalent to x α for α dense in [1 , ∞ ). Let L n and C n be groupsconstructed above. By construction, L n is the amalgamation H ∗ F S where H is a hyperbolicgroup (see Lemma 5.10) and F is a retract of H . Moreover, the Dehn function of S is equivalentto x α (see [BF17]) and so is bounded above by a polynomial, the Dehn function of H is linear, UPEREXPONENTIAL DEHN FUNCTIONS INSIDE CAT(0) GROUPS 35
Current paper [BF17] paper
CAT(0) group T CAT(0) group G T,n in Theorem 4.5 T = π ( Z ) where Z is n.p.c. G T,n = π ( K T,n ) where K T,n is n.p.c.Convex subgroup H = F ⋊ Z ≤ T Convex subgroup B v ≤ G T,n H = π ( K ) where B v = π ( L ) where K ⊂ Z is convex L ⊂ K T,n is convex (eqn. (4.3))Subgroup S of T Subgroup S T,n of G T,n in Theorem 5.6Subgroup F = S ∩ H Subgroup A v = B v ∩ S T,n
Table 1.
Correspondence between current construction and [BF17]. The no-tation n.p.c. denotes non-positively curved.and the distortion of F in S is equivalent to x α . By Proposition 4.5 and Remark 4.6 theDehn function of L n is dominated by a polynomial. By Proposition 6.1, the CAT(0) group C n contains a subgroup whose Dehn function is exp ( n ) ( x α ).We note that the dimension of the space Z chosen above in this case is 6 (see [BF17]).Therefore, the geometric dimension of C n is at most 6 by Proposition 6.1 and is at least 6 bythe existence of Z subgroups (which are contained in the T vertex groups). Therefore, thegeometric dimension of C n is exactly 6 and the maximal rank of free abelian subgroups in C n is also 6. (cid:3) Note that the CAT(0) ambient groups in the Main Theorem have geometric dimension 4 or6. Also, the maximum rank of a free abelian subgroup is 2 in the exp n ( x ) examples, 4 in theexp n ( x m ) (for integers m ≥
2) examples, and 6 in the exp ( n ) ( x α ) examples. Also, these CAT(0)groups are not relatively hyperbolic with proper peripheral subgroups. These observations (andrelated considerations) prompt the following questions. Question 6.3 (3–dimensions) . The constructions of the CAT(0) spaces in the Main Theoremuse products and so the ambient CAT(0) groups have geometric dimension at least 4.(1) Do there exist CAT(0) groups of geometric dimension 3 which contain finitely presentedsubgroups with superexponential Dehn function?(2) Is there an upper bound on the Dehn functions of finitely presented subgroups of 3–dimensional CAT(0) groups?Corollary 1.3 of [KP20] implies that it is impossible to build 2–dimensional examples; indeed,if a finitely presented group coarsely embeds into a CAT(0) group of geometric dimension 2,then it must satisfy a quadratic isoperimetric inequality. If one is investigating 3–dimensionalconstructions, it is good to be aware of [HMP14] which gives a general treatment of towerarguments and their implications for the structure of finitely presented subgroups in varioussettings.
Question 6.4 (relatively hyperbolic) . The prevalence of product structures in our examplesmean that the groups are not relatively hyperbolic with proper peripheral subgroups. Indeed,by construction, our groups are geometrically thick and so are not relatively hyperbolic (seeCorollary 7.9 in [BDM09]). The following questions are interesting.(1) Are there CAT(0) groups which are relatively hyperbolic with abelian peripheral struc-ture and which contain finitely presented subgroups with superexponential Dehn func-tion?(2) Is there an upper bound on the Dehn functions of finitely presented subgroups of rela-tively hyperbolic CAT(0) with abelian peripheral structure?
UPEREXPONENTIAL DEHN FUNCTIONS INSIDE CAT(0) GROUPS 36
The various definitions of relatively hyperbolic groups involve actions on hyperbolic metricspaces. Furthermore, there is a short path from relatively hyperbolic groups to hyperbolicgroups via hyperbolic Dehn fillings. Perhaps investigations into the previous set of questionswill shed light on the next set.
Question 6.5 (hyperbolic) . The following questions are folklore.(1) Are there hyperbolic groups which contain finitely presented subgroups with Dehn func-tion bounded below by an arbitrary polynomial function?(2) Are there hyperbolic groups which contain finitely presented subgroups with exponentialor superexponential Dehn functions?There are finitely presented subgroups of hyperbolic groups which are not hyperbolic. See[Bra99], [Lod18], and [KG18] for examples. All known examples are not of type FP . TheDehn functions of these subgroups are bounded below by a quadratic function and are boundedabove by a polynomial [GS02]. Question 6.6 (Akermannian) . Are there CAT(0) groups containing finitely presented sub-groups with Akermannian Dehn functions?The hydra groups of [DR13] (and the hyperbolic hydra groups of [BDR13]) are 2-dimensionalCAT(0) groups containing finite rank free subgroups with Akermannian distortion. One canamalgamate a hyperbolic F ℓ ⋊ F k group with ultra-convex F k with a hydra group, identifyingthe F k with the highly distorted free subgroup in the hydra group. This yields a 2-dimensionalCAT(0) group containing a free-by-free subgroup and a convex hydra subgroup. It is temptingto explore if some variation of the constructions in the current paper can produce subgroups ofCAT(0) groups with Ackermannian Dehn functions. A direct application of the constructionin this paper will not work because the highly distorted subgroup of the hydra group is notnormal. Question 6.7 (cubical, RAAG) . The construction in the Main Theorem makes use of amalga-mations along diagonal subgroups of direct products. This suggests that these CAT(0) groupsmight not act properly and cocompactly by isometries on CAT(0) cube complexes. Here arerelated questions.(1) Are there CAT(0) cubical groups containing finitely presented subgroups with superex-ponential Dehn functions?(2) Are there RAAGs containing finitely presented subgroups with superexponential Dehnfunctions?It is known that RAAGs contain finitely presented subgroups with exponential Dehn function[Bri13] and polynomial Dehn function of arbitrary degree [BS19].
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