# Homological and homotopical Dehn functions are different

^{a}Mathematics Department, Washington and Lee University, Lexington, VA 24450;^{b}Department of Mathematics, University of Oklahoma, Norman, OK 73019;^{c}Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803-4918; and^{d}Department of Mathematics, University of Toronto, Toronto, ON, Canada, M5S 2E4

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Edited by Robion C. Kirby, University of California, Berkeley, CA, and approved May 3, 2013 (received for review May 2, 2012)

## Abstract

The homological and homotopical Dehn functions are different ways of measuring the difficulty of filling a closed curve inside a group or a space. The homological Dehn function measures fillings of cycles by chains, whereas the homotopical Dehn function measures fillings of curves by disks. Because the two definitions involve different sorts of boundaries and fillings, there is no a priori relationship between the two functions; however, before this work, there were no known examples of finitely presented groups for which the two functions differ. This paper gives such examples, constructed by amalgamating a free-by-cyclic group with several Bestvina–Brady groups.

## 1. Introduction

The classical isoperimetric problem is to determine the maximum area that can be enclosed by a closed curve of fixed length in the plane. This problem has been generalized in many different ways. For example, in a metric space *X*, one can study the homotopical filling area of a curve *γ*, denoted and defined to be the infimal area of a disk whose boundary is *γ*. This leads to the idea of the *homotopical Dehn function* of *X*, defined as the smallest function such that any closed curve *γ* of length has filling area at most . A remarkable result of Gromov (1) states that if *X* is simply connected and if there is a group *G* that acts on *X* geometrically (i.e., cocompactly, properly discontinuously, and by isometries), then the growth rate of depends only on *G*; indeed, is related to the difficulty of determining whether a product of generators of *G* represents the identity. We can thus define the Dehn function of a group to be the (homotopical) Dehn function of any simply connected space that the group acts on geometrically; this is well-defined up to certain constants (details are provided in *Section 2*).

Another way to generalize the isoperimetric problem is to consider fillings of 1-cycles by 2-chains instead of fillings of curves by disks. If *α* is a 1-cycle in *X*, we can define its homological filling area, , to be the infimal mass of a 2-chain in *X* with integer coefficients whose boundary is *α*. This leads to the *homological Dehn function* , defined as the smallest function such that any 1-chain *α* of mass at most has a homological filling of area at most . Like its homotopical counterpart, can be used to construct a group invariant: If and if there is a group *G* that acts on *X* geometrically, then the growth rate of depends only on *G* and we can define . Again, this is well-defined up to constants.

The exact relationship between these two filling functions has been an open question for some time. The homological Dehn function deals with a wider class of possible fillings (surfaces of arbitrary genus) and a wider class of possible boundaries (sums of arbitrarily many disjoint closed curves), so it is not a priori clear whether is always the same as when they are both defined. Some hints that they may differ come from a construction, due to Bestvina and Brady (2), of groups with unusual finiteness properties. Bestvina and Brady used a combinatorial version of Morse theory to construct a group that is but not finitely presented. Such a group does not act geometrically on any simply connected space but does act geometrically on a space with trivial first homology, so its homological Dehn function is defined but its homotopical Dehn function is undefined.

In this paper, we will construct a family of finitely presented groups such that grows strictly more slowly than . Specifically, we will show the following.

### Theorem 1.1.

*For every d* ∈ ℕ ∪ {∞}, *there is a CAT(0) group G containing a finitely presented subgroup H such that* FA_{H}(ℓ) ≤ ℓ^{5} *and the homotopical Dehn function satisfies*

#### Remark:

Using the methods of Brady, Guralnik, and Lee (3), one can show that in the case, the group *H* constructed in the theorem satisfies .

Our construction uses methods of Brady, Guralnik, and Lee (3) to create a hybrid of a Bestvina–Brady group with a group having a large Dehn function. The resulting group is finitely presented, so both and are defined, and we will show that the unusual finiteness properties coming from the Bestvina–Brady construction lead to a large gap between homological and homotopical filling functions.

Similar results are known for higher dimensional versions of *δ* and FA. One can define *k*-dimensional homotopical and homological Dehn functions by considering fillings of *k*-spheres or *k*-cycles by -balls or -chains; by historical accident, the corresponding homotopical and homological filling functions have come to be called and , respectively. The relationship between and is better understood when , because in this case, the Hurewicz theorem can be used to replace cycles and chains by spheres and balls.

If *X* is *k*-connected and *β* is a -chain with , then the Hurewicz theorem can be used to show that *β* is the image of the fundamental class of a ball under a map with . Thus, if is a map of a sphere and *α* is the image of the fundamental class of under *a*, then , so for (see appendix 2 in ref. 4, 5).

Likewise, if *X* is *k*-connected and *α* is a *k*-cycle for , then the Hurewicz theorem can be used to show that *α* is the image of the fundamental class of a sphere under a map such that [see remark 2.6.(4) in ref. 6]. Consequently, because , we have for .

Thus, if and if *H* is a group that acts geometrically on a *k*-connected complex, the above results imply that . When , Young (7) constructed examples of groups for which . The examples in this paper are the earliest known examples of groups for which .

## 2. Preliminaries

### 2.1. Dehn Functions.

For a full exposition of Dehn functions, we recommend the study by Bridson (8). We will briefly review the definitions that we will need. Let *X* be a simply connected Riemannian manifold or simplicial complex. If is a Lipschitz map, define the *homotopical filling area* of *α* to bewhere *β* ranges over Lipschitz maps which agree with *α* on . Because *X* is simply connected and any continuous map can be approximated by a Lipschitz map, such maps exist. We can define an invariant of *X* by letting

We call this the *homotopical Dehn function* of *X*.

We define a relation on functions by if there is a such that for all *n*,

If and , we write . Thus, distinguishes all functions for , and all functions of the form are equivalent for . Gromov (1) stated [and Bridson (8) proved] that if *H* acts geometrically on *X* (for instance, if and for some compact *M*), then is determined up to -equivalence by *H*. If *H* is finitely presented, then *H* acts geometrically on the universal cover of a presentation complex, which is a 2-complex with . Thus, is well-defined up to .

To define the homological invariant , suppose that *X* is a polyhedral complex with . If *α* is a 1-cycle in *X*, we letwhere is defined to be if is a sum of 2-simplices in *X* with integer coefficients. We can define an invariant of *X* by letting

We call this the *homological Dehn function* of *X*. Like the homotopical Dehn function, if *H* acts geometrically on *X*, then is determined up to by *H*, and if is a presentation complex for a finitely presented group *H*, we define .

### 2.2. Right-Angled Artin Groups.

If is a simple graph (i.e., without loops or multiple edges), we can define a *right-angled Artin group* (RAAG) based on . If and are the vertex set and edge set of , respectively, we definewhere are the functions taking an edge to its start and end. We say that is the *defining graph* of . These RAAGs generalize free groups and free abelian groups; if is a complete graph, there is an edge between every pair of vertices, so every pair of generators of commutes and is free abelian. On the other hand, if has no edges, then is a free group.

A full exposition of RAAGs can be found in the paper by Charney (9). One important fact that we will use is that for every , there is a one-vertex locally CAT(0) cube complex with ; this is called the *Salvetti complex*. This complex can be built directly from the graph : it has one vertex, one edge for every vertex of , one square for each edge of , and in general one *n*-cube for each *n*-vertex clique in .

Bestvina and Brady (2) used RAAGs to construct subgroups of nonpositively curved groups with unusual finiteness properties. They defined a homomorphism which sends each generator of to 1, and, viewing as the 0-skeleton of , they extended to a map . This map is linear on each cube of , so the level set can be given the structure of a polyhedral complex. The subgroup acts vertex-transitively on , so if is connected, the 1-skeleton of is a Cayley graph for . In this case, we can construct a generating set for explicitly: each edge of is a diagonal of a square of , so has a generating set consisting of elements of the form , where *a* and *b* are generators of .

Recall that a complex is *flag* if every clique of *n* vertices spans an -dimensional simplex. Bestvina and Brady (2) proved that the topology of determines the topology of .

### Theorem 2.1 (Bestvina and Brady).

*If* *is the 1-skeleton of a flag complex Y*, *and* *are the maps defined above*, *then* *acts on the complex* , *which is homotopy equivalent to a wedge product of infinitely many copies of Y*, *indexed by the vertices in* . *In fact*, *is a union of infinitely many scaled copies of Y*.

The main tool used to prove this theorem is a combinatorial version of Morse theory. If *X* is a complex, is a vertex, and is a function that is linear on each cell and is not constant on any edge, one may define subcomplexes and of the link called the *ascending* and *descending links* of *x*. To define these, we identify the vertices of with the neighbors of *x*. The ascending link is the full subcomplex spanned by vertices *y* such that ; likewise, the descending link is the full subcomplex spanned by vertices *y* such that . These ascending and descending links play a role similar to the ascending and descending manifolds in classical Morse theory.

If *X* has one vertex, then all vertices of have the same link, so we will write , , and . The link has two vertices for each generator *s* of ; the ascending link is spanned by the ’s, and the descending link is spanned by the ’s. If *Y* is the flag complex with 1-skeleton , then and are isomorphic to *Y*.

### 2.3. Labeled Oriented Graph Groups.

We can also construct groups using *labeled oriented graphs* (LOGs). A LOG on a set *S* is a directed multigraph with vertex set *S* and a labeling of the edges given by ; loops and multiple edges are allowed. We say that presents the group:where the notation represents the conjugation and, again, are the end points of *e*. Because each relation has length 4, the presentation 2-complex of is a two-dimensional cube complex.

Note that although is possible, we may assume that because, otherwise, we could contract such an edge without changing the group. This implies that contains no loops or edges of the form .

As with RAAGs, we can apply Morse theory to LOG groups. Let be the homomorphism mapping each to 1. This homomorphism can be extended linearly over each cell of to get a map . Consider the level set . As in the RAAG case, the group acts on vertex-transitively, so if is connected, then its 1-skeleton is a Cayley graph for . Edges in are diagonals of squares in , so each orbit of squares labeled contributes a generator that can be written as or (Fig. 1).

As was the case with RAAGs, the link has two vertices for each vertex *s* of . The ascending link is the full subcomplex of spanned by the ’s, and the descending link is spanned by the ’s. Brady (10) showed the following.

### Theorem 2.2 (Brady).

*Suppose* *is a group presented by a LOG* *such that*:

•

*The ascending and descending links**and**are trees.*•

*The full link**has girth at least 4.*

*Then*, (1) *is locally CAT(0), hence a* ; (2) *the level set* *is a tree; and* (3) *is isomorphic to the free-by-cyclic group* , *where* .

Brady, Guralnik, and Lee (3) used these groups to construct Stallings-type examples of groups that are of type but not of type and that have Dehn functions with prescribed polynomial or exponential growth rates.

## 3. Main Theorem

To understand our construction, first consider the problem of constructing a space where the homological and homotopical filling functions differ. Suppose *W* is a simply connected space with a large Dehn function and *α* is a closed curve in *W*. To reduce the homological filling area but not the homotopical filling area of *α*, we could attach a 2-complex *Z* to *α* in which *α* is the boundary of a 2-chain but not the boundary of a disk. If , the resulting space is still simply connected. By attaching copies of *Z* to infinitely many closed curves, we can obtain a complex that has large *δ* but small FA.

Our construction will be based on a graph of groups with each vertex labeled by one of two groups, *A* and *Q*. The first group, *A*, will be a RAAG with a kernel that is but not finitely presented. This subgroup acts geometrically on a space that has trivial and nontrivial , which will provide the *Z*’s in the construction.

We define a *Thompson complex* to be a connected, finite, two-dimensional flag complex *Y* whose fundamental group is a simple group with an element of infinite order. (The name comes from the earliest known group with these properties, Thompson’s group *T*.)

Let *Y* be a Thompson complex (for example, a triangulation of a presentation complex for Thompson’s group). Note that because is simple, , and every normally generates all of . Let be an element of infinite order. By gluing an annulus to *Y*, we may assume that there is a path of length 4 in the 1-skeleton of *Y* which represents *g*. We label the vertices of this path , and we label the rest of the vertices of *Y* by . Since *Y* is flag, the subcomplex spanned by must be a cycle of length 4. We will consider the RAAG , where is the 1-skeleton of *Y*.

As we will not need to refer to explicitly, we drop it from the notation and set . We denote the associated homomorphism by , its extension to a Morse function by , the level set by , and so on. By results of Bestvina and Brady (2), the group is but not finitely presented.

The second group, *Q*, will be a product of a LOG group and a free group. Suppose we are given a LOG that satisfies the hypotheses of *Theorem 2.2*. We may form a new LOG by adding an isolated vertex *s* to , adding a loop connecting a vertex *t* of to itself, and labeling the new edge by *s*. This corresponds to adding a generator *s* and a relation to . We call a LOG obtained this way a *special LOG* (SLOG), and the corresponding group a *SLOG group*. Note that still satisfies the hypotheses of *Theorem 2.2*.

As with , we will often omit from the notation when it is easily understood. We will abbreviate by *B*, by , by , and so on.

If *B* is a SLOG group, then by *Theorem 2.2*, it can be written as a free-by-cyclic group . (The notation indicates a rank *n* free group; if we want to emphasize a particular free basis , we will write .) We define to be the distortion of inside *B*; precisely:where is the word length of *g* in the subscripted group. Brady, Guralnik, and Lee (3) give constructions of SLOG groups with , and also with for all integers *d* ≥ 3. The Bieri–Stallings double of *B*, denoted , has large Dehn function resulting from this distortion. Specifically, Bridson and Haefliger (theorem III.Γ.6.20 of ref. 11) showed the following.

### Theorem 3.1 (Bridson and Haefliger).

*If B and D are as above*, *then*

The group *D* will serve as *W* in our construction; because its Dehn function is large, it has many curves that are difficult to fill by disks. By an embedding trick appearing in a paper by Baumslag, Bridson, Miller, and Short (12), *D* can be viewed as a subgroup of the product ; in fact, we will see that *D* is the kernel of a map .

We will construct a finitely presented CAT(0) group *G* as a graph product of *Q* with several copies of *A*. The subgroup *H* will be the kernel of a map , and *H* will have the structure of a graph product of copies of *D* and . We will show that attaching to *D* does not affect *δ* but that the copies of *Y* that lie in can be used to replace fillings by disks with more efficient fillings by chains.

### Theorem 3.2.

*Let A be a RAAG based on a Thompson complex as described above*, *and let* *be a SLOG group*. *Then there exists a finitely presented CAT*(*0*) *group G containing A and B such that the homomorphisms* *and* *extend to* *and such that* *is finitely presented and satisfies*:

Using the examples of SLOG groups constructed by Brady, Guralnik, and Lee (3), this implies Theorem 1.1.

## 4. Constructing *G* and *H*

In this section, we construct the groups *G* and *H* of *Theorem 3.2*. The construction is similar to the perturbed RAAGs in the paper by Brady, Guralnik, and Lee (3), but we glue several RAAGs (rather than just one) to a free-by-cyclic group.

Throughout this paper, if *g* is a group element, will represent its inverse.

### 4.1. The SLOG Piece.

Let be as in *Theorem 3.2*. The first step of the construction is to use *B* to construct a group with large Dehn function. The group *Q* will contain several copies of the group , and we will attach RAAGs to *Q* along some of these groups. The result of this gluing will be *G*.

Since is a SLOG, it contains an isolated vertex *s*, which is the label of a single loop in . Call the vertex of that loop *t*. Call the rest of the vertices . We have two presentations for *B*, namely, the SLOG presentation with generating set and a free-by-cyclic presentation. For the latter, we may take as a generating set, where for and (Fig. 1). Thus, .

Let *D* be the double , where the is generated by the . *Theorem 3.1* implies that *D* has Dehn function at least as large as . By results of Baumslag, Bridson, Miller, and Short (12), *D* is isomorphic to the subgroupof the group . Furthermore, if is the group homomorphism taking the elements to , then the kernel of is precisely *D*.

Because is a SLOG, the group contains many copies of . Recall that the presentation 2-complex of *B* is a locally CAT(0) two-dimensional cube complex and thus a . For any *i*, consider the subgroup of *B* generated by and *s*. It is easy to check that any two of the vertices and in the link of are separated by a distance of at least 2 in . Consequently, the Cayley graph of the subgroup generated by and *s* is convexly embedded in . It is therefore a copy of .

Define , where is a wedge of two circles, so that is a . This is locally CAT(0), and by the argument above, for all *i*, the subgroup generated by is a convexly embedded copy of .

### 4.2. Attaching the RAAG Pieces.

Let be a RAAG constructed from a Thompson complex *Y* as in *Section 3*. Thus, has infinite order and is represented by a path in the defining graph for *A*. Let be the Salvetti complex of *A*. Because , and *v* span a square in *Y*, *A* has a convex subgroup isomorphic to , generated by . Let .

We form the group *G* by gluing copies of *A* to *Q* along copies of *E*. Specifically, consider a graph of groups with vertex groups , where , and with each vertex connected to *Q* by an edge. Each edge group will be isomorphic to *E*. As noted above, for each *i*, the elements generate a copy of *E*; denote this copy by . We identify with the copy of *E* in by , , , . Let *G* be the fundamental group of this graph of groups. This is a group generated by

We can define subgroups *Q*, , and , where , , and

The homomorphisms and agree on the edge groups, so we can extend them to a function .

Let .

## 5. Finite Presentability

In this section, we will construct a space on which *H* acts and consider its topology. This will let us prove that *H* is finitely presented and will help us bound the Dehn functions of *H*.

We can realize the above construction of *G* geometrically to construct a as follows. Let , where is the wedge of two circles; this is a , and each edge group corresponds to copies of in and . Each of these copies of is convex, so we can glue copies of to along the ’s to obtain a locally CAT(0) cube complex that we call . This is a , and the 1-skeleton of its universal cover is a Cayley graph of *G*. In particular, we have the following.

### Lemma 5.1.

*The group G is CAT*(*0*). □

Now, *H* is the kernel of the homomorphism . As before, the vertices of are in correspondence with the elements of *G*, so by viewing *h* as a function on the vertices of , we may extend *h* linearly over cubes to obtain a Morse function . Let .

Because *h* cuts cubes of “diagonally,” is a polyhedral 2-complex whose cells are slices of cubes. The subgroup *H* acts freely on , and since the vertices of are in one-to-one correspondence with the elements of *H*, the action is cocompact and thus geometric. We now show the following.

### Lemma 5.2.

*is simply connected and thus* *is finitely presented*.

#### Proof:

Because is contractible and *h* is a Morse function on , theorem 4.1 in ref. 2 implies that it is enough to show that the ascending and descending links of any vertex in are simply connected. Since has only one vertex, it is enough to show this for that vertex.

Since the 1-skeleton of is a Cayley graph of *G*, we can label the vertices of by , where *g* ranges over the generating set *S*. The link of the vertex of is obtained by gluing and the various . For each , the links and each contain a subcomplex with vertices , , , and , and gluing the links along these subcomplexes gives .

Likewise, we can form by gluing , , to along subcomplexes spanned by , and . We claim that is simply connected. Since is a tree by hypothesis, is the suspension of a tree (with suspension points and ), and thus, it is simply connected. Since is a RAAG with defining complex *Y*, each is isomorphic to *Y*, and each is a square such that the normal closure of in is all of . By the Seifert–van Kampen theorem,so the ascending link is simply connected. The same argument with +'s changed to −'s shows that the descending link is also simply connected, so is simply connected. Therefore, and *H* is finitely presented. □

For an alternate description of *H*, recall that the group *G* is the fundamental group of a graph of groups, with one vertex labeled *Q* connected to vertices labeled by edges labeled . Since , *G* induces a graph of groups structure on *H*. Indeed, *G* acts on a tree *T* whose vertices correspond to the cosets of *Q* and , whose edges correspond to cosets of , and whose quotient is a star with edges. We can restrict the action of *G* on *T* to an action of *H*, and since any coset of *Q*, , or has nontrivial intersection with *H*, the orbit of any vertex or edge under *H* is the same as its orbit under *G*. Therefore, *H* acts on *T* with vertex stabilizers conjugate to and , edge stabilizers conjugate to , and quotient . This shows that *H* is the fundamental group of the graph of groups with a central vertex labeled connected to vertices labeled by edges labeled .

The level set , however, is *not* the universal cover of a corresponding graph of spaces. To describe , we define , , and . These level sets have geometric actions by , , and , respectively. We can write the quotient as with the attached along copies of , but because and are not simply connected, and ; this is a graph of spaces for a different graph of groups.

The fact that and are not simply connected will be important in the rest of this paper, so we will go into some more detail. All the ’s and all the ’s are isometric, so when *i* is unimportant, we will denote them by and , respectively. To understand the topology of and , consider them as subsets of . According to Bestvina and Brady (2), is a union of scaled copies of *Y*, indexed by vertices in . Likewise, is composed of scaled copies of a square, which we denote , indexed by vertices in . Translating by elements of gives infinitely many disjoint copies of inside .

By a result of Bestvina and Brady (theorem 8.6 in ref. 2), is homotopy equivalent to an infinite wedge sum of copies of *Y* and is homotopy equivalent to an infinite wedge sum of copies of , so and have infinitely generated . Generators of can be filled in two ways. First, each generator can be freely homotoped into some copy of . Each copy of is contained in some , and because is simply connected, each generator of is filled by a disk in one of the copies of . Second, although is infinitely generated, is trivial; thus, any curve in can be filled by some 2-chain entirely inside . Our goal in the rest of this paper is to use these two types of fillings to show that the homological and homotopical Dehn functions of are different.

## 6. Upper Bound on the Homological Dehn Function

In this section, we prove the following.

### Proposition 6.1.

*With H as above*, *we have* .

#### Proof:

We show that any 1-cycle in of mass at most can be filled by a 2-chain of mass . Since is a superadditive function, it is enough to prove this for loops in .

consists of copies of and glued together along copies of . We first show how to homologically fill loops that lie in a single copy of or , and we then use these fillings to fill arbitrary loops. Note that each 1-cell of is a diagonal of some square in , so each 1-cell corresponds to a product , where *x* and *y* are (certain) generators of *G*.

Consider a loop *α* of length that lies in a copy of . Recall that is a level set of the Morse function . Because is CAT(0), there exists a 2-chain *β* with boundary equal to *α* and mass . Further, *β* lies in for some (13; cf. proposition 2.2 in ref. 14). We will use *β* to produce a filling of *α* in using a pushing map as in the paper by Abrams, Brady, Dani, Duchin, and Young (14).

Let *Z* be the space obtained by deleting open neighborhoods of the vertices of outside , with the induced cell structure; that is, . According to Abrams, Brady, Dani, Duchin, and Young (theorem 4.2 in ref. 14), there is an -equivariant locally Lipschitz retraction (pushing map) such that the Lipschitz constant grows linearly with distance from . Furthermore, if is the boundary of one of the deleted neighborhoods, the image of is a copy of *Y* with the metric scaled by a factor of . In particular, if *γ* is a 1-cycle in of length , then is a 1-cycle in a scaled copy of *Y*. Since and *Y* is compact, the corresponding 1-cycle in *Y* has homological filling area . Therefore, the original cycle, , has homological filling area .

Consider the restriction of *β* to *Z*; this is a 2-chain in *Z*, and where is a 1-cycle in . We can construct a filling of *α* in by combining the image with fillings of each of the ’s. Since lies in , the restriction of to has Lipschitz constant, and

Each 2-cell of *β* contributes at most four 1-cells to the ’s; thus, , and we have .

Next, we produce a quartic mass filling of any loop that lies entirely in a copy of . Such a loop *α* is labeled by generators ofwhere for and . In this section, we will use the notation and . Let *w* denote the word labeling *α*, where

Here, the are generators and *w* represents the identity.

Let send to and each other generator to itself. The word lies in a free-by-cyclic group which is CAT(0) and therefore has quadratic Dehn function. Thus, to fill our loop with quartic mass, it will be enough to reduce *w* to in such a way that the reduction takes quartic mass.

We will achieve this reduction by first decomposing *w* into subwords as follows. Let be the projection map. Defineso that . Now, decompose *w* as follows:We can reduce *w* to by reducing each subword in this decomposition to its image under *ϑ*. If , then is freely equal to the identity and no reduction is necessary. Otherwise,. Thus, it suffices to reduce words of the form , with , to or, equivalently, to fill loops with labels of the form .

Write , where alternates between 1 and 2 and . We proceed by induction on *m*.

If and , there is nothing to do. If , then can be written as the sum of four 1-cycles as shown in Fig. 2. Writing and , the words labeling the 1-cycles are , , , and . The first and last are words representing the identity in the CAT(0) group , and thus can be filled with quadratic mass. The middle two are generators of and can be filled in with a scaled copy of *Y* with quadratic mass. These four fillings fit together to give a filling of *v* with quadratic mass.

If , then . Let and . Let and note that . As in the case, we can reduce to using quadratic area. This immediately lets us reduceto

Because we use *m* steps, and , it takes area (and linear genus) to reduce to .

Since each of the subwords in the decomposition of *w* above can be reduced to its image under *ϑ* using mass , the word *w* can be reduced to with mass .

Finally, we consider curves that travel through multiple copies of and . We will need to make arguments based on the graph product decomposition of *H*, so it will be helpful to have a slightly different complex *L* on which *H* acts. We construct *L* by “stretching” each copy of in into a product . Let *Z* be the complex obtained by gluing and copies of to copies of according to the graph product decomposition of *G*, and let be the universal cover of *Z*. Then, the homotopy equivalence that collapses each copy of to a copy of lifts to a homotopy equivalence . Then, acts geometrically on the level set , because *H* acts geometrically on . The following lemma describes the structure of *L*.

### Lemma 6.2.

*The level set L intersects each vertex space of* *of the form* *in a copy of* . *Likewise*, *L intersects each vertex space* *in a copy of* *and each edge space* *in a copy of* . □

Each edge in either lies in a copy of , lies in a copy of for some *i*, or crosses from to ; thus, we can classify the edges as *Q*-edges, *A*-edges, or *E*-edges. Consider a path of edges in of length and call it *α*.

By standard arguments (i.e., the normal form theorem for graphs of groups), *α* must have an “innermost piece,” i.e., a subpath that enters a copy of or through a copy of and then leaves through the same . We write this as , where *t* and lie in the same copy of and where *γ* is either a path of *A*-edges or a path of *Q*-edges.

Without loss of generality, suppose that the endpoints of *γ* lie in . Call them and , and let be a geodesic in from (*v*, 0) to (*w*, 0). Then, *p* and form a loop *θ*, and because they lie in the union of a copy of and a copy of or , there is a 2-chain filling *θ* whose mass is . But is undistorted in *L*, because is undistorted in and is convex in , so and the filling area of *θ* is .

Repeating this process for the loop inductively, we obtain a filling of *α*. Each time we repeat the process, the number of *E*-edges in *α* decreases by 2, and we use a filling of mass , so the total filling area of *α* is .□

## 7. Lower Bound on Homotopical Dehn Function

Recall the situation we are in: The group *G* is a graph product of groups *Q* and (copies of) *A* along edge groups *E*, and are contractible spaces on which , respectively, act geometrically. Meanwhile is a graph product of *D* (the double of a SLOG group *B*) and copies of (the Bestvina–Brady group associated with the RAAG *A*). In this section, we prove the following.

### Theorem 7.1.

*With G* (*hence*, ) *as above*, *we have**where* *is as in Section 3*.

To prove this, we will need the following refinement of *Theorem 3.1*.

### Lemma 7.2.

*For all* , *there is a curve* *of length* *such that if* *is a filling of γ*, *then**where* *is the support of* Δ.

We can take *γ* to be the curve used by Bridson and Haefliger (proof of theorem III..6.20 in ref. 11); their proof shows that the image of any disk filling *γ* has to have area , but the same bound applies to any chain filling *γ* as well.

#### Proof of Theorem 7.1:

Let *L* be the complex constructed in the previous section, on which *H* acts; this is made up of copies of and , joined along copies of . The translates of the set separate *L* into infinitely many components, each of which is either a copy of glued to copies of or a copy of glued to copies of . Let and be complexes isometric to each type of piece. We will refer to the union of the copies of that lie in and as and .

Fix a “root” copy of in *L*. Let *γ* be a curve in *L*_{0} as in *Lemma 7.2*, and let be its fundamental class. Let τ:*D*^{2} → *L* be a filling of γ in *L*. We will show that there is a chain that fills and is supported on . Then by Lemma 7.2, which proves the theorem.

First, we use the structure of *L* to break a disk in *L* into punctured disks whose images lie in copies of and . Homotope *τ* so that it is transverse to each copy of . The preimages of the copies of then divide into *pieces* , and for each *i*, is a punctured disk in a copy of or . Each has a distinguished boundary component that is homotopic to in , and we call that component the *outer boundary* of , denoted ; we call the other boundary curves *inner boundaries*. Each boundary curve of either coincides with or lies in one of the preimages.

Next, we claim that if *M* is a punctured disk in with boundary in , the topology of places strong restrictions on *M*. More precisely, we show the following.

### Lemma 7.3.

*Suppose that* *is a punctured disk in* *with boundary in* . *Suppose that M has a distinguished boundary component* *and other boundary components* . *If* *is the fundamental class of* , *then there are* *such that*

*Furthermore, we may assume that* *only if* *and* *lie in the same copy of* .

We will prove this lemma at the end of the section after we use it to construct . Let be one of the pieces of , and suppose that *τ* takes to a curve in . (For example, take such that .) Then is either a punctured disk in or a punctured disk in a copy of that neighbors . Let be the disk bounded by . We claim that there is a chain in such that .

We proceed by induction on the number *n* of pieces of contained in . If , then . Then, as before, is either a disk in or a disk in a copy of that neighbors . In the first case, we can take . In the second case, the lemma implies that , so we can take .

Suppose that the claim is true for and let be a piece of such that and is comprised of *n* pieces of . If is a punctured disk in , then *τ* takes each inner boundary of to a curve in and each inner boundary bounds a disk in with at most pieces. By induction, each bounds a chain in . Consequently, we can get the required filling of as

On the other hand, if is a punctured disk in a copy of , then *Lemma 7.3* implies that we can writewhere each is an inner boundary component of that lies in . By induction, each of these can be filled by a chain in , and ifthen fills .

Therefore, supports a chain filling *γ*.□

#### Proof of Lemma 7.3:

Let *Y* be the complex used in the construction of *A* and let be as in *Section 5*. Choose a basepoint such that . Then, as discussed in Section 5, is homotopy equivalent to an infinite wedge sum of copies of *Y*,where is the set of vertices in . Similarly, is made up of disjoint copies of ; call these , where . Each of these is homotopy equivalent to a wedge sum indexed by a subset ,

The ’s partition into disjoint sets. We can consider each of the ’s as a subset of , and we can define a homotopy equivalence that restricts to a homotopy equivalence on each .

Let be the basepoint of the wedge sum. Letbe a map that differs from by a small homotopy such that is a graph in *M*. We can further require that if *λ* is a boundary component of *M* and , then . Then cuts *M* into punctured disks , and for each *i*, we can choose an such that and .

Consider *M* as a subset of , embedded so that is the outer boundary of the subset. For each *i*, let be the disk bounded by . This embedding lets us choose an outer boundary component for each . Furthermore, each closed curve *λ* in *M* has an inside, and we can use this to put a partial ordering on the boundary curves of the ’s. If *λ* and are two boundary curves, we write if the inside of *λ* is a subset of the inside of .

If are the inner boundary components of , we claim that is a linear combination of the ’s. All these chains are in fact 1-cycles in , and because , it is enough to show that if , then one of the ’s is nonzero too. However, if for all *j*, then is a null-homotopic curve for all *j*, and so is the boundary of a disk in . Since the inclusion is -injective, this means that if for all *j*, then as well, which proves the claim.

Next, we claim that if *λ* is a boundary curve of some , then is a linear combination of the ’s. We proceed by induction on the number of boundary curves with . If this number is 0, then *λ* is one of the ’s and there is nothing to prove. Otherwise, the inside of *λ* is a union of ’s and ’s, so is a sum of ’s and ’s. By induction, if is inside *λ* and , then is a linear combination of the ’s, so is a linear combination of the ’s too.

Therefore, there are such thatwhere the equality is taken in , soin . Because is one-dimensional, the equality in fact holds in as well. Finally, , and if we project the above equation to the factor, we getas desired.□

## Acknowledgments

We thank Dan Guralnik and Sang Rae Lee for helpful discussions and the American Institute of Mathematics for supporting this research. We gratefully acknowledge additional funding from National Science Foundation Award DMS-0906962 (to N.B.), the Louisiana Board of Regents Support Fund Contract LEQSF(2011-14)-RD-A-06 (to P.D.), and the Natural Sciences and Engineering Research Council of Canada (to R.Y.).

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. E-mail: abramsa{at}wlu.edu.

Author contributions: A.A., N.B., P.D., and R.Y. performed research and wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

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