## Dehn functions of subgroups of CAT(0) groups

This is a post based on a guest lecture by Pallavi Dani.

Introduction

In this post, we will discuss Dehn functions of subgroups of non-positively curved groups. Let ${G}$ be a finitely presented group and let ${K_G}$ be a presentation complex for some presentation of ${G}$. If ${l}$ is a loop in ${\widetilde{K}^{(1)}_G}$, we define the filling area of ${l}$ to be

$\displaystyle \textup{FArea}(l)=\min \{\textup{Area} (D)| D=\text{disk in } \widetilde{K}^{(1)}_G, \partial D=l\}.$

Furthermore, we define the Dehn function of a finite presentation for ${G}$ by

$\displaystyle \delta_G(x)=\sup\{\textup{FArea}(l)|\; l \text{ is a loop of length } x\}.$

In these definitions length and area are defined combinatorially, that is, by counting cells.

As we have seen in previous posts [1] and [2], the Dehn functions of hyperbolic and CAT(0) groups are quite well understood:

1. ${G}$ is hyperbolic if and only if ${\delta_G(x)\simeq x}$, or equivalently if and only if ${\delta_G(x)}$ is subquadratic. This result is due to Gromov; the absence of Dehn functions between linear and quadratic is called the “Gromov Gap”.
2. If ${G}$ is CAT(0) then ${\delta_G(x)\preceq x^2}$. (On account of the Gromov gap, this means that ${\delta_G(X) \simeq x}$ or ${x^2}$.)

Thus, the possibilities for Dehn functions of hyperbolic and CAT(0) groups are very limited, and it is natural to ask whether subgroups of such groups inherit this property. More specifically:

1. Which functions arise as Dehn functions of subgroups of CAT(0) groups?
2. Are there gaps in the corresponding isoperimetric spectrum?
3. If so, do they correspond to something algebraic/geometric?

Subgroups of CAT(0) groups with large Dehn functions

It is possible to construct subgroups of CAT(0) groups with “large” Dehn functions. Here, by “large” we mean strictly greater than quadratic. We will discuss a 3-step scheme for constructing such subgroups, called the Bieri doubling trick. This idea originated in the work of Bieri, and it was explored extensively by Baumslag-Bridson-Miller-Short in the context of automatic groups.

The construction involves first finding distorted subgroups of CAT(0) groups. We denote the distortion of a subgroup ${H}$ in ${G}$ by ${\text{disto}_H^G(x)}$.

Step 1. Find ${H such that ${1\rightarrow H\rightarrow G\rightarrow \mathbb{Z}\rightarrow 1}$ is a short exact sequence, ${G}$ is CAT(0), and ${H}$ is distorted, with ${\text{disto}_H^G(x)\succcurlyeq x^2}$.

Example. Here is an example of such a pair constructed by N. Brady (for details, see section 2.3.3 of The Geometry of the Word Problem for Finitely Generated Groups). The group ${G}$ is isomorphic to ${F_2 \rtimes \mathbb{Z}}$ and is defined by ${G=\langle a,b,t\,|\, a^t=\phi (a), b^t=\phi(b)\rangle}$. The automorphism ${\phi\colon F(a,b)\rightarrow F(a,b)}$ is given by ${a\mapsto a}$ and ${b\mapsto ab}$. Let ${H}$ be ${F(a,b)}$, the free group generated by ${a}$ and ${b}$. You can see that ${\text{disto}_{H}^G(x)\succcurlyeq x^2}$ by considering the family of words ${t^nb^nt^{-n}}$ (with length on the order of ${n}$ in ${G}$) and noting that

$\displaystyle t^nb^nt^{-n}=\phi^n(b^n)=\phi^{n-1}(ababab\ldots ab)=$

$\displaystyle =\phi^{n-2}(aabaabaab\ldots aab)=aaa\ldots aabaaa\ldots aab\ldots\ldots aaa\ldots aab.$

The final word has ${n}$ ${a}$‘s in between every pair of consecutive ${b}$‘s, giving a total of about ${n^2}$ letters, and it is clearly reduced in ${H}$. Hence, the distortion is at least ${x^2}$.

To see that ${G}$ is CAT(0), note that ${G}$ is isomorphic to ${\langle \alpha, \beta, t|\; \alpha^t=\alpha, t^{\beta}=\alpha\rangle}$, where ${\alpha=at^{-1}, \beta=bt}$. The corresponding presentation complex built out of squares can be shown to satisfy the link condition.

Step 2. Form the double ${\Delta (G,H)}$ of ${G}$ along ${H}$ by amalgamating two copies of ${G}$ along ${H}$. So ${\Delta =\Delta(G,H) :=G\ast_{H} G}$.

Bridson showed that ${\delta_{\Delta}(x)\succeq \text{disto}_H^G(x)}$. Alternatively, see Bridson-Haefliger.

Example. In fact, for ${G}$ and ${H}$ in Brady’s example above, we can do better than Bridson’s lower bound: we have ${\delta_{\Delta (G,H)}(x)\succeq x^3}$, for a proof see Theorem 6.20 in Bridson-Haefliger.

To see this, write ${\Delta=\langle a,b,t,s\;|\;a^t=a^s=\phi(a), b^t=b^s=\phi(b)\rangle}$. Now consider the following family of embedded disks in ${\widetilde {K}_\Delta}$:

These disks have boundary length on the order of ${n}$, and Area ${\succeq n^3}$. Moreover, the boundary loops of these disks do not admit fillings with smaller area, since ${\widetilde {K}_\Delta}$ is ${2}$-dimensional and aspherical. (In such spaces the embedded filling is the most efficient among the fillings of any loop.) This shows that ${x^3}$ is a lower bound on the Dehn function.

Step 3. The double ${\Delta(G,H)}$ embeds in ${G\times F_2}$. Since ${G}$ and ${F_2}$ are CAT(0), so is ${G\times F_2}$.

Example. We illustrate this for our example. Write ${G\times F_2}$ as ${(F(a,b)\rtimes \langle t \rangle) \times F(u,v)}$. Then the subgroup ${\langle a, b , tu, tv\rangle}$ of ${G\times F_2}$ is isomorphic to ${\Delta (G, F(a,b))}$. (It is easy to see that the relations ${(tu)a(tu)^{-1}=\phi(a)}$ and ${(tv)a(tv)^{-1}=\phi(a)}$, and the corresponding relations involving ${b}$, are satisfied, since ${u}$ and ${v}$ commute with the other generators. Further, one can show that there are no additional relations among these generators — see Baumslag-Bridson-Miller-Short.)

Our example shows that ${x^3}$ occurs as the Dehn function of a subgroup of a CAT(0) group. By varying the pair ${(G,H)}$ in Step 1, one can construct other examples.

Sources for Step 1 and results.

1. Brady constructed a family of pairs ${F_n < G_n}$, where ${G_n\cong F_n\rtimes \mathbb{Z}}$ is a CAT(0) group, and ${\text{disto}_{F_n}^G(x) \simeq x^n}$. (The example above is the case n=2.) Then the double ${\Delta}$ embeds in ${G_n \times F_2}$, and ${\delta_{\Delta}(x)\simeq x^{n+1}}$. Thus, ${x^n}$ occurs as the Dehn function of a subgroup of a CAT(0) group for all integers ${n \geq 3}$.

2. Let ${M^3}$ be a hyperbolic surface bundle over a circle. Then one has a short exact sequence ${1\rightarrow \pi_1(S)\rightarrow \pi_1(M^3)\rightarrow \mathbb{Z}\rightarrow 1}$, and ${\pi_1(M^3)}$ is a CAT(0) group. Bridson and Haefliger prove that if ${H}$ is a finitely generated, infinite index subgroup of a hyperbolic group ${G}$ then ${\text{disto}_H^G(x)\succcurlyeq e^x}$. Hence, ${\Delta(\pi_1(M^3),\pi_1(S))\hookrightarrow \pi_1(M^3)\times F_2}$ and ${\delta_{\Delta}(x)\succeq e^x}$. In fact it is ${\simeq e^x}$. Thus, ${e^x}$ occurs as the Dehn function of a subgroup of a CAT(0) groups.

There are other sources of examples of ${H \leq G}$ to use in Step 1, for example due to Barnard-Brady and Samuelson. However, these do not appear to lead to additional examples of Dehn functions. For example, it remains unknown whether ${x^{\alpha}}$ for ${\alpha \notin \mathbb{Z}}$ or functions growing faster than ${e^x}$ can occur.

Dehn functions of kernels

The Bestvina-Brady groups, discussed in previous posts, have been a good source of counterexamples, and so one might expect them to be candidates for subgroups of CAT(0) groups with large Dehn functions. It turns out that their Dehn functions are relatively well-behaved:

• W. Dison showed that if ${H}$ is a Bestvina-Brady group then ${\delta_{H}(x)\preceq x^4}$.
• N. Brady constructed a Bestvina-Brady group ${H}$ with ${\delta_{H}(x)\succeq x^4}$. The lower bound was proved in a manner similar to the example above, by constructing a sequence of embedded diagrams in the level set for ${H}$, which is ${2}$-dimensional and contractible. Abrams-Brady-Dani-Duchin-Young generalized this example to produce a large class of Bestvina-Brady groups whose Dehn functions attain the general upper bound of ${x^4}$.
• For a while it was thought that Stallings group might have Dehn function of the form ${x^\alpha}$, where ${\alpha}$ is not an integer, but Dison-Elder-Riley-Young showed that its Dehn function is in fact quadratic.

One can also consider other kernels of particular classes of homomorphisms from particular classes of CAT(0) groups. For instance, Dison studies kernels of maps from direct products of free groups to free abelian groups. He shows, for example, that the kernel of a certain surjective homomorphism ${\phi\colon F_2\times F_2\times F_2 \rightarrow \mathbb{Z}\times\mathbb{Z}\times\mathbb{Z}}$ has Dehn function ${\succeq x^3}$.

Brady-Guralnik-Lee define a new class of CAT(0) groups, which they call perturbed right-angled Artin groups, and consider kernels of homomorphisms to ${\mathbb Z}$. Using these in Step 1 of the Bieri doubling trick, they produce a class of examples of subgroups of CAT(0) groups with Dehn functions of the form ${x^n}$, where ${n\geq 4}$ is an integer. The kernels are of type ${F_2}$ but not ${F_3}$, unlike the original Bieri doubles, which all have finite ${2}$-dimensional ${K(G,1)}$‘s. Another interesting direction is to study higher dimensional Dehn functions of CAT(0) groups and their subgroups. We won’t go into the technicalities here, but just say that the ${k}$-dimensional Dehn function ${\delta^{(k)}}$ measures the difficulty of filling ${k}$-cycles with ${(k+1)}$-chains (or ${k}$-spheres with ${(k+1)}$-balls) in a suitable model space for ${G}$.

By a result of Gromov, if ${G}$ is a CAT(0) group, then ${\delta_G^{(k)}\preceq x^{(k+1)/k}}$. (As in the case ${k=1}$, the group ${\mathbb Z^n}$ attains this general upper bound.) Abrams-Brady-Dani-Duchin-Young generalized Dison’s result above and showed that for a Bestvina-Brady group ${H}$, one has

$\displaystyle x^{(k+1)/k}\preceq \delta_H^{(k)}(x)\preceq x^{(2k+2)/k}.$

Moreover, these inequalities are sharp.

Hyperbolic groups

In all the examples above, the ambient CAT(0) groups are not hyperbolic. In fact hardly anything is known about Dehn functions of subgroups of hyperbolic groups. The problem is that there is essentially only one example (due to N. Brady) of a finitely presented subgroup of a hyperbolic group which is not itself hyperbolic. (A variant of this example with infinitely many conjugacy classes of finite order elements with was discussed in this post.) The Dehn function of Brady’s example is not linear, since the group is not hyperbolic. It follows from a result of Gersten-Short that its Dehn function is bounded above by a polynomial. However, the Dehn function is not known explicitly.