3.1. Van Kampen diagrams, corridors, and Dehn functions

These topics feature in many surveys, for instance, Section III.H.2 in [Reference Bridson and Haefliger9]. Here are the essentials.

Suppose $G=\langle X \! \mid \! R \rangle$ is a finitely presented group (so $R$ is a finite set of words on a finite alphabet $X$ and its inverse letters). Suppose $w$ is a word on $X \cup X^{-1}$ such that $w=1$ in $G$ . A van Kampen diagram $\Delta$ for $w$ is a simply connected planar 2-complex with edges labeled by elements of $X$ and directed so that the following holds. When traversing $\partial \Delta$ counterclockwise from some base vertex, we read off $w$ , and around the boundary of each 2-cell in one direction or the other and from a suitable base vertex, we read an element of $R$ . (If an edge is traversed in the direction of its orientation, the positive generator is implied, and if against its orientation, the inverse of the generator.) The 1-skeleton $\Delta ^{(1)}$ of $\Delta$ has the path metric in which every edge has length 1. The area of $\Delta$ is the number of 2-cells it has. $\textrm{Area}(w)$ denotes the minimum area among all van Kampen diagrams with boundary word $w$ .

The Dehn function $\delta \;:\;\mathbb{N} \to \mathbb{N}$ of $\langle X \! \mid \! R \rangle$ is $ \delta (n) \;:\!=\; \max \{\textrm{Area}(w) \mid \left |w\right | \leq n \text{ and } w =1 \text{ in } G \}$ .

Up to the equivalence relation $\simeq$ defined in Section 1, the Dehn function does not depend on the choice of finite presentation for $G$ and, moreover, is a quasi-isometry invariant among finitely presented groups. In particular, we will need:

Corridors appear in van Kampen diagrams over a presentation $\langle X \! \mid \! R \rangle$ when there is some $a \in X$ such that all relators $r \in R$ in which $a$ appears can be expressed as $w_1a^{\pm 1}w_2a^{\mp 1}w_3$ where $w_1$ , $w_2$ , and $w_3$ are words not containing $a^{\pm 1}$ . Such presentations naturally arise for HNN-extensions, with $a$ being the stable letter. Suppose $\Delta$ is a van Kampen diagram for a word $w$ over such a presentation. If there is an $a$ -edge (an edge labeled $a$ ) in $\Delta$ and there is a 2-cell in $\Delta$ with that edge in its boundary, then that 2-cell will have exactly one other $a$ -edge, and that $a$ -edge will either be in the boundary or will be in the boundary of another 2-cell. Concatenations of 2-cells in $\Delta$ across $a$ -edges in this manner are called corridors. A corridor either connects a pair of $a$ -edges in the boundary of $\Delta$ or closes up to form an annulus. The $a$ and $a^{-1}$ that label edges in the boundaries of the two-dimensional parts (Figure 1b) of $\Delta$ are paired off and connected by corridors. The number of 2-cells involved is the length of the corridor. An $a$ -corridor is reduced if it contains no back-to-back canceling pair of 2-cells—that is, no two 2-cells sharing an $a$ -edge for which the word around the boundary of their union is freely reducible to the identity in the group. If an $a$ -corridor is the concatenation of $2$ -cells labeled $u_1 a v_1^{-1} a^{-1}$ , …, $u_r a v_r^{-1} a^{-1}$ , then the top (respectively, bottom) of that corridor is the path that is labeled $v_1 \cdots v_r$ (respectively, $u_1 \cdots u_r$ ) and passes through the terminal (respectively, initial) vertices of the $a$ -edges—see Figure 1c.

Figure 1. Corridors.

Suppose $\Delta$ is a van Kampen diagram with $N$ $a$ -corridors. Then $N$ is at most half the length of the boundary (at most half the number of $a^{\pm 1}$ in $w$ ). Since $a$ -corridors cannot cross, removing all the $a$ -corridors leaves $N+1$ connected subdiagrams called $a$ -complementary regions. The words around the perimeters of each of these regions contain no $a^{\pm 1}$ . Therefore, analysis of the lengths of the $a$ -corridors and of the areas of the $a$ -complementary regions can lead to estimates on the area of $\Delta$ .

The dual tree to the set of $a$ -corridors has vertices corresponding to $a$ -complementary regions and has an edge between two vertices when an $a$ -corridor borders the two corresponding $a$ -complementary regions. (There is no vertex corresponding to the outside of the van Kampen diagram.)

An $a$ in the boundary of a van Kampen diagram will either be connected by a full $a$ -corridor to another edge labeled by $a$ in the boundary, or it begins a partial $a$ -corridor ending at one of the capping faces. An $a$ -edge on a capping face is connected via a partial $a$ -corridor (possibly of length zero) either to the boundary or to an $a$ -edge of another capping face.

Like standard corridors, partial corridors cannot cross. However, there is no immediate control on the number of partial corridors in terms of $|w|$ , since they may begin and end in capping faces within the diagram.

3.2. General bounds on Dehn functions of mapping tori of RAAGs

RAAGs are (bi)automatic [Reference Hermiller and Meier17, Reference VanWyk25], so have either linear or quadratic Dehn functions. A finitely presented group is hyperbolic if and only if it has linear Dehn function. Finite-rank free groups are hyperbolic. Non-free RAAGs have $\mathbb{Z}^2$ subgroups and so are not hyperbolic (e.g. [Reference Bridson and Haefliger9]). So RAAGs have either linear or quadratic Dehn functions, the linear case only occurring for free RAAGs. This will be useful for the following lemma.

Proof. Suppose $G$ is a non-free RAAG. So $G$ has a finite presentation $\langle X \! \mid \! R \rangle$ derived from a graph with at least one edge. Then $G$ and hence $M_{\Psi }$ will contain a $\mathbb{\mathbb{Z}}^2$ subgroup. This implies that $M_{\Psi }$ is not hyperbolic and therefore $n^2 \preceq \delta (n)$ (again, e.g. [Reference Bridson and Haefliger9]).

A word $w$ on the generators of

\begin{equation*}M_{\Psi } \ = \ \langle X, t \mid {R}, \ t^{-1}xt= \Psi (x),\, \forall \, x \in X \rangle \end{equation*}

can be expressed as $t^{k_0}a_1t^{k_1}\cdots a_mt^{k_m}$ for some $a_1, \ldots, a_m \in X^{\pm 1}$ and some $k_1, \ldots, k_m \in \mathbb{Z}$ . Suppose $w$ represents the identity in $M_{\Psi }$ . Then shuffling all the $t^{\mp 1}$ to the right, replacing each $a_i$ by the freely reduced word representing $\Psi ^{\pm 1}(a_i)$ does not change the element of $M_{\Psi }$ represented. Eventually, we arrive at $u t^{k_0 + \cdots + k_m}$ where $u$ is a word on $X^{\pm 1}$ that represents $1$ in $G$ and $k_0 + \cdots + k_m =0$ . Applying $\Psi ^{\pm 1}$ to a letter in $X^{\pm 1}$ increases its length by at most the factor $C\;:\!=\;\max _{a \in X} |\Psi ^{\pm 1}(a)|$ . So $m C^{\left |k_0\right |+ \cdots + \left |k_m\right |} \leq \left |w\right | C^{\left |w\right |}$ is an upper bound for both $\left |u\right |$ and for the number of relation applications needed to convert $w$ to $u$ .

The Dehn function of $G$ is at most quadratic, so $u$ can be reduced to the empty word using at most a constant times $\left |u\right |^2$ defining relations. Thus $\textrm{Area}(w)$ is at most a constant times $\left |w\right | C^{\left |w\right |} + (\left |w\right | C^{\left |w\right |})^2$ , and therefore (since $\alpha ^n \simeq \beta ^n$ for all $\alpha, \beta \gt 1$ ) we deduce $\delta (n) \preceq 2^n$ .

Our next lemma is the special case of Theorem 4.1 of [Reference Bridson and Gersten7] in which, in the notation of [Reference Bridson and Gersten7], $G=H$ and $K$ is quasi-isometrically embedded. We will call on it repeatedly to establish lower bounds on the Dehn functions.

Lemma 3.5 (Adapted from Theorem 4.1 of Bridson–Gersten [Reference Bridson and Gersten7]). Suppose $K = \langle k_1, \dots, k_m \rangle$ is a quasi-isometrically embedded infinite abelian subgroup of a finitely presented group $G$ . If $\Phi \in{\textrm{Aut}}(G)$ and $\Phi (K) = K$ , then the Dehn function $\delta$ of $\langle G, t\, |\, g^t=\Phi (g) \rangle$ satisfies

\begin{equation*}{n^2\max _{1 \leq i \leq m}{ \left |\Phi ^{\pm n}(k_i)\right |} \ \preceq \ \delta (n)}.\end{equation*}

Equivalently, suppose $\phi = \Phi \left |_{K} \right .$ is associated to the matrix $A$ ; then

1. 1. If $\phi$ has an eigenvalue $\lambda$ such that $|\lambda | \neq 1$ , then $M_{\Phi }$ has exponential Dehn function.

2. 2. If $\phi$ only has eigenvalues $\lambda$ such that $|\lambda |=1$ , then $n^{c+1} \preceq \delta (n)$ , where the size of the largest Jordan block for $A$ is $c \times c$ .

The following lemma allows us to specialize to convenient $\Psi$ when analyzing the Dehn functions of mapping tori. We include the proof because it is brief and the result is vital to this paper.

Proof, based on [Reference Bogopolski, Martino and Ventura4]. Map $M_{\Psi }$ onto $\langle t \rangle = \mathbb{Z}$ by killing $G$ and then onto $\mathbb{Z}/n\mathbb{Z}$ by the natural quotient map. The kernel of this composition is the index- $n$ subgroup $M_{\Psi ^n}$ . By Proposition 3.1, $M_{\Psi }$ and $M_{\Psi ^n}$ have equivalent Dehn functions.

As $w^t = \Psi (w)$ for all $w \in G$ , it follows that $w^{t^{-1}} = \Psi ^{-1}(w)$ , so mapping $t \mapsto t^{-1}$ and fixing $G$ gives an isomorphism $M_{\Psi } \to M_{\Psi ^{-1}}$ . Thus, $M_{\Psi }$ and $M_{\Psi ^{-1}}$ have equivalent Dehn functions.

If $\Psi _1$ and $\Psi _2$ are conjugate in $\textrm{Out}(G)$ , there exists $\eta \in{\textrm{Aut}}(G)$ and $h\in G$ such that $\Psi _2(g) = \eta ^{-1}(\Psi _1(\eta (g^h)))$ for all $g \in G$ . We will show that $M_{\Psi _1}$ and $M_{\Psi _2}$ are isomorphic. Consider $F\;:\;M_{\Psi _2} \to M_{\Psi _1}$ given by $x \mapsto \eta (x)$ for $x \in G$ and $t \mapsto t\,\hat{h}$ , where $\hat{h}\;:\!=\;\Psi _1(\eta (h))$ . It is a homomorphism because the relators $(g^{-1})^t\Psi _2(g)$ for $g \in G$ are mapped to the identity in $M_{\Psi _1}$ , since $(w^{-1})^t\Psi _1(w)=1$ in $M_{\Psi _1}$ for $w \in G$ . Indeed,

\begin{eqnarray*} F\left ((g^{-1})^t\Psi _2(g)\right ) \ = \ F\left ((g^{-1})^t\eta ^{-1}(\Psi _1(\eta (g^h)))\right ) \ = \ \eta (g^{-1})^{t\hat{h}}\Psi _1( \eta (g^h)) \\[5pt] \ = \ (\eta (g^{-1}))^{t\hat{h}} \Psi _1(\eta (g))^{\hat{h}} \ = \ \left ((w^{-1})^{t} \Psi _1(w)\right )^{\hat{h}} \ = \ 1^{\hat{h}} \ = \ 1. \end{eqnarray*}

where $w=\eta (g)$ . It is certainly onto. This homomorphism has inverse given by $x \mapsto \eta ^{-1}(x)$ for $x \in G$ and $t \mapsto t\,\eta ^{-1}(\hat{h}^{-1})$ , so it is an isomorphism.

3.3. Growth and automorphisms of $\mathbb{Z}^2$

Let $||A||$ denote the maximum of the absolute values of the entries in a matrix $A \in \textrm{GL}(2, \mathbb{Z}) ={\textrm{Aut}}(\mathbb{Z}^2)$ . We say $A$ has linear growth when the function $\mathbb{N} \to \mathbb{N}$ mapping $n \mapsto ||A^n||$ is $\simeq$ -equivalent to $n \mapsto n$ .

The following lemmas will allow us to specialize to convenient cases of $\Phi$ when analyzing Dehn functions of mapping tori $M_{\Phi }$ of $F_2 \times \mathbb{Z}$ and $\mathbb{Z}^2 \ast \mathbb{Z}$ .

Proof. As $A$ has linear growth, Theorem 2.1 of [Reference Bridson and Gersten7] tells us that there exists an integer $k \gt 0$ such that $A^k$ is $I+N$ for some nonzero matrix $N$ such that $N^2=0$ . As $N^2=0$ , the trace of $N$ is zero, and $N=\left ( \begin{smallmatrix} a & b \\[5pt] c & -a\\[5pt] \end{smallmatrix}\right )$ for some integers $a, b, c$ not all zero such that $a^2 = -bc$ . If $a=c=0$ , then the result holds with $\alpha =b$ . So assume they are not both zero. Notice that $N \left (\begin{smallmatrix} a \\[5pt] c \end{smallmatrix}\right ) = \left (\begin{smallmatrix} 0 \\[5pt] 0 \end{smallmatrix}\right )$ . So there are coprime integers $p$ and $q$ (in particular not both zero) with $N \left (\begin{smallmatrix} p \\[5pt] q \end{smallmatrix}\right ) = \left (\begin{smallmatrix} 0 \\[5pt] 0 \end{smallmatrix}\right )$ . By Bézout, there are $r, s \in \mathbb{Z}$ such that $ps-qr=1$ , and so $B\;:\!=\; \left (\begin{smallmatrix} p & r \\[5pt] q & s \end{smallmatrix}\right )$ is in $\textrm{SL}(2, \mathbb{Z})$ . And then ${B}^{-1}NB = \left (\begin{smallmatrix} 0 & \alpha \\[5pt] 0 & 0 \end{smallmatrix}\right )$ where $\alpha = 2ars +bs^2 -cr^2$ , and the result follows.

Proof. As $A \in \textrm{SL}(2, \mathbb{Z})$ we can say $\det (A)=1 =(x+yi)(x-yi)=x^2+y^2$ , so $|\lambda |=1$ and $\lambda ^2-\mbox{tr}(A) \lambda +1 =0$ . Since $\lambda$ is not real, the discriminant $\mbox{tr}(A)^2-4\lt 0$ , and as $A$ has only integer entries, $\mbox{tr}(A)\in \{0, \pm 1\}$ . Then, as $\mbox{tr}(A)= 2x$ , we find $x \in \{0, \pm \frac{1}{2}\}$ . It follows that $A$ is conjugate in $\textrm{SL}(2,{\mathbb{C}})$ to $\left ( \begin{smallmatrix} e^{i\theta } & 0 \\[5pt] 0 & e^{-i \theta } \\[5pt] \end{smallmatrix} \right )$ , where $\theta$ is $\pm \pi/2$ , $\pm 2\pi/3$ , or $\pm \pi/3$ , and so $A$ has order dividing $6$ .

Proof. If $A \in \textrm{SL}(2, \mathbb{Z})$ has real non-unit eigenvalues, then $A$ has two eigenvalues $\lambda$ and $\lambda ^{-1}$ , where $|\lambda |\gt 1$ . It follows that $A$ is conjugate in $\textrm{SL}(2,{\mathbb{C}})$ to $\left ( \begin{smallmatrix} \lambda & 0 \\[5pt] 0 & \lambda ^{-1} \\[5pt] \end{smallmatrix} \right )$ . By taking powers of the diagonalization in $\textrm{SL}(2,{\mathbb{C}})$ , $A^k=PD^kP^{-1} =\left ( \begin{smallmatrix} a & b \\[5pt] c & d \\[5pt] \end{smallmatrix} \right )\left ( \begin{smallmatrix} \lambda ^k & 0 \\[5pt] 0 & \lambda ^{-k} \\[5pt] \end{smallmatrix} \right )\frac{1}{ad-bc}\left ( \begin{smallmatrix} d & -b \\[5pt] -c & a \\[5pt] \end{smallmatrix} \right )$ , we can see that $A$ has exponential growth.

For the converse, suppose $A$ grows exponentially. By Lemma 3.8, it has real eigenvalues. Its eigenvalues cannot be both $1$ or both $-1$ because then $A^2$ would be conjugate in $\textrm{SL}(2,{\mathbb{C}})$ to a matrix of linear growth (as in Lemma 3.7).

5.1. Automorphisms of $F_2 \times \mathbb{Z}$

Recall the notation of Theorem A: $\Psi \in{\textrm{Aut}}(F_2 \times \mathbb{Z})$ induces $\psi \in{\textrm{Aut}}(F_2)$ , $\psi$ induces $\psi _{ab} \in \textrm{GL}(2,\mathbb{Z})$ via the abelianization map $F_2 \to \mathbb{Z}^2$ , $g \mapsto g_{ab}$ , and $p\;:\;F_2 \times \mathbb{Z} \to \mathbb{Z} = \langle c \rangle$ is projection onto the second factor. Let $\lambda ^{\pm 1}$ be the (complex) eigenvalues of $\psi _{ab}$ . We will prove Theorem A by separately addressing three comprehensive and mutually exclusive cases.

1. 1. $\left |\lambda \right |\neq 1$ ,

2. 2. $\left |\lambda \right | = 1$ and there exists $g \in F_2$ and $m \in \mathbb{N}$ such that $\psi _{ab}^m(g_{ab}) = g_{ab}$ and $p(\Psi ^m(g)) \neq 0$ —equivalently, $g_{ab} = [\psi ^m(g)]_{ab}$ and $\Psi ^m(g) = \psi ^m(g)c^k$ for some $k \neq 0$ .

3. 3. all other cases—that is, $\left |\lambda \right | = 1$ and for all $g \in F_2$ and all $m \in \mathbb{N}$ , if $p(\Psi ^m(g)) \neq 0$ then $\psi _{ab}^m(g_{ab}) \neq g_{ab}$ .

In Section 5.2, we will prove that the Dehn function of the mapping torus $M_{\Psi }$ of $F_2 \times \mathbb{Z} = \langle a,b \rangle \times \langle c \rangle$ is quadratic in case 1. In Section 5.3, we will prove that it is cubic in case 2 and is quadratic in case 3. First, we narrow the family of automorphisms $\Psi$ that must be explored.

Proof. ${\textrm{Aut}}(F(a,b))$ is generated by the following five elementary Nielsen transformations: $(a,b)$ maps to $(a^{-1}, b)$ , $(a, b^{-1})$ , $(b, a)$ , $(ab, b)$ , or $(a, ba)$ . Each sends $[a,b]$ to a conjugate of $[a,b]^{\pm 1}$ .

To prove Theorem A, it will suffice to focus only on the mapping tori of the form $M_{\Phi }$ described in the next lemma.

Lemma 5.2. Let $G = F_2 \times \mathbb{Z} = \langle a, b \rangle \times \langle c \rangle$ . Given $\Psi \in{\textrm{Aut}}(G)$ , there exists $\Phi \in{\textrm{Aut}}(G)$ such that

• there is $k \geq 0$ such that $[\Phi ] = [\Psi ^k]$ in $\textrm{Out}(F_2\times \mathbb{Z})$ ,

• the Dehn functions of $M_{\Psi }$ and $M_{\Phi }$ are equivalent,

• $\phi ([a,b]) = [a,b]$ ,

• $M_{\Phi }$ is a central extension of $M_{\phi } = \langle a, b, t \mid a^{t}=\phi (a), \ b^{t}=\phi (b) \rangle$ by $\mathbb{Z} = \langle c \rangle$ ,

where $\phi \in{\textrm{Aut}}(F(a,b))$ is the map induced from $\Phi$ by killing $c$ . Thus, there exist $k_a, k_b \in \mathbb{Z}$ such that

\begin{equation*}M_{\Phi } \ = \ \langle a, b, c, t \mid a^{t}=\phi (a)c^{k_a}, \ b^{t}=\phi (b)c^{k_b},\ c^{t}=c, \ [a,c]=1, \ [b,c]=1 \rangle .\end{equation*}

Additionally, by replacing $\Phi$ by $\Phi ^l$ for a suitable $l$ , we can further achieve that the map $\phi _{ab}$ induced by $\phi$ via abelianizing $F(a,b)$ to $\mathbb{Z}^2$ has determinant $1$ and its eigenvalues are real and positive.

Finally, each of conditions 1, 2, and 3 from the start of this section hold for $\Phi$ exactly when they hold for $\Psi$ .

Proof. The center $\langle c \rangle$ of $G$ , being characteristic, is preserved by $\Psi$ , so $\Psi (c) = c^{\pm 1}$ , and $\Psi ^2(c) =c$ . On killing $c$ , $\Psi$ induces some $\psi \in{\textrm{Aut}}(F(a,b))$ , whereby the mapping torus $M_{\Psi ^2} = G \rtimes _{\Psi ^2} \mathbb{Z}$ is

\begin{equation*}\langle a, b,c, t \mid a^{t}=\psi ^2(a)\,c^{k_a}, \ b^{t}=\psi ^2(b)\,c^{k_b},\ c^{t}=c, \ [a,c]=1, \ [b,c]=1\rangle \end{equation*}

for some $k_a, k_b \in \mathbb{Z}$ . By Lemma 5.1, $\psi ^2([a,b]) =[a,b]^h$ for some $h \in F(a,b)$ .

How we will define $\Phi$ will depend on which of the cases 1, 2, and 3 from the start of this section, $\Psi$ falls into, as well as whether $\psi _{ab}$ is finite order or has linear growth.

In case 1, define $\Phi = \iota _{h^{-1}}\circ \Psi ^2$ where $\iota _{h^{-1}}$ denotes conjugation by $h^{-1}$ . Then $\phi _{ab}$ has determinant 1 as $(\psi _{ab})^2$ has determinant 1, and $\Phi$ satisfies the properties above by definition and by Lemma 3.6. Because the eigenvalues of $\psi _{ab}$ were real, the eigenvalues of $\phi _{ab}$ are real and positive.

In cases 2 and 3, $\psi _{ab}$ has unit eigenvalues. If these eigenvalues are not real, Lemma 3.8 tells us that $\psi _{ab}$ has order dividing $6$ . Define $\Phi$ to be $\Psi ^6$ composed with an appropriate inner automorphism so that $\phi = \textrm{id}$ . Otherwise, $\phi _{ab}$ has real eigenvalues of $\pm 1$ . Define $\Phi$ to be $\Psi ^2$ , composed with the inner automorphism guaranteed by Lemma 5.1 so that $\phi ([a,b]) = [a,b]$ . In both cases, $\phi$ and $\Phi$ satisfy all the required properties (again using Lemma 3.6).

Here is why conditions 1, 2, and 3 hold for $\Phi$ exactly when they hold for $\Psi$ . Suppose that $\Psi$ satisfies condition 1, that $\psi _{ab}$ has a non-unit eigenvalue. Then $\Phi = \iota _{h^{-1}}\circ \Psi ^2$ for some $h \in F(a,b)$ , and $\psi _{ab}$ has a non-unit eigenvalue if and only if $\phi _{ab}= \psi ^2_{ab}$ does. Suppose that $\Psi$ satisfies condition 2 or 3, and $\psi _{ab}$ has complex eigenvalues. Then $\Phi = \iota _{h^{-1}}\circ \Psi ^6$ for some $h\in F(a,b)$ . If for some $g \in F(a,b)$ , we have that $g_{ab}$ is a fixed point of $\psi _{ab}^m$ , then it is also a fixed point of $\phi _{ab}^m=\psi _{ab}^{6m}$ . If there is $m\in \mathbb{N}$ so that $\phi _{ab}^m$ has a fixed point, then $\psi _{ab}^{6m}$ will also have a fixed point. Moreover, $p(\Phi ^{m}(g)) = 6p(\Psi ^m(g))$ , so $p(\Phi ^m(g))=0$ if and only if $p(\Psi ^m(g))=0$ . Thus, $\Phi$ and $\Psi$ either both satisfy 2 or both satisfy 3. The real case follows by a similar argument.

We are now ready to deduce:

Proof. Bridson and Groves [Reference Bridson and Groves8] prove that for all $\phi \in{\textrm{Aut}}(F_2)$ , $F_2 \rtimes _{\phi } \mathbb{Z}$ has a quadratic Dehn function, so Corollary 4.3 applies and allows us to use Theorem 4.1 to deduce that every central extension of $F_2 \rtimes _{\phi } \mathbb{Z}$ has at most cubic Dehn function. Lemma 5.2 then tells us that $M_{\Phi }$ has at most a cubic Dehn function.

In Sections 5.2 and 5.3, we will refine this method to improve the upper bound from cubic to quadratic in special cases. In Section 6, we will adapt the arguments to certain examples which fall short of being central extensions.

5.2. Theorem A when $\psi _{ab}$ has non-unit eigenvalues

The primary tool for this section is relative hyperbolicity, a concept introduced by Gromov, and then developed by Bowditch, Farb, Osin, and others [Reference Bowditch5, Reference Farb14, Reference Osin21].

Suppose $M_{\phi }$ is a group presented by:

\begin{equation*}\mathcal {P}_1 \ \;:\!=\; \ \langle a, b, t \mid a^{t}=\phi (a), \ b^{t}=\phi (b) \rangle \end{equation*}

where $\phi \in{\textrm{Aut}}(F(a,b))$ such that $\phi ([a,b]) = [a,b]$ .

Lemma 5.4. If $\phi _{ab}$ has non-unit eigenvalues, $M_{\phi }$ is strongly hyperbolic relative to the subgroup:

\begin{equation*}H\,\;:\!=\;\,\langle [a,b], \ t \rangle \,\cong \,\mathbb {Z}^2.\end{equation*}

Proof. $M_{\phi }$ is the fundamental group of a finite-volume hyperbolic once-punctured torus bundle. In Theorem 4.11 of [Reference Farb14], Farb showed that such groups are strongly hyperbolic relative to their cusp subgroups. In our case, that is the subgroup $\langle [a,b], \ t \rangle$ (see also Section 4 of [Reference Button and Kropholler12] for a survey of when mapping tori of free groups are relatively hyperbolic and acylindrically hyperbolic).

Consider the presentation:

\begin{equation*}\mathcal {P}_2 \ \;:\!=\; \ \langle a, b, z, t \mid a^{t}=\phi (a), \ b^{t}=\phi (b), \ z=[a,b], \ z^t=z \rangle \end{equation*}

for $M_{\phi }$ obtained from $\mathcal{P}_1$ by adding an extra generator $z$ , an extra relation which declares that $z$ equals $[a,b]$ in the group, and a further extra relation which declares that $[a,b]$ commutes with $t$ (which is a consequence of the other defining relations since $\phi ([a,b]) = [a,b]$ , but we include it nevertheless). Then, $\langle t, z \rangle \cong \mathbb{Z}^2$ is the subgroup $H$ of Lemma 5.4. Refer to faces of a van Kampen diagram over $\mathcal{P}_2$ as $\mathbb{Z}^2$ -faces when they correspond to the relation $z^t=z$ , and refer to the remaining faces as $\mathcal{R}$ -faces.

Proof. Let $\mathcal{A}_H=\left \{t,z\right \} \cup \{h_{ij} \mid i,j \in \mathbb{Z}, \ (i,j) \neq (0,0), (1, 0), (0,1) \}$ be an alphabet, with a letter for each non-identity element of the subgroup $\langle t, z \rangle \cong \mathbb{Z}^2$ of $M_{\phi }$ . Here, $h_{ij}$ corresponds to the element represented by $t^{i}z^{j}$ . Let $S$ denote the set of words in $\mathcal{A}_H^{\ast }$ that represent the identity in $M_{\phi }$ . For example, $S$ includes the word $[z,t]$ and $h_{ij}z^{-j}t^{-i}$ for all $(i,j) \neq (0,0), (1, 0), (0,1)$ .

The presentation

\begin{equation*} \mathcal {P}_3 \ \;:\!=\; \ \langle a, b, \mathcal {A}_H \mid a^{t} = \phi (a), \ b^{t} = \phi (b), \ z=[a,b], \ S \rangle \end{equation*}

again gives $M_{\phi }$ . Note that the elements $t$ and $z$ appear in $\mathcal{A}_H$ , and the defining relation $z^t=z$ appears in $S$ . Again, we will refer to van Kampen diagram faces that correspond to elements of $S$ as $\mathbb{Z}^2$ -faces.

Then $\mathcal{P}_3$ is a finite relative presentation for $M_{\phi }$ with respect to the subgroup $H$ , as per Definitions 2.2 and 2.3 of Osin in [Reference Osin21]. Theorem 1.5 in [Reference Osin21] says (in particular) that a finitely generated group which is hyperbolic relative to a subgroup in the sense of Farb, as is the case for $M_{\phi }$ relative to $H$ by Lemma 5.4, has a linear relative Dehn function. What this means (as unpacked per Definitions 2.26, 2.31, and 2.32 of [Reference Osin21]) is that there exists $C\gt 0$ such that for every word $w$ on $\{a,b,t\}^{\pm 1}$ representing the identity, there is a van Kampen diagram $\hat{\Delta }$ over $\mathcal{P}_3$ whose number of $\mathcal{R}$ -faces is at most $C |w|$ .

Osin proves further facts that we will need concerning the geometry of $\hat{\Delta }$ . A diagram for the word $w$ is of minimal type over all diagrams for $w$ if under lexicographic ordering it minimizes

\begin{align*} (N_{\mathcal{R}} = \#\ \text{of}\ \mathcal{R}-\text{faces}, \ \ N_{\mathbb{Z}^2} = \# \ \text{of} \ \mathbb{Z}^2-\text{faces}, \ \ E= \text{total} \# \text{of edges}). \end{align*}

Choose $\hat{\Delta }$ be of minimal type.

Let $M$ be the maximum length of the relators $a^{t}\phi (a)^{-1}$ , $b^{t}\phi (b)^{-1}$ , $z[a,b]^{-1}$ , and $z^tz^{-1}$ . Call an edge of $\hat{\Delta }$ internal to the $\mathbb{Z}^2$ -faces when it has $\mathbb{Z}^2$ -faces (or a $\mathbb{Z}^2$ -face) on both sides. Osin (Lemma 2.15 of [Reference Osin21]) tells us that if $\hat{\Delta }$ is of minimal type then it has no edges which are internal to the $\mathbb{Z}^2$ -faces and deduces (Corollary 2.16) that the sum of the lengths of the perimeters of $\mathbb{Z}^2$ -faces in $\hat{\Delta }$ is at most $|w| + M N_{\mathcal{R}}$ .

Suppose that $w$ is a word on $\{a,b, t\}^{\pm 1}$ and take $\hat{\Delta }$ to be a diagram of minimal type for $w$ over $\mathcal{P}_3$ .

The words around $\mathcal{R}$ -faces only include the letters $t, z, a$ and $b$ , so they can overlap $\mathbb{Z}^2$ -faces only in edges labeled by $t$ and $z$ . Therefore, the word around each $\mathbb{Z}^2$ -face is a word on $\{t, z\}^{\pm 1}$ since every edge in the boundary of a $\mathbb{Z}^2$ -face is either in $\partial \hat{\Delta }$ or is also in the boundary of an $\mathcal{R}$ -face. Let $\Delta$ be a diagram obtained from $\hat{\Delta }$ by excising all $\mathbb{Z}^2$ -faces and replacing each $\mathbb{Z}^2$ -face with the appropriate minimal area diagram over $\langle t, z \mid z^t =z \rangle$ . So $\Delta$ is a van Kampen diagram over $\mathcal{P}_2$ . By Osin’s Theorem 1.5, as discussed above, $\Delta$ satisfies (1). As the Dehn function of $\langle t, z \mid z^t =z \rangle$ enjoys a quadratic upper bound, and, given the bound on the lengths of the boundaries of $\mathbb{Z}^2$ -faces explained in the previous paragraph, $\Delta$ also satisfies (2).

Because of the minimality assumption on the number of $\mathbb{Z}^2$ -faces, no two $\mathbb{Z}^2$ -faces will have a vertex in common in $\hat{\Delta }$ : two $\mathbb{Z}^2$ -faces with a vertex in common could be replaced by a single $\mathbb{Z}^2$ -face. Also the boundary circuit of any $\mathbb{Z}^2$ -face in $\hat{\Delta }$ will be a simple loop. This is because $E(\hat{\Delta })$ is minimal: a $\mathbb{Z}^2$ -face with a non-simple loop as its boundary circuit could be excised and a $\mathbb{Z}^2$ -face with a shorter and simple boundary loop inserted in its place. Thus, the $\mathbb{Z}^2$ -faces form disjoint islands in $\hat{\Delta }$ and there are no $\mathcal{R}$ -faces enclosed within these islands. In the light of this, (3) follows from (1).

Proof of Theorem A in Case 1. We suppose $\Psi \in{\textrm{Aut}}(F_2 \times \mathbb{Z})$ . By Lemma 5.2, there exists $\Phi$ so that $M_{\Phi }$ and $M_{\Psi }$ have equivalent Dehn function, and $M_{\Phi }$ has presentation:

\begin{equation*}\langle a, b,c, t \mid a^{t}=\phi (a)c^{k_a}, \ b^{t}=\phi (b)c^{k_b},\ c^{t}=c, \ [a,c]=1, \ [b,c]=1\rangle,\end{equation*}

which is a central extension of

\begin{equation*} \langle a, b, t \mid a^{t}=\phi (a), \ b^{t}=\phi (b) \rangle \end{equation*}

where $\phi \in{\textrm{Aut}}(F(a,b))$ has the property that $\phi ([a,b]) = [a,b]$ . If $z = [a,b]$ , then in $M_{\Phi }$

\begin{equation*}z^t \ = \ [a^t,b^t] \ = \ [\Phi (a),\Phi (b)] \ = \ [\phi (a)c^{k_a}, \phi (b)c^{k_b}] \ = \ [\phi (a), \phi (b)] \ = \ \phi ([a,b]) \ = \ [a,b] \ = \ z.\end{equation*}

So $M_{\Phi }$ also can be presented as:

\begin{equation*}\mathcal {Q} \ \;:\!=\; \ \langle a, b,c, t, z \mid a^{t}=\phi (a)c^{k_a}, \ b^{t}=\phi (b)c^{k_b},\ c^{t}=c, \ [a,c]=1, \ [b,c]=1, \ z = [a,b], \ z^t=z \rangle,\end{equation*}

which reveals it to be a central extension of

\begin{equation*} \mathcal {P}_2 \ = \ \langle a, b, t, z \mid a^{t}=\phi (a), \ b^{t}=\phi (b), \ z=[a,b], \ z^t =z \rangle \end{equation*}

by $\mathbb{Z} = \langle c \rangle$ .

Suppose $w$ is a word in $\{a,b, c, t\}^{\pm 1}$ of length $n$ representing the identity in $\mathcal{Q}$ . Let $\overline{w}$ be $w$ with all $c^{\pm 1}$ deleted.

Let $\overline{\Delta }$ be a van Kampen diagram for $\overline{w}$ as per Lemma 5.5. Given (3) of that lemma, there is a forest $\mathcal{F}$ in the 1-skeleton of the union of the $\mathcal{R}$ -faces in $\overline{\Delta }$ joining every vertex of an $\mathcal{R}$ -face to $\partial \overline{\Delta }$ by a path of length at most $Cn$ .

Charge $\overline{\Delta }$ . Given that the defining relation $z^t =z$ is unchanged on lifting to the central extension, the $\mathbb{Z}^2$ -faces of Lemma 5.5 (2) are unchanged. There are $Cn^2$ such $\mathbb{Z}^2$ faces. Let $m = \max \{|k_a|, |k_b|\}$ . The remaining $Cn$ $\mathcal{R}$ -faces of Lemma 5.5 (1) each acquire at most $m$ charges. These are discharged by adding partial $c$ -corridors that follow the forest $\mathcal{F}$ to the boundary and then around the boundary to a base vertex. Each partial $c$ -corridor has length at most $(C+1)n$ : the length of the path to the boundary is at most $Cn$ by Lemma 5.5 (3) and the length of the path to the base vertex is at most $n$ . In total then, $c$ -partial corridors contribute at most $(C+1)^{2}mn^2$ 2-cells to the new diagram. The result is a diagram over $\mathcal{Q}$ of area at most $((C+1)^{2}m +C) n^2$ for a word $\overline{w}c^k$ , which has length less than $n$ . By adding in an annular region to rearrange $\overline{w}c^k$ to $w$ , as per the electrostatic model of Section 4, it follows that $w$ has a diagram over $\mathcal{Q}$ of area at most $({(C+1)}^{2}m +C+1) n^2$ .

5.3. Theorem A in the case where all eigenvalues of $\psi_{\boldsymbol{ab}}$ are unit

We begin by arguing that for the purpose of determining Dehn functions, we can further specialize the family of presentations as follows.

Lemma 5.6. Suppose that $\Phi \in{\textrm{Aut}}(G)$ is as per Lemma 5.2, and that eigenvalues of $\phi _{ab}$ are $1$ . Then there exists $\Xi \in{\textrm{Aut}}(G)$ such that the eigenvalues of $\xi _{ab}$ are also $1$ , the Dehn functions of $M_{\Phi }$ and $M_{\Xi }$ are equivalent, and

\begin{equation*}M_{\Xi }=\langle a, b, c, t \mid a^t=ab^{\beta }c^{k_a}, b^t= bc^{k_b}, c^t =c, \ [a,c]=1, \ [b,c]=1\rangle \end{equation*}

for some $\beta \in \mathbb{Z}$ . Moreover, conditions 2 and 3 of Section 5.1 hold for $\Psi$ exactly when they hold for $\Xi$ , and they are characterized by $k_b \neq 0$ and $k_b = 0$ , respectively.

Proof. We have $\Phi$ per Lemma 5.2. So $\Phi (a) = \phi (a)c^{k_a'}, \ \Phi (b) = \phi (b)c^{k_b'}, \ \Phi (c) =c$ , for some $k_a', k_b' \in \mathbb{Z}$ and some $\phi \in{\textrm{Aut}}(F(a,b))$ such that $\phi _{ab}$ has determinant $1$ . Lemma 5.2 implied that the eigenvalues of $\phi _{ab}$ are 1 for automorphisms of types 2 and 3.

We will show that there is $\Xi \in{\textrm{Aut}}(F_2 \times \mathbb{Z})$ such that for some $\kappa \geq 0$ , $[\Phi ^\kappa ]$ and $[\Xi ]$ are conjugate in $\textrm{Out}(F_2 \times \mathbb{Z})$ and $\xi = \Xi \!\! \upharpoonright _{F(a,b)}$ maps $b \mapsto b$ .

As $\phi _{ab}$ has only eigenvalue 1, it is either the identity or it has linear growth. Lemma 3.7, implies that for some $\kappa$ , $\phi ^\kappa _{ab}$ is conjugate in $\textrm{SL}(2, \mathbb{Z})$ to $\left (\begin{smallmatrix} 1 & \beta \\[5pt] 0 & 1 \end{smallmatrix}\right )$ for some $\beta \in \mathbb{Z}$ . Define $\Phi ^\prime =\Phi ^\kappa$ , with restriction $\phi ^\prime$ . On account of the standard isomorphism between $\textrm{Out}(F_2)$ and $\textrm{GL}(2,\mathbb{Z})$ , $[\phi ^\prime ]$ is conjugate in $\textrm{Out}(F_2)$ to $[\xi ]$ where $\xi (a) = ab^{\beta }$ and $\xi (b) = b$ . So $\xi = f^{-1}\circ \phi ^\prime \circ f\circ \iota _g$ for some $f \in{\textrm{Aut}}(F_2)$ and some $\iota _g \in \textrm{Inn}(F_2)$ . We lift $f, \xi, \iota _g \in{\textrm{Aut}}(F_2)$ to $F, \Xi, \hat{\iota }_g \in{\textrm{Aut}}(F_2 \times \mathbb{Z})$ by defining $F(gc^k) = f(g)c^k$ for $g \in F_2$ and $k \in \mathbb{Z}$ , by taking $\hat{\iota }_g$ to be conjugation by $g$ , and by defining $\Xi \;:\!=\; F^{-1} \circ \Phi ^\prime \circ F \circ \hat{\iota }_g$ . Because $c$ is central, $\hat{\iota }_g(c) = c$ . In particular,

\begin{equation*}\Xi \;:\; a \mapsto ab^{\beta }c^{k_a}, \ \ b \mapsto bc^{k_b}, \ \ c \mapsto c,\end{equation*}

for some $k_a, k_b \in \mathbb{Z}$ . (Note that $p( \Phi ^{\prime }(b))$ and $p(\Xi (b)) = k_b$ may not be equal, as $\Phi ^\prime (f(b)^{g^{-1}})$ , and $\Phi ^\prime (b)$ will not generally have the same index sum of $c$ letters.)

Therefore, $M_{\Xi }$ has the presentation claimed and $\xi _{ab}$ has only $1$ as an eigenvalue, as required. By Lemma 3.6, the mapping tori $M_{\Phi }$ and $M_{\Xi }$ have equivalent Dehn functions.

Next we will show that

1. 1. If $\phi ^{\prime }_{ab} \neq \textrm{id}$ , then $p(\Xi (b))=k_b\neq 0$ if and only if there exists $w \in F_2$ such that $\phi ^{\prime }_{ab}(w_{ab})= w_{ab}$ and $p(\Phi ^{\prime }(w)) \neq 0$ .

2. 2. If $\phi ^{\prime }_{ab} = \textrm{id}$ , exactly one of the following holds:

   1. (a) $p(\Phi ^\prime (x)) = 0$ for all $x \in \langle a,b \rangle$ , in which case $\Xi =\textrm{Id}$ ,

   2. (b) $p(\Phi ^\prime (x)) \neq 0$ for some $x$ , in which case $p(\Xi (a))$ or $p(\Xi (b))$ is nonzero.

Moreover, this implies that $\Phi$ , $\Phi '$ and $\Xi$ either all satisfy condition 2 or all satisfy condition 3 of Section 5.1.

Proof of 1: We wish to compare $p(\Phi ^\prime (w))$ and $p(\Xi (b))$ . Let $w^{\prime } = f(b)$ . The following calculation shows that $w_{ab}^{\prime }$ is another fixed point of $\phi ^\prime _{ab}$ and that $p(\Phi ^\prime (w^{\prime })) =p(\Xi (b))= k_b$ :

\begin{equation*}\Phi ^\prime (w^{\prime }) \ = \ F\circ \Xi \circ \iota _{g^{-1}}\circ F^{-1}(f(b)) \ = \ F(\Xi (b^{g^{-1}})) \ = \ F(b^{g^{-1}}c^{p(\Xi (b))}) \ = \ f(b)^{g^{-1}}c^{p(\Xi (b))} \ = \ (w')^{g^{-1}}c^{p(\Xi (b))}.\end{equation*}

If $\phi ^{\prime }_{ab} \neq \mbox{id}$ , then since $w_{ab}$ and $w^{\prime }_{ab}$ are both fixed by $\phi ^{\prime }_{ab}$ , $w_{ab} = dw^{\prime }_{ab}$ for some $d\neq 0$ , and therefore $p(\Phi ^{\prime }(w)) = dp(\Phi ^{\prime }(w^{\prime })) = dk_b$ . So $p(\Xi (b))=k_b\neq 0$ if and only if $p(\Phi ^{\prime }(w)) \neq 0$ . In case 1, $\Phi ^{\prime }$ and $\Xi$ either both satisfy condition 2 or both satisfy condition 3, and so this holds for $\Psi$ and $\Xi$ by Lemma 5.2.

Proof of 2: $\phi ^{\prime } \in \textrm{Inn}(F_2)$ since

\begin{equation*}1\to \textrm {Inn}(F_2) \to {\textrm {Aut}}(F_2) \to \textrm {SL}(2,\mathbb {Z})\to 1\end{equation*}

is exact and $\phi ^{\prime }_{ab}=\textrm{id}$ . Therefore, $\Xi$ maps $a \mapsto ac^{k_a}$ , $b \mapsto bc^{k_b}$ , and $c \mapsto c$ for some $k_a, k_b \in \mathbb{Z}$ , and so $\xi _{ab} = \mbox{id}$ also. This suffices to show 2.

When $\phi ^{\prime }_{ab}= \mbox{id}$ , condition 2 amounts to “there exists $g \in F_2$ such that $p(\Phi ^{\prime }(g)) \neq 0$ ”. This holds for $\Phi ^{\prime }$ if and only if it holds for $\Xi$ , by 2. Again $\Phi$ , $\Phi '$ and $\Xi$ either all satisfy condition 2 or all satisfy condition 3, and so this holds for $\Psi$ and $\Xi$ by Lemma 5.2.

Proof of Theorem A in Case 2. By Lemmas 5.2 and 5.6, for the purpose of calculating the Dehn function we may work with

\begin{equation*}M_{\Xi }=\langle a, b, c, t \,|\, a^t=ab^{\beta }c^{k_a},\, b^t= bc^{k_b},\, c^t =c, \ [a,c]=1, [b,c]=1\rangle \end{equation*}

where $k_b \neq 0$ . The subgroup $K \;:\!=\; \langle b, c \mid [b,c] \rangle \cong \mathbb{Z}^2$ quasi-isometrically embeds in $F_2 \times \langle c \rangle$ and $\Xi (K) = \langle bc^{k_b}, c \rangle = K$ . So, by Lemma 3.5, $n \mapsto n^2 \max \{|\Xi ^n(b)|, |\Xi ^n(c)|\}= n^2(k_bn+1)$ is a lower bound for the Dehn function of $M_{\Xi }$ . This lower bound is cubic (as $k_b \neq 0$ ), matching our upper bound from Corollary 5.3, so the claim is established.

We now turn to case 3. This time, Lemmas 5.2 and 5.6 allow us to work with $M_{\Xi }$ which has the form:

\begin{equation*}M_{\Xi }=\langle a, b, c, t \,|\, a^t=ab^{\beta }c^{k_a}, \,b^t= b,\, c^t =c, \ [a,c]=1, [b,c]=1\rangle \end{equation*}

where, $\beta$ is nonzero. (The case $\beta =0$ and $k_a \neq 0$ is covered by Theorem A in case 2—the Dehn function of this mapping torus is cubic. If $\beta =0$ and $k_a =0$ , then $M_{\Xi } = \langle a, b, c \mid [a,c]=1, [b,c]=1 \rangle \times \langle t \rangle$ , which has quadratic Dehn function.)

The methods of case 1 cannot be used here. Indeed, Button and R. Kropholler in Theorem 4.4 [Reference Button and Kropholler12] have shown that for $\xi$ with this form, $M_{\xi }$ is not strongly hyperbolic relative to any finitely generated proper subgroup, so van Kampen diagrams over $M_{\xi }$ do not decompose into uncharged islands with linear-area complement. Instead will use a variant of the electrostatic model whereby the diagram will be discharged along partial corridors (see Section 3.1) in a manner controlled by an application of Hall’s Marriage Theorem, which we now review.

A subgraph $F$ of a graph $\Gamma$ is a 1-factor for $\Gamma$ if it contains all vertices of $\Gamma$ and each vertex meets precisely one edge of $F$ . In other words, a 1-factor pairs every vertex with a neighbor. We will be interested in the following special case:

This is a consequence of Hall’s Marriage Theorem. See [Reference Diestel13] for a proof.

Proof of Theorem A in Case 3. By Lemma 5.6, it suffices to prove that $M_{\Phi }$ , presented by:

\begin{equation*}\mathcal {P}= \langle a, b, t, c \mid a^t=ab^{\beta }c^{k_a}, \ b^t=b,\ ac=ca,\ bc=cb, \ ct=tc \rangle,\end{equation*}

has quadratic Dehn function. If $k_a =0$ , then $M_{\Phi }\cong M_{\phi }\times \langle c \rangle$ and so the Dehn functions of $M_{\Phi }$ and $M_{\phi }$ agree and are quadratic. Therefore, we may restrict our attention to the case where $\beta$ and $k_a$ are both nonzero.

Van Kampen diagrams over $\mathcal{P}$ have both partial $b$ -corridors and partial $c$ -corridors. $M_{\Phi }$ is a central extension of $M_{\phi }$ by $\langle c \rangle$ , where $M_{\phi }$ is presented by:

\begin{equation*}\mathcal {Q} \ = \ \langle a, b, t \mid a^t=ab^{\beta }, b^t=b \rangle .\end{equation*}

Van Kampen diagrams over $\mathcal{Q}$ may have partial $b$ -corridors.

Suppose $w$ is a word of length $n$ representing the identity in $\mathcal{P}$ . Let $\overline{w}$ be $w$ with all $c^{\pm 1}$ removed. Then $w = \overline{w}c^m$ in $M_{\Phi }$ for some $m \in \mathbb{Z}$ and $|\overline{w}| \leq |w|$ . Since $\mathcal{Q}$ is free-by-cyclic and non-hyperbolic, it has a quadratic Dehn function. So there exists a minimal area diagram $\overline{\Delta }$ for $\overline{w}$ over $\mathcal{Q}$ such that $\textrm{Area}(\overline{\Delta }) \leq C |\overline{w}|^2$ . We charge $\overline{\Delta }$ by replacing 2-cells in $\overline{\Delta }$ with 2-cells labeled by the defining relators from $\mathcal{P}$ , as in the first steps of the electrostatic model (see Section 4). What follows is a scheme for adding in 2-cells to “discharge” $\overline{\Delta }$ so as to create a diagram for $\overline{w}c^m$ over $\mathcal{P}$ .

The idea is that if we can pair off oppositely oriented capping faces that are joined by partial $b$ -corridors, then we can add in partial $c$ -corridors following the $b$ -corridors, as in Figure 5, in order to discharge the $c$ -edges in our diagram. As $c$ is central in $\mathcal{P}$ , partial $c$ -corridors can be run alongside this partial $b$ -corridor, and the word one reads along both the top and bottom of the $c$ -corridor will be the same as that word along the top and bottom of the $b$ -corridor, namely some power of $t$ . We wish to find a consistent way of partnering vertices so that we can replicate the picture in Figure 5, adding in partial $c$ -corridors to discharge between partners throughout the van Kampen diagram, with no leftover charges to consider.

Figure 5. If a partial $b$ -corridor joins two capping faces in $\overline{\Delta }$ , their $c$ -charges can be discharged by adding partial $c$ -corridors “following” that partial $b$ -corridor.

I. Modeling $\overline{\Delta }$ with a graph. Construct a planar graph with multi-edges, $\Gamma$ , from $\overline{\Delta }$ as illustrated in Figure 6: $\Gamma$ has a black vertex for each capping face in $\overline{\Delta }$ ; whenever two capping faces are connected by a partial $b$ -corridor, possibly of length zero, an edge connects the corresponding vertices (two vertices may share multiple edges); we also add an edge and a white vertex to $\Gamma$ for each partial $b$ -corridor that goes to the boundary. Every black vertex in the graph $\Gamma$ is degree $|\beta |$ and every white vertex has degree 1.

Figure 6. From capping faces and partial corridors in $\overline{\Delta }$ , construct a graph $\Gamma$ . Black vertices correspond to capping faces, white vertices correspond to 1-cells labeled $b$ in $\partial \overline{\Delta }$ , and the edges correspond to partial $b$ -corridor.

The graph $\Gamma$ is bipartite (but not generally black–white bipartite, as you can see in Figure 6): partition the black vertices according to whether they correspond to capping faces with clockwise or anticlockwise oriented $b$ -edges and extend this partition to the white vertices.

II. Building a regular bipartite graph. We would like to apply Lemma 5.7, but $\Gamma$ may not be regular: black vertices have degree $\left |\beta \right |$ , but white vertices have degree $1$ . So, as illustrated in Figure 7a, we construct a regular graph $\hat{\Gamma }$ which has $\Gamma$ as a subgraph. Take $|\beta |$ many copies of $\Gamma$ , and identify the white vertices in each of the copies. That is,

\begin{equation*}\hat {\Gamma } \ \;:\!=\; \ \displaystyle \left (\bigsqcup _{i=1}^{|\beta |} \Gamma \times \{i\}\right )/ \sim,\end{equation*}

where $(v,i) \sim (v,j)$ for all $i,j$ when $v$ is a white vertex. White vertices are degree one, so the identification of $|\beta |$ copies of $\Gamma$ forces $\hat{\Gamma }$ to be a $|\beta |$ -regular graph. If $\Gamma$ is bipartite with respect to a partition $A \sqcup B$ of its vertices, then $\hat{\Gamma }$ is bipartite with respect to

\begin{equation*}\left ( \bigcup _{i=1}^{|\beta |}A \times \left \{i\right \} \right )/ \sim \ \bigsqcup \ \left ( \bigcup _{i=1}^{|\beta |}B \times \left \{i\right \} \right )/ \sim .\end{equation*}

Figure 7. Finding neighbor partners for $\Gamma$ via Hall’s Marriage Theorem.

III. Finding pairing partners for $b$ - and $c$ -corridors. Lemma 5.7 tells us that $\hat{\Gamma }$ has a 1-factor. This partners each vertex $v \in \hat{\Gamma }$ with an adjacent vertex $v^{\prime }$ . View the image of $\Gamma \times \{1\}$ in $\hat{\Gamma }$ as $\Gamma$ , sitting as a subgraph in $\hat{\Gamma }$ . In the example of Figure 7c, $\Gamma$ is the gray subgraph at the back. If $v \in \Gamma$ is a black vertex, its partner $v'$ is also a vertex of $\Gamma$ , but this may fail for white vertices.

IV. Completing to a van Kampen diagram. If $v$ and $v^{\prime }$ are partnered black vertices in $\Gamma$ , then the corresponding capping faces are connected by at least one partial $b$ -corridor (possibly of length zero). In $\overline{\Delta }$ , the capping faces $f$ and $f'$ corresponding to $v$ and $v^{\prime }$ have $|k_a|$ many oppositely oriented charges. We will connect these charges with $|k_a|$ partial $c$ -corridors, as in Figure 5. Choose one of the partial $b$ -corridors joining $f$ to $f'$ (there is at least one). Run all of the partial $c$ -corridors for one capping face alongside the partial $b$ -corridor. If a black vertex $v$ is paired with a white vertex $v'$ in $\Gamma$ , run all of the partial $c$ -corridors alongside the partial $b$ -corridor to the boundary. Two white vertices will never be paired. At the ends of partial $b$ -corridors on capping faces, it may be necessary to insert rectangles in which the $b$ -and $c$ -corridors cross, as in Figure 8, but this requires no more than $|\beta ||k_a|\textrm{Area}(\overline{\Delta })$ additional 2-cells. The total number of 2-cells added to $\overline{\Delta }$ in this process is no more than $(|\beta |+1)|k_a|\textrm{Area}(\overline{\Delta })$ .

Figure 8. Partnering in $\Gamma$ gives a consistent way to discharge $c$ -charges.

V. Correcting the boundary. Partial $c$ -corridors follow partial $b$ -corridors to the boundary in groups of $|k_a|$ . The new diagram has boundary length between $|\overline{w}|$ and $(|k_a|+1)|\overline{w}|$ and is a van Kampen diagram over $\mathcal{P}$ for some word $w^{\prime }$ in the pre-image of $\overline{w}$ . Deleting all $c^{\pm 1}$ from $w^{\prime }$ produces $\overline{w}$ , but the arrangement of the $c^{\pm 1}$ letters in $w^{\prime }$ may differ from that in $w$ . As was described in Section 4, we glue around the outside of this diagram an annular diagram with the word $w^{\prime }$ along the inner boundary component and the word $w$ along the outer boundary component. Together, they form $\Delta$ , a van Kampen diagram for $w$ over $\mathcal{P}$ . This annular diagram has area at most $(|k_a|+1)^2|\overline{w}|^2$ , and summing our area estimates, $\Delta$ has area no more than $(1+ (|\beta |+1)|k_a|)\textrm{Area}(\overline{\Delta }) + (|k_a|+1)^2|\overline{w}|^2$ . Since $\textrm{Area}(\overline{\Delta }) \leq C|\overline{w}|^2$ , it follows that there is constant $A\gt 0$ such that for any given word $w$ in the generators of $\mathcal{P}$ that represents the identity, this construction produces a van Kampen diagram of area at most $A|w|^2$ .

6.1. Automorphisms of $\mathbb{Z}^2 \ast \mathbb{Z}$

Servatius [Reference Servatius24] and Laurence [Reference Laurence18] found a generating set for the automorphism group of a RAAG $A(\Gamma )$ based on the underlying graph $\Gamma$ (see Lemma 7.2). For

\begin{equation*}\mathbb {Z}^2 \ast \mathbb {Z} \ = \ \langle a, b \mid [a,b] \rangle \ast \langle c \rangle,\end{equation*}

it consists of the inner automorphisms, inversions, the one nontrivial graph isomorphism ( $a \mapsto b$ , $b \mapsto a$ , and $c \mapsto c$ ), and the four transvections

\begin{equation*}\begin {array}{l@{\quad}l@{\quad}l@{\quad}l} \tau _a \;:\;& a \mapsto ab, & b \mapsto b, & c \mapsto c, \\[5pt] \tau _b \;:\;& a \mapsto a, & b \mapsto ba, & c \mapsto c, \\[5pt] \psi _a \;:\;& a \mapsto a, & b \mapsto b, & c \mapsto ca, \\[5pt] \psi _b \;:\;& a \mapsto a, & b \mapsto b, & c \mapsto cb. \end {array}\end{equation*}

The following lemma and proposition are steps toward Theorem B in that they let us focus on particular presentations for the purposes of classifying Dehn functions of mapping tori of $\mathbb{Z}^2 \ast \mathbb{Z}$ . Recall that $\iota _h$ denotes the inner automorphism $x \mapsto h^{-1} x h$ .

Lemma 6.1. For all $\Psi \in{\textrm{Aut}}( \mathbb{Z}^2 \ast \mathbb{Z})$ , there exist $\Phi \in{\textrm{Aut}}( \mathbb{Z}^2 \ast \mathbb{Z})$ , $\phi \in{\textrm{Aut}}(\mathbb{Z}^2)$ , and words $w$ and $x$ on $a$ and $b$ such that

\begin{equation*}\Phi \;:\; a \mapsto \phi (a), \ \ b \mapsto \phi (b), \ \ c \mapsto wc^{\pm 1}x\end{equation*}

and $[\Phi ] = [\Psi ]$ in $\textrm{Out}(\mathbb{Z}^2 \ast \mathbb{Z})$ . Explicitly, if $\Psi (a) = u_1c^{\epsilon _1}\dots u_n c^{\epsilon _n}u_{n+1}$ , where each $\epsilon _i \neq 0$ and each $u_i \in \langle a, b\rangle$ , and $u_2, \ldots, u_n \neq 1$ , then $n=2m$ is even and for $g \;:\!=\; c^{\epsilon _{m+1}}u_{m+2}\dots u_{2m} c^{\epsilon _{2m}}u_{2m+1}$ , the map $\iota _{g^{-1}}\circ \Psi$ satisfies the properties required of $\Phi$ .

Moreover, $M_{\Psi }$ and $M_{\Phi }$ have equivalent Dehn functions for any such $\Phi$ .

Proof. Since $\textrm{Inn}(\mathbb{Z}^2 \ast \mathbb{Z}) \trianglelefteq{\textrm{Aut}}(\mathbb{Z}^2 \ast \mathbb{Z})$ , all automorphisms $\Psi \in{\textrm{Aut}}(\mathbb{Z}^2 \ast \mathbb{Z})$ can be written as $\iota _h \circ \Phi$ where $\iota _h$ denotes conjugation by some $h \in \mathbb{Z}^2 \ast \mathbb{Z}$ and $\Phi$ is some product of the inversions, transvections, and graph isomorphisms in the generating set above. These inversions, transvections, and graph isomorphisms restrict to automorphisms of the subgroup $\langle a, b \mid [a,b] \rangle$ and map the subset $\langle a, b \rangle c^{\pm 1} \langle a, b \rangle$ to itself. So

\begin{equation*}\Phi \;:\;a \mapsto \phi (a), \ \ b \mapsto \phi (b), \ \ c \mapsto wc^{\pm 1}x,\end{equation*}

for some $\phi \in{\textrm{Aut}}(\mathbb{Z}^2)$ and some words $w$ and $x$ on $a^{\pm 1}$ and $b^{\pm 1}$ . This proves the existence of a $\Phi$ with the required properties. We turn next to how to find such a $\Phi$ explicitly.

Suppose $\Psi (a)$ is as per the statement. For $h \in \mathbb{Z}^2 \ast \mathbb{Z}$ as above, we have that $\Psi (a) \in \iota _{h}( \langle a, b \rangle )$ . So

\begin{equation*} \Psi (a) \ = \ u_1c^{\epsilon _1}\dots u_n c^{\epsilon _n}u_{n+1} \ \in \ h^{-1} \langle a, b \rangle h.\end{equation*}

But, given the free product structure of $\mathbb{Z}^2 \ast \mathbb{Z}$ , that implies that $n=2m$ is even and

\begin{equation*}h \ = \ v c^{\epsilon _{m+1}}u_{m+2}c^{\epsilon _{m+2}}\cdots u_{2m}c^{\epsilon _{2m}}u_{2m+1} \ = \ vg\end{equation*}

where $v$ is some element of $\langle a, b \rangle$ and $g$ is as defined in the statement.

It follows then that $\iota _{g^{-1}} \circ \Psi = \iota _{v} \circ \iota _{h^{-1}} \circ \Psi = \iota _{v} \circ \Phi$ and maps $a \mapsto \phi '(a), \ \ b \mapsto \phi '(b), \ \ c \mapsto w'c^{\pm 1}x'$ for some $\phi ' \in{\textrm{Aut}}(\mathbb{Z}^2)$ and some words $w'$ and $x'$ on $a^{\pm 1}$ and $b^{\pm 1}$ .

By Lemma 3.6, $M_{\Psi }$ and $M_{\Phi }$ have equivalent Dehn functions.

Proposition 6.2. Given $\Phi$ as per Lemma 6.1, there exist $\,\Xi \in{\textrm{Aut}}( \mathbb{Z}^2 \ast \mathbb{Z})$ , $\xi \in{\textrm{Aut}}(\mathbb{Z}^2)$ , and $z \in \langle a, b \rangle$ such that

\begin{equation*}\Xi \;:\; a \mapsto \xi (a), \ \ b \mapsto \xi (b), \ \ c \mapsto cz\end{equation*}

and $M_{\Phi }$ and $M_{\Xi }$ have equivalent Dehn functions. Moreover, Conditions 1, 2, and 3 of Theorem B apply to $\Xi$ exactly when they apply to $\Phi$ . Additionally,

• when $\xi$ has finite order (Condition 1 of Theorem B), we may further assume $\xi \;:\;a \mapsto a, \ b \mapsto b$ , so that

  \begin{equation*}M_{\Xi } \ = \ \langle a,b,c,t \ \mid \ [a,b]=[a,t]=[b,t]=1, \ c^t=ca^kb^l \rangle,\end{equation*}

• when $\xi$ is of infinite order and has only unit eigenvalues (Condition 3 of Theorem B), we may further assume $\xi \;:\;a \mapsto ab^k, \ b \mapsto b$ for some $k \neq 0$ , so that for some $l,m \in \mathbb{Z}$ ,

  \begin{equation*} M_{\Xi } \ = \ \langle a, b,c, t \mid [a,b]=1, \ a^{t}=ab^{k}, \ b^{t}=b,\ c^{t}=c a^lb^m \rangle .\end{equation*}

Proof. How we will define $\Xi$ will depend on the form of $\Phi (c)$ . Recall $\Phi (c)= wc^{\pm 1}x$ as per Lemma 6.1. Define $\Xi _1 \;:\!=\; \iota _{w} \circ \Phi$ and $\Xi _2 \;:\!=\; \iota _{\phi (w)x^{-1}}\circ \Phi ^2$ . If $\Phi (c) = wcx$ , then $\Xi _1(c) = \iota _w \circ \Phi (c) =cz$ where $z=xw$ . If $\Phi (c) = wc^{-1}x$ , then $\Xi _2(c) = \iota _{\phi (w)x^{-1}} \circ \Phi ^2(c) = cz$ where $z=w^{-1}\phi (xw)x^{-1}$ . For $i=1,2$ , let $\xi _i \;:\!=\; \Xi _i \left |_{\langle a, b \rangle } \right .$ , the restriction of $ \Xi _i$ to the $\mathbb{Z}^2$ factor.

Suppose $\xi _i$ has exponential growth (so has a non-unit eigenvalue as per Lemma 3.9 and Condition 2 of Theorem B). Define $\Xi \;:\!=\;\Xi _i$ . Then $\Xi$ has the general form claimed in the proposition, and since $[\Xi _1] = [\Phi ]$ and $[\Xi _2] = [\Phi ]^2$ , Lemma 3.6 implies that $M_{\Phi }$ and $M_{\Xi }$ have equivalent Dehn functions.

Suppose $\xi _i \in{\textrm{Aut}}(\mathbb{Z}^2)$ has finite order $n$ (i.e. $\xi _i$ has trivial growth). Define $\Xi = \Xi _i^n$ . This has the promised form: its restriction to $\langle a,b \rangle$ is the identity and $\Xi (c) = cz$ for some $z \in \langle a,b \rangle$ . Lemma 3.6 implies that $M_{\Xi }$ and $M_{\Phi }$ have equivalent Dehn functions.

Finally, suppose $\xi _i$ is of infinite order and has only unit eigenvalues. Lemma 3.7 implies that for some power $n$ , $(\xi _i)^{n}$ is conjugate in ${\textrm{Aut}}(\mathbb{Z}^2)$ to the automorphism $\xi \;:\;a \mapsto ab^{k}$ , $b \mapsto b$ , for some $k \neq 0$ . Therefore, for some $f \in{\textrm{Aut}}(\mathbb{Z}^2)$ , ${\xi } = f^{-1}\circ{(\xi _i)}^n \circ f$ . Let $F$ be the automorphism that restricts to $f$ on $\mathbb{Z}^2$ and maps $c \mapsto c$ . Define $\Xi \;:\!=\; F^{-1}\circ (\Xi _i)^n \circ F$ . By Lemma 3.6, $M_{\Phi }$ and $M_{\Xi }$ have equivalent Dehn functions. Let $z^{\prime } = z\xi _i(z) \dots (\xi _i)^{n-1}(z)$ and $z^{\prime \prime } = f^{-1}(z^{\prime })$ . Both are elements of $\langle a,b \rangle$ . The map $\Xi$ has the desired form: