# Subgroup Distortion and the Relative Dehn Functions of Metabelian Groups

Wenhao Wang  Department of Mathematics
Vanderbilt University
Nashville, TN 37240<EMAIL_ADDRESS>
###### Abstract.

We show the connection between the relative Dehn function of a finitely
generated metabelian group and the distortion function of a corresponding
subgroup in the wreath product of two free abelian groups of finite rank.
Further, we show that if a finitely generated metabelian group $G$ is an
extension of an abelian group by $\mathbb{Z}$ the relative Dehn function of
$G$ is polynomially bounded. Therefore, if $G$ is finitely presented, the Dehn
function is bounded above by the exponential function up to equivalence.

##  0\. Notation

Let $G$ be a group. For elements $x,y\in G,n\in\mathbb{N}$, our conventions
are $x^{ny}=y^{-1}x^{n}y,[x,y]=x^{-1}y^{-1}xy$. We use double bracket
$\langle\langle\cdot\rangle\rangle_{G}$ to denote the normal closure of a set
in the group $G$. Sometimes we omit the subscript when there is no
misunderstanding in the context.

In addition, for a group $G$ and a commutative ring $K$ with $1\neq 0$, we let
$KG$ be the group ring of $G$ over $K$. An element $\lambda\in KG$ is usually
denoted as $\lambda=\sum_{g\in G}\alpha_{g}g,\alpha_{g}\in K$ where all but
finitely many $\alpha_{g}$’s are 0. We also regard $\lambda$ as a function
$\lambda:G\to K$ with finite support, where $\lambda(g)=\alpha_{g}$. We let
$|\lambda|=\sum_{g\in G}|\alpha_{g}|$.

For groups $A$ and $T$, let $B=\oplus_{t\in T}A^{t}$ be the direct sum of
copies of $A$ indexed by elements in $T$. Then _the wreath product_ $A\wr T$
is defined to be the semidirect product $B\rtimes T$ where $T$ acts on $B$ by
$t\circ(a_{\omega})=(a_{t^{-1}\omega})$. The subgroup $B$ is called the base
group of the wreath product.

Suppose $G$ is an extension of $A$ by $T$ where both $A$ and $T$ are abelian.
$A$ has a natural module structure over the group ring $\mathbb{Z}T$, and the
action of $T$ on $A$ is given by conjugation. In this case, we also say that
$G$ is an extension of a $T$-module $A$ by $T$. In this paper, we will use the
following notation for actions of $\mathbb{Z}T$ on $A$. Let
$\lambda=\sum_{t\in T}\alpha_{t}t\in\mathbb{Z}T$. Then for $a\in A$, we define

$a^{\lambda}:=\prod_{t\in T}a^{\alpha_{t}t}.$

##  1\. Introduction and Results

In this paper, we study two different functions associates with finitely
generated metabelian groups, both of which are introduced to describe some
geometric properties of groups, the (relative) Dehn function and the subgroup
distortion function. They are naturally related since for a finitely presented
group the Dehn function can be regarded as the distortion function of a
subgroup in a free group over an infinite generating set (the set of all
conjugates of relators). Asides from the canonical connection, we investigate
the connection between the subgroup distortion function (of a special type of
finitely generated subgroups) in the wreath product to the Dehn function
relative to the variety of metabelian groups, which it interesting because it
provides an estimation of the usual Dehn function of a finitely presented
metabelian group [Wan20].

We first define the (relative) Dehn function. Recall that a _variety_ of
groups is a class of groups that closed under taking subgroups, epimorphic
images, and unrestricted direct products. Inside a variety, we can talk about
relative free groups and relative presentations, just like in the usual sense
(the variety of all groups). Let $\mathcal{V}$ be a variety. The free group
relative to the variety $\mathcal{V}$ on a set $X$, denoted by $\tilde{F}(X)$,
is a group in $\mathcal{V}$ satisfying the following property: it is equipped
with a map $\theta:X\to\tilde{F}(X)$ such that for every group $G$ in
$\mathcal{V}$ and every set map $\sigma:X\to G$ there exists a unique
homomorphism $\varphi:\tilde{F}(X)\to G$ such that the following diagram
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In particular, a group in $\mathcal{V}$ generated by $|X|$ elements is a
epimorphic image of $\tilde{F}(X)$. A relative finite presentation
$\mathcal{P}=\langle X\mid R\rangle_{\mathcal{V}}$ of $G$ consists of two
finite sets $X$ and $R$ where $R$ is a subset of $\tilde{F}(X)$ such that
there exists an epimorphism $\varphi:\tilde{F}(X)\to G$ and the kernel of
$\varphi$ is $\langle\langle R\rangle\rangle_{\tilde{F}(X)}$. Let $w$ be a
word in $G$ such that $w=_{G}1$. Then $w$ lies in the normal closure of $R$.
Thus $w$ can be written as

$w=_{\tilde{F}(X)}\prod_{i=1}^{l}f_{i}^{-1}r_{i}{f_{i}}\text{ where }r_{i}\in
R\cup R^{-1},f_{i}\in\tilde{F}(X).$

The smallest possible $l$ is called the relative area of $w$, denoted by
$\widetilde{\operatorname{Area}}_{\mathcal{P}}(w)$. Consequently, the Dehn
function relative to $\mathcal{V}$ with respect to the presentation
$\mathcal{P}$ is defined as

$\tilde{\delta}_{\mathcal{P}}(n)=\sup\\{\widetilde{\operatorname{Area}}_{\mathcal{P}}(w)\mid|w|_{X}\leqslant
n\\}.$

Here $|\cdot|_{X}$ is the word length in alphabet $X$.

Throughout this paper, we focus on two varieties of groups: the variety of all
groups and the variety of metabelian groups, where the later will be denoted
by $\mathcal{S}_{2}$.

If $\mathcal{V}$ is the variety of all groups, then the Dehn function relative
to $\mathcal{V}$ is just the usual Dehn function.

Dehn functions are defined up to an asymptotic equivalence $\approx$ taken on
functions $\mathbb{N}\to\mathbb{N}$ by $f\approx g$ if and only if
$f\preccurlyeq g$ and $g\preccurlyeq f$ where $f\preccurlyeq g$ if and only if
there exists $C>0$ such that $f(n)\leqslant Cg(Cn)+Cn+C$ for all
$n\in\mathbb{N}$. One can verify that $\approx$ is an equivalence relation.
This relation preserves the asymptotic nature of a function. For example, it
distinguishes polynomials of different degrees and likewise polynomials and
the exponential function. It also distinguishes functions like $n^{p}$ and
$n^{p}\log n$ for $p>1$. On the other hand, it identifies all polynomials of
the same degree, and likewise all single exponentials, i.e., $a^{n}\approx
b^{n}$ for $a,b>1$.

Despite the dependence of Dehn function on finite presentations of a group,
all Dehn functions of the same finitely presented group are equivalent under
$\approx$ [Gro93], i.e., given a finitely presented group $G$ with finite
presentations $\mathcal{P}$ and $\mathcal{P}^{\prime}$, one can show that
$\delta_{\mathcal{P}}\approx\delta_{\mathcal{P}^{\prime}}$. Thus, we define
the _Dehn function_ of a finitely presented group $G$, $\delta_{G}$, as the
Dehn function of any of its finite presentation.

If $\mathcal{V}$ is the variety of metabelian groups $\mathcal{S}_{2}$. It has
been shown that the relative Dehn functions are also independent of the choice
of finite presentations up to equivalence [Fuh00]. Therefore it is valid to
denote the relative Dehn function of a finitely generated metabelian group $G$
by $\tilde{\delta}_{G}$. One non-trivial property of metabelian groups is that
all finitely generated metabelian groups are relatively finitely presentable
in $\mathcal{S}_{2}$ [Hal54].

In what follows, the Dehn function of a finitely presented group $G$ and the
area of a word $w$ in $G$ relative to the variety of all groups will be
denoted by $\delta_{G}(n)$ and $\operatorname{Area}(w)$ respectively, while
the Dehn function of a finitely generated group $G$ and the area of a word $w$
in $G$ relative to the variety of metabelian groups will be denoted by
$\tilde{\delta}_{G}(n)$ and $\widetilde{\operatorname{Area}}(w)$ respectively.

Next, let us talk about the distortion function. Let $G$ be a finitely
generated group with a finite generating set $X$ and $H$ be a a subgroup of
$G$ with finite generating set $Y$. The _distortion function_ of $H$ in $G$ is

$\Delta_{H}^{G}(n)=\sup\\{|w|_{Y}\mid w\in H,|w|_{X}\leqslant n\\}.$

We consider a slightly different equivalence relation for distortion
functions. For non-decreasing functions $f$ and $g$ on $\mathbb{N}$, we say
that $f\preceq g$ if there exists a constant $C$ such that $f(n)\leqslant
Cg(Cn)$. Hence we say that two functions $f$ and $g$ are equivalent, written
$f\asymp g$, if $f\preceq g$ and $g\preceq f$. As expected, the distortion
function is independent of the choice of the finite generating set under this
equivalence relation [DK18, Proposition 8.98]. The reason we consider $\asymp$
rather than $\approx$ is that if the subgroup is infinite then the distortion
function is at least linear. We say a subgroup is _undistorted_ if the
distortion function is equivalent to a linear function.

Let $A$ and $T$ be free abelian groups with bases
$\\{a_{1},a_{2},\dots,a_{m}\\}$ and $\\{t_{1},t_{2},\dots,t_{k}\\}$
respectively. Consider the wreath product $W:=A\wr T$. The base group
$B:=\langle\langle A\rangle\rangle$ is a $T$-module. For a finite subset
$\mathcal{X}=\\{f_{1},f_{2},\dots,f_{l}\\}$ of $B$, let $H$ be the subgroup of
$W$ generated by $\mathcal{X}\cup\\{t_{1},t_{2},\dots,t_{k}\\}$ and $G$ be the
group $W/\langle\langle\mathcal{X}\rangle\rangle$.

Our main result is the following:

######  Theorem A (Theorem 4.1).

Let $W,H,G$ be groups defined as above, then

$\Delta_{H}^{W}(n)\preccurlyeq\tilde{\delta}_{G}^{k}(n)+n^{k},\tilde{\delta}_{G}(n)\preccurlyeq\max\\{n^{3},(\Delta_{H}^{W}(n^{2}))^{3}\\}.$

In particular, if $k=1$,

$\Delta_{H}^{W}(n)\preccurlyeq\tilde{\delta}_{G}(n).$

Theorem 4.1 leads to some interesting examples.

######  Corollary (Corollary 4.3).

For each $l\in\mathbb{N}$, there exists a finitely generated metabelian group
such that its relative Dehn function is asymptotically greater or equal to
$n^{l}$.

The distortion function of subgroups in $A\wr\mathbb{Z}$ has been studied
extensively by Davis and Olshanskiy [DO11]. Combining their result with
Theorem 4.1, we immediately have

######  Theorem B (Theorem 4.5).

Let $G$ be a finitely generated metabelian group such that $G$ is an extension
of an abelian group $A$ by a virtually cyclic abelian group $T$. Then the
relative Dehn function of $G$ is polynomially bounded. If in addition $G$ is
finitely presented, the Dehn function of $G$ is asymptotically bounded above
by the exponential function.

This theorem gives the exponential upper bound of Dehn functions for many
examples including the metabelian Baumslag-Solitar groups $BS(1,n)$ and
$\mathbb{Z}^{n}\rtimes_{\phi}\mathbb{Z}$ where $\phi\in GL(n,\mathbb{Z})$ (See
in [BG96]). And it also improves the main results in [Wan20].

Moreover, we estimate the relative Dehn function of various examples.

######  Theorem C.

  1. (1)

(Proposition 3.3) The metabelianized Baumslag-Solitar group
$\tilde{BS}(n,m)=\langle a,t\mid(a^{n})^{t}=a^{m}\rangle_{\mathcal{S}_{2}}$
has at most cubic relative Dehn function when $n\neq m$ and has at most
quartic relative Dehn function when $n=m$.

  2. (2)

(Corollary 3.4) The metabelianized Baumslag-Solitar group
$\tilde{BS}(n,m)=\langle
a,t\mid(a^{n})^{t}=a^{m}\rangle_{\mathcal{S}_{2}},m>2,m=n+1$ has at most
quadratic relative Dehn function.

  3. (3)

(Proposition 3.5) The lamplighter groups $L_{m}$ have at most cubic relative
Dehn function.

  4. (4)

(Proposition 3.6) The lamplighter group $L_{2}$ has linear relative Dehn
function.

_The structure of this paper._ In Section 2 we will state some preliminaries
on the topic of Dehn function of a module and the relative Dehn function of a
finitely generated metabelian group. Next in Section 3, we estimate the
relative Dehn function for different examples including the Baumslag-Solitar
groups and Lamplighter groups. Finally in Section 4 we prove the main theorem
about distortion function and the relative Dehn function.

_Acknowledgement._ I would like to thank my advisor Mark Sapir who points out
to me the study of distortion functions for subgroups in $A\wr\mathbb{Z}$
where $A$ is abelian.

##  2\. Preliminaries

###  2.1. The Dehn function of a Module

Let $T$ be a free abelian group of rank $k$ and $R=\mathbb{Z}T$ the group ring
of $T$. In what follows, we will only discuss modules over $R$.

Similar to groups, we have free modules and hence we can define the
presentation of a module. A subset $\\{f_{1},f_{2},\dots,f_{l}\\}$ of a
$R$-module $M$ is called a _generating set_ if every $f\in M$ is the linear
combination of them, i,e, there exists
$\alpha_{1},\alpha_{2},\dots,\alpha_{l}\in R$ such that

$f=\alpha_{1}f_{1}+\alpha_{2}f_{2}+\dots+\alpha_{l}f_{l}.$

A set of elements $\\{f_{1},f_{2},\dots,f_{l}\\}$ of a module $M$ is called
_independent_ if no nontrivial linear combination is zero, that is,

$\text{If }\alpha_{1}f_{1}+\alpha_{2}f_{2}+\dots+\alpha_{l}f_{l}=0,\text{ then
}\alpha_{i}=0,\text{ for }i=1,2,\dots,l.$

A _basis_ is an independent generating set.

One immediate example for a $R$-module is $R^{m}$. The addition and scalar
multiplication on $R^{m}$ are the following, respectively:

$\displaystyle(a_{1},a_{2},\dots,a_{m})+(b_{1},b_{2},\dots,b_{m})$
$\displaystyle=(a_{1}+b_{1},a_{2}+b_{2},\dots,a_{m}+b_{m}),$ $\displaystyle
r(a_{1},a_{2},\dots,a_{m})$ $\displaystyle=(ra_{1},ra_{2},\dots,ra_{m}).$

The module $R^{m}$ is called a _free $R$-module of rank $m$_. The canonical
basis of $R^{m}$ is $\\{e_{1},e_{2},\dots,e_{m}\\}$ where
$e_{i}=(0,\dots,1,\dots,0)$ with all but the $i$-th entry is 0.

A submodule of the free module $R^{1}$ is an ideal in the ring $R$.

Given a free $R$-module $M$ of finite rank and a submodule $S$ generated by a
finite set $\\{f_{1},f_{2},\dots,f_{l}\\}$, the membership of a submodule $S$
we are considering in this thesis is the following

######  Problem 2.1.

Given an element $f$ in $M$, decide whether $f$ in $S$, i.e., if there exists
elements $\alpha_{1},\alpha_{2},\dots,\alpha_{l}$ such that

$f=\alpha_{1}f_{1}+\alpha_{2}f_{2}+\dots+\alpha_{l}f_{l}.$

A _homomorphism_ $\varphi:M\to N$ of $R$-modules is a map which is compatible
with the laws of composition:

$\varphi(f+f^{\prime})=\varphi(f)+\varphi(f^{\prime}),\varphi(rf)=r\varphi(f)$

for all $f,f^{\prime}\in M,r\in R$. A bijective homomorphism is called an
_isomorphism_.

Last we define the concept of quotient modules. Let $R$ be a ring, and let $S$
be a submodule of an $R$-module $M$. The quotient $M/S$ is the additive group
of cosets $\bar{f}=f+S$. And the scalar multiplication is defined by

$r\bar{f}=\overline{rf}.$

Thus $M/S$ is made an $R$-module.

Conversely, let $A$ be a finitely generated $R$-module, then there exists a
free $R$-module $M$ with basis $\\{a_{1},a_{2},\dots,a_{m}\\}$ and a submodule
$S$ of $M$ such that $A\cong M/S$. Since $R$ is a Noetherian ring, $S$ is
finitely generated, and hence we assume its generating set is
$\\{f_{1},f_{2},\dots,f_{l}\\}$. Therefore we have a module presentation of
$A$ as the following:

$A=\langle a_{1},a_{2},\dots,a_{m}\mid f_{1},f_{2},\dots,f_{l}\rangle.$

For an element $f=\mu_{1}a_{1}+\mu_{2}a_{2}+\dots+\mu_{m}a_{m}$ in $M$ we
define its _length_ , denoted by $\|f\|$, to be the following:

$\|f\|=\sum_{i=1}^{l}|\mu_{i}|+\mathrm{reach}(f),$

where $\mathrm{reach}(f)$ is the minimal length over the lengths of close
loops that starts at $1$ and passes through all points in
$\cup_{i=1}^{l}\operatorname{supp}{\mu_{i}}$ in the Cayley graph of $T$.
Another way to think of this length $\|\cdot\|$ is that it is the minimal
length of a group word among words that are rearranges of all conjugates of
elements in $\mathcal{A}$ in $a_{1}^{\mu_{1}}a_{2}^{\mu_{2}}\dots
a_{m}^{\mu_{m}}$. For example, suppose $m=k=1$, we have

$\|(t_{1}^{n}+t_{1}^{n-1}+\dots+t_{1}+1)a_{1}\|=(n+1)+2n=3n+1,$

because the minimal length of a loop passing $\\{1,t,t^{2},\dots,t^{n}\\}$ is
$2n$. Note that
$a_{1}^{t_{1}^{n}+t_{1}^{n-1}+\dots+t_{1}+1}=t_{1}^{-n}a_{1}t_{1}a_{1}\dots
a_{1}t_{1}a_{1}$ is a group word of length $3n+1$ in the alphabet
$\\{a_{1}\\}\cup\\{t_{1}\\}$.

Then for every element $f$ in $S$, there exists
$\alpha_{1},\alpha_{2},\dots,\alpha_{l}\in R$ such that

$f=\alpha_{1}f_{1}+\alpha_{2}f_{2}+\dots+\alpha_{l}f_{l}.$

We denote by $\widehat{\operatorname{Area}}_{A}(f)$ the minimal possible
$\sum_{i=1}^{l}|\alpha_{i}|$ ($|\cdot|$ is defined in Section ). Then _the
Dehn function_ of the $R$-module $A$ is defined to be

$\hat{\delta}_{A}(n)=\max\\{\widehat{\operatorname{Area}}_{A}(f)\mid\|f\|\leqslant
n\\}.$

As expected, the Dehn function of a module is also independent from the choice
of the finite presentation [Fuh00].

######  Remark.

Now we have three different types of Dehn functions in this paper: the Dehn
function, the relative Dehn function and the Dehn function of a module. They
are similar and we distinguish them by the notation. We denote by
$\delta_{G}(n),\operatorname{Area}(w)$ the Dehn function of $G$ and the area
of a word $w$; $\tilde{\delta}_{G}(n),\widetilde{\operatorname{Area}}(w)$ the
relative Dehn function and the relative area of a word $w$,
$\hat{\delta}_{A}(n),\widehat{\operatorname{Area}}(f)$ the Dehn function of
the module $A$ and the area of a module element $f$.

The membership problem 2.2 can be regarded as the word problem of the quotient
$M/S$.

######  Problem 2.2.

Given an element $f$ in $A$, decide whether $f$ represents the trivial element
in $A$.

It turns out that the Dehn function of a module plays an essential role for
understanding the relative Dehn function for finitely generated metabelian
group [Fuh00].

Last, let us define a well-order $\prec$ on the ring $R=\mathbb{Z}T$. On
$\mathbb{Z}$, we define an order $\prec_{\mathbb{Z}}$ as following

$0\prec_{\mathbb{Z}}1\prec_{\mathbb{Z}}2\prec_{\mathbb{Z}}\dots\prec_{\mathbb{Z}}-1\prec_{\mathbb{Z}}-2\prec_{\mathbb{Z}}\dots.$

For monomials in $R$, we use the degree lexicographical order (also called
shortlex or graded lexicographical order) $\prec_{R}$ which is defined with
respect to the convention $t_{1}\succ t_{1}^{-1}\succ\dots\succ t_{k}\succ
t_{k}^{-1}$, i.e. for
$\mu_{1}=t_{1}^{n_{1}}t_{1}^{-n_{2}}t_{2}^{n_{3}}t_{2}^{-n_{4}}\dots
t_{k}^{n_{2k-1}}t_{k}^{-n_{2k}},\mu_{2}=t_{1}^{m_{1}}t_{1}^{-m_{2}}t_{2}^{m_{3}}t_{2}^{-m_{4}}\dots
t_{k}^{m_{2k-1}}t_{k}^{-m_{2k}}$

$\mu_{1}\prec_{R}\mu_{2}\text{ if
}\sum_{i=1}^{2k}|n_{i}|<\sum_{i=1}^{2k}|m_{i}|\text{ or
}\sum_{i=1}^{2k}|n_{i}|=\sum_{i=1}^{2k}|m_{i}|,\mu_{1}\prec_{lex}\mu_{2},$

where $\prec_{lex}$ is the usual lexicographical order which is defined in the
following way

$t_{1}^{n_{1}}t_{1}^{-n_{2}}t_{2}^{n_{3}}t_{2}^{-n_{4}}\dots
t_{k}^{n_{2k-1}}t_{k}^{-n_{2k}}\prec_{lex}t_{1}^{m_{1}}t_{1}^{-m_{2}}t_{2}^{m_{3}}t_{2}^{-m_{4}}\dots
t_{k}^{m_{2k-1}}t_{k}^{-m_{2k}}$

if $n_{i}<m_{i}$ for the first $i$ where $n_{i}$ and $m_{i}$ differ. Note that
$\prec_{R}$ on $\mathcal{X}$ in fact is a well-oder while $\prec_{lex}$ might
not be (See in [BN98]).

Finally we set $\prec$ on $R=\mathbb{Z}T$ to be the lexicographical order
based on $T\succ\mathbb{Z}$. It is not hard to verify that $\prec$ is a well-
order on $\mathcal{T}$. The degree $\deg\mu$ of $\mu\in\mathbb{Z}T$ is define
to be the degree of its leading monomial and the degree $\deg f$ of $finM$ is
define to be the maximal degree of coefficients of basis.

###  2.2. Relative Dehn Functions of Finitely Generated Metabelian Groups

As we discuss above, the Dehn function for a finitely generated metabelian
group relative to $\mathcal{S}_{2}$ is always defined. So it provides a
convenient tool to study finitely generated metabelian groups.

Let $G$ be a finitely generated metabelian group. Then $G$ sits inside a short
exact sequence.

$1\to A\to G\to T\to 1,$

where $A$ and $T$ are abelian. Since $G$ is finitely generated, $T$ is
finitely generated abelian group. We now choose $A$, $T$ among all such short
exact sequences such that the torsion-free rank of $T$ is minimized. We denote
by $\operatorname{rk}(G)$ the minimal torsion-free rank of $T$.

Now let $k=\operatorname{rk}(G)$ and $\pi:G\to T$ be the canonical quotient
map. It is not hard to show that there exists a subgroup $G_{0}$ of finite
index in $G$ such that $\pi(G_{0})=\mathbb{Z}^{k}$. It has been shown that the
Dehn function, the relative Dehn function and $\operatorname{rk}(G)$ are all
preserved (up to equivalence) under taking finite index subgroups [Gro93],
[Wan20]. Thus, in what follows, we always assume that $T$ is a free abelian
group.

For the finitely generated metabelian group $G$, let
$\\{t_{1},t_{2},\dots,t_{k}\\}$ be a subset of $G$ such that its image in $T$
form a basis. Since $A$ is a normal subgroup in $G$, by [Hal54], it is a
normal closure of a finite set. Let
$\mathcal{A}=\\{a_{1},a_{2},\dots,a_{m}\\}$ be a subset of $A$ satisfying: (1)
$\langle\langle\mathcal{A}\rangle\rangle=A$; (2) $\mathcal{A}$ contains all
commutators of $\\{t_{1},t_{2},\dots,t_{k}\\}$, i.e., for any pair $1\leqslant
i<j\leqslant k$, there exists $l(i,j)\in\\{1,2,\dots,m\\}$ such that
$a_{l(i,j)}=[t_{i},t_{j}]$, $l(i,j)=l(i^{\prime},j^{\prime})$ if and only if
$i=i^{\prime},j=j^{\prime}$.

We associate $G$ with an auxiliary group $\tilde{G}$, which has the following
relative presentation:

$\displaystyle\tilde{G}=$ $\displaystyle\langle
a_{1},a_{2},\dots,a_{m},t_{1},t_{2},\dots,t_{k}\mid a_{l(i,j)}=[t_{i},t_{j}],$
$\displaystyle[a,b]=1,[a,b^{t}]=1,1\leqslant i<j\leqslant
k,a,b\in\mathcal{A},t\in\mathcal{T}\rangle_{\mathcal{S}_{2}}$

Relations $\\{[a,b]=1,[a,b^{t}]=1,a,b\in\mathcal{A},t\in\mathcal{T}\\}$ is
enough along with all metabelian relations inherited from the free metabelian
groups. And the area of $[a,b^{u}]$ is linearly controlled by the length of
$u$, that is,

######  Lemma 2.3 (Wang [Wan20]).

$\\{[a,b]=1,[a,b^{t}]=1,a,b\in\mathcal{A},t\in\mathcal{T}\\}$ generates all
commutative relations $[a,b^{u}]=1,a,b\in\mathcal{A},u\in F(\mathcal{T})$ in
the presentation relative to the variety of metabelian groups. Moreover, the
relative area of $[a,b^{u}]$ is bounded by $4|u|-3$.

All relations in $\tilde{G}$ are also represent the identity in $G$. It
follows that the identity map on $\mathcal{A}\cup\mathcal{T}$ induces an
epimorphism $\varphi:\tilde{G}\to G$. The kernel $\ker\varphi$ is a normal
subgroup in $\langle\langle\mathcal{A}\rangle\rangle_{\tilde{G}}$, since
$\varphi$ induces an isomorphism on $T$. Let $\\{f_{1},f_{2},\dots,f_{l}\\}$
be a subset of $\langle\langle\mathcal{A}\rangle\rangle_{\tilde{G}}$ such that
$\langle\langle
f_{1},f_{2},\dots,f_{l}\rangle\rangle_{\tilde{G}}=\ker\varphi$. Thus we obtain
a relative presentation of $G$.

$\displaystyle G=$ $\displaystyle\langle\tilde{G}\mid
f_{1},f_{2},\dots,f_{l}\rangle_{\mathcal{S}_{2}}$ $\displaystyle=$
$\displaystyle\langle a_{1},a_{2},\dots,a_{m},t_{1},t_{2},\dots,t_{k}\mid
f_{1},f_{2},\dots,f_{l},a_{l(i,j)}=[t_{i},t_{j}],$
$\displaystyle[a,b]=1,[a,b^{t}]=1,1\leqslant i<j\leqslant
k,a,b\in\mathcal{A},t\in\mathcal{T}\rangle_{\mathcal{S}_{2}}.$

We focus on the module structure on
$\langle\langle\mathcal{A}\rangle\rangle_{\tilde{G}}$, which is a free
$T$-module generated by the basis $\mathcal{A}$ [Wan20]. Let us define the
ordered form of an element $f$, denoted by $\operatorname{OF}(f)$, in
$\langle\langle\mathcal{A}\rangle\rangle_{\tilde{G}}$. $f$ can be written as
$a_{1}^{\alpha_{1}}a_{2}^{\alpha_{2}}\dots a_{m}^{\alpha_{m}}$ as a group
element or $\alpha_{1}a_{1}+\alpha_{2}a_{2}+\dots+\alpha_{m}a_{m}$ as a module
element. Let $\prec$ be the well-order we construct in Section 2.1. The
ordered form $\operatorname{OF}(f)$ is of the form
$a_{1}^{\mu_{1}}a_{2}^{\mu_{2}}\dots a_{m}^{\mu_{m}}$ such that

  1. (1)

$\mu_{i}\in\mathbb{Z}T$ for $1\leqslant i\leqslant m$, and each $\mu_{i}$ is
of the form $\mu_{i}=\sum_{j=1}^{n_{j}}c_{ij}u_{ij}$ such that
$c_{ij}\in\mathbb{Z},u_{ij}\in\bar{F}$ and $u_{i1}\succ u_{i2}\succ\dots\succ
u_{in_{i}}$;

  2. (2)

$f=_{\tilde{G}}\operatorname{OF}(f)$,

where

$\bar{F}=\\{t_{1}^{m_{1}}t_{2}^{m_{2}}\dots t_{k}^{m_{k}}\mid
m_{1},\dots,m_{k}\in\mathbb{Z}\\}.$

It has been shown that the ordered form is well-defined and for any
$f,g\in\langle\langle\mathcal{A}\rangle\rangle_{\tilde{G}}$, $f=_{\tilde{G}}g$
if and only if
$\operatorname{OF}(f)=_{F(\mathcal{A}\cup\mathcal{T})}\operatorname{OF}(g)$
[Wan20]. For an element $f$ in $\langle\langle\mathcal{A}\rangle\rangle_{G}$,
we define the ordered form of $f$ by lifting $f$ to $\tilde{G}$. The ordered
form is useful for estimating the relative area of a word.

Let $M$ be the free $T$-module with basis $\\{a_{1},a_{2},\dots,a_{m}\\}$ and
$S$ be a submodule of $M$ generated by $\\{f_{1},f_{2},\dots,f_{l}\\}$. Then
the $T$-module $A$ is isomorphic to $M/S$. We have the following connection
between the Dehn function of the module $A$ and the relative Dehn function of
$G$:

######  Proposition 2.4 (Wang [Wan20]).

Let $G$ be a finitely generated metabelian group and $A$ is defined as above,
then

$\hat{\delta}_{A}(n)\preccurlyeq\tilde{\delta}_{G}(n)\preccurlyeq\max\\{\hat{\delta}_{A}^{3}(n^{3}),n^{6}\\}.$

If $G$ is a semidirect product, the result can be slightly improved.

######  Proposition 2.5.

Let $T$ be a finitely generated abelian group and let $A$ be a finitely
generated $T$-module. Form the semidirect product

$G=A\rtimes T.$

Then $\delta_{G}(n)\preccurlyeq\max\\{\hat{\delta}_{A}^{3}(n^{2}),n^{3}\\}.$

If $G$ happens to be finitely presented, we have

######  Theorem 2.6.

Let $G$ be a finitely presented metabelian group. Then

$\tilde{\delta}_{G}(n)\preccurlyeq\delta_{G}(n)\preccurlyeq\max\\{\tilde{\delta}_{G}^{3}(n^{3}),2^{n}\\}.$

##  3\. Estimate the Relative Dehn Function

Computing the relative Dehn function is harder than computing the Dehn
function. Many techniques are no longer useful for the relative case. For the
variety of metabelian groups, fortunately, the structure of groups in it is
not complicated. The key is to understand the natural module structure of a
finitely generated metabelian group.

First, let us list some known results for relative Dehn functions, they are
computed by Fuh in her thesis [Fuh00]. Note that most of them only give the
upper bound of the relative Dehn function.

######  Theorem 3.1 (Fuh [Fuh00]).

  1. (1)

The realative Dehn function of a wreath product of two finitely generated
abelian groups is polynomially bounded.

  2. (2)

The Baumslag-Solitar group $BS(1,2)$ has linear Dehn function.

  3. (3)

Let $G=\tilde{BS}(n,m)=\langle
a,t\mid(a^{n})^{t}=a^{m}\rangle_{\mathcal{S}_{2}}$ where $m>2,m=n+1$. Then
$\tilde{\delta}_{G}(n)\preccurlyeq n^{3}$.

Now let us estimate the relative Dehn function from above for some concrete
examples. By the _cost_ of converting $w_{1}$ to $w_{2}$ ($w_{2}$ to $w_{1}$)
in $G$ we mean the relative area of $w_{2}^{-1}w_{1}$ (resp.
$w_{1}^{-1}w_{2}$) in $G$. If $w_{2}$ happens to be the identity, then the
cost of converting $w_{1}$ to $w_{2}$ coincides with the area of $w_{1}$. By
the definition of the area, it is not hard to see that if
$w_{1}=_{G}w_{2}=_{G}w_{3}$ and the cost of converting $w_{1}$ to $w_{2}$,
$w_{2}$ to $w_{3}$ is $N_{1}$ and $N_{2}$ respectively then the cost of
converting $w_{1}$ to $w_{3}$ is at most $N_{1}+N_{2}$. Essentially, to
estimate the relative Dehn function from above we need to estimate the cost of
converting a word to the identity.

To begin with, we consider the metabelianized Baumslag-Solitar group

$\tilde{BS}(n,m)=\langle a,t\mid(a^{n})^{t}=a^{m}\rangle_{\mathcal{S}_{2}}.$

The normal subgroup generated by $a$ is a $\mathbb{Z}\langle t\rangle$-module.
In this case, i.e., when the module is over the ring of Laurent polynomial of
one variable and is generated by one variable, the Dehn function of the module
is well-studied. The following theorem from Davis and Olshanskiy [DO11] shows
that the Dehn function of a finitely generated $\langle t\rangle$-module is a
polynomial.

######  Theorem 3.2 (Davis, Olshanskiy [DO11, Theorem 8.6]).

Let $M=\langle a\rangle$ is the free module of rank one over the group ring
$\mathbb{Z}\langle t\rangle$. Let $f=h(t)a$ where $h(x)$ is a polynomial of
the form $d_{n}x^{n}+d_{n-1}x^{n-1}+\dots+d_{0}$. Then the Dehn function of
the $\langle t\rangle$-module $M/\langle f\rangle$ is a polynomial.
Furthermore, the degree of this polynomial is exactly one plus the maximal
multiplicity of a (complex) root of $h(x)$ having modulus one.

Thus we have

######  Proposition 3.3.

The metabelianized Baumslag-Solitar group $\tilde{BS}(n,m)=\langle
a,t\mid(a^{n})^{t}=a^{m}\rangle_{\mathcal{S}_{2}}$ has at most cubic relative
Dehn function when $n\neq m$ and has at most quartic relative Dehn function
when $n=m$.

###### Proof.

Note that in this case we have $|\mathcal{A}|=|\mathcal{T}|=1$, which
simplifies the process a lot. Given a word $w=_{G}1$ of length $l$.

First we claim that converting $w$ to the ordered form $\operatorname{OF}(w)$
(defined in Section 2.2) costs at most $(4l-3)l^{2}$.

Suppose $w=t^{n_{1}}a^{m_{1}}t^{n_{2}}a^{m_{2}}\dots
t^{n_{s}}a^{m_{s}}t^{n_{s+1}},n_{i},m_{i}\in\mathbb{Z}$ for all $i$ and only
$n_{1},n_{s+1}$ can be zero. Since $w=1$, we have that

$\sum_{i=1}^{s+1}n_{i}=0,\sum_{i=1}^{s}(|n_{i}|+|m_{i}|)+|n_{s+1}|=l.$

The first conditions comes from the fact that the image of $w$ is 0 in $T$ and
the second condition comes from $|w|=l$.

Thus we can rewrite $w$ by inserting trivial words $tt^{-1}$ to the form

$w=a^{t^{-n_{1}}}a^{t^{-(n_{1}+n_{2})}}\dots
a^{t^{-(n_{1}+n_{2}+\dots+n_{s})}}=a^{\mu},$

where
$\mu=t^{-n_{1}}+t^{-(n_{1}+n_{2})}+\dots+t^{-(n_{1}+n_{2}+\dots+n_{s})}\in\mathbb{Z}T.$
We immediately have that $\deg(\mu)\leqslant l$, $|\mu|=s\leqslant l$, and
$\|w\|\leqslant l$ by the definition.

The cost of converting $w$ to $a^{t^{-n_{1}}}a^{t^{-(n_{1}+n_{2})}}\dots
a^{t^{-(n_{1}+n_{2}+\dots+n_{s})}}$ is zero since we only insert trivial words
in the absolute free group.

Next we will convert $a^{t^{-n_{1}}}a^{t^{-(n_{1}+n_{2})}}\dots
a^{t^{-(n_{1}+n_{2}+\dots+n_{s})}}$ to the ordered form of $w$. To do this, we
have to rearrange conjugates of $a$ such that exponents are ordered by $\prec$
from high to low. In order to commute two conjugates in $w$, we have to insert
commutator of the form

$[a^{t^{-l_{i}}},a^{t^{-l_{j}}}]=[a,a^{t^{l_{i}-l_{j}}}],\text{ where
}l_{i}=n_{1}+n_{2}+\dots+n_{i}.$

By Lemma 2.3, the area is bounded by $4l-3$. To rearrange $s$ conjugates of
$a$ we need to insert at most $s^{2}$ many such commutators. Thus the cost of
converting $w$ to $\operatorname{OF}(w)$ is bounded by $(4l-3)l^{2}$. The
claim is proved.

Suppose $OF(w)=a^{\mu}$, where $|\mu|\leqslant l,\deg\mu\leqslant l$. We can
conjugate $w$ by $t^{l}$ such that $\mu$ only have positive powers of $t$.
Thus we assume that $|\mu|\leqslant l,\deg\mu\leqslant 2l$. Further, the
length of $\mu$ is bounded by $l$ by definition.

In this case, the module $A$ is isomorphic to $M/S$ where $M$ is a free
$T$-module with basis $a$ and $S$ is its submodule generated by
$\\{(nt-m)a\\}$. Consider the polynomial ring $R=\mathbb{Z}[t,t^{-1}]$ and its
ideal $I=\langle nt-m,tt^{-1}-1\rangle$. We have that $A\cong R/I$. The
Gröbner basis of $I$ is $\\{tt^{-1}-1,nt-m,mt^{-1}-n\\}$. If we regard $\mu$
as an element in $I$, it can only be reduced by $nt-m$ since it only has
positive power of $t$. It follows that there exists a polynomial $\nu$, which
only consists of the power of $t$, such that

$\mu=(nt-m)\nu.$

This equality also holds in the polynomial ring $\mathbb{Z}[t]$. When $n\neq
m$, the Dehn function of $\langle t\rangle$-module $\mathbb{Z}T/\langle
nt-m\rangle$ is linear, by Theorem 3.2. Thus there exists $C$ such that
$|\nu|\leqslant C\|\mu\|+C$. We have that

$a^{\mu}=_{G}(a^{mt-n})^{\nu}.$

The area of the right hand side is at most $Cl+C$. Converting the right hand
side to its ordered form costs at most $(4l-3)((m+n)(Cl+C))^{2}$ since the
degree is less than $l$ and we have $(m+n)(Cl+C)$ many conjugates to
rearrange. Thus the upper bound of $\widetilde{\operatorname{Area}}(w)$ is at
most $l^{3}$ up to equivalence when $n\neq m$.

When $n=m$, the Dehn function of $\langle t\rangle$-module
$\mathbb{Z}T/\langle nt-m\rangle$ is quadratic. Following the same process, we
have that the upper bound of $\widetilde{\operatorname{Area}}(w)$ is at most
$l^{4}$ up to equivalence when $n\neq m$. This finishes the proof. ∎

For the case $n=1$, the group $\tilde{B}S(1,n)\cong BS(1,n)$ is finitely
presented. Following from Theorem 2.6, the Dehn function of $BS(1,n)$ is at
most exponential. We will extend this idea of using relative Dehn function to
estimate the Dehn function in the next Section.

One special case Fuh [Fuh00, Theorem 6.1] concerned is when $m>2,m=n+1$. In
this case, we have that $a=[a^{n},t]$. Since $a$ itself is a commutator, it
follows that the relative area of words like $[a^{t^{k}},a]$ is at most 4
instead of linearly depending on $k$. Therefore we can improve the result in
[Fuh00, Theorem 6.1] by the following corollary of Proposition 3.3.

######  Corollary 3.4.

The metabelianized Baumslag-Solitar group $\tilde{BS}(n,m)=\langle
a,t\mid(a^{n})^{t}=a^{m}\rangle_{\mathcal{S}_{2}},m>2,m=n+1$ has at most
quadratic relative Dehn function.

The lamplighter groups are another interesting class of infinite presented
metabelian groups with a simple module structure. We have

######  Proposition 3.5.

The lamplighter groups $L_{m},m\geqslant 2$ have at most cubic relative Dehn
function.

###### Proof.

Consider the lamplighter group $L_{m}$ with the standard presentation.

$L_{m}=\langle a,t\mid a^{m}=1,[a,a^{t^{n}}]=1,n\in\mathbb{N}\rangle.$

By the discussion in Section 2.2, we have a finite relative presentation as
the following

$L_{m}=\langle a,t\mid a^{m}=1,[a,a^{t}]=1\rangle_{\mathcal{S}_{2}}.$

The rest of the proof is the same as the proof of Proposition 3.3. The only
difference is that in this case the submodule is generated by $\\{m\\}$. ∎

This slightly improves the estimation in [Fuh00, Theorem B2].

In particular, when $m=2$, for the case of $L_{2}$, we are able to improve the
upper bound to linear.

######  Proposition 3.6.

The lamplighter groups $L_{2}$ has linear relative Dehn function.

###### Proof.

The linear lower bound is given by Theorem 3.2.

We choose the following relative presentation of $L_{2}$:

$L_{2}=\langle a,t\mid a^{2}=1,[a,a^{t}]=1\rangle_{\mathcal{S}_{2}}.$

For the upper bound, consider a word $w\in L_{2}$ that represents the
identity. Thus $w$ has the form

$w=t^{n_{1}}at^{n_{2}}a^{n_{3}}\dots t^{n_{2k}}at^{n_{2k+1}},\text{ where
}n_{2},n_{3},\dots,n_{2k}\neq 0.$

Suppose the length of $w$ is $n$, combining the fact that $w=1$, we have

$2k+\sum_{i=1}^{2k+1}|n_{i}|=n,\sum_{i=1}^{2k+1}n_{i}=0.$

Inserting $tt^{-1}$ or $t^{-1}t$, we can rewrite $w$ as the following form:

$w=a^{t^{-n_{1}}}a^{t^{-(n_{1}+n_{2})}}\dots
a^{t^{-(n_{1}+n_{2}+\dots+n_{2k})}}.$

Thus $w$ represents an element in $\oplus_{i\in\mathbb{Z}}\mathbb{Z}_{2}$,
where the $a^{t^{i}}$ is the generator of the $i$-th copy of $\mathbb{Z}_{2}$.
Since $w=1$, then every element in the set
$\\{-n_{1},-(n_{1}+n_{2}),\dots,-(n_{1}+n_{2}+\dots+n_{2k})\\}$ occurs even
many times in the sequence
$-n_{1},-(n_{1}+n_{2}),\dots,-(n_{1}+n_{2}+\dots+n_{2k})$. Our goal is to
gather the conjugates of $a$ of the same exponents together at a linear cost
with respect to $n$.

Since $a^{-1}=a$, we notice that

$a^{t^{s}}a^{t^{l}}=(aa^{t^{l-s}})^{t^{s}}=[a,t^{l-s}]^{t^{s}},l,s\in\mathbb{Z}.$

Thus any pair of two consecutive conjugates of $a$ is a commutator. It follows
that any such pair commutes with any other pair of this form without any cost
inside the variety of metabelian groups.

For convenience, let $m_{i}=\sum_{i=1}^{2k}-n_{i}$. We now turn the problem of
estimating the relative area of $w$ to a problem of cancelling numbers in a
sequence and estimating the cost. Consider a sequence of number

$m_{1},m_{2},\dots,m_{2k}.$

The goal is to cancel all the pairs of the same value. We have three
operations allowed:

  1. (i)

Cancel two consecutive numbers of the same value without any cost.

  2. (ii)

Commute a pair of consecutive numbers with another pair of consecutive numbers
next to it without any cost.

  3. (iii)

Commute two consecutive numbers $c,d$ with a cost of $|c-d|$.

Applying all three operations to the original sequence many times, the result
might seems chaotic. To analyze the process, for a sequence of numbers, we
define the $\iota(m_{i})$ be the position of $m_{i}$ in the sequence. At the
beginning, $\iota(m_{i})=i$. Then we define
$\sigma(m_{i},m_{j})=|\iota(m_{i})-\iota(m_{j})|\mod 2$. So
$\sigma(m_{i},m_{j})=0$ if $m_{i}$ and $m_{j}$ are even positions apart and
$\sigma(m_{i},m_{j})=1$ if $m_{i}$ and $m_{j}$ are odd positions apart. We
notice that

  1. (a)

operations from (i) and (ii) do not change $\sigma(m_{i},m_{j})$;

  2. (b)

if $m_{i}$ is next to $m_{j}$, applying the operation (iii) to commute $m_{i}$
and $m_{j}$ will change all values of
$\sigma(m_{i},m_{l}),\sigma(m_{j},m_{l})$ for $l\neq i,j$ but all other values
of $\sigma$ remain the same.

From the above observation, we have that

  1. (1)

if $\sigma(m_{i},m_{j})=0$ and $i<j$, $m_{j}$ can be moved to the position
next to $m_{i}$ just using operations from (ii).

  2. (2)

if $\sigma(m_{i},m_{j})=0$, $i<j$ and $m_{i}=m_{j}$, then $m_{i}$ and $m_{j}$
can be cancelled using just operations from (i) and (ii).

  3. (3)

for $m_{i},m_{j},m_{l}$ such that $m_{i}=m_{j}$,
$\sigma(m_{i},m_{j})=1,\sigma(m_{i},m_{l})=0$, we can cancel $m_{i},m_{j}$
with the cost of $|m_{i}-m_{l}|$.

(1) can be achieved by commuting two consecutive pairs of numbers. (2) is a
direct consequence of (1). Let us show how to achieve (3). By (1), we can move
$m_{l}$ next to $m_{i}$. Then by using operation (ii), the pair $m_{i}m_{l}$
(or $m_{l}m_{i}$) can be moved to the position next to $m_{j}$, resulting the
form of $m_{i}m_{l}m_{j}$ or $m_{j}m_{l}m_{i}$. Finally, we commute $m_{i}$
and $m_{j}$ using operation (iii) at a cost of $|m_{i}-m_{j}|$ and cancel
$m_{i}m_{j}$.

Now we are ready to estimate the cost to cancel the sequence
$m_{1},m_{2},\dots,m_{2k}$ to the empty sequence. By $(2)$, we can assume that
we have already cancelled all the pairs $m_{i},m_{j}$ where
$\sigma(m_{i},m_{j})=0,m_{i}=m_{j}$ using operations (i) and (ii). This step
costs nothing and does not change any $\sigma(m_{i},m_{j})$ for $m_{i},m_{j}$
remaining in the resulting sequence. Let the remaining elements after
cancellations be $m_{i(1)},m_{i(2)},\dots,m_{i(4s)}$ for some $2s\leqslant k$
and $i(1)<i(2)<\dots<i(4s)$. The reason we will have even pairs numbers left
is that if the number of pairs is odd, there must be a pair of number of the
same value that is even positions apart. The remaining sequence satisfies the
following properties:

  1. (a)

$\sigma(m_{i(s)},m_{i(l)})=i(s)-i(l)\mod 2$,

  2. (b)

if $m_{i(s)}=m_{i(l)}$ then $\sigma(m_{i(s)},m_{i(l)})=1$,

  3. (c)

$\sigma(m_{i(s)},m_{i(l)})=\sigma(m_{i(s^{\prime})},m_{i(l^{\prime})})$ for
$m_{i(s)}=m_{i(s^{\prime})},m_{i(l)}=m_{i(l^{\prime})}$.

Here the property (a) is true because in the original sequence
$\iota(m_{i(s)})=i(s)$ and we only use operation (i) and (ii) which do not
change $\sigma(m_{i(s)},m_{i(l)})$. (b) and (c) follow from the definition of
$\sigma$ and the remaining sequence.

Figure 1. the corresponding graph of the sequence 2,3,5,3,5,8,2,8

We define the weighted graph $\Gamma_{0}$ associated with
$(m_{i(1)},m_{i(2)},\dots,m_{i(4s)})$ where the vertex set is
$\\{m_{i(1)},m_{i(2)},\dots,m_{i(4s)}\\}$ and there is an edge with weight
$|m_{i(s)}-m_{i(l)}|$ connects $m_{i(s)},m_{i(l)}$ if
$\sigma(m_{i(s),i(l)})=0$. Note that this graph is invariant under operations
(ii) and may have multi-edge.

By (3), we are allowed to cancel $m_{i(s)},m_{i(l)}$ at a cost of
$|m_{i(s)}-m_{i(j)}|$ for some $m_{i(j)}$ that $\sigma(m_{i(s)},m_{i(j)})=0$.
After the cancellation, since we use operation (iii) once,
$\sigma(m_{i(j)},m_{i(j^{\prime})})$ change to 0 for some $m_{i(k^{\prime})}$
that $m_{i(j)}=m_{i(j^{\prime})}$. Therefore we can then cancel
$m_{i(j)},m_{i(j^{\prime})}$ without any cost. In summary, we have

  1. (4)

for $m_{i(s)}\neq m_{i(j)}$ that $\sigma(m_{i(s)},m_{i(j)})=0$, we can cancel
a pair of number $m_{i(s)}$ and a pair of number $m_{i(j)}$ at a cost of
$|m_{i(s)}-m_{i(j)}|$ where $\sigma(m_{i},m_{j})$ remains the same for numbers
that have not been cancelled.

(4) will delete an edge of $(m_{i(s)},m_{i(j)})$ in the graph. If no edges
connecting $m_{i(s)}$ and $m_{i(j)}$, we delete the two vertices
$m_{i(s)},m_{i(j)}$. The cost is the weight of that edge. Let $\mathcal{C}$ be
a cancellation of $\Gamma_{0}$ where $\mathcal{C}$ consists of an ordered
sequence of edges in $\Gamma_{0}$, where we cancel the edge by the order of
the sequence. Thus the total cost of a cancellation $\mathcal{C}$ to the empty
graph is just a sum of the weight of edges in $\mathcal{C}$. Every
cancellation can be associated with a path $p_{\mathcal{C}}$ where the path
passes through the sequence of edges in $\mathcal{C}$ in the same order.

Now we delete edges in the following way. We first delete one edge
$(m_{i(1)},m_{i(2)})$ since $\sigma(m_{i(1)},m_{i(2)})=0$. We let the resulted
graph to be $\Gamma_{1}$.

Figure 2. two different cancellations $\mathcal{C},\mathcal{C}^{\prime}$ and
their corresponding $p_{\mathcal{C}},p_{\mathcal{C}^{\prime}}$. The total cost
of $\mathcal{C}$ is 3 and the total cost of $\mathcal{C}^{\prime}$ is 8.

Inductively, $\Gamma_{i+1}$ is obtained by deleting an edge
$(m_{i(s)},m_{i(j)})$ where $i,j$ are the smallest numbers remained in
$\Gamma_{i}$. $\Gamma_{s}$ will be an empty graph since every time we delete
four numbers from the sequence.

Let us estimate the cost from $\Gamma_{0}$ to $\Gamma_{s}$. Since every time
we cancel pairs of numbers based on the order of the original sequence
$m_{1},m_{2},\dots,m_{2k}$ (always cancel the first two numbers remained). The
cost is bounded by

$\sum_{i=1}^{2k-1}|m_{i+1}-m_{i}|=\sum_{i=2}^{2k}|n_{i}|<n.$

Let inequality can also be realized by the following interpretation: the
sequence $m_{i(1)},m_{i(2)},\dots,m_{i(4s)}$ defines a path $p$ in
$\Gamma_{0}$ (since $\sigma(m_{i(i)},m_{i(j+1)})=0$) that $p(j)=m_{i(j)}$, the
weight of the path $p$ is bounded by $n$ by the definition of $m_{i}$. $p$
happens to be the path associated with this cancellation. It follows that the
cost of the cancellation is bounded by the total weight of $p$. Thus the total
cost is bounded by $n$.

By Lemma 2.3, the total cost of converting

$a^{t^{m_{1}}}a^{t^{m_{2}}}\dots a^{t^{m_{2k}}}$

to $0$ is bounded by $4n-3$. We finish the proof.∎

##  4\. Relative Dehn Functions and Subgroup Distortions

So far for all the examples considered in [Fuh00] and Section 3, only the
upper bounds of their relative Dehn functions are estimated. Similar to the
case of the Dehn function, it is genuinely much harder to estimate the lower
bound. In this section, we will connect the relative Dehn function of a
finitely generated metabelian group to the subgroup distortions in a wreath
product of two free abelian groups. This connection provides a new method to
estimate the lower bound for the relative Dehn function and yields a sequence
of examples of finitely generated metabelian groups with relative Dehn
function larger that $n^{k}$ for arbitrary $k\in\mathbb{N}$.

Let $G$ be a finitely generated group with a finite generating set $X$ and $H$
be a a subgroup of $G$ with finite generating set $Y$. The _distortion
function_ of $H$ in $G$ is

$\Delta_{H}^{G}(n)=\sup\\{|w|_{Y}\mid w\in H,|w|_{X}\leqslant n\\}.$

For example, the subgroup $\langle a\rangle$ in the Baumslag-Solitar group
$\langle a,t\mid a^{t}=a\rangle$ is exponentially distorted since
$a^{t^{n}}=a^{2^{n}}$. And it not hard to check that infinite subgroups of a
finitely generated abelian group are undistorted.

Let $A$ and $T$ be free abelian groups with bases
$\\{a_{1},a_{2},\dots,a_{m}\\}$ and $\\{t_{1},t_{2},\dots,t_{k}\\}$
respectively. Consider the wreath product $W:=A\wr T$. The base group
$B:=\langle\langle A\rangle\rangle$ is a $T$-module. For a finite subset
$\mathcal{X}=\\{f_{1},f_{2},\dots,f_{l}\\}$ of $B$, let $H$ be the subgroup of
$W$ generated by $\mathcal{X}\cup\\{t_{1},t_{2},\dots,t_{k}\\}$ and $G$ be the
group $W/\langle\langle\mathcal{X}\rangle\rangle$. We denote by
$\pi:W\twoheadrightarrow T$ the canonical quotient map.

######  Theorem 4.1.

Let $W,H,G$ be groups defined as above, then

$\Delta_{H}^{W}(n)\preccurlyeq\tilde{\delta}_{G}^{k}(n)+n^{k},\tilde{\delta}_{G}(n)\preccurlyeq\max\\{n^{3},(\Delta_{H}^{W}(n^{2}))^{3}\\}.$

In particular, if $k=1$,

$\Delta_{H}^{W}(n)\preccurlyeq\tilde{\delta}_{G}(n).$

###### Proof.

First we show the following lemma.

######  Lemma 4.2.

let $M$ be the $T$-module $B/\langle\langle\mathcal{X}\rangle\rangle$. Then
$\hat{\delta}_{M}(n)\preccurlyeq\Delta_{H}^{W}(n)\preccurlyeq\hat{\delta}_{M}^{k}(n)+n^{k}$.

###### Proof.

Let $g\in H$. Note that $g$ can be written as $g_{0}t$, by adding $t:=\pi(g)$
to the end, where $g_{0}\in B,t\in T$. Since
$|\pi(t)|_{T}\leqslant|g|_{W}\leqslant n$, $|g_{0}|_{W}\leqslant 2|g|_{W}$.
Thus, we have

$|g|_{H}=|g_{0}t|_{H}\leqslant|g_{0}|_{H}+|t|_{H}\leqslant|g|_{W}+|g_{0}|_{H}.$

Assume that the ordered form of $OF(g_{0})$ is
$a_{1}^{\mu_{1}}a_{2}^{\mu_{2}}\dots a_{m}^{\mu_{m}}$, let us estimate
$|g_{0}|_{H}$. First note that $\deg\mu_{i}\leqslant|g|_{W}$ for all $i$. Let
$\alpha_{1},\alpha_{2},\dots,\alpha_{l}$ be elements in $\mathbb{Z}T$ such
that $g_{0}=f_{1}^{\alpha_{1}}f_{2}^{\alpha_{2}}\dots f_{l}^{\alpha_{l}}$ and
$\sum_{i=1}^{l}|\alpha_{i}|$ is minimized. By Theorem 3.4 in [DO11],

$|g_{0}|_{H}=\sum_{i=1}^{l}|\alpha_{i}|+\mathrm{reach}(g_{0}),$

where $\mathrm{reach}(g_{0})$ is the length of the shortest loop starting at 0
in the Cayley graph of $T$ that passing through all points in the set
$\cup_{i=1}^{l}\operatorname{supp}\alpha_{i}$. By [Wan20, Lemma 6.7], for all
$i$, $\deg(\alpha_{i})\leqslant|g|_{W}+C\sum_{i=1}^{l}|\alpha_{i}|$ for some
constant $C$. It follows that $\cup_{i=1}^{l}\operatorname{supp}\alpha_{i}$
lies in Ball $B_{0}(|g|_{W}+C\sum_{i=1}^{l}|\alpha_{i}|)$ of radius
$|g|_{W}+C\sum_{i=1}^{l}|\alpha_{i}|$ centered at 0 in the Cayley graph of
$T$. Since there exists a path of length
$(2(|g|_{W}+C\sum_{i=1}^{l}|\alpha_{i}|)+1)^{k}$ passing through all the
points in $B_{0}(|g|_{W}+C\sum_{i=1}^{l}|\alpha_{i}|)$,

$\mathrm{reach}(g_{0})\leqslant(2(|g|_{W}+C\sum_{i=1}^{l}|\alpha_{i}|)+1)^{k}.$

Therefore, we have

$\sum_{i=1}^{l}|\alpha_{i}|\leqslant|g|_{H}\leqslant|g|_{W}+\sum_{i=1}^{l}|\alpha_{i}|+2^{k}(|g|_{W}+C\sum_{i=1}^{l}|\alpha_{i}|)^{k}.$

Since $\sum_{i=1}^{l}|\alpha_{i}|=\widehat{\operatorname{Area}}_{M}(g_{0})$ by
definition and $\|g_{0}\|\leqslant 2|g|_{W}$, we have the following
estimation:

$\widehat{\operatorname{Area}}(2|g|_{W})\leqslant|g|_{H}\leqslant|g|_{W}+\widehat{\operatorname{Area}}(2|g|_{W})+2^{k}(|g|_{W}+C\widehat{\operatorname{Area}}(2|g|_{W}))^{k}.$

∎

By Proposition 2.4, we have

$\Delta_{H}^{W}(n)\preccurlyeq\tilde{\delta}_{G}^{k}(n)+n^{k}.$

Last, by Proposition 2.5,

$\tilde{\delta}_{G}(n)\preccurlyeq\max\\{n^{3},(\Delta_{H}^{W}(n^{2}))^{3}\\}.$

∎

Theorem 4.1 connects the subgroup distortion function and the relative Dehn
function, as it provides a way to estimate the relative Dehn function from the
bottom. One special case is that both $A$ and $T$ are free abelian group of
rank 1. Davis and Olshanskiy [DO11] show that subgroups in $W=\langle
a\rangle\wr\langle t\rangle$ have polynomial distortion functions and moreover
for each $l\in\mathbb{Z}$, a subgroup of the form
$H_{l}:=\langle[\dots,[a,t],t],\dots,t],t\rangle$, where the commutator is
$(l-1)$-fold, is isomorphic to $\mathbb{Z}\wr\mathbb{Z}$ with $n^{l}$
distortion. It follows immediately that

######  Corollary 4.3.

Let $W=\langle a\rangle\wr\langle t\rangle$ be the wreath product of two
infinite cyclic group. For each $l\in\mathbb{N}$, let
$w_{l}=[\dots,[a,t],t],\dots,t]$ be the $(l-1)$-fold commutator. Finally let
$H_{l}=W/\langle\langle w_{l}\rangle\rangle$. Then we have

$\tilde{\delta}_{H_{l}}\succcurlyeq n^{l}.$

Let us consider the case when the rank of $T$ is 1, that is, when $k=1$. The
following result characterizes the distortion function of subgroups when
$k=1$.

######  Theorem 4.4 (Davis, Olshanskiy, [DO11, Theorem 1.2]).

Let $A$ be a finitely generated abelian group.

  1. (1)

For any finitely generated infinite subgroup $H\leqslant A\wr\mathbb{Z}$ there
exists $l\in\mathbb{N}$ such that the distortion of $H$ in $A\wr\mathbb{Z}$ is

$\delta_{H}^{A\wr\mathbb{Z}}(n)\asymp n^{l}.$

  2. (2)

If $A$ is finite, then $l=1$; that is, all subgroup are undistorted.

  3. (3)

If $A$ is infinite, then for every $l\in\mathbb{N}$, there is a 2-generated
subnormal subgroup $H$ of $A\wr\mathbb{Z}$ having distortion function

$\Delta_{H}^{A\wr\mathbb{Z}}(n)\asymp n^{l}.$

It follows that

######  Theorem 4.5.

Let $G$ be a finitely generated metabelian group such that $rk(G)=1$. Then the
relative Dehn function of $G$ is polynomially bounded. If in addition $G$ is
finitely presented, the Dehn function of $G$ is asymptotically bounded above
by the exponential function.

###### Proof.

By passing to a finite index subgroup, we can assume that there exists a short
exact sequence

$1\to A\to G\to\mathbb{Z}\to 1,$

where $A$ is abelian.

We denote by $T=\langle t\rangle$ the $\mathbb{Z}$ in the short exact
sequence. Since every short exact sequence $1\to A\to G\to\mathbb{Z}\to 1$
splits, $G$ is isomorphic to the semidirect product $A\rtimes T$.

Note that $A$ is a normal subgroup of $G$, then it is finitely generated as a
$T$-module. Thus, there exists a free $T$-module $M$ of rank $m$ and a
submodule $S=\langle f_{1},f_{2},\dots,f_{l}\rangle$ such that $A\cong M/S$.
We have that

$G\cong(M/S)\rtimes T\cong(M\rtimes T)/\langle\langle
f_{1},f_{2},\dots,f_{l}\rangle\rangle.$

Let $\bar{A}$ be a free abelian group of rank $m$ and $W:=\bar{A}\wr T$ be the
wreath product of $\bar{A}$ and $T$. Then there is an isomorphism
$\varphi:M\rtimes T\to W$. We have

$G\cong
W/\langle\langle\varphi(f_{1}),\varphi(f_{2}),\dots,\varphi(f_{l})\rangle\rangle.$

Let $H$ be the subgroup in $W$ generated by
$\\{\varphi(f_{1}),\varphi(f_{2}),\dots,\varphi(f_{l}),t\\}$. By Theorem 4.1,
we have that

$\tilde{\delta}_{G}(n)\preccurlyeq\max\\{n^{3},(\Delta_{H}^{W}(n^{2}))^{3}\\}.$

By Theorem 4.4, $\Delta_{H}^{W}(n)$ is a polynomial. Therefore the relative
Dehn funcion $\tilde{\delta}_{G}(n)$ of $G$ is polynomially bounded. We are
done for the relative Dehn function case.

If $G$ is finitely presented, by Theorem 2.6, if the relative Dehn function is
polynomially bounded, $\delta_{G}(n)$ is bounded above by the exponential
function. ∎

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