# Divergence of finitely presented subgroups
of CAT(0) groups

Noel Brady  University of Oklahoma, Norman, OK 73019-3103, USA<EMAIL_ADDRESS>and  Hung Cong Tran  University of Oklahoma, Norman, OK 73019-3103, USA
<EMAIL_ADDRESS>
###### Abstract.

We construct families of $\operatorname{CAT}(0)$ groups containing finitely
presented groups whose divergence functions are of the form $r^{\alpha}$ for a
dense set of exponents $\alpha\in[2,\infty)$ and $r^{q}\log(r)$ for integers
$q\geq 2$. The same construction also yields examples of
$\operatorname{CAT}(0)$ groups containing contracting elements which are not
contracting in certain finitely presented subgroups.

##  1\. Introduction

In classical geometry, one can distinguish between euclidean space and
hyperbolic space by measuring the rates at which geodesics diverge apart. In
particular, in the zero curvature (euclidean) case the divergence is linear
and in the negative curvature (hyperbolic) case the divergence is exponential.
In [Gro93] Gromov discussed geodesic divergence for more general non-
positively curved metric spaces and asked (6.B2(h), page 90) if such
divergence is always linear or exponential. In [Ger94b] Gersten gave the first
examples of $\operatorname{CAT}(0)$ groups whose geodesic divergence is
quadratic. In the same paper, Gersten defined divergence as a quasi-isometry
invariant of geodesic metric spaces and of finitely generated groups.

Roughly speaking, Gersten’s notion of divergence is a function $\delta(r)$
which measures the diameter of the sphere of radius $r$ in the path-metric
inherited from the complement of the open ball of radius $r$. In order to
obtain a quasi-isometry invariant, one needs to consider the path-metric
inherited from the complement of an open ball of radius $\rho r$ for some
$0<\rho\leq 1$ and one works with equivalence classes of one-parameter
families of functions $\\{\delta_{\rho}(r)\\}$. Precise definitions are given
in section 2.

In [Mac02] Macura studied divergence of free-by-cyclic groups with
polynomially growing monodromies and gave the first example of a
$\operatorname{CAT}(0)$ group with cubic divergence and the first examples of
finitely presented groups with polynomial divergence of arbitrary degree.
Examples of $\operatorname{CAT}(0)$ groups with polynomial divergence of
arbitrary degree are provided in [Mac13] and [BD14].

There has been intense study of divergence functions for particular families
of finitely presented groups over the years. All these examples have
divergence functions which are either polynomial or exponential. The first
examples of finitely presented groups whose divergence is neither polynomial
nor exponential appear in [BT]. The groups in [BT] make essential use of
distorted infinite cyclic subgroups and so cannot be subgroups of
$\operatorname{CAT}(0)$ groups. The research in the current paper was
motivated by the following open question (paraphrased from [BD14]).

######  Question 1.1.

Can the divergence of a $\operatorname{CAT}(0)$ group be other than a
polynomial or an exponential function?

This question has been answered in the negative for certain collections of
$\operatorname{CAT}(0)$ groups; namely, right-angled Artin groups (see
[BC12]), right-angled Coxeter groups (see [DT15, Lev18, BHS17, Sis, Lev],
$\operatorname{CAT}(0)$ $3$–manifold groups (see [Ger94a]), and
$\operatorname{CAT}(0)$ free-by-cyclic groups (see [Mac02]). However, there is
no current strategy for handling $\operatorname{CAT}(0)$ groups in general.

The main result of this paper (stated in Theorem 5.7 below) is the
construction of $\operatorname{CAT}(0)$ groups containing finitely presented
subgroups that exhibit a wide range of divergence behavior beyond polynomial
and exponential.

######  Theorem 5.7.

There exist $\operatorname{CAT}(0)$ groups containing finitely presented
subgroups whose divergence is equivalent to $r^{\alpha}$ for a dense set of
exponents $\alpha\in[2,\infty)$ and to $r^{q}\log(r)$ for integers $q\geq 2$.

These subgroups provide further instances of the remarkable diversity of
geometric behavior of subgroups of $\operatorname{CAT}(0)$ groups.

There are two very useful properties of geodesics in negatively curved spaces.
The first is that quasi-geodesics lie in bounded neighborhoods of geodesics
between their endpoints; this was a critical ingredient in the proof of Mostow
rigidity. This has been generalized to give the notions of Morse quasi-
geodesic in a space and Morse element in a group. The second is the strong
control on the diameter of images under nearest point projection onto
geodesics. This has been generalized in [ABD] to give the notion of
contracting quasi-geodesic in a metric space and contracting element in a
group. These group theory generalizations are related as follows. Every
contracting element is Morse. Moreover, these notions are equivalent in
$\operatorname{CAT}(0)$ and hyperbolic groups. In Proposition 4.28 we show
that the finitely presented subgroups in Theorem 5.7 are not
$\operatorname{CAT}(0)$ because they contain Morse elements which are not
contracting.

######  Proposition 4.28 (paraphrased).

The finitely presented subgroups in Theorem 5.7 contain Morse elements which
are not contracting. In particular, they are not $\operatorname{CAT}(0)$.

In Remark 5.10 we use Dehn functions to give an alternate reason why the
subgroups in Theorem 5.7 are not $\operatorname{CAT}(0)$ themselves.

It is not hard to see from the definition of contracting elements that if $H$
is an undistorted subgroup of a group $G$ then all group elements in $H$ which
are contracting in $G$ are also contracting in $H$. However, we prove that
this fact is not true for distorted subgroups of $\operatorname{CAT}(0)$
groups. These are the first examples in which the property of being a
contracting element is not inherited on passing to a finitely presented
subgroup.

######  Proposition 5.9.

There exist $\operatorname{CAT}(0)$ groups containing contracting elements
which are not contracting in some finitely presented subgroups.

The construction in this paper is similar in outline to that of [BT], although
there are significant technical differences. The key geometric idea behind the
construction in [BT] is to start with a base $\mathbb{Z}^{2}=\langle
a_{0},a_{1}|[a_{0},a_{1}]=1\rangle$ which is amalgamated along the $\langle
a^{-1}_{0}a_{1}\rangle$ subgroup to a suitable group in order to distort the
metric along this direction. An important feature is that the $a_{0}$ and
$a_{1}$ lines remain undistorted in the amalgam. Then one builds a sequence of
groups as iterated HNN extensions in a similar fashion to the examples of
Gersten and Macura. This construction gives rise to examples of finitely
presented groups with new divergence behavior; namely, $r^{\alpha}$ and
$r^{q}\log(r)$.

The main challenge is to find a suitable replacement for the base plane with
distorted direction of [BT]. We construct an analogue of this base plane using
products of finite rank free groups in such a way that the distorted paths are
far from periodic and so that infinite cyclic subgroups remain undistorted.
These can then be embedded into $\operatorname{CAT}(0)$ groups. These
embeddings rely on the fact that there are $\operatorname{CAT}(0)$ free-by-
cyclic groups with distorted fibers.

The analogue of the base plane of [BT] is a direct product of three free
groups with certain diagonal subgroups taking the place of the $a_{0}$,
$a_{1}$ directions and a skew-diagonal group as the transverse direction. This
construction has a combinatorial shortcoming: analogues of triangles built
from $a_{0}$, $a_{1}$ and transverse direction sides can only be constructed
using palindromic words in the free groups. This is not a serious shortcoming
since power function distortions of palindromic paths can be provided using
the snowflake construction of [BF17] where palindromic automorphisms play a
central role, and exponential distortions can be provided using a free-by-
cyclic group with palindromic monodromy of exponential growth.

A significant portion of the paper is devoted to proving the following
technical divergence result. The product of three free groups (the analogue of
the base plane in [BT]) appears in the definition of $G_{1}$; so does the
skew-diagonal free group analogue of $a_{0}^{-1}a_{1}$. The diagonal free
group analogues of $a_{0}$ and $a_{1}$ appear in the definition of $G_{2}$.
Note that a key contribution to the divergence comes from the inverse of a
distortion function.

######  Theorem 2.13.

Let $H$ be a finitely presented group that contains a free subgroup $F$ of
rank $p$ generated by $d_{i}$ for $1\leq i\leq p$. Assume that the distortion
function $\operatorname{Dist}_{F}^{H}$ admits an exponentially bounded
sequence of palindromic certificates in $F$. Let $F_{x}$, $F_{y}$, and $F_{z}$
be other free groups of rank $p$ generated by $x_{i}$, $y_{i}$, and $z_{i}$
for $1\leq i\leq p$ respectively.

For each integer $m\geq 1$ there are finitely presented one-ended groups

$\displaystyle G_{1}\;=\;\bigl{[}H\ast_{\langle
d_{i}=x_{i}y_{i}^{-1}\rangle}(F_{x}\times F_{y}\times
F_{z})\bigr{]}\times{\langle s_{1}\rangle};$ $\displaystyle G_{2}\;=\;\langle
G_{1},s_{2}|s_{2}^{-1}(x_{i}z_{i})s_{2}=y_{i}z_{i}\rangle;$ and $\displaystyle
G_{m}\;=\;\langle
G_{m-1},s_{m}|s_{m}^{-1}s_{1}s_{m}=s_{m-1}\rangle\quad\quad{\hbox{ for $m\geq
3$}}$

with the following properties. The subgroup $\langle s_{m}\rangle$ is
undistorted and the following table holds.

$m$ |  ${\rm Div}_{\langle s_{m}\rangle}^{G_{m}}$ |  ${\rm Div}_{G_{m}}$
---|---|---
$2$ |  $r(\operatorname{Dist}_{F}^{H})^{-1}(r)$ |  $r^{2}$
$\geq 3$ |  $r^{m-1}(\operatorname{Dist}_{F}^{H})^{-1}(r)$ |  $r^{m-1}(\operatorname{Dist}_{F}^{H})^{-1}(r)$

Using $\operatorname{CAT}(0)$ free-by-cyclic groups as building blocks, we
construct a sequence of $\operatorname{CAT}(0)$ groups $B_{m}$ which mirrors
the construction of the groups $G_{m}$ above and we prove that the groups
$B_{m}$ contain the groups $G_{m}$. We refer the reader to section 5 for the
constructions of $\operatorname{CAT}(0)$ groups $B_{m}$. For the $r^{\alpha}$
divergence (resp. $r^{q}\log{r}$) in Theorem 5.7 we use the snowflake groups
of [BF17] (resp. certain free-by-cyclic group with palindromic automorphism of
exponential growth) as the subgroup $H$ in the theorem above.

This paper is organized as follows. Section 2 contains definitions and a
statement of the main divergence theorem. Section 3 includes fundamental
geometric and algebraic properties of the groups $G_{m}$ which play a key role
in the proof of the main divergence theorem. The proof of this theorem is
given in section 4. In section 5 the sequence of $\operatorname{CAT}(0)$
groups $(B_{m})$ is defined and the inclusions $G_{m}\leq B_{m}$ are
established.

### Acknowledgements

The first author was supported by Simons Foundation collaboration grant
#430097. The second author was supported by an AMS-Simons Travel Grant.

##  2\. Definitions and statement of the divergence results

######  Convention 2.1.

Let $\mathcal{M}$ be the collection of all functions from positive reals to
positive reals. Let $f$ and $g$ be arbitrary elements of $\mathcal{M}$. We say
that $f$ is _dominated_ by $g$, denoted $f\preceq g$, if there are positive
constants $A,B,C$ such that $f(x)\leq g(Ax)+Bx$ for all $x>C$. We say that $f$
is _equivalent_ to $g$, denoted $f\sim g$, if $f\preceq g$ and $g\preceq f$.
This defines an equivalence relation on $\mathcal{M}$.

One can define an equivalence relation between indexed families of functions
as follows. Let $\\{\delta_{\rho}\\}$ and $\\{\delta^{\prime}_{\rho}\\}$ be
two families of functions of $\mathcal{M}$, indexed over $\rho\in(0,1]$. The
family $\\{\delta_{\rho}\\}$ is _dominated_ by the family
$\\{\delta^{\prime}_{\rho}\\}$, denoted
$\\{\delta_{\rho}\\}\preceq\\{\delta^{\prime}_{\rho}\\}$, if there exists
constant $L\in(0,1]$ such that $\delta_{L\rho}\preceq\delta^{\prime}_{\rho}$
for all $\rho\in(0,1]$. We say $\\{\delta_{\rho}\\}$ is _equivalent_ to
$\\{\delta^{\prime}_{\rho}\\}$, denoted
$\\{\delta_{\rho}\\}\sim\\{\delta^{\prime}_{\rho}\\}$, if
$\\{\delta_{\rho}\\}\preceq\\{\delta^{\prime}_{\rho}\\}$ and
$\\{\delta^{\prime}_{\rho}\\}\preceq\\{\delta_{\rho}\\}$. If $f$ is a function
in $\mathcal{M}$, the family $\\{\delta_{\rho}\\}$ is _equivalent_ to $f$ if
there is $b\in(0,1]$ such that $\delta_{\rho}$ is equivalent to $f$ for each
$\rho\in(0,b]$.

We now recall Gersten’s definition of divergence from [Ger94b]. Let $X$ be a
geodesic space and $x_{0}$ one point in $X$. Let $d_{r,x_{0}}$ be the induced
length metric on the complement of the open ball with radius $r$ about
$x_{0}$. If the point $x_{0}$ is clear from context, we will use the notation
$d_{r}$ instead of $d_{r,x_{0}}$.

######  Definition 2.2 (Group divergence).

Let $X$ be a geodesic space with a fixed point $x_{0}$. For each
$\rho\in(0,1]$ we define a function $\delta_{\rho}\\!:(0,\infty)\to(0,\infty)$
as follows. For each $r>0$, let $\delta_{\rho}(r)=\sup d_{\rho
r}(x_{1},x_{2})$ where the supremum is taken over all $x_{1}$ and $x_{2}$ on
the sphere $S(x_{0},r)$ such that $d_{\rho r}(x_{1},x_{2})<\infty$. The family
of functions $\\{\delta_{\rho}\\}$ is the _divergence_ of $X$.

Using Convention 2.1 one can show that up to equivalence the divergence of $X$
does not depend on the choice of $x_{0}$ and it is a quasi-isometry invariant
(see [Ger94b]). The _divergence_ of a finitely generated group $G$, denoted
${\rm Div}_{G}$, is the divergence of the Cayley graph $\Gamma(G,S)$ for some
(any) finite generating set $S$.

We now recall the definition of quasi-geodesic divergence and undistorted
cyclic subgroup divergence.

######  Definition 2.3 (Quasi-geodesic divergence).

Let $X$ be a geodesic space and $\alpha\\!:(-\infty,\infty)\to X$ be an
$(L,C)$–bi-infinite quasi-geodesic. The _divergence_ of $\alpha$ in $X$ is the
function ${\rm Div}_{\alpha}\\!:(0,\infty)\to(0,\infty)$ defined as follows.
Let $r>0$ be an arbitrary number. If there is no path outside the open ball
$B(\alpha(0),r/L-C)$ connecting $\alpha(-r)$ and $\alpha(r)$, we define ${\rm
Div}_{\alpha}(r)=\infty$. Otherwise, we define ${\rm Div}_{\alpha}(r)$ to be
the infimum over the lengths of all paths outside the open ball
$B(\alpha(0),r/L-C)$ connecting $\alpha(-r)$ and $\alpha(r)$.

######  Definition 2.4 (Cyclic subgroup divergence).

Let $G$ be a finitely generated group and $\langle c\rangle$ be an
undistorted, infinite cyclic subgroup of $G$. Let $S$ be a finite generating
set of $G$ that contains $c$. Since $\langle c\rangle$ is undistorted, every
bi-infinite path with edges labeled by $c$ is a quasi-geodesic in the Cayley
graph $\Gamma(G,S)$. The _divergence_ of the cyclic subgroup $\langle
c\rangle$ in $G$, denoted ${\rm Div}_{\langle c\rangle}^{G}$, is defined to be
the divergence of such a bi-infinite quasi-geodesic.

Using Convention 2.1 the divergence of the cyclic subgroup $\langle c\rangle$
in $G$ does not depend on the choice of finite generating set $S$. We leave
the proof of this fact as an exercise for the reader.

We now review the concept of contracting quasi-geodesics and contracting
elements in a finitely generated group.

######  Definition 2.5 (Contracting quasi-geodesic).

Let $X$ be a geodesic space. A bi-infinite quasi-geodesic $\alpha$ in $X$ is
_contracting_ in $X$ if there exist a map $\pi_{\alpha}\\!:X\to\alpha$ and
constants $0<A<1$ and $D\geq 1$ satisfying:

  1. (1)

$\pi_{\alpha}$ is $(D,D)$–coarsely Lipschitz (i.e., $\forall x_{1},x_{2}\in
X,d(\pi_{\alpha}(x_{1}),\pi_{\alpha}(x_{2}))\leq
Dd\bigl{(}x_{1},x_{2}\bigr{)}+D$).

  2. (2)

For any $y\in\alpha$, $d\bigl{(}y,\pi_{\alpha}(y)\bigr{)}\leq D$.

  3. (3)

For all $x\in X$, if we set $R=Ad(x,\alpha)$, then
$\operatorname{diam}\bigl{(}\pi_{\alpha}\bigl{(}B(x,R)\bigr{)}\bigr{)}\leq D$.

######  Definition 2.6 (Contracting element).

Let $G$ be a finitely generated group and $g$ an infinite order element in
$G$. Let $S$ be a finite generating set of $G$ that contains $g$. Let $\alpha$
be a bi-infinite path in the Cayley graph $\Gamma(G,S)$ with edges labeled by
$g$. The element $g$ in $G$ is _contracting_ if the path $\alpha$ is a
contracting quasi-geodesic.

It is not hard to see that the notion of contracting quasi-geodesic is a
quasi-isometry invariant. In particular, the concept contracting elements in a
finitely generated group do not depend on the choice of finite generating set
$S$. The following proposition shows that the divergence of a contracting
quasi-geodesic is always at least quadratic.

######  Proposition 2.7 (See Proposition 5.3 in [BT]).

Let $\alpha\\!:(-\infty,\infty)\to X$ be an $(L,C)$–quasi-geodesic in a
geodesic space $X$. Assume that $\alpha$ is also an $(A,D)$–contracting quasi-
geodesic. Then the divergence of $\alpha$ is at least quadratic.

######  Definition 2.8 (Lipschitz equivalence of functions).

Let $f,g\\!:[0,\infty)\to[0,\infty)$ be two non-decreasing function. We say
that $f$ and $g$ are _Lipschitz equivalent_ if there is a constant $M\geq 1$
such that for each $r\geq 1$ we have

$g(r)\leq Mf(Mr)\text{ and }f(r)\leq Mg(Mr).$

######  Definition 2.9 (Distortion function).

Let $H$ be a finitely generated group with a finite generating set $T$ and let
$F$ be a subgroup of $H$ with a generating set $R$. The _distortion function_
$\operatorname{Dist}_{F}^{H}\\!:\mathbb{N}\to\mathbb{N}$ is defined as
follows:

$\operatorname{Dist}_{F}^{H}(n)=\max\\{\,{|g|_{R}}\mid{g\in F,|g|_{T}\leq
n}\,\\}\text{ for $n\in\mathbb{N}$}.$

We note that the function $\operatorname{Dist}_{F}^{H}$ is non-decreasing. We
can also consider the function $\operatorname{Dist}_{F}^{H}$ as a non-
decreasing function from $[0,\infty)$ to $[0,\infty)$ by linear interpolation
over the open unit intervals.

The function $\operatorname{Dist}_{F}^{H}$ is said to _admit an exponentially
bounded sequence of certificates_ $(g_{m})$ in $F$ if there is a constant
$C>1$ such that the following hold:

  1. (1)

$|g_{m}|_{R}\to\infty$ as $m\to\infty$;

  2. (2)

$|g_{m+1}|_{R}\leq C|g_{m}|_{R}$ for each $m$; and

  3. (3)

$\operatorname{Dist}_{F}^{H}\bigl{(}|g_{m}|_{T}/C\bigr{)}\leq C|g_{m}|_{R}$
for each $m$.

The certificate elements $g_{m}$ will be used to certify upper bounds on group
divergence functions later on.

Up to Lipschitz equivalence, the distortion function
$\operatorname{Dist}_{F}^{H}$ does not depend on the choice of finite
generating sets $R$ and $T$. Furthermore, the fact that a distortion function
admits an exponentially bounded sequence of certificates also does not depend
on the choice of finite generating sets $R$ and $T$.

Note that the function $\operatorname{Dist}_{F}^{H}(r)$ grows at least
linearly and so is Lipschitz equivalent to the monotonic increasing function
$\operatorname{Dist}^{H}_{F}(r)+r$. Henceforth, we will always assume that
$\operatorname{Dist}^{H}_{F}(r)$ is a bijective function.

######  Remark 2.10.

The reader may recall that the distortion of $F_{\\{a,b\\}}$ in
$F_{\\{a,b\\}}\rtimes_{\varphi}\mathbb{Z}$ where $\varphi$ has exponential
growth is exponential. The upper bound for this distortion function is proven
algebraically using successive applications of Britton’s lemma for HNN
extensions. The sequence of elements $(\varphi^{n}(a))$ is exponentially
bounded in the $F_{\\{a,b\\}}$–metric (and is linearly bounded in the
$F_{\\{a,b\\}}\rtimes_{\varphi}\mathbb{Z}$–metric) and provides a lower bound
for the distortion function.

This same sequence $(\varphi^{n}(a))$ is an exponentially bounded sequence of
certificates in the sense of Definition 2.9. However, note that the inequality
in item (3) of that definition is the opposite of what would be used if we
were establishing a lower bound for the distortion function. This is
important. As was stated in Definition 2.9, this sequence will be used later
on to certify an upper bound for a divergence function. The next lemma
reformulates conditions (1)–(3) of that definition in a format which will be
easy to apply to give upper bounds on group divergence.

######  Lemma 2.11.

Let $H$ be a finitely generated group with a finite generating set $T$ and let
$F$ be a subgroup of $H$ with a generating set $R$. Up to the Lipschitz
equivalence relation we can choose the distortion
$\operatorname{Dist}_{F}^{H}$ such that

$\operatorname{Dist}_{F}^{H}(r)\geq r\text{ for each }r\geq 1\text{ and
}(\operatorname{Dist}_{F}^{H})^{-1}\bigl{(}|g|_{R}\bigr{)}\leq|g|_{T}\text{
for each }g\in F.$

Moreover, if $\operatorname{Dist}_{F}^{H}$ admits an exponentially bounded
sequence of certificates $(g_{m})$ in $F$, then there are constants $D>1$ and
$r_{0}\geq 1$ such that for each $r\geq r_{0}$ there is a group element $u$ in
the sequence $(g_{m})$ satisfying

$r/D\leq|u|_{R}\leq r\text{ and }|u|_{T}\leq
D~{}(\operatorname{Dist}_{F}^{H})^{-1}(r).$

###### Proof.

We note that the function $\operatorname{Dist}_{F}^{H}(r)$ grows at least
linearly and so is Lipschitz equivalent to the monotonic increasing function
$\operatorname{Dist}^{H}_{F}(r)+r$. Replacing $\operatorname{Dist}_{F}^{H}(r)$
by $\operatorname{Dist}^{H}_{F}(r)+r$ (if necessary), we can assume that
$\operatorname{Dist}_{F}^{H}(r)\geq r$ for each $r\geq 1$. We also observe
that

$\operatorname{Dist}_{F}^{H}\bigl{(}|g|_{T}\bigr{)}\geq|g|_{R}\text{ for each
$g\in F$}.$

Therefore,
$(\operatorname{Dist}_{F}^{H})^{-1}\bigl{(}|g|_{R}\bigr{)}\leq|g|_{T}$.

We now prove second statement. Since the distortion function
$\operatorname{Dist}_{F}^{H}$ admits the exponentially bounded sequence of
certificates $(g_{m})$, there is a constant $C>1$ such that the following
hold:

  1. (1)

$|g_{m}|_{R}\to\infty$ as $m\to\infty$;

  2. (2)

$|g_{m+1}|_{R}\leq C|g_{m}|_{R}$ for each $m$; and

  3. (3)

$\operatorname{Dist}_{F}^{H}\bigl{(}|g_{m}|_{T}/C\bigr{)}\leq C|g_{m}|_{R}$
for each $m$.

Let $r_{0}=C|g_{1}|_{R}\geq 1$ and $D=C^{3}>1$. Let $r\geq r_{0}$ be an
arbitrary number. Then there is a positive integer $n$ such that

$C^{n}|g_{1}|_{R}\leq r\leq C^{n+1}|g_{1}|_{R}.$

Since $|g_{m}|_{R}\to\infty$ as $m\to\infty$ and $|g_{m+1}|_{R}\leq
C|g_{m}|_{R}$ for each $m$, we can also choose $u=g_{m_{0}}$ for some $m_{0}$
such that

$C^{n-2}|g_{1}|_{R}\leq|u|_{R}\leq C^{n-1}|g_{1}|_{R}.$

By the choice of $r_{0}$, $D$, and $n$ we observe that

$|u|_{R}\geq C^{n-2}|g_{1}|_{R}\geq(C^{n+1}|g_{1}|_{R})/C^{3}\geq r/C^{3}\geq
r/D$

and

$|u|_{R}\leq C^{n-1}|g_{1}|_{R}\leq(C^{n}|g_{1}|_{R})/C\leq r/C.$

Moreover,

$\operatorname{Dist}_{F}^{H}\bigl{(}|u|_{T}/D\bigr{)}\leq\operatorname{Dist}_{F}^{H}\bigl{(}|u|_{T}/C\bigr{)}\leq
C|u|_{R}\leq C(r/C)\leq r$

Therefore, $|u|_{T}/D\leq(\operatorname{Dist}_{F}^{H})^{-1}(r)$ or
$|u|_{T}\leq D~{}(\operatorname{Dist}_{F}^{H})^{-1}(r)$.

∎

######  Definition 2.12.

Let $F$ be a free group of rank $p$ with a generating set
$R=\\{\,{d_{i}}\mid{1\leq i\leq p}\,\\}$. A word in $R$ is _palindromic_ if it
reads the same left-to-right as right-to-left. A group element $g\in F$ is
_palindromic_ with respect to $R$ if the reduced word in $R$ that represents
$g$ is palindromic.

We now state the main divergence result of the paper. The reason for the
technical conditions in this statement will become clear as we fill in details
of the proof in the next two sections.

######  Theorem 2.13.

Let $H$ be a finitely presented group that contains a free subgroup $F$ of
rank $p$ generated by $d_{i}$ for $1\leq i\leq p$. Assume that the distortion
function $\operatorname{Dist}_{F}^{H}$ admits an exponentially bounded
sequence of palindromic certificates in $F$. Let $F_{x}$, $F_{y}$, and $F_{z}$
be other free groups of rank $p$ generated by $x_{i}$, $y_{i}$, and $z_{i}$
for $1\leq i\leq p$ respectively.

For each integer $m\geq 1$ there are finitely presented one-ended groups

$\displaystyle G_{1}\;=\;\bigl{[}H\ast_{\langle
d_{i}=x_{i}y_{i}^{-1}\rangle}(F_{x}\times F_{y}\times
F_{z})\bigr{]}\times{\langle s_{1}\rangle};$ $\displaystyle G_{2}\;=\;\langle
G_{1},s_{2}|s_{2}^{-1}(x_{i}z_{i})s_{2}=y_{i}z_{i}\rangle;$ and $\displaystyle
G_{m}\;=\;\langle
G_{m-1},s_{m}|s_{m}^{-1}s_{1}s_{m}=s_{m-1}\rangle\quad\quad{\hbox{ for $m\geq
3$}}$

with the following properties. The subgroup $\langle s_{m}\rangle$ is
undistorted and the following table holds.

$m$ |  ${\rm Div}_{\langle s_{m}\rangle}^{G_{m}}$ |  ${\rm Div}_{G_{m}}$
---|---|---
$2$ |  $r(\operatorname{Dist}_{F}^{H})^{-1}(r)$ |  $r^{2}$
$\geq 3$ |  $r^{m-1}(\operatorname{Dist}_{F}^{H})^{-1}(r)$ |  $r^{m-1}(\operatorname{Dist}_{F}^{H})^{-1}(r)$

The proof of this theorem occupies the next two sections. In section 3 we
establish some algebraic properties of $G_{m}$; the reason for the palindromic
condition becomes clear in Lemma 3.8. In section 4 we establish upper bounds
and lower bounds for $\operatorname{Div}_{G_{m}}$ and for the divergence of
$\langle s_{m}\rangle$ in $G_{m}$.

##  3\. Algebraic and geometric properties of the groups $G_{m}$

We fix generating sets $S_{m}$ for the groups $G_{m}$ in Theorem 2.13.

######  Definition 3.1 (Generating sets).

The group $G_{1}$ is defined in the Theorem 2.13 as an amalgam

$G_{1}\;=\;\bigl{[}H\ast_{\langle d_{i}=x_{i}y_{i}^{-1}\rangle}(F_{x}\times
F_{y}\times F_{z})\bigr{]}\times{\langle s_{1}\rangle};$

where group $H$ has finite generating set $T$, three free groups $F_{x}$,
$F_{y}$, and $F_{z}$ of rank $p$ generated by the finite generating sets
$R_{x}=\\{\,{x_{i}}\mid{1\leq i\leq p}\,\\}$, $R_{y}=\\{\,{y_{i}}\mid{1\leq
i\leq p}\,\\}$, and $R_{z}=\\{\,{z_{i}}\mid{1\leq i\leq p}\,\\}$ respectively.
Group $H$ contains the free subgroup $F$ with the generating set
$R=\\{\,{d_{i}}\mid{1\leq i\leq p}\,\\}$ and we assume that $T$ contains $R$.

For each $1\leq i\leq p$ we let $a_{i}=x_{i}z_{i}$ and $b_{i}=y_{i}z_{i}$.
Then two subgroups $F_{xz}$ and $F_{yz}$ generated by two sets
$R_{xz}=\\{\,{a_{i}}\mid{1\leq i\leq p}\,\\}$ and
$R_{yz}=\\{\,{b_{i}}\mid{1\leq i\leq p}\,\\}$ respectively are free of rank
$p$. Let $S_{1}=T\cup R_{x}\cup R_{y}\cup R_{z}\cup R_{xz}\cup
R_{yz}\cup\\{s_{1}\\}$. Then $S_{1}$ is a finite generating set for $G_{1}$.
Moreover, we also can write

$\displaystyle G_{1}\;=\;\bigl{[}H\ast_{\langle
d_{i}=a_{i}b_{i}^{-1}\rangle}(F_{x}\times F_{y}\times
F_{z})\bigr{]}\times{\langle s_{1}\rangle};$ $\displaystyle G_{2}\;=\;\langle
G_{1},s_{2}|s_{2}^{-1}a_{i}s_{2}=b_{i}\rangle.$

For each integer $m\geq 3$ the groups $G_{m}$ are defined recursively by

$G_{m}\;=\;\langle G_{m-1},s_{m}\,|\,s_{m}^{-1}s_{1}s_{m}=s_{m-1}\rangle$

and so have recursively defined finite generating sets
$S_{m}=S_{m-1}\cup\\{s_{m}\\}$ for groups $G_{m}$ for each $m\geq 2$.

The following lemma will be used repeatedly to prove the group divergence by
induction in Theorem 2.13. In particular, the last two inequalities in the
following lemma will be used to show that certain paths lies outside certain
open balls.

######  Lemma 3.2.

The inclusion maps
$i_{0}\\!:\Gamma(F_{xz},R_{xz})\hookrightarrow\Gamma(G_{1},S_{1})$ and
$i_{1}\\!:\Gamma(F_{yz},R_{yz})\hookrightarrow\Gamma(G_{1},S_{1})$ are
isometric embedding maps. Moreover, if $g_{0}$, $g_{1}$, and $h$ are group
elements in $F_{xz}$, $F_{yz}$, and $H$ respectively, then

$|g_{0}h|_{S_{1}}\geq|g_{0}|_{{S_{1}}}\text{ and
}|g_{1}h|_{S_{1}}\geq|g_{1}|_{{S_{1}}}.$

###### Proof.

We observe that there is a group homomorphism $\Psi\\!:G_{1}\to F_{z}$ taking
each $z_{i}$ to itself and taking other generators to the identity. We observe
that $\Psi_{|F_{xz}}$ be a group monomorphism taking each $a_{i}$ to $z_{i}$
and $\Psi(F_{xz})=F_{z}$. Similarly, $\Psi_{|F_{yz}}$ is also a group
monomorphism taking each $b_{i}$ to $z_{i}$ and $\Psi(F_{yz})=F_{z}$.
Therefore, the inclusion maps
$i_{0}\\!:\Gamma(F_{xz},R_{xz})\hookrightarrow\Gamma(G_{1},S_{1})$ and
$i_{1}\\!:\Gamma(F_{yz},R_{yz})\hookrightarrow\Gamma(G_{1},S_{1})$ are
isometric embedding maps.

Now let $g_{0}$, $g_{1}$, and $h$ be group elements in $F_{xz}$, $F_{yz}$, and
$H$ respectively. Then,

$|g_{0}h|_{S_{1}}\geq|\Psi(g_{0}h)|_{R_{z}}\text{ and
}|g_{1}h|_{S_{1}}\geq|\Psi(g_{1}h)|_{R_{z}}.$

Also,
$|\Psi(g_{0}h)|_{R_{z}}=|\Psi(g_{0})|_{R_{z}}=|g_{0}|_{R_{xz}}=|g_{0}|_{S_{1}}$
and
$|\Psi(g_{1}h)|_{R_{z}}=|\Psi(g_{1})|_{R_{z}}=|g_{1}|_{R_{yz}}=|g_{0}|_{S_{1}}$.
Therefore,

$|g_{0}h|_{S_{1}}\geq|g_{0}|_{S_{1}}\text{ and
}|g_{1}h|_{S_{1}}\geq|g_{1}|_{S_{1}}.$

∎

######  Lemma 3.3.

Let $m\geq 2$ be an integer. Then the inclusion map
$i\\!:\Gamma(G_{m-1},S_{m-1})\hookrightarrow\Gamma(G_{m},S_{m})$ is an
isometric embedding.

###### Proof.

For $m=2$ the group $G_{2}$ is an isometric HNN extension in the sense of
[Bri98], with base group $G_{1}$ and edge groups $F_{xz}$ and $F_{yz}$. By
Lemma 3.2 two groups $F_{xz}$ and $F_{yz}$ are isometrically embedded into
$G_{1}$ with the given generating sets. By Lemma 2.2(2) in [Bri98] (taking
$G=H=G_{1}$) the inclusion $G_{1}\hookrightarrow G_{2}$ is an isometric
embedding.

For each $m\geq 3$ the group $G_{m}$ is also an isometric HNN extension, with
base group $G_{m-1}$ and edge groups $\langle s_{1}\rangle$ and $\langle
s_{m-1}\rangle$. The homomorphism $G_{m-1}\to{\mathbb{Z}}$ taking all the
$s_{j}$ ($1\leq j\leq m-1$) to a generator of ${\mathbb{Z}}$ and all other
generators of $G_{m-1}$ to the identity shows that each $\langle s_{j}\rangle$
is a retract of $G_{m-1}$ and so are isometrically embedded subgroups. By
Lemma 2.2(2) in [Bri98] again (taking $G=H=G_{m-1}$) the inclusion
$G_{m-1}\hookrightarrow G_{m}$ is an isometric embedding. ∎

The following lemma will be used to define the concepts of $1$–rays,
$(1,k)$–rays, and $k$–corners (see Definition 4.9 and Definition 4.10) which
appear in the proof of the lower bound of group divergence in Theorem 2.13.

######  Lemma 3.4.

Let $1\leq i<j\leq m$ be integers. Let $p,q$ be arbitrary integers. Then
$|s_{i}^{p}s_{j}^{q}|_{S_{m}}=|p|+|q|$.

###### Proof.

There is a group homomorphism $\Psi\\!:G_{j}\to\mathbb{Z}^{2}$ taking $s_{j}$
to the generator $(0,1)$, taking each $s_{k}$ to the generator $(1,0)$ for
$1\leq k\leq j-1$, and taking each element in
$S_{j}-\\{s_{1},s_{2},\cdots,s_{j}\\}$ to the identity $(0,0)$. Therefore,
$\Psi(s_{i}^{p}s_{j}^{q})=(p,q)$. This implies that

$|s_{i}^{p}s_{j}^{q}|_{S_{j}}\geq|\Psi(s_{i}^{p}s_{j}^{q})|=|p|+|q|.$

Since $s_{i}$ and $s_{j}$ are also elements in the finite generating set
$S_{j}$, we have

$|s_{i}^{p}s_{j}^{q}|_{S_{j}}=|p|+|q|.$

Also, the inclusion map
$\Gamma(G_{j},S_{j})\hookrightarrow\Gamma(G_{m},S_{m})$ is an isometric
embedding by Lemma 3.3. Therefore,
$|s_{i}^{p}s_{j}^{q}|_{S_{m}}=|s_{j}^{p}s_{j}^{q}|_{S_{j}}=|p|+|q|$. ∎

The next lemma is used in Lemma 4.13 to help determine the structure of
$k$–corners.

######  Lemma 3.5.

Let $1\leq i<j<k\leq m$ be integers. Let $p,q,r$ be arbitrary integers such
that $s_{i}^{p}s_{j}^{q}s_{k}^{r}=e$ in $G_{m}$. Then $p=q=r=0$.

###### Proof.

We first prove that $r=0$. In fact, there is a group homomorphism
$\Phi\\!:G_{k}\to\mathbb{Z}$ taking $s_{k}$ to $1$ and each generators in
$S_{k}-\\{s_{k}\\}$ to $0$. Therefore,
$r=\Phi(s_{i}^{p}s_{j}^{q}s_{k}^{r})=\Phi(e)=0$. Thus, $s_{i}^{p}s_{j}^{q}=e$
which implies that $p=q=0$ by Lemma 3.4. ∎

######  Remark 3.6.

Recall from Definition 3.1 that $F$ is a free subgroup of rank $p$ of $H$ with
a finite generating set $R=\\{\,{d_{i}}\mid{1\leq i\leq p}\,\\}$ and that $R$
is a subset of the finite generating set $T$ of $H$. We choose the distortion
$\operatorname{Dist}_{F}^{H}$ as in Lemma 2.11 and let
$f=(\operatorname{Dist}_{F}^{H})^{-1}$. Then it is clear that $f(r)\leq r$ for
each $r\geq 1$. By Lemma 2.11 we also have

  1. (1)

$f\bigl{(}|g|_{R}\bigr{)}\leq|g|_{T}$ for each $g\in F$; and

  2. (2)

There is constants $D>1$ and $r_{0}\geq 1$ such that for each $r\geq r_{0}$
there is a palindromic group element $u$ satisfying $r/D\leq|u|_{R}\leq r$ and
$|u|_{T}\leq Df(r)$.

We will fix the positive numbers $r_{0}$, $D$, and the function
$f=(\operatorname{Dist}_{F}^{H})^{-1}$ in the above statement for the
remaining of the paper.

Two elements $g_{0}\in F_{xz}$ and $g_{1}\in F_{yz}$ determine a pair of
geodesic edge-paths based at the identity (one from $e$ to $g_{0}$ and the
other from $e$ to $g_{1}$); we call this pair of edge-paths a _corner_. The
next lemma establishes a lower bound on the divergence of such corners in the
group $G_{1}$.

######  Lemma 3.7.

Let $g_{0}$ be a group element in $F_{xz}$ and let $g_{1}$ be a group element
in $F_{yz}$. Then

$|g_{1}^{-1}g_{0}|_{S_{1}}\geq f\bigl{(}|g_{0}|_{S_{1}}\bigr{)}.$

###### Proof.

There is a group homomorphism $\Psi\\!:G_{1}\to H$ taking each $x_{i}$ to
$d_{i}$, taking each $y_{i}$, $z_{i}$ and $s_{1}$ to the identity $e$, and
taking each element in $T$ to itself. Thus, $\Psi$ takes each $a_{i}$ to
$d_{i}$ and takes each $b_{i}$ to the identity $e$. Therefore,
$\Psi(g_{1}^{-1}g_{0})=\Psi(g_{0})$. Also $\Psi|F_{xz}$ is a monomorphism and
$\Psi(F_{xz})=F$. Therefore, $|\Psi(g_{0})|_{R}=|g_{0}|_{R_{xz}}$. We remind
the reader that $R=\\{\,{d_{i}}\mid{1\leq i\leq p}\,\\}$ is a finite
generating set $F$ and $R_{xz}=\\{\,{a_{i}}\mid{1\leq i\leq p}\,\\}$ is a
finite generating set $F_{xz}$. Since the finite generating set $S_{1}$ of
$G_{1}$ containing the finite generating set $R_{xz}$ of $F_{xz}$, we have
$|\Psi(g_{0})|_{R}=|g_{0}|_{R_{xz}}\geq|g_{0}|_{S_{1}}$. Also, $\Psi(g_{0})$
is a group element in $F$. Therefore, we have

$|\Psi(g_{0})|_{T}\geq f\bigl{(}|\Psi(g_{0})|_{R}\bigr{)}\geq
f\bigl{(}|g_{0}|_{S_{1}}\bigr{)}.$

This implies that

$|g_{1}^{-1}g_{0}|_{S_{1}}\geq|\Psi(g_{1}^{-1}g_{0})|_{T}=|\Psi(g_{0})|_{T}\geq
f\bigl{(}|g_{0}|_{S_{1}}\bigr{)}.$

∎

The next lemma establishes the existence of corners in $G_{1}$ whose
divergences agree with the lower bounds of Lemma 3.7.

######  Lemma 3.8.

For each positive integer $r\geq r_{0}$ there are elements $g_{0}$ in $F_{xz}$
and $g_{1}$ in $F_{yz}$ such that the following hold:

  1. (1)

$s_{2}^{-1}g_{0}s_{2}=g_{1}$;

  2. (2)

$r/D\leq|g_{0}|_{R_{xz}}\leq r$ and $r/D\leq|g_{1}|_{R_{yz}}\leq r$;

  3. (3)

There is a path $\gamma$ connecting $g_{0}$ and $g_{1}$ with length at most
$Df(r)$ and each vertex in $\gamma$ is a group element $g_{0}h$ for $h\in H$;
in particular, $\gamma$ avoids the $r/D$–ball about $e$.

###### Proof.

Let $u$ be a palindromic group element in $F$ such that $r/D\leq|u|_{R}\leq r$
and $|u|_{T}\leq Df(r)$. Then $u$ is represented by a reduced palindromic word
$w$ as follows

$w=d_{i_{1}}^{\epsilon_{1}}d_{i_{2}}^{\epsilon_{2}}\cdots
d_{i_{\ell}}^{\epsilon_{\ell}}\text{ where }\epsilon_{i}\neq 0\text{ and
}|\epsilon_{1}|+|\epsilon_{2}|+\cdots+|\epsilon_{\ell}|=|u|_{R}.$

We define

$g_{0}=a_{i_{1}}^{\epsilon_{1}}a_{i_{2}}^{\epsilon_{2}}\cdots
a_{i_{\ell}}^{\epsilon_{\ell}}\text{ and
}g_{1}=b_{i_{1}}^{\epsilon_{1}}b_{i_{2}}^{\epsilon_{2}}\cdots
b_{i_{\ell}}^{\epsilon_{\ell}}.$

Then $g_{0}$ and $g_{1}$ are group elements in $F_{xz}$ and $F_{yz}$
respectively, and

$r/D\leq|g_{0}|_{R_{xz}}=|g_{1}|_{R_{yz}}=|u|_{R}\leq r.$

Since $s_{2}^{-1}a_{i_{j}}s_{2}=b_{i_{j}}$ for $1\leq j\leq\ell$, we have
$s_{2}^{-1}g_{0}s_{2}=g_{1}$.

We note that $x_{i_{j}}$ commutes $y_{i_{k}}$ for $1\leq k,l\leq\ell$ and
$d_{i_{j}}=x_{i_{j}}y_{i_{j}}^{-1}$ for $1\leq j\leq\ell$. Therefore,

$u=d_{i_{1}}^{\epsilon_{1}}d_{i_{2}}^{\epsilon_{2}}\cdots
d_{i_{\ell}}^{\epsilon_{\ell}}=(x_{i_{1}}y_{i_{1}}^{-1})^{\epsilon_{1}}(x_{i_{2}}y_{i_{2}}^{-1})^{\epsilon_{2}}\cdots(x_{i_{\ell}}y_{i_{\ell}}^{-1})^{\epsilon_{\ell}}=(x_{i_{1}}^{\epsilon_{1}}x_{i_{2}}^{\epsilon_{2}}\cdots
x_{i_{\ell}}^{\epsilon_{\ell}})(y_{i_{1}}^{-\epsilon_{1}}y_{i_{2}}^{-\epsilon_{2}}\cdots
y_{i_{\ell}}^{-\epsilon_{\ell}}).$

Similarly, we also have

$g_{0}=(x_{i_{1}}^{\epsilon_{1}}x_{i_{2}}^{\epsilon_{2}}\cdots
x_{i_{\ell}}^{\epsilon_{\ell}})(z_{i_{1}}^{\epsilon_{1}}z_{i_{2}}^{\epsilon_{2}}\cdots
z_{i_{\ell}}^{\epsilon_{\ell}})\text{ and
}g_{1}=(y_{i_{1}}^{\epsilon_{1}}y_{i_{2}}^{\epsilon_{2}}\cdots
y_{i_{\ell}}^{\epsilon_{\ell}})(z_{i_{1}}^{\epsilon_{1}}z_{i_{2}}^{\epsilon_{2}}\cdots
z_{i_{\ell}}^{\epsilon_{\ell}}).$

Since $y_{i_{1}}^{\epsilon_{1}}y_{i_{2}}^{\epsilon_{2}}\cdots
y_{i_{\ell}}^{\epsilon_{\ell}}$ is a palindromic word, we have
$(y_{i_{1}}^{\epsilon_{1}}y_{i_{2}}^{\epsilon_{2}}\cdots
y_{i_{\ell}}^{\epsilon_{\ell}})(y_{i_{1}}^{-\epsilon_{1}}y_{i_{2}}^{-\epsilon_{2}}\cdots
y_{i_{\ell}}^{-\epsilon_{\ell}})=e$. This implies that $g_{0}=g_{1}u$ or
$g_{1}=g_{0}u^{-1}$.

Let $\gamma_{0}$ be a geodesic in the Cayley graph
$\Gamma(H,T)\subset\Gamma(G_{1},S_{1})$ connecting $e$ and $u^{-1}$. Then

$\ell(\gamma_{0})=|u|_{T}\leq Df(r).$

Moreover, the path $\gamma=g_{0}\gamma_{0}$ connects $g_{0}$ and $g_{1}$, the
length of $\gamma$ is at most $Df(r)$, and each vertex in $\gamma$ is a group
element $g_{0}h$ for $h\in H$. By Lemma 3.2 the path avoid $\gamma$ is
$r/D$–ball about $e$.

∎

The last lemma in this section is very straightforward; it will be used
multiple times in section 4.

######  Lemma 3.9.

Let $m$ be a positive integer and let $n$ be non-zero integer. Then

  1. (1)

There is a path in $\Gamma(G_{m},S_{m})$ which connects $s_{1}^{-n}$ and
$s_{1}^{n}$, lies outside the open ball $B(e,|n|)$, and has length at most
$4|n|$.

  2. (2)

For each element $s\in S_{1}\cup S_{1}^{-1}$, two group elements $s_{1}^{2n}$
and $ss_{1}^{2n}$ can be connected by an edge labeled $s$ in
$\Gamma(G_{m},S_{m})$.

###### Proof.

By Lemma 3.3 we only need to prove Statement (1) for $m=1$. In fact, Statement
(1) for $m=1$ can be proved from the fact the $\mathbb{Z}^{2}$-group generated
by $a_{1}$ and $s_{1}$ is isometrically embedded in $G_{1}$. Statement (2)
holds due to the fact that $s_{1}$ and $s$ commutes. ∎

##  4\. Divergence in $G_{m}$

The purpose of this section is twofold. In the first two subsections upper and
lower bounds for the divergence of $G_{m}$ are established, thus proving the
claims in the right column of the table in Theorem 2.13. The third subsection
is devoted to establishing the divergence of the cyclic subgroups $\langle
s_{m}\rangle$ and $\langle s_{1}s_{2}\rangle$ in $G_{m}$. The former completes
the proof of Theorem 2.13 and is also used in subsection 5.3 to prove
Proposition 5.9. In the case $F$ has nonlinear distortion in $H$, the latter
is used to prove that the $G_{m}$ are not $\operatorname{CAT}(0)$ groups for
$m\geq 2$.

###  4.1. The upper bound

In this subsection we establish upper bounds for the divergence of $G_{m}$. A
fundamental observation (Lemma 3.3) is that the group $G_{i}$ isometrically
embeds into $G_{i+1}$. Therefore paths which avoid the open $r$–ball (are open
$r$–ball avoidant) about the identity in $G_{i}$ remain open $r$–ball avoidant
in $G_{i+1}$. This allows us to prove upper bounds on the divergence of
$G_{m}$ by induction on $m$.

The following four lemmas will be used in the proof of the upper bound of the
group divergence in Theorem 2.13. The upper bounds are established in
Proposition 4.5 for $m\geq 3$ and in Proposition 4.6 for $m=2$.

######  Lemma 4.1 (see Lemma 4.1 in [BT]).

Let $X$ be a geodesic space and $x_{0}$ be a point in $X$. Let $r$ be a
positive number and let $x$ be a point on the sphere $S(x_{0},r)$. Let
$\alpha_{1}$ and $\alpha_{2}$ be two rays with the same initial point $x$ such
that $\alpha_{1}\cup\alpha_{2}$ is a bi-infinite geodesic. Then either
$\alpha_{1}$ or $\alpha_{2}$ has an empty intersection with the open ball
$B(x_{0},r/2)$.

The second lemma provides an upper bound on the divergence of $s_{1},s_{2}$
corners in $G_{2}$.

$e$$s_{2}^{n_{2}}$$s_{1}^{n_{1}}$$s_{2}^{i}$$\beta_{i}$$\beta_{1}$$\beta$$g_{1}$$g_{0}$$e_{1}$$e_{i}$$e_{i+1}$$e_{r}$$g_{0}$$g_{1}$$g_{1}$$g_{0}$$g_{0}$$\gamma$$\gamma^{\prime}$
Figure 1. The $r$–ball avoiding path (in blue) in the proof of Lemma 4.2

######  Lemma 4.2.

Let $r_{0}\geq 1$ be the constant in Remark 3.6. There is a positive constant
$N_{2}$ with the following property. For all integers $r\geq r_{0}$, there
exist paths in $\Gamma(G_{2},S_{2})$ connecting each of $s_{1}^{\pm r}$ to
each of $s_{2}^{\pm r}$ which lie outside of the open ball $B(e,r)$ and whose
lengths are bounded above by $N_{2}r(f(N_{2}r)+1)$.

###### Proof.

For concreteness, we connect $s_{1}^{n_{1}}$ to $s_{2}^{n_{2}}$ where
$|n_{1}|=|n_{2}|=r$ by paths avoiding the open ball $B(e,r)$ in the case
$n_{2}>0$. The path $\overline{\gamma}$ which avoids the $r$–ball is
constructed from three parts:
$\overline{\gamma}=\beta\cup\gamma^{\prime}\cup\gamma$. The reader can refer
to Figure 1 while reading this proof.

_The sub-path $\gamma^{\prime}$._ Let $D$ be the constant in Remark 3.6 and
let $m=4Dr$. By Lemma 3.8, there are group elements $g_{0}$ in $F_{xz}$ and
$g_{1}$ in $F_{yz}$ such that the following hold:

  1. (1)

$s_{2}^{-1}g_{0}s_{2}=g_{1}$;

  2. (2)

$m/D\leq|g_{0}|_{R_{xz}}\leq m$ and $m/D\leq|g_{1}|_{R_{yz}}\leq m$;

  3. (3)

There is a path $\gamma$ connecting $g_{0}$ and $g_{1}$ with the length at
most $Df(m)$ and each vertex in $\gamma$ is a group element $g_{0}h$ for $h\in
H$;

By Lemma 3.2 and Lemma 3.3 we have
$|g_{0}|_{R_{xz}}=|g_{0}|_{S_{1}}=|g_{0}|_{S_{2}}$ and
$|g_{1}|_{R_{yz}}=|g_{1}|_{S_{1}}=|g_{1}|_{S_{2}}$. Since $m/D=4r$, we have

$4r\leq|g_{0}|_{S_{2}}\leq 4Dr\text{ and }4r\leq|g_{1}|_{S_{2}}\leq 4Dr.$

Another application of Lemma 3.2 and Lemma 3.3 implies that the path $\gamma$
lies outside the open ball $B(e,4r)$ in the group $G_{2}$.

For $i\in\\{1,2,\cdots,r\\}$ let $x_{i}=s_{2}^{i-1}g_{0}$ and let
$y_{i}=s_{2}^{i}g_{1}$. Since $s_{2}^{-1}g_{0}s_{2}=g_{1}$, we have

$y_{i}=s_{2}^{i}g_{1}=s_{2}^{i-1}(s_{2}g_{1})=s_{2}^{i-1}(g_{0}s_{2})=(s_{2}^{i-1}g_{0})s_{2}=x_{i}s_{2}.$

Therefore, we can connect $x_{i}$ and $y_{i}$ by an edge $e_{i}$ labeled by
$s_{2}$. Since

$d_{S_{2}}(y_{i},s_{2}^{i})=|g_{1}|_{S_{2}}\geq 4r\text{ and
}d_{S_{2}}(e,s_{2}^{i})=i\leq r,$

the edge $e_{i}$ lies outside the open ball $B(e,3r-1)$ (therefore outside the
open ball $B(e,r)$).

Since $y_{i}=s_{2}^{i}g_{1}$ and $x_{i+1}=s_{2}^{i}g_{0}$ for each
$i\in\\{1,2,\cdots,i-1\\}$, the path $\beta_{i}=s_{2}^{i}\gamma$ (by which we
mean the left translate of the path $\gamma$ by $s_{2}^{i}$) lies outside the
open ball $B(s_{2}^{i},4r)$, connects $y_{i}$ and $x_{i+1}$, and has the
length at most $Df(m)$. Also, since $d_{S_{2}}(e,s_{2}^{i})=i\leq r$, each
path $\beta_{i}$ lies outside the open ball $B(e,3r)$ and hence outside of
$B(e,r)$. Let

$\gamma^{\prime}=(e_{1}\cup\beta_{1})\cup(e_{2}\cup\beta_{2})\cup\cdots\cup(e_{r-1}\cup\beta_{r-1})\cup
e_{r}.$

Then $\gamma^{\prime}$ is a path outside the open ball $B(e,r)$ that connect
$x_{1}=g_{0}$ and $y_{n_{2}}=s_{2}^{n_{2}}g_{1}$ and has the length at most
$r\bigl{(}Df(m)+1\bigr{)}$.

_The sub-path $\beta$._ Let $m_{1}=|g_{0}|_{R_{xz}}$. Then

$4r=m/D\leq m_{1}\leq m.$

Since $g_{0}$ is a group element in $F_{xz}$ and subgroup $F_{xz}\times\langle
s_{1}\rangle$ is isometrically is embedded into $G_{2}$, there is a path
$\beta$ outside the open ball $B(e,r)$ which connects $s_{1}^{n_{1}}$ and
$g_{0}$ and has length exactly $m_{1}+|n_{1}|$.

_The sub-path $\eta$._ Let $\eta$ be the path labeled by the reduced word in
$R_{yz}$ representing $g_{1}$ which connects $s_{2}^{n_{2}}$ and $y_{n_{2}}$.
Then the length of $\eta$ is exactly $|g_{1}|_{R_{yz}}\leq m$. We observe that
each vertex in $\eta$ is a group element $s_{2}^{n_{2}}u$ for some $u\in
F_{yz}$. Also the group automorphism $G_{2}\to{\mathbb{Z}}$ taking all the
$s_{2}$ to a generator of ${\mathbb{Z}}$ and all other generators of $G_{2}$
to the identity shows that $s_{2}^{n_{2}}u$ lies outside the open ball
$B(e,n_{2})$(=$B(e,r)$). In other words, $\eta$ lies outside the open ball
$B(e,r)$.

Let $\overline{\gamma}=\beta\cup\gamma^{\prime}\cup\eta$. Then
$\overline{\gamma}$ is a path outside the open ball $B(e,r)$ that connects
$s_{1}^{n_{1}}$ and $s_{2}^{n_{2}}$. Moreover,

$\displaystyle\ell(\overline{\gamma})$
$\displaystyle=\ell(\beta)+\ell(\gamma^{\prime})+\ell(\eta)$
$\displaystyle\leq(m_{1}+|n_{1}|)+r\bigl{(}Df(m)+1\bigr{)}+m$
$\displaystyle\leq(4Dr+r)+r\bigl{[}Df\bigl{(}4Dr\bigr{)}+1\bigr{]}+(4Dr).$

Since $f$ is a non-decreasing function, we can choose a number $N_{2}$
depending on $D$ such that the length of $\overline{\gamma}$ is at most
$N_{2}r\bigl{(}f(N_{2}r)+1\bigr{)}$.

∎

######  Lemma 4.3.

Let $n$ and $r$ be non-zero integers such that $|n|=r$ and let
$\epsilon\in\\{-1,+1\\}$. Then there is a path in $\Gamma(G_{2},S_{2})$ which
connects $s_{1}^{2n}$ and $s_{2}^{\epsilon}s_{1}^{2n}$, lies outside the open
ball $B(e,r)$, and has length at most $8r+1$.

###### Proof.

We prove the lemma for the case $\epsilon=1$; the proof for the case
$\epsilon=-1$ is similar. Since $s_{2}b_{1}^{2n}=a_{1}^{2n}s_{2}$, we can
connect $a_{1}^{2n}$ and $s_{2}b_{1}^{2n}$ by an edge $e_{1}$ labeled by
$s_{2}$ in $\Gamma(G_{2},S_{2})$. Since the $\mathbb{Z}^{2}$-group generated
by $a_{1}$ and $s_{1}$ is isometrically embedded into $G_{2}$, we can connect
$s_{1}^{2n}$ and $a_{1}^{2n}$ by a path $\alpha$ outside the open ball
$B(e,2r)$ with length at most $4|n|=4r$. Similarly, we can connect
$s_{1}^{2n}$ and $b_{1}^{2n}$ by a path $\beta_{0}$ outside the open ball
$B(e,2r)$ with length at most $4r$. Therefore, $\beta=s_{2}\beta_{0}$ is a
path outside the open ball $B(e,r)$ that connects $s_{2}s_{1}^{2n}$ and
$s_{2}b_{1}^{2n}$ and has the length at most $4r$. This implies that
$\gamma=\alpha\cup e_{1}\cup\beta$ is a path outside the open ball $B(e,r)$
that connects $s_{1}^{2n}$ and $s_{2}s_{1}^{2n}$ and has the length at most
$8r+1$. ∎

Say that a path is _open $r$–ball avoidant_ if it does not intersect the open
ball $B(e,r)$.

######  Lemma 4.4.

For each integer $m\geq 3$ there are constants $M_{m}$ and $N_{m}$ such that
the following two statements hold:

  * $(P_{m})$

Given integers $n$ and $r$ satisfying $|n|=r\geq r_{0}$ and
$\epsilon\in\\{-1,1\\}$, there exists an open $r$-ball avoidant path in
$\Gamma(G_{m},S_{m})$ connecting $s_{1}^{2n}$ and $s_{m}^{\epsilon}s_{1}^{2n}$
which has length at most $M_{m}r^{m-2}\bigl{(}f(M_{m}r)+1\bigr{)}$.

  * $(Q_{m})$

Given integers $n_{1}$, $n_{2}$ and $r$ satisfying $|n_{1}|=|n_{2}|=r\geq
r_{0}$, there exists an open $r$–ball avoidant path in $\Gamma(G_{m},S_{m})$
connecting $s_{1}^{n_{1}}$ and $s_{m}^{n_{2}}$ which has length at most
$N_{m}r^{m-1}\bigl{(}f(N_{m}r)+1\bigr{)}$.

$e$$s_{1}^{n}$$s_{m}^{n}$$s_{1}^{4n}$$s_{m}^{n}s_{1}^{4n}$$(P_{m})$
estimates$(P_{m})\Rightarrow(Q_{m})$$(Q_{m})\Rightarrow(P_{m+1})$$e$$s_{m+1}^{\epsilon}$$s_{1}^{2n}$$s_{m+1}^{\epsilon}s_{1}^{2n}$$(Q_{m})$
estimate Figure 2. Inductive steps in proof of Lemma 4.4

###### Proof.

We prove the above lemma by induction on $m$ using an analogous strategy as in
the proof of Lemma 4.2 in [BT]. We first prove Statement $P_{3}$. Then we
prove the implication $(P_{m})\Rightarrow(Q_{m})$ for each $m\geq 3$. Finally,
we prove the implication $(Q_{m})\Rightarrow(P_{m+1})$ for each $m\geq 3$.

The proofs of the implications $(P_{m})\Rightarrow(Q_{m})$ for each $m\geq 3$
and $(Q_{m})\Rightarrow(P_{m+1})$ for each $m\geq 3$ are analogous to the
proofs of the corresponding implications in Lemma 4.2 in [BT] and we leave
these to the reader. The diagrams in Figure 2 are included to facilitate the
translation of the proofs of the corresponding statements in Lemma 4.2 of [BT]
to the current setting.

We now prove Statement $(P_{3})$ for the case of $\epsilon=1$; the proof for
the case of $\epsilon=-1$ is very similar. Since
$s_{3}s_{2}^{2n}=s_{1}^{2n}s_{3}$, we can connect $s_{1}^{2n}$ and
$s_{3}s_{2}^{2n}$ by an edge $e_{1}$ outside the open ball $B(e,r)$. By Lemma
4.2 there is a path $\gamma_{0}$ outside the open ball $B(e,2r)$ that connects
$s_{2}^{2n}$ and $s_{1}^{2n}$ and has length at most
$2N_{2}r\bigl{(}f(2N_{2}r)+1\bigr{)}$. Therefore, $\gamma_{1}=s_{3}\gamma_{0}$
is a path outside the open ball $B(e,r)$ that connects $s_{3}s_{2}^{2n}$ and
$s_{3}s_{1}^{2n}$ and has length at most
$2N_{2}r\bigl{(}f(2N_{2}r)+1\bigr{)}$. This implies that
$\gamma=e_{1}\cup\gamma_{1}$ is a path outside the open ball $B(e,r)$ that
connects $s_{1}^{2n}$ and $s_{3}s_{1}^{2n}$ and has length at most
$2N_{2}r\bigl{(}f(2N_{2}r)+1\bigr{)}+1$. Therefore, $M_{3}=2N_{2}+1$ is a
desired constant. ∎

The following two propositions give the upper bounds for the divergence of the
group $G_{m}$ for $m\geq 2$. The proof of these propositions is analogous to
the proof of Proposition 4.3 and Proposition 4.4 in [BT]. We only need to
replace Lemma 3.2, Lemma 3.7, Lemma 4.1, and Lemma 4.2 in [BT] by Lemma 3.3,
Lemma 3.9, Lemma 4.1, Lemma 4.3, and Lemma 4.4. We note that the linear upper
bound in Lemma 4.3 in this paper is different from the sublinear upper bound
of Statement ($P_{2}$) of Lemma 4.2 in [BT]. However, this difference does not
affect the proof of Proposition 4.5 and Proposition 4.6 in this paper.

The following proposition establishes the upper bound for the divergence of
the group $G_{m}$ for $m\geq 3$.

######  Proposition 4.5 (Upper bound for $\operatorname{Div}_{G_{m}}$, $m\geq
3$).

Let $\\{\delta_{\rho}\\}$ be the divergence of $\Gamma(G_{m},S_{m})$ for
$m\geq 3$. Then

$\delta_{\rho}(r)\preceq r^{m-1}f(r)$

for each $\rho\in(0,1/2]$.

$e$$s_{1}^{r}$$s_{1}^{-r}$$s_{1}^{4r}$Length $\leq 4r$$x$$xs_{1}^{4r}$Bounded
by Lemma 4.4 Figure 3. Divergence upper bound in the case that the positive
$s_{1}$-ray from $x$ avoids the open $r/2$–ball.

###### Proof.

We will prove that there is a constant $A_{m}$ such that we have

$\delta_{\rho}(r)\leq A_{m}r^{m-1}\bigl{(}f(A_{m}r)+1\bigr{)}$

for each $\rho\in(0,1/2]$ and each $r\geq r_{0}$. Here $r_{0}$ is the constant
from Remark 3.6.

We can assume that $r$ is an integer. By Lemma 3.9, there is a path which
connects $s_{1}^{-r}$ and $s_{1}^{r}$, lies outside the open ball $B(e,r)$,
and has length at most $4r$. It suffices to show there is a constant $B_{m}$
depending only on $m$ such that for each point $x$ on the sphere $S(e,r)$ we
can either connect $x$ to $s_{1}^{r}$ or connect $x$ to $s_{1}^{-r}$ by a path
outside the open ball $B(e,r/2)$ with the length at most
$B_{m}r^{m-1}\bigl{(}f(B_{m}r)+1\bigr{)}$. Now the proposition follows
choosing a suitable constant $A_{m}\geq 2B_{m}+4$.

Now we establish the $B_{m}r^{m-1}\bigl{(}f(B_{m}r)+1\bigr{)}$ bound. Let
$\alpha$ be a bi-infinite geodesic which contains $x$ and has edges labeled by
$s_{1}$. Then $\alpha$ is the union of two rays $\alpha_{1}$ and $\alpha_{2}$
that share the initial point $x$. Assume that $\alpha_{1}$ traces each edge of
$\alpha$ in the positive direction and $\alpha_{2}$ traces each edge of
$\alpha$ in the negative direction. By Lemma 4.1, either $\alpha_{1}$ or
$\alpha_{2}$ (say $\alpha_{1}$) lies outside the open ball $B(e,r/2)$. We now
construct a path $\eta$ which connects $x$ and $s_{1}^{r}$, lies outside the
open ball $B(e,r/2)$, and has the length at most
$B_{m}r^{m-1}\bigl{(}f(B_{m}r)+1\bigr{)}$. We note that we can use an
analogous argument to construct a similar path connecting $x$ and $s_{1}^{-r}$
in the case $\alpha_{2}$ lies outside the open ball $B(e,r/2)$.

Constructing the comb and connecting endpoints of successive teeth. First, we
connect $s_{1}^{r}$ and $s_{1}^{4r}$ by the geodesic $\eta_{0}$ labeled by
$s_{1}^{3r}$. Then $\eta_{0}$ lies outside the open ball $B(e,r)$ and has
length exactly $3r$. Since $|x|_{S_{m}}=r$, we can write $x=u_{1}u_{2}\cdots
u_{r-1}u_{r}$ where $u_{i}\in S_{m}\cup S_{m}^{-1}$. Let $R_{m}\geq 17$ be a
constant which is greater than $4^{m}$ times of each constant $M_{j}$ for
$3\leq j\leq m$ in Lemma 4.4. By Lemma 3.3, Lemma 3.9, Lemma 4.3, and Lemma
4.4 we can connect $s_{1}^{4r}$ and $u_{1}s_{1}^{4r}$ by a path $\eta_{1}$
which lies outside the open ball $B(e,2r)$ and has length at most
$R_{m}r^{m-2}\bigl{(}f(R_{m}r)+1\bigr{)}$.

Similarly for $2\leq i\leq r$ we can also connect $(u_{1}u_{2}\cdots
u_{i-1})s_{1}^{4r}$ and $(u_{1}u_{2}\cdots u_{i})s_{1}^{4r}$ by a path
$\eta_{i}$ which lies outside the open ball $B(u_{1}u_{2}\cdots u_{i-1},2r)$
and has length at most $R_{m}r^{m-2}\bigl{(}f(R_{m}r)+1\bigr{)}$. Finally, we
connect $xs_{1}^{4r}$ and $x$ using the subsegment $\eta_{r+1}$ of
$\alpha_{1}$. Then $\eta_{r+1}$ lies outside the open ball $B(e,r/2)$ and has
length exactly $4r$.

Let
$\eta=\eta_{0}\cup\eta_{1}\cup\eta_{2}\cup\cdots\cup\eta_{r}\cup\eta_{r+1}$.
Then the path $\eta$ connects $s_{1}^{r}$ and $x$, lies outside the open ball
$B(e,r/2)$, and has length at most
$R_{m}r^{m-1}\bigl{(}f(R_{m}r)+1\bigr{)}+7r$. Since $f$ is a non-decreasing
function, the length of $\eta$ is bounded above by
$B_{m}r^{m-1}\bigl{(}f(B_{m}r)+1\bigr{)}$ by some appropriate choice of
$B_{m}\geq R_{m}+7$. ∎

The following proposition gives the quadratic upper bound for the divergence
of the group $G_{2}$. The proof of this proposition proceeds exactly as the
proof of Proposition 4.5 and we leave it to the reader. The main difference is
that we only need the upper bounds $16r+1$ on lengths on paths avoiding
$B(e,2r)$ which connect $s_{1}^{4r}$ with $s_{2}s_{1}^{4r}$ or
$s_{2}^{-1}s_{1}^{4r}$ from Lemma 4.3. The other length estimates are
$s$-edges outside $B(e,2r)$ which connect $s_{1}^{4r}$ with $ss_{1}^{4r}$ for
$s$ a generator of $G_{1}$ (see Lemma 3.9).

######  Proposition 4.6 (Quadratic upper bound for
$\operatorname{Div}_{G_{2}}$).

Let $\\{\delta_{\rho}\\}$ be the divergence of $\Gamma(G_{2},S_{2})$. Then

$\delta_{\rho}(r)\preceq r^{2}$

for each $\rho\in(0,1/2]$.

Finally, we note that the groups $G_{m}$ are one-ended. The proof of this
statement is exactly as the proof of Proposition 4.5 in [BT].

######  Proposition 4.7.

For each $m\geq 1$ the group $G_{m}$ is one-ended.

###### Proof.

$G_{1}$ is one-ended because it is a direct product of two infinite groups.
For $m\geq 2$ note that (by retracting along geodesic rays based at $e$) it is
sufficient to prove that any two points on the sphere $S(e,r)$ can be
connected by a path avoiding $B(e,r/2)$ for large integers $r$. In the proof
of Proposition 4.5 and Proposition 4.6 we construct paths which avoid
$B(e,r/2)$ and which connect an arbitrary $x\in S(e,r)$ to one of $s_{1}^{r}$
or $s_{1}^{-r}$, and we can connect $s_{1}^{r}$ to $s_{1}^{-r}$ by a path
outside of $B(e,r/2)$ in the $a_{1}s_{1}$-plane. Therefore, the group $G_{m}$
is one-ended. ∎

###  4.2. The lower bound

####  4.2.1. Lower bound strategy

In this subsection, we prove the lower bound for the group divergence of
$G_{m}$. The idea is to work by induction on $m$ using the fact that $G_{i}$
isometrically embeds into $G_{i+1}$. However, one has to proceed carefully.
There are more paths connecting points in the complement of the open $r$–ball
in $G_{i+1}$ than in the complement of the open $r$–ball in $G_{i}$, and so
lower bounds on path lengths in the latter do not _a priori_ imply the same
bounds on path lengths in the former. Furthermore, there are more points in
the $r$–sphere in $G_{i+1}$ than in the $r$–sphere in $G_{i}$.

We shall establish lower bound estimates for the lengths of open $r$–ball
avoidant paths connecting two points on the $r$–sphere. These lower bound
estimates are obtained by cutting these paths into pieces using
$s_{m}$-hyperplanes and estimating the lengths of these pieces. We work with a
$2$–dimensional geometric model $X_{m}$ for $G_{m}$.

####  4.2.2. Lower bound details

Let $Y_{1}$ be the standard presentation $2$–complex for the group $G_{1}$
with the finite generating set $S_{1}$. We construct 2-complexes $Y_{m}$ for
the groups $G_{m}$ inductively as follows.

The complex $Y_{2}$ is the total space of the graph of spaces based on a
circle with one vertex. The vertex space is $Y_{1}$ and the edge space is
$\Sigma_{2}$, a bouquet of oriented circles labeled $e_{i}$ for $1\leq i\leq
p$. There are two maps $\Sigma_{2}\to Y_{1}$, one sending $e_{i}$ to $a_{i}$
for $1\leq i\leq p$ and the other sending $e_{i}$ to $b_{i}$ for $1\leq i\leq
p$. The total space of this graph of spaces (as defined in [SW79])

$Y_{2}\;=\;[Y_{1}\,\cup\,(\Sigma_{2}\times[0,1])]/\sim$

has one extra 1-cell which we label $s_{2}$ and 2-cells with boundary words
$s_{2}^{-1}a_{i}s_{2}b_{i}^{-1}$ for $1\leq i\leq p$.

For $m\geq 3$ the complex $Y_{m}$ is the total space of the graph of spaces
based on a circle with one vertex. The vertex space is $Y_{m-1}$ and the edge
space is $\Sigma_{m}$, an oriented circle labeled $e$. There are two maps
$\Sigma_{m}\to Y_{m-1}$, one sending $e$ to $s_{1}$ and the other sending $e$
to $s_{m-1}$. The total space

$Y_{m}\;=\;[Y_{m-1}\,\cup\,(\Sigma_{m}\times[0,1])]/\sim$

has one extra 1-cell which we label $s_{m}$ and a 2-cell with boundary word
$s_{m}^{-1}s_{1}s_{m}s_{m-1}^{-1}$.

Throughout this section, $X_{m}$ denotes the universal cover of $Y_{m}$. We
define the notions of $s_{m}$-hyperplane, $k$–rays and $(1,k)$–rays,
$k$–corners, and $r$–avoidant paths over corners in $X_{m}$. These play a
fundamental role in the proof of the lower bounds.

######  Definition 4.8 ($s_{m}$-hyperplanes).

Let $\widetilde{\Sigma}_{m}$ denote the universal cover of the circle
$\Sigma_{m}$ consisting of one 0–cell and one 1–cell. The cell structure of
$\widetilde{\Sigma}_{m}$ agrees with the standard 1–dimensional cubing of
$\mathbb{R}$ with 0–cells at $\mathbb{Z}$. The space
$\widetilde{\Sigma}_{m}\times[0,1]$ has the product square structure. The map

$\widetilde{\Sigma}_{m}\times[0,1]\,\to\,\Sigma_{m}\times[0,1]\,\to\,[Y_{m-1}\,\cup\,(\Sigma_{m}\times[0,1])]/\sim\,\;=\;Y_{m}$

lifts to the universal cover $X_{m}$. These lifts are indexed by the edges of
the Bass-Serre tree of the HNN description of $G_{m}$. Each image of
$\widetilde{\Sigma}_{m}\times\\{1/2\\}$ in $X_{m}$ is called an _$s_{m}$
-hyperplane_. We say that the $s_{m}$-hyperplane, $H$, is _dual_ to the edges
of $X_{m}$ which it intersects; these edges are all labeled $s_{m}$. Note that
each $s_{m}$-hyperplane $H$ separates $X_{m}$ into two components. We will
make use of open and closed star neighborhoods of $H$ in the cell structure on
$X_{m}$. Note that $star(H)$ (resp. $ostar(H)$) is the image of
$\widetilde{\Sigma}_{m}\times[0,1]$ (resp.
$\widetilde{\Sigma}_{m}\times(0,1)$) in $X_{m}$.

######  Definition 4.9 ($k$–rays and $(1,k)$–rays).

Let $m\geq 1$ be an integer. A geodesic ray $\alpha$ in the complex $X_{m}$ is
a _$k$ –ray_ for $1\leq k\leq m$ if all edges of $\alpha$ are labeled by
$s_{k}$. A geodesic ray $\beta$ in the complex $X_{m}$ is a _$(1,k)$ –ray_ for
$2\leq k\leq m$ if one of the following holds:

  1. (1)

$\beta$ is a $1$–ray;

  2. (2)

$\beta$ is a concatenation of $\sigma_{1}\sigma_{2}$, where $\sigma_{1}$ is a
non-degenerate segment with edges labeled by $s_{1}$ and $\sigma_{2}$ is a
$k$–ray.

Other than the fact that the rays are geodesics, we do not make any
assumptions on the orientations of edges in $k$–rays or in $(1,k)$–rays.

We define $k$–corners for $k\geq 2$ as follows.

######  Definition 4.10 ($k$–corner for $k\geq 2$).

Let $2\leq k\leq m$ be integers. Let $\alpha$ be a $(1,k)$–ray and $\beta$ be
a $k$–ray in the complex $X_{m}$ which share the same initial point $x$. Then
$(\alpha,\beta)_{x}$ is called a _$k$ –corner_ at $x$.

The following definition recalls Macura’s notion of detour paths over corners
in [Mac13].

######  Definition 4.11 (An $r$–avoidant path over a $k$–corner).

Let $2\leq k\leq m$ be integers and let $(\alpha,\beta)_{x}$ be a $k$–corner
in $X_{m}$. For each $r>0$ a path which lies outside the open ball $B(x,r)$ in
$X_{m}$ connecting a vertex of $\alpha$ to a vertex of $\beta$ is called an
_$r$ –avoidant path over the $k$–corner $(\alpha,\beta)_{x}$_ in $X_{m}$.

Next, we investigate how hyperplanes intersect corners. We start with a basic
result about the intersection of hyperplanes and rays.

######  Lemma 4.12.

Let $k\geq 2$ and $H$ be an $s_{k}$-hyperplane in $X_{k}$. Then $H$ intersects
each $k$–ray at most once.

###### Proof.

Suppose $H$ intersects a $k$–ray at two points. Then there is a path in
$star(H)-ostar(H)$ which connects two vertices of the $k$–ray. In the case
$k\geq 3$, this means that a non-zero power of $s_{k}$ is equal to a power of
either $s_{1}$ or $s_{k-1}$. This contradicts Lemma 3.4. In the case $k=2$,
this intersection implies that a non-zero power of $s_{2}$ is equal to an
element of the free group generated by $\\{a_{1},\ldots,a_{p}\\}$ or
$\\{b_{1},\ldots,b_{p}\\}$. But this is impossible because these two elements
have distinct normal forms in the HNN extension structure of $G_{2}$. ∎

The following lemma is analogous to Lemma 4.11 in [BT]. It states that an
$s_{k}$-hyperplane can only intersect a $k$–corner in $X_{k}$ in a very
restricted way.

######  Lemma 4.13.

Let $k\geq 3$ and $(\alpha,\beta)_{x}$ be a $k$–corner in $X_{k}$. If $H$ is
an $s_{k}$-hyperplane which intersects both $\alpha$ and $\beta$, then $H$
intersects the first edge of $\beta$.

###### Proof.

Let $i$ be the smallest positive integer such that the $s_{k}$-hyperplane
$H_{i}$ dual to the $i^{th}$ edge of $\beta$ intersects $\alpha$. We observe
that if $i\geq 2$, the $s_{k}$-hyperplane $H_{i-1}$ dual to the $(i-1)^{th}$
edge of $\beta$ cannot intersect $\alpha$ by the choice of $i$ and cannot
intersect $\beta$ twice by Lemma 4.12, and so must intersect $H_{i}$ which
contradicts the fact that $s_{k}$-hyperplanes are disjoint. Therefore, $i$
must equal $1$.

Assume that some hyperplane $H_{j}$ dual to the $j^{th}$ edge of $\beta$ for
$j\geq 2$ intersects $\alpha$. Arguing as in the preceding paragraph, we see
that the hyperplane $H_{2}$ dual to the second edge of $\beta$ must intersect
$\alpha$. We note that for $i=1,2$ each $star(H_{i})-ostar(H_{i})$ consists of
two bi-infinite geodesics: one has edges labeled by $s_{1}$ and the other has
edges labeled by $s_{k-1}$. This gives a loop based at the vertex between the
first and second edge of $\beta$ which is labeled by
$s_{k-1}^{n}s_{k}^{p}s_{1}^{m}$ for $m\neq 0$ and $n\neq 0$. This contradicts
Lemma 3.5.

This implies that all the $s_{k}$-hyperplanes dual to the second or subsequent
edges of $\beta$ cannot meet $\alpha$, and the lemma is proved. ∎

######  Remark 4.14.

One can prove that the $s_{k}$-hyperplane $H$ in the previous lemma must
intersect the ray $\alpha$ in the first occurrence of the edge $s_{k}$.
However, we do not need this feature of $H$.

The intersection of $s_{2}$-hyperplanes with $2$–corners in $X_{2}$ is even
more restricted.

######  Lemma 4.15.

Let $(\alpha,\beta)_{x}$ be a $2$–corner in $X_{2}$. There is no
$s_{2}$-hyperplane in $X_{2}$ that intersects both $\alpha$ and $\beta$.

###### Proof.

Assume that some hyperplane $H$ dual to an edge of $\beta$ intersects
$\alpha$. We note that $star(H)-ostar(H)$ consists of two infinite trees; one
with edge labels from $\\{a_{1},\ldots,a_{p}\\}$ and the other with edge
labels from $\\{b_{1},\ldots,b_{p}\\}$. Thus there is a loop based at a point
in $(star(H)-ostar(H))\cap\beta$ which is labeled by
$s_{2}^{m}s_{1}^{n}s_{2}^{q}w$ where $n\neq 0$ and $w$ is a word representing
a group element $g_{0}$ in $F_{xz}$. This implies that
$s_{2}^{m}s_{1}^{n}s_{2}^{q}g_{0}=e$. We observe that there is a group
automorphism $\Psi:\\!G_{2}\to\mathbb{Z}$ taking $s_{1}$ to $1$ and other
generators of $G_{2}$ to $0$. Since
$\Psi(s_{2}^{m}s_{1}^{n}s_{2}^{q}g_{0})=n\neq 0$, we have
$s_{2}^{m}s_{1}^{n}s_{2}^{q}g_{0}\neq e$ which is a contradiction. Therefore,
there is no hyperplane in $X_{2}$ that intersects both $\alpha$ and $\beta$. ∎

The following lemma is analogous to Lemma 4.13 in [BT]. It states that
$s_{1}$-geodesic segments can be modified relative to their endpoints to
become ball-avoidant paths with linearly bounded lengths. The modifications
can be made in the the appropriate coset of the isometrically embedded rank-2
free abelian group $\langle a_{1},s_{1}\rangle$ in $G_{m}$. The reader may
refer to [BT, Lemma 4.13] for the proof.

######  Lemma 4.16.

Let $m\geq 1$ and $\alpha$ be a geodesic segment in $X_{m}$ with edges labeled
by $s_{1}$ and endpoints $x$, $y$. Let $z$ be a vertex of $X_{m}$ and assume
that $r=\min\\{d_{S_{m}}(x,z),d_{S_{m}}(y,z)\\}>0$. Then there exists a path
$\beta$ connecting $x$ and $y$ with edges labeled by $s_{1}$ and $a_{1}$ such
that $\beta$ lies outside the open ball $B(z,r/2)$ and $\ell(\beta)\leq
11\ell(\alpha)$.

The goal of the next three lemmas is to establish lower bounds of
$r^{n-1}f(r)$ on the lengths of $r$–avoidant paths over $n$–corners in
$X_{n}$. The precise bound is given in Lemma 4.19. We prove this bound by
proving a stronger statement: namely, $r^{n-1}f(r)$ is a lower bound for the
lengths of $r$–avoidant paths over $n$–corners in the larger space $X_{n+1}$.
This is achieved by proving the following more general version of the latter
statement by induction on $d$:

> For each $n\geq d$ the lengths of $r$–avoidant paths over $n$–corners in
> $X_{n+1}$ are bounded below by $r^{d-1}f(r)$.

Note that setting $d=n$ yields the strong statement above. The induction step
is proven in Lemma 4.18. The base case of the induction breaks into two parts;
one for $2$–corners in $X_{3}$ and the other for $m$–corners in $X_{m+1}$ when
$m\geq 3$.

######  Lemma 4.17 (Base case).

For each integer $m\geq 2$ the length of $r$–avoidant paths over $m$–corners
in $X_{m+1}$ is at least $(r/16)f(r/4)$ for $r$ sufficiently large.

$s_{m}^{j}$$\beta_{j}$$\alpha_{j}$$u^{\prime}_{j+1}$$u_{j+1}$$v^{\prime}_{j}$$v_{j}$$\beta$$\alpha$$\tau_{\ell(j)}$$\tau_{\ell(j+1)}$$\eta_{1}$$\eta_{2}$
Figure 4. The paths $\alpha_{j}$ and $\beta_{j}$ determined by consecutive
$s_{m}$-hyperplanes.

###### Proof.

Suppose that $\gamma$ is an $r$–avoidant path of minimal length over all
$m$–corners in $X_{m+1}$. By left translating if necessary, assume that
$\gamma$ is $r$–avoidant over an $m$–corner $(\alpha,\beta)_{e}$ based at the
identity. Now, $\gamma\cap\alpha=u$ is a vertex of the form
$s_{1}^{s}s_{m}^{t}$ for some integers $s\neq 0$ and $t$ such that
$|s|+|t|\geq r$ and $\gamma\cap\beta=v$ is a vertex of the form $s_{m}^{p}$
for some integer $p$ such that $|p|\geq r$. We assume that $p>0$; the proof
for $p<0$ is similar.

The intersection $\gamma\cap X_{m}$ is a union of closed, connected sub-paths
of $\gamma$ some of which may be single vertices. These sub-paths together
with the closures of components of $\gamma-X_{m}$ mean that $\gamma$ can be
written as a concatenation

$\sigma_{1}\tau_{1}\sigma_{2}\tau_{2}\cdots\sigma_{\ell}\tau_{\ell}\sigma_{\ell+1}$

such that:

  1. (1)

Each $\tau_{i}$ intersects $X_{m}$ only at its endpoints
$x_{i}=\sigma_{i}\cap\tau_{i}$ and $y_{i}=\tau_{i}\cap\sigma_{i+1}$;

  2. (2)

Each $\sigma_{i}$ lies completely in the $1$–skeleton of $X_{m}$.

We observe that each $x_{i}^{-1}y_{i}$ is a group element in the cyclic
subgroup $\langle s_{m}\rangle$ or in the cyclic subgroup $\langle
s_{1}\rangle$. If $x_{i}^{-1}y_{i}$ is a group element in the cyclic subgroup
$\langle s_{m}\rangle$, then we replace $\tau_{i}$ by a geodesic
$\tau^{\prime}_{i}$ labeled by $s_{m}$. Otherwise, $x_{i}^{-1}y_{i}$ is a
group element in the cyclic subgroup $\langle s_{1}\rangle$, then we replace
$\tau_{i}$ by a geodesic $\tau^{\prime}_{i}$ labeled by $s_{1}$. The new path
$\gamma^{\prime}=\sigma_{1}\tau^{\prime}_{1}\sigma_{2}\tau^{\prime}_{2}\cdots\sigma_{\ell}\tau^{\prime}_{\ell}\sigma_{\ell+1}$
lies completely in the $1$–skeleton of $X_{m}$, has the same initial and end
points as $\gamma$, and $\ell(\gamma^{\prime})\leq\ell(\gamma)$. Note that the
new path $\gamma^{\prime}$ may intersect the open ball $B(e,r)$. We call each
subpath $\tau^{\prime}_{i}$ of $\gamma^{\prime}$ a short-cut segment.

We claim that $\gamma^{\prime}\cap\beta$ does not contain an edge. Assume to
the contrary that $\gamma^{\prime}\cap\beta$ contains an edge. Then some
short-cut segment $\tau^{\prime}_{i}$ of $\gamma^{\prime}$ with edges labeled
by $s_{m}$ must contain an edge of $\beta$. Therefore, the endpoint $x_{i}$ of
$\tau^{\prime}_{i}$ has the form $s_{m}^{q}$ for some $|q|\geq r$. This
implies that the subpath $\zeta$ of $\gamma$ connecting $u$ and $x_{i}$ is
also an $r$–avoidant path over an $m$–corner in the complex $X_{m+1}$. Also,
$|\zeta|<|\gamma|$ which contradicts to the minimality of $\gamma$.

For each positive integer $j$ we call $e_{j}$ the $j^{th}$ edge of $\beta$. We
know that all edges $e_{j}$ are labeled by $s_{m}$. Let $H_{j}$ be the
$s_{m}$-hyperplane of the complex $X_{m}$ dual to the edge $e_{j}$. We now
consider the edges $e_{j}$ for $r/8\leq j\leq r/4$. Assume that $r\geq 16$ so
that $j\geq 2$. Therefore, by Lemma 4.13 (in the case $m\geq 3$) or Lemma 4.15
(in the case $m=2$) each hyperplane $H_{j}$ will not intersect $\alpha$ and so
must intersect $\gamma^{\prime}$. Let $m_{j}$ be the point in the intersection
$H_{j}\cap\gamma^{\prime}$ such that the subpath of $\gamma^{\prime}$
connecting $m_{j}$ and $v$ does not intersect $H_{j}$ at point other than
$m_{j}$. Then $m_{j}$ is an interior point of an edge $f_{j}$ labeled by
$s_{m}$ in $\gamma^{\prime}$. Since $\gamma^{\prime}\cap\beta$ does not
contain an edge, two edges $e_{j}$ and $f_{j}$ are distinct.

Let $\beta_{j}$ be the path in $star(H_{j})-ostar(H_{j})$ that connects the
terminal $s_{m}^{j}$ of the edge $e_{j}$ to some endpoint $v^{\prime}_{j}$ of
the edge $f_{j}$. Then edges of $\beta_{j}$ are labeled by $s_{m-1}$ (in the
case $m\geq 3$) or by generators of the free group $F_{yz}$ (in the case
$m=2$). If $f_{j}$ is not an edge of a short-cut segment of $\gamma^{\prime}$,
then we let $v_{j}=v^{\prime}_{j}$. Otherwise, $f_{j}$ is an edge in some
short-cut segment $\tau^{\prime}_{\ell_{j}}$ of $\gamma^{\prime}$. In this
case, we let $v_{j}$ be the endpoint $y_{\ell_{j}}$ of
$\tau^{\prime}_{\ell_{j}}$. Let $\alpha_{j}$ be the path in
$star(H_{j+1})-ostar(H_{j+1})$ that connects the initial endpoint $s_{m}^{j}$
of the edge $e_{j+1}$ to some endpoint $u^{\prime}_{j+1}$ of the edge
$f_{j+1}$. Then edges of $\alpha_{j}$ are labeled by $s_{1}$ (in the case
$m\geq 3$) or by generators of $F_{xz}$ (in the case $m=2$). If $f_{j+1}$ is
not an edge of a short-cut segment of $\gamma^{\prime}$, then we let
$u_{j+1}=u^{\prime}_{j+1}$. Otherwise, $f_{j+1}$ is an edge in some
$\tau^{\prime}_{\ell_{j+1}}$. In this case, we let $u_{j+1}$ be the endpoint
$x_{\ell_{j+1}}$ of $\tau^{\prime}_{\ell_{j+1}}$. We refer the reader to
Figure 4 to help visualize the construction.

Let $\gamma^{\prime}_{j}$ be the subpath of $\gamma^{\prime}$ that connects
$v^{\prime}_{j}$ and $u^{\prime}_{j+1}$. We will prove that the length of
$\gamma^{\prime}_{j}$ is at least $f(r/4)$. By the construction, the path
$\gamma^{\prime}_{j}$ contains two subpaths $\eta_{1}$ and $\eta_{2}$ of
$\gamma^{\prime}$, where $\eta_{1}$ (resp. $\eta_{2}$) is the (possibly
degenerate) subsegment of $\gamma^{\prime}_{j}$ connecting $v^{\prime}_{j}$
and $v_{j}$ (resp. $u^{\prime}_{j+1}$ and $u_{j+1}$). If either the length of
$\eta_{1}$ or the length of $\eta_{2}$ is greater than $r/4$, then the length
of $\gamma^{\prime}_{j}$ is greater than $r/4$ and therefore greater than
$f(r/4)$.

We now assume that the lengths of $\eta_{1}$ and $\eta_{2}$ are both less than
or equal to $r/4$. In this case

$d_{S_{m}}(v^{\prime}_{j},v_{j})\leq r/4\text{ and
}d_{S_{m}}(u^{\prime}_{j+1},u_{j+1})\leq r/4.$

Also, $v_{j}$ and $u_{j+1}$ lies outside the open ball $B(e,r)$. Then
$v^{\prime}_{j}$ and $u^{\prime}_{j+1}$ lies outside the open ball
$B(e,3r/4)$. At this point the proof splits into two cases; the case $m\geq 3$
and the case $m=2$.

_Case $m\geq 3$._ In this case, we note that
$v^{\prime}_{j}=s_{m}^{j}s_{m-1}^{q_{1}}$ and
$u^{\prime}_{j+1}=s_{m}^{j}s_{1}^{q_{2}}$ for some integers $q_{1}$ and
$q_{2}$. Moreover, $d_{S_{m}}(e,s_{m}^{j})=j\leq r/4$. Therefore, we have
$|q_{1}|\geq r/2$ and $|q_{2}|\geq r/2$. This implies that

$\ell(\gamma^{\prime}_{j})\geq
d_{S_{m}}(v^{\prime}_{j},u^{\prime}_{j+1})=|s_{m-1}^{-q_{1}}s_{1}^{q_{2}}|_{S_{m}}=|q_{1}|+|q_{2}|\geq
r\geq r/4\geq f(r/4).$

_Case $m=2$._ In this case, we note that $v^{\prime}_{j}=s_{2}^{j}g_{1}$ for
some $g_{1}\in F_{yz}$ and $u^{\prime}_{j+1}=s_{2}^{j}g_{0}$ for some
$g_{0}\in F_{xz}$. Moreover, $d_{S_{2}}(e,s_{2}^{j})=j\leq r/2$. Therefore, we
have $|g_{0}|_{S_{2}}\geq r/4$ and $|g_{1}|_{S_{2}}\geq r/4$. By Lemma 3.3 and
Lemma 3.7 we have

$\ell(\gamma^{\prime}_{j})\geq
d_{S_{2}}(v^{\prime}_{j},u^{\prime}_{j+1})=|g_{1}^{-1}g_{0}|_{S_{2}}\geq
f(r/4).$

Therefore, in both the $m=2$ and $m\geq 3$ cases we have

$\ell(\gamma)\geq\ell(\gamma^{\prime})\geq\sum_{r/8\leq j\leq
r/4}\ell(\gamma^{\prime}_{j})\geq(r/16)f(r/4).$

∎

The following lemma is analogous to Proposition 4.14 in [BT]. The key
technical ingredients Lemma 4.11 and Lemma 4.13 used in the proof of
Proposition 4.14 in [BT] are replaced in this paper by Lemma 4.13 and Lemma
4.16 respectively. We give the full details of the proof of Lemma 4.18 for the
convenience of the reader.

######  Lemma 4.18 (The induction step).

Let $d\geq 2$ be an integer and $g\\!:(0,\infty)\to(0,\infty)$ be a function.
Assume that for all $m\geq d$ all $r$–avoidant paths over an $m$–corner in
$X_{m+1}$ have length at least $g(r)$ for $r$ sufficiently large. Then for all
$n\geq d+1$ all $r$–avoidant paths over an $n$–corner in the complex $X_{n+1}$
have length at least $(r/180)g(r/4)$ for $r$ sufficiently large.

###### Proof.

Let $n\geq d+1$ and let $\gamma$ be an $r$–avoidant path over an $n$–corner
$(\alpha,\beta)_{x}$ in the complex $X_{n+1}$. We assume $\gamma$ is an
$r$–avoidant path of minimal length over all $n$–corners in $X_{n+1}$. This
implies that $\gamma\cap\alpha$ is a vertex $u$ and $\gamma\cap\beta$ is a
vertex $v$. By translating the corner if necessary, we assume that $x$ is the
identity $e$. Therefore, the vertex $u$ has the form $s_{1}^{s}s_{n}^{t}$ for
some integers $s\neq 0$ and $t$ such that $|s|+|t|\geq r$. Similarly, the
vertex $v$ has the form $s_{n}^{p}$ for some integer $p$ such that $|p|\geq
r$. We give the proof in the case $p>0$; the proof for $p<0$ is similar.

As in the proof of Lemma 4.17, $\gamma$ can be written as a concatenation

$\sigma_{1}\tau_{1}\sigma_{2}\tau_{2}\cdots\sigma_{\ell}\tau_{\ell}\sigma_{\ell+1}$

such that:

  1. (1)

Each $\tau_{i}$ intersects $X_{n}$ only at its endpoints
$x_{i}=\sigma_{i}\cap\tau_{i}$ and $y_{i}=\tau_{i}\cap\sigma_{i+1}$;

  2. (2)

Each $\sigma_{i}$ lies completely in the $1$–skeleton of $X_{n}$.

We also observe that $x_{i}^{-1}y_{i}$ is a group element in the cyclic
subgroup $\langle s_{n}\rangle$ or in the cyclic subgroup $\langle
s_{1}\rangle$ for each $i$.

If $x_{i}^{-1}y_{i}$ is a group element in the cyclic subgroup $\langle
s_{n}\rangle$, then we replace $\tau_{i}$ by a geodesic $\tau^{\prime}_{i}$
labeled by $s_{n}$. Call this path $\tau^{\prime}_{i}$ an $X_{n}$-replacement
segment of type 1. Note that $\tau^{\prime}_{i}$ in this case may have non-
empty intersection with the open ball $B(e,r/2)$. These $X_{n}$-replacement
segments of type 1 will be handled in Case 2 below; see also Figure 6. The
technical formulation of the induction statement is needed to handle the
situation where these $X_{n}$-replacement segments of type 1 meet $B(e,r/2)$.

If $x_{i}^{-1}y_{i}$ is a group element in the cyclic subgroup $\langle
s_{1}\rangle$, then by Lemma 4.16 we can replace $\tau_{i}$ a path
$\tau^{\prime}_{i}$ with edges labeled by $s_{1}$ and $a_{1}$ such that
$\tau^{\prime}_{i}$ lies outside the open ball $B(e,r/2)$ and
$\ell(\tau^{\prime}_{i})\leq 11\text{ }d_{S_{n}}(x_{i},y_{i})\leq
11\ell(\tau_{i})$. Call this path an $X_{n}$-replacement segment of type 2.

The new path
$\gamma^{\prime}=\sigma_{1}\tau^{\prime}_{1}\sigma_{2}\tau^{\prime}_{2}\cdots\sigma_{\ell}\tau^{\prime}_{\ell}\sigma_{\ell+1}$
lies completely in the $1$–skeleton of $X_{n}$, shares two endpoints $u$ and
$v$ with $\gamma$, and $\ell(\gamma^{\prime})\leq 11\ell(\gamma)$. We claim
that $\gamma^{\prime}\cap\beta$ does not contain an edge. In fact, assume by
the way of contradiction that $\gamma^{\prime}\cap\beta$ contains an edge.
Then some $X_{n}$-replacement segment $\tau^{\prime}_{i}$ of type $1$ must
contain an edge of $\beta$. Therefore, the endpoint $x_{i}$ of
$\tau^{\prime}_{i}$ has the form $s_{n}^{q}$ for some $|q|\geq r$. Therefore,
the subpath $\zeta$ of $\gamma$ connecting $u$ and $x_{i}$ is also an
$r$–avoidant path over an $n$–corner in the complex $X_{n+1}$. Also,
$|\zeta|<|\gamma|$ which contradicts to the choice of $\gamma$.

For each positive integer $j$ we call $e_{j}$ the $j^{th}$ edge of $\beta$. We
know that all edges $e_{j}$ are labeled by $s_{n}$. Let $H_{j}$ be the
$s_{n}$-hyperplane of the complex $X_{n}$ that corresponds to the edge
$e_{j}$. We now consider $r/8\leq j\leq r/4$. Assume that $r\geq 16$. Then
$j\geq 2$. Therefore by Lemma 4.13 each hyperplane $H_{j}$ must intersect
$\gamma^{\prime}$. Let $m_{j}$ be the point in the intersection
$H_{j}\cap\gamma^{\prime}$ such that the subpath of $\gamma^{\prime}$
connecting $m_{j}$ and $v$ does not intersect $H_{j}$ at point other than
$m_{j}$. Then $m_{j}$ is an interior point of an edge $f_{j}$ labeled by
$s_{n}$ in $\gamma^{\prime}$. Since $\gamma^{\prime}\cap\beta$ does not
contain an edge, two edges $e_{j}$ and $f_{j}$ are distinct.

Let $\beta_{j}$ be the path in $star(H_{j})-ostar(H_{j})$ that connects the
terminal $s_{n}^{j}$ of the edge $e_{j}$ to some endpoint $v^{\prime}_{j}$ of
the edge $f_{j}$. Then $\beta_{j}$ is a part of an $(n-1)$–ray. If $f_{j}$ is
not an edge of an $X_{n}$-replacement segment of $\gamma^{\prime}$, then we
let $v_{j}=v^{\prime}_{j}$. Otherwise, $f_{j}$ is an edge in some
$X_{n}$-replacement segment $\tau^{\prime}_{\ell_{j}}$ of type 1 of
$\gamma^{\prime}$. In this case, we let $v_{j}$ be the endpoint $y_{\ell_{j}}$
of $\tau^{\prime}_{\ell_{j}}$. Let $\alpha_{j}$ be the path in
$star(H_{j+1})-ostar(H_{j+1})$ that connects the initial endpoint $s_{n}^{j}$
of the edge $e_{j+1}$ to some endpoint $u^{\prime}_{j+1}$ of the edge
$f_{j+1}$. Then $\alpha_{j}$ is a part of an $1$–ray. If $f_{j+1}$ is not an
edge of an $X_{n}$-replacement segment of $\gamma^{\prime}$, then we let
$u_{j+1}=u^{\prime}_{j+1}$. Otherwise, $f_{j+1}$ is an edge in some
$\tau^{\prime}_{\ell_{j+1}}$. In this case, we let $u_{j+1}$ be the endpoint
$x_{\ell_{j+1}}$ of $\tau^{\prime}_{\ell_{j+1}}$.

$s_{n}^{j}$$\beta_{j}$$\alpha_{j}$$u^{\prime}_{j+1}$$u_{j+1}$$v^{\prime}_{j}$$v_{j}$$\beta$$\alpha$$\tau_{\ell(j)}$$\tau_{\ell(j+1)}$
Figure 5. $(\alpha_{j},\beta_{j})_{s_{n}^{j}}$ is a part of a $(n-1)$–corner
at $s_{n}^{j}$

We see that $(\alpha_{j},\beta_{j})_{s_{n}^{j}}$ is a part of an
$(n-1)$–corner at $s_{n}^{j}$ (see Figure 5). Let $\gamma^{\prime}_{j}$ be the
subpath of $\gamma^{\prime}$ that connects $v^{\prime}_{j}$ and
$u^{\prime}_{j+1}$. Let $\gamma^{\prime\prime}_{j}$ be the subpath of
$\gamma^{\prime}$ that connects $v_{j}$ and $u_{j+1}$. Let $\gamma_{j}$ be the
subpath of $\gamma$ that connects $v_{j}$ and $u_{j+1}$. Then by the
construction of $\gamma^{\prime}$ we have

$\ell(\gamma^{\prime\prime}_{j})\leq 11\ell(\gamma_{j}).$

Also, $\gamma^{\prime}_{j}=\eta_{1}\cup\gamma^{\prime\prime}_{j}\cup\eta_{2}$,
where $\eta_{1}$ (resp. $\eta_{2}$) is the (possibly degenerate) subsegment of
$\gamma^{\prime}_{j}$ connecting $v^{\prime}_{j}$ and $v_{j}$ (resp.
$u^{\prime}_{j+1}$ and $u_{j+1}$). Therefore,

$\ell(\gamma^{\prime\prime}_{j})=\ell(\gamma^{\prime}_{j})-\bigl{(}\ell(\eta_{1})+\ell(\eta_{2})\bigr{)}.$

Since $\eta_{1}$ and $\eta_{2}$ are subpaths of $X_{n}$-replacement segments
of type 1 which are also geodesics, we have

$\ell(\eta_{1})=d(v^{\prime}_{j},v_{j})\text{ and
}\ell(\eta_{2})=d(u^{\prime}_{j+1},u_{j+1}).$

This implies that

$\ell(\gamma^{\prime}_{j})-\bigl{(}d(v^{\prime}_{j},v_{j})+d(u^{\prime}_{j+1},u_{j+1})\bigr{)}\leq
11\ell(\gamma_{j}).$

Therefore,

$d(v^{\prime}_{j},v_{j})+\ell(\gamma_{j})+d(u^{\prime}_{j+1},u_{j+1})\geq\ell(\gamma^{\prime}_{j})/11.$

We note that each $X_{n}$-replacement segment $\tau^{\prime}$ of type $2$ of
$\gamma^{\prime}$ lies outside the open ball $B(e,r/2)$ by the construction.
Therefore, $\tau^{\prime}$ also lies outside each open ball $B(s_{n}^{j},r/4)$
for $r/8\leq j\leq r/4$. We now consider the case of $X_{n}$-replacement
segments $\tau^{\prime}$ of type 1 and we have two cases:

Case 1: For each $X_{n}$-replacement segment $\tau^{\prime}$ of type $1$ of
$\gamma^{\prime}$ such that
$\tau^{\prime}\cap\gamma^{\prime}_{j}\neq\varnothing$ the intersection
$\tau^{\prime}\cap\gamma^{\prime}_{j}$ lies outside the open ball
$B(s_{n}^{j},r/4)$. Then, $\gamma^{\prime}_{j}$ is an $(r/4)$–avoidant path
over the $(n-1)$–corner containing $(\alpha_{j},\beta_{j})_{s_{n}^{j}}$ in
$X_{n}$. Also, $n-1\geq d$. Therefore, $\ell(\gamma^{\prime}_{j})\geq g(r/4)$
for $r$ sufficiently large by the induction hypothesis. This implies that

$d(v^{\prime}_{j},v_{j})+\ell(\gamma_{j})+d(u^{\prime}_{j+1},u_{j+1})\geq
g(r/4)/11.$

$s_{n}^{j}$$r/4$$s_{n}^{*}$$s_{1}^{*}$$s_{n}^{*}$$w$$w^{\prime}$$\widetilde{w}$$f$$f^{\prime}$$u_{j+1}$$v_{j}$$\widetilde{v}$$\eta$
Figure 6. Some $X_{n}$-replacement segment of $\gamma^{\prime}_{j}$ intersect
the open ball $B(s_{n}^{j},r/4)$ and the subsegment $\eta$ of $\gamma_{j}$
that connects $\widetilde{v}$ and $\widetilde{w}$ is an $(r/4)$–avoidant path
over an $n$–corner in $X_{n+1}$.

Case 2: We now assume that there is an $X_{n}$-replacement segment
$\tau^{\prime}$ of type $1$ of $\gamma^{\prime}$ such that
$\tau^{\prime}\cap\gamma^{\prime}_{j}\neq\varnothing$ and it intersects the
open ball $B(s_{n}^{j},r/4))$ (see Figure 6). We will prove that some
subsegment of $\gamma_{j}$ is an $(r/4)$–avoidant path over an $n$–corner in
$X_{n+1}$. Let $f$ be an edge of $\tau^{\prime}\cap\gamma^{\prime}_{j}$ that
lies inside the open ball $B(s_{n}^{j},(r/4)+1))$. Then $f$ is labeled by
$s_{n}$. Let $H$ be the hyperplane in $X_{n}$ that is dual the edge $f$. Then
$H$ must intersect $\gamma^{\prime}_{j}$. Let $x$ be the point in the
intersection $H\cap\gamma^{\prime}_{j}$. Then $x$ is an interior point of an
edge $f^{\prime}$ labeled by $s_{n}$ in $\gamma^{\prime}_{j}$. Let
$\alpha^{\prime}$ be the path labeled by $s_{1}$ in $star(H)-ostar(H)$ that
connect a vertex $w$ of $f$ to a vertex $w^{\prime}$ of $f^{\prime}$. If
$f^{\prime}$ is not an edge in any $X_{n}$-replacement segment of
$\gamma^{\prime}$, then $w^{\prime}$ is a vertex of $\gamma$. In this case, we
let $\widetilde{w}=w^{\prime}$ and $\widetilde{\alpha}=\alpha^{\prime}$. In
the case $f^{\prime}$ is an edge in some $X_{n}$-replacement segment
$\tau^{\prime\prime}$ of $\gamma^{\prime}$, we let $\widetilde{w}$ is an
endpoint of $\tau^{\prime\prime}$ that belongs to $\gamma^{\prime}_{j}$, let
$\alpha^{\prime\prime}$ be the subsegment $\tau^{\prime\prime}$ connecting
$w^{\prime}$ and $\widetilde{w}$, and let
$\widetilde{\alpha}=\alpha^{\prime}\cup\alpha^{\prime\prime}$. Therefore,
$\widetilde{\alpha}$ is a part of an $(1,n)$–ray. Let $\widetilde{v}$ be the
endpoint of $\tau^{\prime}$ that belongs to $\gamma_{j}$ and let
$\widetilde{\beta}$ is a subsegment of $\tau^{\prime}$ that connects $w$ and
$\widetilde{v}$. Then $\widetilde{\beta}$ is a part of an $n$–ray and
$(\widetilde{\alpha},\widetilde{\beta})_{w}$ is a part of an $n$–corner.

Let $\eta$ be the subsegment of $\gamma_{j}$ that connects $\widetilde{v}$ and
$\widetilde{w}$. We note that $\eta$ lies outside the open ball $B(e,r)$ in
$X_{n+1}$ and therefore it lies outside the open ball $B(s_{n}^{j},3r/4)$ in
$X_{n+1}$. Also, $d(s_{n}^{j},w)<r/4+1\leq r/2$ if we assume that $r>4$.
Therefore, $\eta$ lies outside the open ball $B(w,r/4)$. Thus, $\eta$ is an
$(r/4)$–avoidant path over an $n$–corner in $X_{n+1}$. Also, $n\geq d+1>d$.
Therefore,

$\ell(\gamma_{j})\geq\ell(\eta)\geq g(r/4).$

In both cases we have shown that

$d(v^{\prime}_{j},v_{j})+\ell(\gamma_{j})+d(u^{\prime}_{j+1},u_{j+1})\geq
g(r/4)/11.$

Therefore,

$\ell(\gamma)\geq\sum_{r/8\leq j\leq
r/4}\bigl{(}d(v^{\prime}_{j},v_{j})+\ell(\gamma_{j})+d(u^{\prime}_{j+1},u_{j+1})\bigr{)}\geq(\frac{r}{16})\bigl{(}\frac{g(r/4)}{11}\bigr{)}\geq\frac{r}{180}g(r/4).$

∎

######  Lemma 4.19.

For each integer $d\geq 2$ there is a positive number $n_{d}$ such that the
following holds. Let $k\geq d$ be an integer and let $(\alpha,\beta)_{x}$ be a
$k$–corner and let $\gamma$ be an $r$–avoidant path over the $k$–corner
$(\alpha,\beta)_{x}$ in the complex $X_{k}$. Then the length of $\gamma$ is at
least $(r^{d-1}/n_{d})f(r/{n_{d}})$ for $r$ sufficiently large.

###### Proof.

We first prove the following claim. For each integer $d\geq 2$ there is a
positive number $n_{d}$ such that the following holds. Let $k\geq d$ be
integers and let $(\alpha,\beta)_{x}$ be a $k$–corner. Let $\gamma$ be an
$r$–avoidant path over the $k$–corner $(\alpha,\beta)_{x}$ in the complex
$X_{k+1}$. Then the length of $\gamma$ is at least
$(r^{d-1}/n_{d})f(r/{n_{d}})$ for $r$ sufficiently large.

The above claim can be proved by induction on $d$. In fact, the claim is true
for the base case $d=2$ due to Lemma 4.17. Then Lemma 4.18 establishes the
inductive step and the above claim is proved. By Lemma 3.3, we observe that if
$\gamma$ is an $r$–avoidant path over a $k$–corner in the complex
$X_{k}\subset X_{k+1}$ then $\gamma$ is also an $r$–avoidant path over the
same $k$–corner in the complex $X_{k+1}$. Therefore, the lemma follows from
the above claim. ∎

We now prove the lower bound for the divergence of the groups $G_{m}$ for
$m\geq 3$.

######  Proposition 4.20 (Lower bound for $\operatorname{Div}_{G_{m}}$, $m\geq
3$).

Let $m\geq 3$ be an integer. Let $\\{\delta_{\rho}\\}$ be the divergence of
the Cayley graph $\Gamma(G_{m},S_{m})$. Then
$r^{m-1}f(r)\preceq\delta_{\rho}(r)$ for each $\rho\in(0,1/2]$.

###### Proof.

Let $n_{m}$ be the positive number in Lemma 4.19. We will prove that

$\delta_{\rho}(r/\rho)\geq(r^{m-1}/n_{m})/f(r/n_{m})$

for $r$ sufficiently large. Let $\alpha$ be a $1$–ray and let $\beta$ be a
$m$–ray such that they share the initial point at the identity $e$. Then
$(\alpha,\beta)_{e}$ is an $m$–corner. Let $\gamma$ be an arbitrary path which
connects $\alpha(r/\rho)$ and $\beta(r/\rho)$ and lies outside the open ball
$B(e,r)$. Then by Lemma 4.19, the length of the path $\gamma$ is bounded below
by $(r^{m-1}/n_{m})/f(r/n_{m})$ for $r$ sufficiently large. This implies that
$\delta_{\rho}(r/\rho)\geq(r^{m-1}/n_{m})/f(r/n_{m})$ for $r$ sufficiently
large. Therefore, $r^{m-1}f(r)\preceq\delta_{\rho}(r)$ for each
$\rho\in(0,1/2]$. ∎

The group $G_{2}$ is a special case. Although the upper bound on $r$–avoidant
paths over 2–corners in $X_{2}$ is $rf(r)$ (this can be seen by arguing
similar to Lemma 4.2), there are pairs of rays in $X_{2}$ which diverge at a
quadratic rate. The next result describes two such rays and establishes a
quadratic lower bound on their divergence.

######  Lemma 4.21.

Let $n$ be an arbitrary integer greater than $16$ and $s$ be a generator of
$G_{2}$ in $H-F$. Let $\gamma$ be a path with endpoints $(s_{2}s)^{-n}$ and
$(s_{2}s)^{n}$ which avoids the open ball $B(e,n)$. Then the length of
$\gamma$ is at least $n^{2}/16$.

###### Proof.

For each $0\leq i\leq n/8$ let $e_{i}$ be an edge labeled by $s_{2}$ with
endpoints $(s_{2}s)^{i}$ and $(s_{2}s)^{i}s_{2}$. Then the hyperplane $H_{i}$
of the complex $X_{2}$ that corresponds to $e_{i}$ intersects $\gamma$. Let
$u_{i}$ be the point in this intersection such that the subpath of $\gamma$
connecting $u_{i}$ and $(a_{2}s)^{n}$ does not intersect $H_{i}$ at point
other than $u_{i}$. Then $u_{i}$ is the midpoint of an edge $f_{i}$ of
$\gamma$.

Let $\alpha_{i}$ be the path in $star(H_{i})-ostar(H_{i})$ that connects
$(s_{2}s)^{i}$ to an endpoint of $f_{i}$. Therefore, $\alpha_{i}$ traces a
word representing a group element $g_{i}\in F_{xz}$ and the endpoint $v_{i}$
of $\alpha_{i}$ in $\gamma$ has the form $(s_{2}s)^{i}g_{i}$. Since
$|(s_{2}s)^{i}|_{S_{2}}\leq 2i\leq n/2$ and $(s_{2}s)^{i}g_{i}$ lies outside
the open ball $B(e,n)$, then $|g_{i}|_{S_{2}}\geq n-n/2\geq n/2$. Let
$\beta_{i}$ be the path in $star(H_{i})-ostar(H_{i})$ that connects
$(s_{2}s)^{i}s_{2}$ to an endpoint of $f_{i}$. Therefore, $\beta_{i}$ traces a
word representing a group element $g^{\prime}_{i}\in F_{yz}$ and the endpoint
$w_{i}$ of $\beta_{i}$ in $\gamma$ has the form
$(s_{2}s)^{i}s_{2}g^{\prime}_{i}$. Since $|(s_{2}s)^{i}s_{2}|_{S_{2}}\leq
2i+1\leq n/2$ and $(s_{2}s)^{i}s_{2}g^{\prime}_{i}$ lies outside the open ball
$B(e,n)$, then $|g^{\prime}_{i}|_{S_{2}}\geq n-n/2\geq n/2$.

For each $1\leq i\leq n/8$ let $\gamma_{i}$ be the subpath of $\gamma$ that
connects $w_{i-1}$ and $v_{i}$. Therefore, the length of $\gamma_{i}$ is at
least $d_{S_{2}}(w_{i-1},v_{i})$. Also,
$d_{S_{2}}(w_{i-1},v_{i})=|w_{i-1}^{-1}v_{i}|_{S_{2}}=|g_{i-1}^{\prime-1}sg_{i}|_{S_{2}}$
and the length of the element $g_{i-1}^{\prime-1}sg_{i}$ in $G_{2}$ is
$|g^{\prime}_{i-1}|_{S_{1}}+1+|g_{i}|_{S_{1}}$. We see this as follows

$\displaystyle d_{S_{2}}(e,g_{i-1}^{\prime-1}sg_{i})$ $\displaystyle=$
$\displaystyle d_{S_{1}}(e,g_{i-1}^{\prime-1}sg_{i})$ $\displaystyle=$
$\displaystyle d_{S^{\prime}_{1}}(e,g_{i-1}^{\prime-1}sg_{i})$
$\displaystyle\geq$ $\displaystyle
d_{S^{\prime}_{1}}(e,g_{i-1}^{\prime-1}F)+d_{S^{\prime}_{1}}(F,sF)+d_{S^{\prime}_{1}}(F,g_{i})$
$\displaystyle=$ $\displaystyle|g^{\prime}_{i-1}|_{S_{1}}+1+|g_{i}|_{S_{1}},$

where $S^{\prime}_{1}=S_{1}-\\{s_{1}\\}$ is the generating set of the
amalgamation factor

$H\ast_{\langle d_{i}=a_{i}b_{i}^{-1}\rangle}(F_{x}\times F_{y}\times F_{z})$

of $G_{1}$. The first two equalities hold because the subgroup inclusions are
isometric embeddings with the respective generating sets. The inequality holds
from Bass-Serre theory (of free products with amalgamation). For the last
equality, we have $d_{S^{\prime}_{1}}(F,sF)=1$ because $s\not\in F$. The
remaining parts are easily seen from Lemma 3.2, the fact $F$ is a subgroup of
$H$, and the fact $H\ast_{\langle d_{i}=a_{i}b_{i}^{-1}\rangle}(F_{x}\times
F_{y}\times F_{z})$ is isometrically embedded into $G_{1}$.

Therefore,

$\ell(\gamma_{i})\geq|g^{\prime}_{i-1}|_{S_{1}}+1+|g_{i}|_{S_{1}}=|g^{\prime}_{i-1}|_{S_{2}}+1+|g_{i}|_{S_{2}}\geq
n/2+n/2+1\geq n.$

This implies that

$\ell(\gamma)\geq\sum_{1\leq i\leq n/8}\ell(\gamma_{i})\geq n^{2}/16.$

∎

We now prove the quadratic lower bound for the divergence of the group
$G_{2}$.

######  Proposition 4.22 (Quadratic lower bound for
$\operatorname{Div}_{G_{2}}$).

Let $\\{\delta_{\rho}\\}$ be the divergence of the Cayley graph
$\Gamma(G_{2},S_{2})$. Then $r^{2}\preceq\delta_{\rho}(r)$ for each
$\rho\in(0,1/2]$.

###### Proof.

We will prove that $\delta_{\rho}(r/\rho)\geq r^{2}/256-2r/\rho$ for $r$
sufficiently large. Let $s$ be a generator of $G_{2}$ in $H-F$. Let $\alpha$
be bi-infinite geodesic containing the identity element $e$ with edges labeled
by $s_{2}$ and $s$ alternately. Let $x$ and $y$ be the two points in the
intersection $\alpha\cap S(e,r/\rho$). We assume that the subsegment of
$\alpha$ from $e$ to $x$ traces each edge of $\alpha$ in the positive
direction and the subsegment of $\alpha$ from $e$ to $y$ traces each edge of
$\alpha$ in the negative direction. Let $\beta$ be an arbitrary path with
endpoints $x$ and $y$ that lies outside the ball $B(e,r)$. Let $n$ be the
largest integer such that $n\leq r/2$. Therefore, $n\geq r/2-1\geq r/4$ for
$r$ sufficiently large. Let $\alpha_{1}$ be a subsegment of $\alpha$ that
connects $(s_{2}s)^{n}$ to $x$. Therefore, $\alpha_{1}$ lies outside the open
ball $B(e,n)$ and has the length bounded above by $r/\rho$. Similarly, let
$\alpha_{2}$ be a subsegment of $\alpha$ that connects $(s_{2}s)^{-n}$ to $y$.
Therefore, $\alpha_{2}$ lies outside the open ball $B(e,n)$ and has the length
bounded above by $r/\rho$. Let $\gamma=\alpha_{1}\cup\beta\cup\alpha_{2}$.
Then, $\gamma$ is a path with endpoints $(s_{2}s)^{-n}$ and $(s_{2}s)^{n}$
which avoids the open ball $B(e,n)$. Therefore, the length of $\gamma$ is at
least $n^{2}/16$ by Lemma 4.21. Therefore,

$\ell(\beta)\geq\ell(\gamma)-2r/\rho\geq n^{2}/16-2r/\rho\geq
r^{2}/256-2r/\rho.$

Thus, $\delta_{\rho}(r/\rho)\geq r^{2}/256-2r/\rho$ for $r$ sufficiently
large. This implies that $r^{2}\preceq\delta_{\rho}(r)$. ∎

###  4.3. Geodesic divergence

In this subsection we establish the geodesic divergence statements in Theorem
2.13; namely, that ${\rm Div}^{G_{m}}_{\langle s_{m}\rangle}$ is equivalent to
$r^{m-1}f(r)$ where $f$ is the inverse of the distortion function. We also
prove that $\operatorname{Div}_{\langle s_{1}s_{2}\rangle}^{G_{m}}$ is
equivalent to $rf(r)$. As an application we have that, for $f(r)\prec r$. the
element $s_{1}s_{2}$ is Morse but not contracting in $G_{m}$.

_The divergence of $\langle s_{m}\rangle$ in $G_{m}$._ The proof of the
following proposition is similar to the proof of Proposition 5.1 in [BT]
although the proof for the case $m=2$ is slightly different.

######  Proposition 4.23.

For each $m\geq 2$, the divergence of $\langle s_{m}\rangle$ in $G_{m}$ is
equivalent to the function $r^{m-1}f(r)$.

###### Proof.

Let $\alpha_{m}$ be a bi-infinite geodesic in the Cayley graph
$\Gamma(G_{m},S_{m})$ corresponding to the subgroup $\langle s_{m}\rangle$.
Without loss of generality we can assume that $\alpha_{m}(0)=e$ and
$\alpha_{m}(1)=s_{m}$. Let $\beta:[0,\infty)\to\Gamma(G_{m},S_{m})$ be a
$1$–ray with $\beta(0)=e$. By Lemma 4.2 and Lemma 4.4 (Statement ($Q_{m}$))
there is a number $M>0$ such that the following hold. Let $r>0$ be an
arbitrary number. There are a path $\gamma_{1}$ outside the open ball
$B(\alpha_{m}(0),r)$ connecting $\alpha_{m}(-r)$ and $\beta(r)$ and a path
$\gamma_{2}$ outside the open ball $B(\alpha_{m}(0),r)$ connecting
$\alpha_{m}(r)$ and $\beta(r)$ such that the lengths of $\gamma_{1}$ and
$\gamma_{2}$ are both bounded above by $Mr^{m-1}\bigl{(}f(Mr)+1\bigr{)}$.
Therefore, the path $\gamma=\gamma_{1}\cup\gamma_{2}$ lies outside the open
ball $B(\alpha_{m}(0),r)$, connects $\alpha_{m}(-r)$ and $\alpha_{m}(r)$, and
has length at most $2Mr^{m-1}\bigl{(}f(Mr)+1\bigr{)}$. This implies that ${\rm
Div}_{\alpha_{m}}(r)\leq 2Mr^{m-1}\bigl{(}f(Mr)+1\bigr{)}$.

We now prove a lower bound for ${\rm Div}_{\alpha_{m}}$. The proof for the
case $m=2$ is analogous to the proof of Lemma 4.17 and we leave it to the
reader for checking the details. We now prove a lower bound for ${\rm
Div}_{\alpha_{m}}$ for $m\geq 3$. Let $n_{m}$ be the constant in Lemma 4.19.
Let $\gamma^{\prime}$ be an arbitrary path outside the open ball
$B(\alpha_{m}(0),r)$ connecting $\alpha_{m}(-r)$ and $\alpha_{m}(r)$. Let
$e_{1}$ be the edge of $\alpha_{m}$ with endpoints $e$ and $s_{m}$. Then the
hyperplane $H$ of the complex $X_{m}$ corresponding to $e_{1}$ must intersect
$\gamma^{\prime}$. Therefore, there is a $1$–ray $\beta_{1}$ with initial
point at $e$ that intersects $\gamma^{\prime}$ at some vertex $v$. This
implies that the subpath $\gamma_{1}$ of $\gamma^{\prime}$ connecting
$\alpha_{m}(r)$ and $v$ is an $r$–avoidant path over the $m$–corner
$({\alpha_{m}}_{|[0,\infty)},\beta_{1})_{e}$. By Lemma 4.19 for $r$
sufficiently large we have

$\ell(\gamma^{\prime})\geq\ell(\gamma_{1})\geq(r^{m-1}/n_{m})f(r/n_{m}).$

This implies that ${\rm Div}_{\alpha_{m}}(r)\geq(r^{m-1}/n_{m})f(r/n_{m})$ for
$r$ sufficiently large. Therefore, the divergence of $\alpha_{m}$ is
equivalent to the function $r^{m-1}f(r)$. ∎

_The divergence of $\langle s_{1}s_{2}\rangle$ in $G_{m}$._ For each $m\geq 2$
we let $\alpha$ be a bi-infinite geodesic with edges labeled $s_{1}$ and
$s_{2}$ in $\Gamma(G_{m},S_{m})$ corresponding to the infinite cyclic subgroup
$\langle s_{1}s_{2}\rangle$. In the remainder of this section, we prove that
the divergence of $\alpha$ in $\Gamma(G_{m},S_{m})$ for $m\geq 2$ is
equivalent to $rf(r)$. This implies that if $F$ is nonlinearly distorted in
$H$ (equivalently $f(r)\prec r$), then the group element $s_{1}s_{2}$ is Morse
but not contracting in $G_{m}$ for $m\geq 2$. Therefore, by Theorem 2.14 of
[CS15], the groups $G_{m}$ are not $\operatorname{CAT}(0)$ for $m\geq 2$.

In what follows, we assume that $\alpha(0)=e$ and $\alpha(1)=s_{1}$. The next
two lemmas establish the upper bound, $\operatorname{Div}_{\langle
s_{1}s_{2}\rangle}^{G_{m}}\preceq rf(r)$.

######  Lemma 4.24.

Let $r_{0}\geq 1$ and $D$ be the constants in Remark 3.6 and let $r\geq r_{0}$
be an arbitrary integer. Then there is a group element $g_{0}$ in $F_{xz}$
such that the following hold.

  1. (1)

$4r\leq|g_{0}|_{S_{2}}\leq 4Dr$; and

  2. (2)

There is a path $\beta$ in $\Gamma(G_{2},S_{2})$ connecting $g_{0}$ and
$s_{2}g_{0}$ with length at most $Df(4Dr)+1$ and avoids the open ball
$B(e,2r)$.

###### Proof.

Let $m=4Dr$. By Lemma 3.8, there are group elements $g_{0}$ in $F_{xz}$ and
$g_{1}$ in $F_{yz}$ such that the following hold:

  1. (1)

$s_{2}^{-1}g_{0}s_{2}=g_{1}$;

  2. (2)

$m/D\leq|g_{0}|_{R_{xz}}\leq m$ and $m/D\leq|g_{1}|_{R_{yz}}\leq m$;

  3. (3)

There is a path $\gamma$ connecting $g_{0}$ and $g_{1}$ with the length at
most $Df(m)$ and each vertex in $\gamma$ is a group element $g_{0}h$ for $h\in
H$;

By Lemma 3.2 and Lemma 3.3 we have
$|g_{0}|_{R_{xz}}=|g_{0}|_{S_{1}}=|g_{0}|_{S_{2}}$ and
$|g_{1}|_{R_{yz}}=|g_{1}|_{S_{1}}=|g_{1}|_{S_{2}}$. Since $m/D=4r$, we have

$4r\leq|g_{0}|_{S_{2}}\leq 4Dr\text{ and }4r\leq|g_{1}|_{S_{2}}\leq 4Dr.$

Another application of Lemma 3.2 and Lemma 3.3 implies that the path $\gamma$
lies outside the open ball $B(e,4r)$ in the group $G_{2}$.

Let $\beta_{1}=s_{2}\gamma$. Then the path $\beta_{1}$ connects two points
$s_{2}g_{1}$ and $s_{2}g_{0}$, lies outside the open ball $B(s_{1},4r)$
(therefore outside the open ball $B(e,2r)$) and has length at most $Df(m)$.
Also, $s_{2}g_{1}=g_{0}s_{2}$. Then we connect $g_{0}$ and $s_{2}g_{1}$ by an
edge $e_{1}$ labeled by $s_{2}$. It is clear that this edge also lies outside
the open ball $B(e,2r)$. Therefore, $\beta=e_{1}\cup\beta_{1}$ is a desired
path. ∎

######  Lemma 4.25.

Let $m\geq 2$ be an integer. Then the divergence of $\alpha$ in
$\Gamma(G_{m},S_{m})$ is dominated by the function $rf(r)$.

###### Proof.

By Lemma 3.3 the path avoids the $r$–ball about $e$ in $G_{2}$ also lies
outside the $r$–ball about $e$ in $G_{m}$ for $m\geq 3$. Therefore, we only
prove the above lemma for the case $m=2$. Let $D$ and $r_{0}$ be constants in
Remark 3.6. We will prove that

$\operatorname{Div}_{\alpha}(r)\leq 2r\bigl{(}Df(4Dr)+1\bigr{)}+8Dr\text{ for
each $r\geq r_{0}$}\,.$

We assume that $r$ is an integer. Let $x_{i}=\alpha(i)$ for each $i\in[-r,r]$.
Let $g_{0}\in F_{xz}$ be a group element in Lemma 4.24. Then

  1. (1)

$4r\leq|g_{0}|_{S_{2}}\leq 4Dr$; and

  2. (2)

There is a path $\beta$ in $\Gamma(G_{2},S_{2})$ connecting $g_{0}$ and
$s_{2}g_{0}$ with length at most $Df(4Dr)+1$ and avoids the open ball
$B(e,2r)$.

We will show that for each $r\in[-r,r-1]$ there is a path $\beta_{i}$ which
connects $x_{i}g_{0}$ and $x_{i+1}g_{0}$, lies outside the open ball $B(e,r)$,
and has length at most $Df(4Dr)+1$. Since edges of $\alpha$ are labeled by
$s_{1}$ and $s_{2}$ alternately, we see that $x_{i+1}=x_{i}s_{1}$ or
$x_{i+1}=x_{i}s_{2}$. If $x_{i+1}=x_{i}s_{1}$, then

$x_{i+1}g_{0}=(x_{i}s_{1})g_{0}=x_{i}(s_{1}g_{0})=x_{i}(g_{0}s_{1})=(x_{i}g_{0})s_{1}\,.$

There we connect $x_{i}g_{0}$ and $x_{i+1}g_{0}$ by an edge $\gamma_{i}$
labelled $s_{1}$ and this edge obviously lies outside the open ball
$B(x_{i},4r-1)$. Since $d_{S_{2}}(e,x_{i})\leq|i|\leq r$, the edge $\beta_{i}$
also lies outside the open ball $B(e,4r-2)$ (therefore outside $B(e,r)$). We
now assume that $x_{i+1}=x_{i}s_{2}$. By Lemma 4.24 there is a path
$\beta_{i}$ (a translate of the edge $\beta$ above by $x_{i}$) connecting
$x_{i}g_{0}$ and $x_{i+1}g_{0}$ with length at most $Df(4Dr)+1$ and avoids the
open ball $B(x_{i},2r)$. Again, we note that $d_{S_{2}}(e,x_{i})\leq|i|\leq
r$. Therefore, $\beta_{i}$ also lies outside the open ball $B(e,r)$.

Let $\gamma_{0}=\bigcup_{-r\leq i\leq r-1}\beta_{i}$. Then $\gamma_{0}$ lies
outside the open ball $B(e,r)$, connects two points $x_{-r}g_{0}$ and
$x_{r}g_{0}$, and has length at most $2r\bigl{(}Df(4Dr)+1\bigr{)}$. Let
$\eta_{1}$ be the path labeled by the word in $R_{xz}$ representing $g_{0}$
which connects $x_{-r}$ and $x_{-r}g_{0}$. Then the length of $\eta_{1}$ is
exactly $|g_{0}|_{R_{xz}}\leq 4Dr$. We observe that each vertex in $\eta_{1}$
is a group element $gu$ where $u\in F_{yz}$ and $g$ is a group element with
length exactly $r$ in the subgroup $K$ generated by $\\{s_{1},s_{2}\\}$. Also
the group automorphism $G_{2}\to K$ taking each $s_{1}$ and $s_{2}$ to itself,
and all other generators of $G_{2}$ to the identity shows that $gu$ lies
outside the open ball $B(e,r)$. In other words, $\eta_{1}$ lies outside the
open ball $B(e,r)$. Similarly, there is a path $\eta_{2}$ which connects
$x_{r}$ and $x_{r}g_{0}$, lies outside the open ball $B(e,r)$, and has length
at most $4Dr$.

Let $\gamma=\eta_{1}\cup\gamma_{0}\cup\eta_{2}$. Then $\gamma$ connects
$\alpha(-r)=x_{-r}$ and $\alpha(r)=x_{r}$, lies outside the open ball
$B(e,r)$, and has length at most $2r\bigl{(}Df(4Dr)+1\bigr{)}+8Dr$. Therefore,

$\operatorname{Div}_{\alpha}(r)\leq 2r\bigl{(}Df(4Dr)+1\bigr{)}+8Dr\,.$

This implies that the divergence of $\alpha$ is dominated by $rf(r)$.

∎

The lower bound, $rf(r)\preceq\operatorname{Div}_{\langle
s_{1}s_{2}\rangle}^{G_{m}}$ is proven by induction on $m$. The next lemma
establishes the base cases $m=3$ and (as a consequence) $m=2$.

######  Lemma 4.26.

Let $\gamma$ be the path in $X_{3}$ which lies outside the open ball $B(e,r)$
and connects a vertex in the ray $\alpha[0,-\infty)$ to a vertex in the ray
$\alpha[0,\infty)$. Then the length of $\gamma$ is at least $(r/16)f(r/4)$ for
$r$ sufficiently large.

###### Proof.

Suppose that $\gamma$ connects a vertex $u$ in $\alpha[0,-\infty)$ to a vertex
$v$ in $\alpha[0,\infty)$. As in the proof of Lemma 4.17, $\gamma$ can be
written as a concatenation

$\sigma_{1}\tau_{1}\sigma_{2}\tau_{2}\cdots\sigma_{\ell}\tau_{\ell}\sigma_{\ell+1}$

such that:

  1. (1)

Each $\tau_{i}$ intersects $X_{2}$ only at its endpoints
$x_{i}=\sigma_{i}\cap\tau_{i}$ and $y_{i}=\tau_{i}\cap\sigma_{i+1}$;

  2. (2)

Each $\sigma_{i}$ lies completely in the $1$–skeleton of $X_{2}$.

We observe that each $x_{i}^{-1}y_{i}$ is a group element in the cyclic
subgroup $\langle s_{2}\rangle$ or in the cyclic subgroup $\langle
s_{1}\rangle$. If $x_{i}^{-1}y_{i}$ is a group element in the cyclic subgroup
$\langle s_{2}\rangle$, then we replace $\tau_{i}$ by a geodesic
$\tau^{\prime}_{i}$ labeled by $s_{2}$. Otherwise, $x_{i}^{-1}y_{i}$ is a
group element in the cyclic subgroup $\langle s_{1}\rangle$, then we replace
$\tau_{i}$ by a geodesic $\tau^{\prime}_{i}$ labeled by $s_{1}$. The new path
$\gamma^{\prime}=\sigma_{1}\tau^{\prime}_{1}\sigma_{2}\tau^{\prime}_{2}\cdots\sigma_{\ell}\tau^{\prime}_{\ell}\sigma_{\ell+1}$
lies completely in the $1$–skeleton of $X_{2}$, has the same initial and end
points as $\gamma$, and $\ell(\gamma^{\prime})\leq\ell(\gamma)$. Note that the
new path $\gamma^{\prime}$ may intersect the open ball $B(e,r)$. We call each
subpath $\tau^{\prime}_{i}$ of $\gamma^{\prime}$ a short-cut segment.

For each positive integer $j$ we call $e_{j}$ the $(2j-1)^{th}$ edge of the
ray $\alpha[0,\infty)$. We know that each edges $e_{j}$ is labeled by $s_{2}$
with endpoints $(s_{1}s_{2})^{j-1}s_{1}$ and $(s_{1}s_{2})^{j}$. Let $H_{j}$
be the $s_{2}$-hyperplane of the complex $X_{2}$ dual to the edge $e_{j}$. We
assume that $r\geq 8$ and we consider the edges $e_{j}$ for $1\leq j\leq r/8$.
Then each hyperplane $H_{j}$ intersect $\gamma^{\prime}$ and we let $m_{j}$ be
the point in the intersection $H_{j}\cap\gamma^{\prime}$ such that the subpath
of $\gamma^{\prime}$ connecting $m_{j}$ and $v$ does not intersect $H_{j}$ at
point other than $m_{j}$. Note that $m_{j}$ is an interior point of an edge
$f_{j}$ labeled by $s_{2}$ in $\gamma^{\prime}$.

Let $\beta_{j}$ be the path in $star(H_{j})-ostar(H_{j})$ that connects the
terminal $(s_{1}s_{2})^{j}$ of the edge $e_{j}$ to some endpoint
$v^{\prime}_{j}$ of the edge $f_{j}$. Then edges of $\beta_{j}$ are labeled by
generators of the free group $F_{yz}$. If $f_{j}$ is not an edge of a short-
cut segment of $\gamma^{\prime}$, then we let $v_{j}=v^{\prime}_{j}$.
Otherwise, $f_{j}$ is an edge in some short-cut segment
$\tau^{\prime}_{\ell_{j}}$ of $\gamma^{\prime}$. In this case, we let $v_{j}$
be the endpoint $y_{\ell_{j}}$ of $\tau^{\prime}_{\ell_{j}}$. Let $\alpha_{j}$
be the path in $star(H_{j+1})-ostar(H_{j+1})$ that connects the initial
endpoint $(s_{1}s_{2})^{j}s_{1}$ of the edge $e_{j+1}$ to some endpoint
$u^{\prime}_{j+1}$ of the edge $f_{j+1}$. Then edges of $\alpha_{j}$ are
labeled by generators of $F_{xz}$. If $f_{j+1}$ is not an edge of a short-cut
segment of $\gamma^{\prime}$, then we let $u_{j+1}=u^{\prime}_{j+1}$.
Otherwise, $f_{j+1}$ is an edge in some $\tau^{\prime}_{\ell_{j+1}}$. In this
case, we let $u_{j+1}$ be the endpoint $x_{\ell_{j+1}}$ of
$\tau^{\prime}_{\ell_{j+1}}$.

Let $\gamma^{\prime}_{j}$ be the subpath of $\gamma^{\prime}$ that connects
$v^{\prime}_{j}$ and $u^{\prime}_{j+1}$. We will prove that the length of
$\gamma^{\prime}_{j}$ is at least $f(r/4)$. By the construction, the path
$\gamma^{\prime}_{j}$ contains two subpaths $\eta_{1}$ and $\eta_{2}$ of
$\gamma^{\prime}$, where $\eta_{1}$ (resp. $\eta_{2}$) is the (possibly
degenerate) subsegment of $\gamma^{\prime}_{j}$ connecting $v^{\prime}_{j}$
and $v_{j}$ (resp. $u^{\prime}_{j+1}$ and $u_{j+1}$). If either the length of
$\eta_{1}$ or the length of $\eta_{2}$ is greater than $r/4$, then the length
of $\gamma^{\prime}_{j}$ is greater than $r/4$ and therefore greater than
$f(r/4)$.

We now assume that the lengths of $\eta_{1}$ and $\eta_{2}$ are both less than
or equal to $r/4$. Therefore,

$d_{S_{2}}(v^{\prime}_{j},v_{j})\leq r/4\text{ and
}d_{S_{2}}(u^{\prime}_{j+1},u_{j+1})\leq r/4.$

Also, $v_{j}$ and $u_{j+1}$ lies outside the open ball $B(e,r)$. Then
$v^{\prime}_{j}$ and $u^{\prime}_{j+1}$ lies outside the open ball
$B(e,3r/4)$. We note that $v^{\prime}_{j}=(s_{1}s_{2})^{j}g_{1}$ for some
$g_{1}\in F_{yz}$ and $u^{\prime}_{j+1}=(s_{1}s_{2})^{j}s_{1}g_{0}$ for some
$g_{0}\in F_{xz}$. Moreover, $d_{S_{2}}(e,(s_{1}s_{2})^{j})=2j\leq r/2$ and
$d_{S_{2}}(e,(s_{1}s_{2})^{j}s_{1})=2j+1\leq r/2$. Therefore, we have
$|g_{0}|_{S_{2}}\geq r/4$ and $|g_{1}|_{S_{2}}\geq r/4$. By Lemma 3.3 and
Lemma 3.7 we have

$\ell(\gamma^{\prime}_{j})\geq
d_{S_{2}}(v^{\prime}_{j},u^{\prime}_{j+1})=|g_{1}^{-1}s_{1}g_{0}|_{S_{2}}=|(g_{1}^{-1}g_{0})s_{1}|_{S_{1}}=|(g_{1}^{-1}g_{0})|_{S_{1}}+1\geq
f(r/4)+1\geq f(r/4)\,.$

Therefore,

$\ell(\gamma)\geq\ell(\gamma^{\prime})\geq\sum_{1\leq j\leq
r/8}\ell(\gamma^{\prime}_{j})\geq(r/16)f(r/4).$

∎

Here is the proof that $rf(r)\preceq\operatorname{Div}_{\langle
s_{1}s_{2}\rangle}^{G_{m}}$ for all $m\geq 2$.

######  Lemma 4.27.

For each integer $m\geq 2$ there is a positive integer $k_{m}$ such that the
following hold. Let $\gamma$ be the path in $X_{m}$ which lies outside the
open ball $B(e,r)$ and connects a vertex in the ray $\alpha[0,-\infty)$ to a
vertex in the ray $\alpha[0,\infty)$. Then the length of $\gamma$ is at least
$(r/k_{m})f(r/k_{m})$.

###### Proof.

We prove the above lemma by induction on $m$. By Lemma 4.26 the above lemma is
true for the case $m=2$ and $m=3$ with $k_{2}=k_{3}=16$. We assume the above
lemma is true for $m\geq 3$ and we will prove that it is also true for $m+1$.

Suppose that $\gamma$ is a path outside then open ball $B(e,r)$ in $X_{m+1}$
which connects a vertex $u$ in $\alpha[0,-\infty)$ to a vertex $v$ in
$\alpha[0,\infty)$. As in the proof of Lemma 4.17, $\gamma$ can be written as
a concatenation

$\sigma_{1}\tau_{1}\sigma_{2}\tau_{2}\cdots\sigma_{\ell}\tau_{\ell}\sigma_{\ell+1}$

such that:

  1. (1)

Each $\tau_{i}$ intersects $X_{m}$ only at its endpoints
$x_{i}=\sigma_{i}\cap\tau_{i}$ and $y_{i}=\tau_{i}\cap\sigma_{i+1}$;

  2. (2)

Each $\sigma_{i}$ lies completely in the $1$–skeleton of $X_{m}$.

We also observe that $x_{i}^{-1}y_{i}$ is a group element in the cyclic
subgroup $\langle s_{m}\rangle$ or in the cyclic subgroup $\langle
s_{1}\rangle$ for each $i$.

If $x_{i}^{-1}y_{i}$ is a group element in the cyclic subgroup $\langle
s_{m}\rangle$, then we replace $\tau_{i}$ by a geodesic $\tau^{\prime}_{i}$
labeled by $s_{m}$. Call this path $\tau^{\prime}_{i}$ an $X_{m}$-replacement
segment of type 1. Note that $\tau^{\prime}_{i}$ in this case may have non-
empty intersection with the open ball $B(e,r/4)$.

If $x_{i}^{-1}y_{i}$ is a group element in the cyclic subgroup $\langle
s_{1}\rangle$, then by Lemma 4.16 we can replace $\tau_{i}$ a path
$\tau^{\prime}_{i}$ with edges labeled by $s_{1}$ and $a_{1}$ such that
$\tau^{\prime}_{i}$ lies outside the open ball $B(e,r/2)$ (therefore outside
the open ball $B(e,r/4)$) and $\ell(\tau^{\prime}_{i})\leq 11\text{
}d_{S_{m}}(x_{i},y_{i})\leq 11\ell(\tau_{i})$. Call this path an
$X_{m}$-replacement segment of type 2.

The new path
$\gamma^{\prime}=\sigma_{1}\tau^{\prime}_{1}\sigma_{2}\tau^{\prime}_{2}\cdots\sigma_{\ell}\tau^{\prime}_{\ell}\sigma_{\ell+1}$
lies completely in the $1$–skeleton of $X_{m}$, shares two endpoints $u$ and
$v$ with $\gamma$, and $\ell(\gamma^{\prime})\leq 11\ell(\gamma)$. We first
assume that $\gamma^{\prime}$ lies outside the open ball $B(e,r/4)$. By
inductive hypothesis we see that

$\ell(\gamma^{\prime})\geq\bigl{(}\frac{r}{4k_{m}}\bigr{)}f\bigl{(}\frac{r}{4k_{m}}\bigr{)}\text{
for $r$ sufficiently large}\,.$

This implies that

$\ell(\gamma)\geq\frac{1}{11}\ell(\gamma^{\prime})\geq\frac{1}{11}\bigl{(}\frac{r}{4k_{m}}\bigr{)}f\bigl{(}\frac{r}{4k_{m}}\bigr{)}\geq\bigl{(}\frac{r}{44k_{m}}\bigr{)}f\bigl{(}\frac{r}{44k_{m}}\bigr{)}\,.$

We now assume that $\gamma^{\prime}$ intersect the open ball $B(e,r/4)$. Then
there is an $X_{m}$-replacement segment $\tau^{\prime}$ of type $1$ of
$\gamma^{\prime}$ that intersects the open ball $B(e,r/4)$. Let $f$ be an edge
of $\tau^{\prime}$ that lies in the open ball $B(e,r/4+1)\subset B(e,r/2)$.
Then $f$ is label by $s_{m}$. Let $H$ be the hyperplane in $X_{m}$ that is
dual the edge $f$. Then $H$ must intersect $\gamma^{\prime}$. Let $x$ be the
point in the intersection $H\cap\gamma^{\prime}$. Then $x$ is an interior
point of an edge $f^{\prime}$ labeled by $s_{m}$ in $\gamma^{\prime}$. Let
$\alpha^{\prime}$ be the path labeled by $s_{1}$ in $star(H)-ostar(H)$ that
connect a vertex $w$ of $f$ to a vertex $w^{\prime}$ of $f^{\prime}$. If
$f^{\prime}$ is not an edge in any $X_{m}$-replacement segment of
$\gamma^{\prime}$, then $w^{\prime}$ is a vertex of $\gamma$. In this case, we
let $\widetilde{w}=w^{\prime}$ and $\widetilde{\alpha}=\alpha^{\prime}$. In
the case $f^{\prime}$ is an edge in some $X_{m}$-replacement segment
$\tau^{\prime\prime}$ of $\gamma^{\prime}$, we let $\widetilde{w}$ is an
endpoint of $\tau^{\prime\prime}$, let $\alpha^{\prime\prime}$ be the
subsegment $\tau^{\prime\prime}$ connecting $w^{\prime}$ and $\widetilde{w}$,
and let $\widetilde{\alpha}=\alpha^{\prime}\cup\alpha^{\prime\prime}$.
Therefore, $\widetilde{\alpha}$ is a part of an $(1,m)$–ray. Let
$\widetilde{v}$ be the endpoint of $\tau^{\prime}$ and let $\widetilde{\beta}$
is a subsegment of $\tau^{\prime}$ that connects $w$ and $\widetilde{v}$. Then
$\widetilde{\beta}$ is a part of an $m$–ray and therefore
$(\widetilde{\alpha},\widetilde{\beta})_{w}$ is a part of an $m$–corner.

Let $\eta$ be the subsegment of $\gamma$ that connects $\widetilde{v}$ and
$\widetilde{w}$. We note that $\eta$ lies outside the open ball $B(e,r)$ in
$X_{m+1}$. Also, $w$ lies in the open ball $B(e,r/2)$. Therefore, $\eta$ lies
outside the open ball $B(w,r/2)$ in $X_{m+1}$. In other words, $\eta$ is an
$(r/2)$–avoidant path over an $m$–corner in $X_{m+1}$. By the proof of Lemma
4.17 we have

$\ell(\gamma)\geq\ell(\eta)\geq\bigl{(}\frac{r}{32}\bigr{)}f\bigl{(}\frac{r}{8}\bigr{)}\geq\bigl{(}\frac{r}{32}\bigr{)}f\bigl{(}\frac{r}{32}\bigr{)}\,.$

for $r$ sufficiently large. Therefore, the lemma is true for $m+1$ with
$k_{m+1}=\max\\{44k_{m},32\\}$ and the lemma is proved. ∎

######  Proposition 4.28.

For each $m\geq 2$ the divergence of the cyclic subgroup $\langle
s_{1}s_{2}\rangle$ in $G_{m}$ is equivalent to $rf(r)$. In particular, if $F$
is distorted in $H$ then the group element $s_{1}s_{2}$ is Morse but not
contracting and therefore $G_{m}$ is not a $\operatorname{CAT}(0)$ group.

###### Proof.

By Lemma 4.25 and Lemma 4.27 the divergence of the cyclic subgroup $\langle
s_{1}s_{2}\rangle$ is equivalent to $rf(r)$. We now assume that $F$ is
distorted in $H$. Therefore, the divergence of the cyclic subgroup $\langle
s_{1}s_{2}\rangle$ is super linear but strictly by a quadratic function. By
Theorem 1.3 in [ACGH17] and Proposition 2.7 the group element $s_{1}s_{2}$ is
Morse but not contracting. This implies that $G_{m}$ is not a
$\operatorname{CAT}(0)$ group by Theorem 2.14 in [CS15]. ∎

######  Remark 4.29.

In section 5 we show that for the particular examples of
$\operatorname{CAT}(0)$ groups $B_{m}$ and subgroups $G_{m}$, none of the
groups $G_{m}$ ($m\geq 1$) are $\operatorname{CAT}(0)$.

##  5\. Embeddings in $\operatorname{CAT}(0)$ groups

The main embedding result of this section relies on $\operatorname{CAT}(0)$
free-by-cyclic groups with palindromic monodromies. These are described in the
first subsection. The second subsection includes the construction of a
sequence of $\operatorname{CAT}(0)$ groups $B_{m}$ which mirrors the
construction of the groups $G_{m}$ in Theorem 2.13. The groups $B_{m}$ are
shown to contain the $G_{m}$ of Theorem 2.13. In the third subsection we
provide explicit examples of $B_{m}$ containing $G_{m}$ with divergence
$r^{\alpha}$ and $r^{q}\log(r)$.

We refer the reader to [BH99] for definitions and background of non-positively
curved (n.p.c.) and $\operatorname{CAT}(0)$ metric spaces and
$\operatorname{CAT}(0)$ groups.

###  5.1. $\operatorname{CAT}(0)$ Free-by-cyclic building blocks

In [Bra95], T. Brady showed that each group of the form
$F_{2}\rtimes_{\varphi}\mathbb{Z}$ is CAT(0) by describing how to build a non-
positively curved, piecewise euclidean 2–complex with this as fundamental
group. Many of the structures in [Bra95] enjoy additional symmetry properties
which reflect symmetries of the corresponding automorphisms of $F_{2}$.

In this section we will make extensive use of the palindromic automorphism

$\varphi:F_{\\{a,b\\}}\to F_{\\{a,b\\}}:a\mapsto aba,\;b\mapsto a$

and the corresponding free-by-cyclic group

$F\rtimes_{\varphi}\langle t_{0}\rangle=\langle
a,b,t_{0}|t_{0}at_{0}^{-1}=aba,t_{0}bt_{0}^{-1}=a\rangle$

as well as its index $n$ subgroup $\langle a,b,t_{0}^{n}\rangle$. In [BF17]
the non-positively curved, piecewise euclidean 2–complex $Z_{0}$ corresponding
to the automorphism $\varphi$ was seen to admit an isometric involution
$\tau_{Z_{0}}\\!:Z_{0}\to Z_{0}$ which induces on automorphism
$(\tau_{Z_{0}})_{\ast}:\pi_{1}(Z_{0})\to\pi_{1}(Z_{0})$ on the fundamental
group taking $a$ to $a^{-1}$, $b$ to $b^{-1}$, and $t_{0}$ to itself. This
lifts to an isometric involution $\tau_{Z}:Z\to Z$ of the $n$–fold cyclic
cover $Z$ of $Z_{0}$ corresponding to the subgroup
$A=F\rtimes_{\varphi^{n}}\langle t\rangle$ where $t=(t_{0})^{n}$. The map
$\tau_{Z}$ induces an automorphism $\tau_{A}:A\,\to\,A:a\mapsto
a^{-1},\,b\mapsto b^{-1},\,t\mapsto t$.

We will build new non-positively curved complexes by isometrically glueing
simpler non-positively curved complexes along isometrically embedded (scaled)
copies of $Z$. Since an isometry $Z_{0}\to Z_{0}$ lifts (modulo choice of
basepoints) to a unique isometry $Z\to Z$, we can describe these glueing maps
on the level of $Z_{0}$. Therefore, it will be useful to have a notation for
different copies of $Z_{0}$. To this end, let $Z_{0,x}$ denote a copy of
$Z_{0}$ with edges relabeled as follows

$a\longrightarrow x_{1}\,,\quad\quad b\longrightarrow x_{2}\,,\quad\quad
t_{0}\longrightarrow t_{0,x}\,.$

This relabeling map determines an isometry $Z_{0}\to Z_{0,x}$. This lifts to a
unique (modulo choice of basepoints) isometry of the $n$–fold cyclic covers
$Z\to Z_{x}$. On the level of fundamental groups this induces the relabeling
isomorphism $A\to A_{x}$ where

$A\;=\;\langle
a,b,t\,|\,tat^{-1}=\varphi^{n}(a),tbt^{-1}=\varphi^{n}(b)\rangle$

and

$A_{x}\;=\;\langle
x_{1},x_{2},t_{x}\,|\,t_{x}x_{1}t_{x}^{-1}=\varphi^{n}(x_{1}),t_{x}x_{2}t_{x}^{-1}=\varphi^{n}(x_{2})\rangle\,.$

Likewise, we define spaces $Z_{y}$, $Z_{z}$, $Z_{d}$, $Z_{xz}$, $Z_{yz}$, and
$Z_{x\overline{y}}$. Double subscript labels are defined as follows

$a\longrightarrow x_{1}z_{1}\,,\quad\quad b\longrightarrow
x_{2}z_{2}\,,\quad\quad t_{0}\longrightarrow t_{0,x}t_{0,z}$

and

$a\longrightarrow x_{1}\overline{y}_{1}\,,\quad\quad b\longrightarrow
x_{2}\overline{y}_{2}\,,\quad\quad t_{0}\longrightarrow t_{0,x}t_{0,y}\,.$

They induce relabeling isomorphisms

$A\;\to\;A_{xz}:\;a\mapsto x_{1}z_{1},\;b\mapsto x_{2}z_{2},\;t\mapsto
t_{x}t_{z}$

and

$A\;\to\;A_{x\overline{y}}:\;a\mapsto x_{1}y_{1}^{-1},\;b\mapsto
x_{2}y^{-1}_{2},\;t\mapsto t_{x}t_{y}\,.$

###  5.2. The $\operatorname{CAT}(0)$ construction

The goal of this subsection is to construct a sequence of
$\operatorname{CAT}(0)$ groups $B_{m}$ which will be shown to contain the
groups $G_{m}$ of Theorem 2.13 as subgroups. The group $B_{m}$ are constructed
inductively via amalgams and HNN extensions, mirroring the construction of the
$G_{m}$. Note that $G_{1}$ is defined by starting with an amalgamation

$H\;\ast_{\langle d_{i}=x_{i}y^{-1}_{i}\rangle}(F_{x}\times F_{y}\times
F_{z})$

where the $F_{d}=\langle d_{1},\ldots,d_{p}\rangle$ subgroup has non-trivial
distortion in $H$. The $\operatorname{CAT}(0)$ analogue will be constructed as
an amalgamation over an undistorted edge subgroup. This is achieved by
embedding $F_{d}$ as a distorted fiber in a $\operatorname{CAT}(0)$ free-by-
cyclic group $A_{d}$ and using $A_{d}$ as the edge group for the
$\operatorname{CAT}(0)$ amalgam. We work in the case $p=2$. Here are the
details.

Recall that the $G_{m}$ are defined inductively via HNN extensions starting
from $G_{1}$. The group $G_{1}$ is built from an amalgam of a group $H$ and a
direct product of three free groups amalgamated over a free group. We describe
$\operatorname{CAT}(0)$ groups containing each of these components.

_The $\operatorname{CAT}(0)$ vertex group containing the product of free
groups._ Let $F_{x}$, $F_{y}$, $F_{z}$ be rank-two free groups generated by
the sets $\\{x_{1},x_{2}\\}$, $\\{y_{1},y_{2}\\}$, and $\\{z_{1},z_{2}\\}$
respectively. Let $\varphi$ be the automorphism of section 5.1, and define
$A_{x}=F_{x}\rtimes_{\varphi^{n}}\langle t_{x}\rangle$,
$A_{y}=F_{y}\rtimes_{\varphi^{n}}\langle t_{y}\rangle$, and
$A_{z}=F_{z}\rtimes_{\varphi^{n}}\langle t_{z}\rangle$. Note that $A_{x}$ is
the fundamental group of the locally $\operatorname{CAT}(0)$ space $Z_{x}$
described in section 5.1, and similarly for $A_{y}$ and $A_{z}$. Note that

$(F_{x}\times F_{y}\times
F_{z})\,\cap\,A_{x\overline{y}}\;=\;F_{x\overline{y}}$ (1)

This identity will be used in proving that $G_{1}\leq B_{1}$ later on.

_The $\operatorname{CAT}(0)$ vertex group containing $H$._ Let $X$ be a
locally $\operatorname{CAT}(0)$ metric space whose fundamental group is $C$.
Assume that $X$ contains an isometric embedding copy $Z_{d}$ of $Z$ with the
fundamental group $A_{d}=F_{d}(d_{1},d_{2})\rtimes_{\varphi^{n}}\langle
t_{d}\rangle$. We also assume that the group $C$ contains the group $H$ in
$G_{1}$ such that

$H\,\cap\,A_{d}\;=\;F_{d}$ (2)

This identity will also be used in the proof that $G_{1}\leq B_{1}$ later on.
In section 5.3 we give two explicit examples of $X$ and $C$. For now the
reader can just work with the assumption that spaces and group with these
properties exist.

_The $\operatorname{CAT}(0)$ edge group._ The $\operatorname{CAT}(0)$ edge
group $A_{d}=F_{d}\rtimes_{\varphi^{n}}\mathbb{Z}$ contains the edge group
$F_{d}$ in the amalgamation in the definition of $G_{1}$. Note that $A_{d}$ is
undistorted in each of the vertex groups above. However, the exponential
distortion of $F_{d}$ in $A_{d}$ gives room for the distortion of $F_{d}$ in
the vertex group $H$.

We define the family of groups $(B_{m})$ by induction on $m$ as follows:

$\displaystyle B_{1}\;=\;\bigl{[}C\ast_{\langle
d_{i}=x_{i}y_{i}^{-1},t_{d}=t_{x}t_{y}\rangle}(A_{x}\times A_{y}\times
A_{z})\bigr{]}\times{\langle s_{1}\rangle};$ $\displaystyle B_{2}\;=\;\langle
B_{1},s_{2}|s_{2}^{-1}(x_{i}z_{i})s_{2}=y_{i}z_{i},s_{2}^{-1}(t_{x}t_{z})s_{2}=t_{y}t_{z}\rangle;$
and $\displaystyle B_{m}\;=\;\langle
B_{m-1},s_{m}|s_{m}^{-1}s_{1}s_{m}=s_{m-1}\rangle\quad\quad{\hbox{ for $m\geq
3$.}}$

The following proposition will be used in the proof that each group $B_{m}$ is
$\operatorname{CAT}(0)$.

######  Proposition 5.1 ([BH99], II.11.13).

Let $X$ and $Y$ be locally $\operatorname{CAT}(0)$ metric spaces. If $Y$ is
compact and $f,g:\\!Y\to X$ are locally isometric immersions, then the
quotient of $X\bigsqcup(Y\times[0,1])$ by the equivalence relation generated
by $(y,0)\sim f(y);(y,1)\sim g(y),\forall y\in Y$ is locally
$\operatorname{CAT}(0)$.

######  Proposition 5.2.

$B_{m}$ is $\operatorname{CAT}(0)$ for $m\geq 1$.

###### Proof.

There are three cases to consider; $m=1$, $m=2$, and $m\geq 3$.

_Case $m=1$._ Let $Z_{x}$, $Z_{y}$, and $Z_{z}$ be copies of $Z$ whose
fundamental groups are $A_{x}$, $A_{y}$, and $A_{z}$ respectively. We equip
$Z_{x}\times Z_{y}\times Z_{z}$ with the product metric. Then $Z_{x}\times
Z_{y}\times Z_{z}$ is a locally $\operatorname{CAT}(0)$ space and
$\pi_{1}(Z_{x}\times Z_{y}\times Z_{z})=A_{x}\times A_{y}\times A_{z}$.

Define a map $g\\!:Z\to Z_{x}\times Z_{y}$ by $g(p)=(p,\tau_{Z}(p))$. We
observe that the induced homomorphism $g_{*}\\!:A\to A_{x}\times A_{y}$ is
given by $a\mapsto x_{1}y_{1}^{-1}$, $b\mapsto x_{2}y_{2}^{-1}$, and $t\mapsto
t_{x}t_{y}$. Metrically, the map $g$ behaves as follows:

$\displaystyle d\bigl{(}g(p),g(q)\bigr{)}$
$\displaystyle=d\bigl{(}(p,\tau_{Z}(p)),(q,\tau_{Z}(q))\bigr{)}$
$\displaystyle=\Bigl{(}\bigl{(}d(p,q)\bigr{)}^{2}+\bigl{(}d(\tau_{Z}(p),\tau_{Z}(q))\bigr{)}^{2}\Bigr{)}^{1/2}=\sqrt{2}d(p,q).$

Hence $g$ induces an isometric embedding of the scaled metric space
$(\sqrt{2})Z$ into $Z_{x}\times Z_{y}\subset Z_{x}\times Z_{y}\times Z_{z}$
and we let $Z_{x\overline{y}}$ denote the image of $(\sqrt{2})Z$ via this
embedding.

Let $X$ be the locally $\operatorname{CAT}(0)$ space we discussed above.
Recall that the fundamental group of $X$ is $C$ and that $X$ contains an
isometrically embedded copy $Z_{d}$ of $Z$ whose fundamental group is the
subgroup $A_{d}=F_{d}(d_{1},d_{2})\rtimes_{\varphi^{n}}\langle t_{d}\rangle$
of $C$. Rescale the metric on $X$ such that $Z_{d}$ is isometric to
$(\sqrt{2})Z$.

Define

$W=\bigl{(}X\bigsqcup(Z_{x}\times Z_{y}\times
Z_{z})\bigsqcup\bigl{(}\sqrt{2})Z\times[0,1]\bigr{)}_{/\sim}$

where the equivalence relation generated by mapping $(\sqrt{2})Z\times\\{0\\}$
isometrically onto $Z_{d}\subset X$ and mapping $(\sqrt{2})Z\times\\{1\\}$
isometrically onto $Z_{x\overline{y}}\subset Z_{0}\times Z_{1}\times Z_{2}$
appropriately. Therefore, $W$ is a locally $\operatorname{CAT}(0)$ space by
Proposition 5.1. By Van Kampen’s theorem, we also have

$\pi_{1}(W)=C\ast_{\langle
d_{i}=x_{i}y_{i}^{-1},t_{d}=t_{x}t_{y}\rangle}(A_{x}\times A_{y}\times
A_{z}).$

Let $S_{1}$ be an oriented circle labeled by $s_{1}$ and let $X_{1}=W\times
S_{1}$. We equip $X_{1}=W\times S_{1}$ with the product metric. Then $X_{1}$
is a locally $\operatorname{CAT}(0)$ space and $\pi_{1}(X_{1})=B_{1}$.

_Case $m=2$._ We first show that $Z_{x}\times Z_{z}$ contains an isometrically
embedded copy $Z_{xz}$ of $(\sqrt{2})Z$ whose fundamental group is the
subgroup $A_{xz}$ of $A_{x}\times A_{z}$ generated by $x_{1}z_{1}$,
$x_{2}z_{2}$, $t_{x}t_{z}$. To see this, define a map $g^{\prime}\\!:Z\to
Z_{x}\times Z_{z}$ by $g^{\prime}(p)=(p,p)$. The induced homomorphism
$g^{\prime}_{*}\\!:A\to A_{x}\times A_{z}$ is given by $a\mapsto x_{1}z_{1}$,
$b\mapsto x_{2}z_{2}$, and $t\mapsto t_{x}t_{z}$. Metrically, the map
$g^{\prime}$ behaves as follows:

$d\bigl{(}g^{\prime}(p),g^{\prime}(q)\bigr{)}\;=\;d\bigl{(}(p,p),(q,q)\bigr{)}\;=\;\Bigl{(}\bigl{(}d(p,q)\bigr{)}^{2}+\bigl{(}d(p,q)\bigr{)}^{2}\Bigr{)}^{1/2}\;=\;\sqrt{2}d(p,q)\,.$

Hence $g^{\prime}$ induces an isometric embedding of the scaled metric space
$(\sqrt{2})Z$ into $Z_{x}\times Z_{z}\subset Z_{x}\times Z_{y}\times Z_{z}$.
Then the image $Z_{xz}$ of $(\sqrt{2})Z$ via this embedding is the desired
subspace of $Z_{x}\times Z_{z}$. Similarly, $Z_{y}\times Z_{z}$ contains an
isometric embedding copy $Z_{yz}$ of $(\sqrt{2})Z$ whose fundamental group is
the subgroup $A_{yz}$ of $A_{y}\times A_{z}$ generated by $y_{1}z_{1}$,
$y_{2}z_{2}$, $t_{y}t_{z}$.

Let
$X_{2}={\bigl{(}X_{1}\bigsqcup\bigl{(}(\sqrt{2})Z\times[0,1]\bigr{)}\bigr{)}}_{/\sim}$,
where the equivalence relation generated by mapping $(\sqrt{2})Z\times\\{0\\}$
isometrically onto $Z_{xz}\subset X_{1}$ and mapping
$(\sqrt{2})Z\times\\{1\\}$ isometrically onto $Z_{yz}\subset X_{1}$
appropriately. Then $X_{2}$ is also a locally $\operatorname{CAT}(0)$ space by
Proposition 5.1 and $\pi_{1}(X_{2})=B_{2}$.

_Case $m\geq 3$._ For each $m\geq 3$ we construct the locally
$\operatorname{CAT}(0)$ space $X_{m}$ by induction on $m$ as follows. Let
$\Sigma_{m}$ be an oriented circle. Let
$X_{m}={\bigl{(}X_{m-1}\bigsqcup\bigl{(}\Sigma_{m}\times[0,1]\bigr{)}\bigr{)}}_{/\sim}$,
where the equivalence relation generated by mapping $\Sigma_{m}\times\\{0\\}$
isometrically onto the oriented circle labeled by $s_{1}$ in $X_{m-1}$ and
mapping $\Sigma_{m}\times\\{1\\}$ isometrically onto the oriented circle
labeled by $s_{m-1}$ in $X_{m-1}$ appropriately. Then $X_{m}$ is a locally
$\operatorname{CAT}(0)$ space by Proposition 5.1 and $\pi_{1}(X_{m})=B_{m}$. ∎

The following lemma is Lemma 5.1 from [BF17]. It is a special case of a result
of Bass [Bas93]. It will be used to prove that $G_{m}\leq B_{m}$ for each
$m\geq 1$.

######  Lemma 5.3 (Injectivity for graphs of groups).

Suppose $\mathcal{A}$ and $\mathcal{B}$ are graphs of groups such that the
underlying graph $\Gamma_{\mathcal{A}}$ of $\mathcal{A}$ is a subgraph of the
underlying graph of $\mathcal{B}$. Let $A$ and $B$ be their respective
fundamental groups. Suppose that there are injective homomorphisms
$\psi_{e}\\!:A_{e}\to B_{e}$ and $\psi_{v}\\!:A_{v}\to B_{v}$ between edge and
vertex groups, for all edges $e$ and vertices $v$ in $\Gamma_{\mathcal{A}}$,
which are compatible with the edge-inclusion maps. That is, whenever $e$ has
initial vertex $v$, the diagram

${A_{e}}$${A_{v}}$${B_{e}}$${B_{v}}$$\scriptstyle{i_{e}}$$\scriptstyle{\psi_{e}}$$\scriptstyle{\psi_{v}}$$\scriptstyle{j_{e}}$

commutes.

If $j_{e}(\psi_{e}(A_{e}))=\psi_{v}(A_{v})\cap j_{e}(B_{e})$ whenever $e$ has
initial vertex $v$, then the induced homomorphism $\psi\\!:A\to B$ is
injective.

######  Proposition 5.4.

$G_{m}$ is a subgroup of $B_{m}$ for $m\geq 1$.

###### Proof.

There are three cases to consider; $m=1$, $m=2$, and $m\geq 3$.

_Case $m=1$._ Recall that

$G_{1}\;=\;\bigl{[}H\ast_{\langle d_{i}=x_{i}y_{i}^{-1}\rangle}(F_{x}\times
F_{y}\times F_{z})\bigr{]}\times{\langle s_{1}\rangle}$

and

$B_{1}\;=\;\bigl{[}C\ast_{\langle
d_{i}=x_{i}y_{i}^{-1},t_{d}=t_{x}t_{y}\rangle}(A_{x}\times A_{y}\times
A_{z})\bigr{]}\times{\langle s_{1}\rangle}\,.$

It sufficient to prove an inclusion $G^{\prime}_{1}\to B^{\prime}_{1}$ for

$G^{\prime}_{1}\;=\;H\ast_{\langle d_{i}=x_{i}y_{i}^{-1}\rangle}(F_{x}\times
F_{y}\times F_{z})$

and

$B^{\prime}_{1}\;=\;C\ast_{\langle