# Divergence property of the Brown-Thompson groups and braided Thompson groups

Xiaobing Sheng  Graduate School of Mathematical Sciences, The University of
Tokyo<EMAIL_ADDRESS>
###### Abstract.

Golan and Sapir proved that Thompson’s groups $F$, $T$ and $V$ have linear
divergence. In the current paper, we focus on the divergence property of
several generalisations of the Thompson groups. We first consider the Brown-
Thompson groups $F_{n}$, $T_{n}$ and $V_{n}$ (also called Brown-Higman-
Thompson group in some other context) and find that these groups also have
linear divergence functions. We then focus on the braided Thompson groups
$BF,$ $\widehat{BF}$ and $\widehat{BV}$ and proved that these groups have
linear divergence. The case of $BV$ has also been done independently by
Kodama.

###### Key words and phrases:

generalised Thompson groups, divergence property

###### 2020 Mathematics Subject Classification:

Primary 20F65,

######

  1. 1 Introduction
    1. 1.1 Background
  2. 2 Basic definitions and notation
    1. 2.1 The group elements, tree pair representations
    2. 2.2 Tree pairs, the Cantor set and the binary words
    3. 2.3 Branches and branches of a tree
    4. 2.4 The estimate of the word lengths via tree pair representations
      1. 2.4.1 The estimate of the word length of elements in $F_{n}$, $T_{n}$
      2. 2.4.2 The definition of $V_{n}$ from a logic viewpoint
      3. 2.4.3 An estimate of the word length of elements in $V_{n}$ from the logic viewpoint
  3. 3 Divergence functions of the generalised Thompson’s groups (the Brown-Thompson groups)
    1. 3.1 The Divergence function
    2. 3.2 Divergence property of $F_{n},T_{n},V_{n}$
  4. 4 Divergence functions for the braided Thompson’s group
    1. 4.1 The braided Thompson groups
    2. 4.2 Divergence property of the braided Thompson groups $BF$, $BV$
    3. 4.3 Divergence property of the geodesics

##  1\. Introduction

Thompson’s groups $F$, $T$, $V$ were first constructed from a logic point of
view while later found to have connections with many other branches of
mathematics such as string rewriting systems, homotopy theory, combinatorics
and dynamical systems.

$F$ and $T$ can be regarded as subgroups of the group of homeomorphisms of the
circle while $V$ can be seen as a subgroup of the group of self-homomorphisms
of the Cantor set. These groups have been generalised to many larger classes
since they were defined. Thompson’s group $V$ has first been generalised to a
family of finitely presented infinite groups by Higman in the 70’s and the
groups $F$, $T$ and $V$ have further been extended to infinite families by
Brown [7] and later being generalised by Stein [27] and at the same time, some
of their homological and simplicity results are obtained[7]. More recently,
the braided version and the higher dimensional version ones are defined and
investigated.

Among these generalisations of Thompson’s groups, most of the early results
are on algebraic and topological properties of groups. The geometry of the
groups are beginning to attract the interest of more geometric group theorists
in the recent years partially because of the still open amenability problem of
Thompson’s group $F.$

We list a selection of the geometric results as follows: the result on the
quadratic Dehn function of $F$ by Guba [23, 22], the construction of the
$CAT(0)$ cube complexes, on which the groups $F$, $T$, $V$ act properly by
Farley [17], some subgroup distortion results by Burillo, et al. [9, 8, 3, 11]
and results by Golan and Sapir [20] on the divergence property of the three
groups. A more recent result on the linear divergence of the braided
Thompson’s group $BV$ has been proved by Kodama [24].

###  1.1. Background

This work focuses on one of the central notion in geometric group theory,
“hyperbolicity” or “the nature of non-positive curvature” of groups and we
consider some generalisations of a particular class of groups, Thompson’s
groups.

Thompson’s groups are not Gromov hyperbolic, but the above geometric results
on the groups suggest that these groups might have some “hyperbolicity”s in a
coarse sense. Then the question is how we are going to measure these
“hyperbolicity”, since there might be many generalisations of the notion of
hyperbolic-type properties. The one that we will be focusing on, in the
current paper, is the divergence property.

The idea of divergence property first appeared in Asymptotic invariants of
infinite groups by Gromov [21], where he suggested that in the non-positively
curved spaces, One expects, that (at least after plusifications) ‘infinitely
close’ geodesic rays issuing from a point, either diverge linearly or
exponentially. The formal definition was first stated by Gersten [19], and the
notion later proved to be a quasi-isometric invariant of metric spaces.

Intuitively, divergence property measures how two geodesics in a metric space
go apart, or in other words, how hard it is to connect points on two distinct
geodesic rays if the backtracking is not allowed. When considering the notion
of the generalised “hyperbolicity ” or the hyperbolic-type properties, in some
particularly “nice” space ( for example, Gromov hyperbolic space, CAT(0)
spaces), the condition of super-linear divergence is equivalent to the notion
of having cut-points in the asymptotic cones in the groups or the “Morse”
property.

When enlarging the scope further, the divergence property also indicates some
large scale geometric properties of the spaces and their associated groups.
Since the notion divergence is a quasi-isometry invariant, we would know
precisely the quasi-isometry classes of the groups.

On the other hand, in Tran’s thesis [28], he investigated the divergence
property of finitely presented groups, and he further defined relative
divergence and the usual definition of divergence in the sense of Gersten [19]
which gives a lower bound on the divergence functions of all the geodesics in
the whole space.

In this paper, we focus on some generalisations of Thompson’s groups and prove
that similar divergence property can be found on geodesics of these groups.
More precisely, we prove that Brown-Thompson groups and the braided Thompson
group $BF$ have linear divergence property. The result for $BF$ together with
the result for $BV$ in [24] can be extended to obtain that $\widehat{BV}$ and
$\widehat{BF}$ also have linear divergence.

##  2\. Basic definitions and notation

Thompson’s groups $F$, $T$ as well as the Brown-Thompson groups $F_{n}$,
$T_{n}$ can be informally considered as subgroups of piecewise-linear
homeomorphisms of the unit interval $[0,1]$ and the circle $S^{1}$,
respectively, while the group $V$ and $V_{n}$ can be regarded as subgroups of
the homeomorphisms of the Cantor set and they all have several different
interpretations. Here, we will be following the convention in [20] and
utilising both combinatorial and analytical definitions.

######  Definition 2.1 (Thompson’s groups $F$, $T$).

Thompson’s groups $F$, $T$, are the groups of orientation-preserving
piecewise-linear homeomorphisms of the unit interval $[0,1]$ and the $S^{1}$
where $S^{1}$ is the unit interval $[0,1]$ identifying the endpoints to
themselves, respectively, which are differentiable except at finitely many
dyadic rationals such that the slope of each subinterval is an integer power
of $2.$

######  Definition 2.2 (The Brown-Thompson groups).

The Brown-Thompson groups $F_{n}$, $T_{n}$, are the groups of orientation-
preserving piecewise-linear homeomorphisms of the unit interval $[0,1]$ and
the circle $S^{1}$, respectively, which are differentiable except at finitely
many $n$-adic rationals such that the slope of each subinterval is an integer
power of $n.$

######  Definition 2.3 (Thompson’s group $V$ and its generalisations $V_{n}$).

$V$ and $V_{n}$ are the groups of the right-continuous bijections from the
unit interval $[0,1]$ onto itself which are differentiable except at finitely
many dyadic and $n$-adic rationals, respectively, such that the slope of each
interval is a power of $2$ and an integer power of $n,$ respectively.

From this standard interpretation of the groups, we define the following
notion: an element $g\in V$ is supported on some interval $(a,b)\subset[0,1]$
meaning that for any point $x\in(a,b),$ we have $g(x)\neq x.$ The collection
of such subintervals in $[0,1]$ are called the support of $g.$ The idea can be
extended to the Brown-Thompson groups as well. The Thompson groups have many
different definitions [25, 14, 8]. Among these different interpretations of
the elements of the Thompson groups, we will mainly use the tree pair
representations.

###  2.1. The group elements, tree pair representations

The tree pair representations come from the fact that intervals or circles
with partitions by dyadic rationals can be identified with trees.

A pair of rooted $n$-ary trees having the same number of leaves with identical
labelings give rise to a representation for an element of $F_{n}$, a pair of
such trees with cyclic labelings represent an element of $T_{n}$ [26] and a
pair with labelings induced from an element of the symmetric group
$S_{l(n-1)+1}$ representing an element of $V_{n},$ here $l\in\mathbb{N}$ and
$l(n-1)+1$ is the number of the leaves in each tree of the pair. Hence, the
elements in Thompson’s groups and the generalisations can be represented by
such labeled tree pairs.

Such pairs of trees can be used to give the unique normal form for elements in
$F,F_{n}$ and a standard form for elements in $T,T_{n}$ with respect to the
infinite standard generating sets. These forms can be used to estimate the
word lengths of the elements of these groups and the bounds of the word
lengths of elements in $V,V_{n}$ by replacing the standard infinite generating
sets by the finite ones. This alternative definition is also given in [8, 11].
Below is a more precise description for the case of the group $F.$

For each unit interval $[0,1]$ with finitely many dyadic breakpoints, the
subintervals are of the form $[\frac{k}{2^{m}},\frac{k+1}{2^{m}}]$ where
$k+1\leq 2^{m}$ and $m,k\in\mathbb{N}\cup\\{0\\}$. A unit interval $[0,1]$
with finitely many breakpoints are called dyadically subdivided interval and
can be associated with a rooted finite binary tree by associating each of
these breakpoints with a $2$-caret, i.e. a rooted binary tree with one vertex
to be the root and two edges attached to it. Then we can associate two
intervals having the same number of dyadic breakpoints with a pair of finite
binary trees with the same number of leaves, namely, we have associated an
element of Thompson’s group $F$ with a pair of rooted finite binary trees.

Here the leaves in each tree of a tree pair correspond to the dyadic
subintervals in each $[0,1]$ and they are labeled by the natural number
$\mathbb{N}$ to indicate which leaf in the source tree maps to which leaf in
the target. It is proved in [14] that there is a uniquely reduced tree pair
representative for each element in $F.$ The above construction can be
generalised to $T$ and $V$ with some variations on the labelings as well as to
the Brown-Thompson groups $F_{n}$, $T_{n}$ and $V_{n}$ by changing binary
trees to the $n$-ary trees.

The Brown-Thompson groups $F_{n}$ $T_{n}$ and $V_{n}$ as the groups $F,$ $T,$
$V$ are proved to be finitely presented [14, 7]. Here we illustrate the tree
pair representations of the group elements in the standard finite generating
sets $\mathcal{C}$ to give a brief idea of this form in Figure 1.

Figure 1. The infinite generating set for $F_{n}$, $T_{n}$ and $V_{n}$

###  2.2. Tree pairs, the Cantor set and the binary words

Alternatively, when we consider such tree pairs, we consider the
correspondence between a rooted finite binary tree and a different rooted
finite binary tree with the same number of leaves which can be regarded as a
partial automorphism of the infinite binary tree [18] such that only the
finite top part from the root of the tree is replaced. In addition, since the
boundary of the infinite binary tree can be identified with the Cantor set, an
element of Thompson’s group $V$ can be regarded as the homeomorphisms of the
Cantor set that preserves finite number of the partitions of the Cantor set.

We consider again the tree pair representations as partial automorphisms of
the rooted infinite binary tree denoted by $\mathcal{T}_{\infty}.$ For
$\mathcal{T}_{\infty},$ each node $\nu$ can be associated with a unique finite
path from the root, and we can read off a binary word from the root along the
path to the node $\nu$ as follows: we identify the root of the binary tree
with the empty word $\varepsilon,$ when the path going down to the lower left
vertex through an edge in a $2$-caret, we add $0$ to the word representing the
path and when the path going down to the lower right vertex in the same
$2$-caret, we concatenate a $1$ to the word representing the path. Then, each
leaf in a rooted finite binary tree can be identified with a finite binary
word in alphabet $\\{0,1\\}.$ The correspondence between a finite binary tree
and another finite binary tree with the same number of the leaves can then be
regarded as the correspondence between a finite set of binary words and
another finite set of binary words of the same order. These lead to the
concept of branches.

###  2.3. Branches and branches of a tree

Following the description in Golan and Sapir [20], we let the symbol
$(\mathrm{T}_{+},\sigma,\mathrm{T}_{-})$ denote a tree pair defining an
element $g\in V$, where $\mathrm{T}_{+}$ and $\mathrm{T}_{-}$ are rooted
finite binary trees with the same number of the leaves and $\sigma$ denotes
the permutation of the order of the leaves.

######  Definition 2.4.

For every leaf in some finite rooted binary tree $\mathrm{T},$ we associate
with it a path labelled by a binary word from the root to $\gamma_{o}$ with
the notation $\ell_{\gamma_{o}}(\mathrm{T}).$ We call the path
$\ell_{\gamma_{o}}(\mathrm{T})$ a branch of the tree.

By identifying a leaf in the finite binary tree with a finite path and hence
with a finite binary word $\omega_{o}$ over the alphabet $\\{0,1\\},$ we could
express the correspondence between a leaf labeled $\omega_{o}$ in source tree
$\mathrm{T}_{+}$ and a leaf labeled $\omega_{\sigma(0)}$ in the target tree
$\mathrm{T}_{-}$ of a tree pair $(\mathrm{T}_{+},\sigma,\mathrm{T}_{-})$
representing the group element $g$ of $V$ by the correspondence between a
finite path $\ell_{\omega_{o}}(\mathrm{T}_{+})$ in the source tree
$\mathrm{T}_{+}$ and a finite path $\ell_{\omega_{\sigma(o)}}(\mathrm{T}_{-})$
in the target tree $\mathrm{T}_{-}.$

######  Definition 2.5.

We denote the correspondence by
$\ell_{\omega_{o}}(\mathrm{T}_{+})\mapsto\ell_{\omega_{\sigma(o)}}(\mathrm{T}_{-})$
for $(\mathrm{T}_{+},\sigma,\mathrm{T}_{-})$ representing $g\in V$ and call it
a branch of tree pair $(\mathrm{T}_{+},\sigma,\mathrm{T}_{-})$ or a branch of
the group element $g$ that the tree pair represents.

We can now generalise the notion to as follows: we denote, similarly, by
$(\mathrm{T}_{n(+)},\sigma,\mathrm{T}_{n(-)}),$ a tree pair where
$\mathrm{T}_{n}$ is an $n$-ary tree, for an element $g\in\mathcal{G}.$ Here
$\mathcal{G}$ could be $F_{n}$, $T_{n}$, $V_{n}$ where $n\in\mathbb{N}$ and
$n\geq 2.$ When $\mathcal{G}=F_{n}$, $\sigma$ represents trivial permutation,
when $\mathcal{G}=T_{n}$, $\sigma$ is a cyclic permutation in the cyclic group
of order $l(n-1)+1$ where $l\in\mathbb{N}$ and $\sigma$ represents a
permutation in the symmetric group $\mathfrak{S}_{l(n-1)+1}$ for some
$l\in\mathbb{N}$ when $\mathcal{G}=V_{n}.$

######  Definition 2.6.

Then we could associate a path labelled by an $n$-ary word from the root of
the tree to some leaf $\omega_{o}$ in the source tree to the notation
$\ell_{\omega_{o}}(\mathrm{T}_{n}).$ Similarly, we denote by
$\ell_{\omega_{o}}(\mathrm{T}_{n(+)})\mapsto\ell_{\omega_{\sigma(o)}}(\mathrm{T}_{n(-)})$
some branch of the element $g\in\mathcal{G}.$

Note that the words corresponding to the leaves are not binary words anymore,
they are now some $n$-ary words.

The branches of a tree pair representation provide some partial information of
a group element. For instant, when considering the group elements of the
Thompson groups to be the maps from the unit interval to itself, a branch of
the unique tree pair representing a group element mapping a branch of the
source tree to a branch of the target tree only gives the map between two
subintervals which reflects the self-similarity properties of the trees. More
precisely, when we identify the unit interval $[0,1]$ with an $n$-nary tree
$\textrm{T}_{n(\pm)}$, a branch of a tree can be interpreted as a subinterval
of $[0,1]$ with $n$-adic bounds and a branch of a tree pair can be seen as the
correspondence between two subintervals of the source interval $[0,1]$ and the
target interval $[0,1]$ respectively.

When given the product of two group elements represented by tree pairs, the
support, branches and the tree pair representation of the product can be
interchanged to provide desired information.

We introduce another notation before we connect the dots together.

######  Definition 2.7.

Let $g\in V_{n},$ represented by tree pair
$(\mathrm{G}_{+},\sigma_{g},\mathrm{G}_{-}).$ We denote by
$g_{[\omega^{\prime}]}$ the tree pair constructed as follows: Take an element
$h\in V_{n}$ represented by the tree pair
$(\mathrm{T}_{+},\sigma,\mathrm{T}_{-})$ containing a branch
$\ell_{\omega^{\prime}}(\mathrm{T}_{+})\mapsto\ell_{\omega^{\prime}}(\mathrm{T}_{-}).$
We attach the tree $\mathrm{G}_{+}$ to the leaf labeled $\omega^{\prime}$ in
the branch $\ell_{\omega^{\prime}}(\mathrm{T}_{+})$ and $\mathrm{G}_{-}$ to
the leaf labeled $\omega^{\prime}$ in the branch
$\ell_{\omega^{\prime}}(\mathrm{T}_{-}).$ More generally, let $t\in V_{n}$ be
represented by $(\mathrm{T}_{+},\sigma,\mathrm{T}_{-})$ and let
$\ell_{o}(\mathrm{T}_{+})\mapsto\ell_{\sigma(o)}(\mathrm{T}_{-})$ be some
branch of the element $t.$ Then we denote by
$g_{[\ell_{o}(\mathrm{T}_{+})\mapsto\ell_{\sigma(o)}(\mathrm{T}_{-})]}$ the
tree pair such that the source tree is the tree $\mathrm{T}_{+}$ with
$\mathrm{G}_{+}$ attached to the leaf $o$ and the target tree is the tree
$\mathrm{T}_{-}$ with $\mathrm{G}_{-}$ attached to the leaf $\sigma(0).$

For example, we could take both the notation $x_{0_{[u+(n-1)]}}$ and
$x_{{n-1}_{[u]}}$ where $u\in\\{0,\cdots,n-2\\}$ to be the same tree pair
representation to represent the same element.

This simply connects the dynamical interpretation and the combinatorial
interpretation of the group elements of Thompson’s groups [3] as well as
provides a method to estimate of the changes in the number of carets.

Combining these different interpretations of the elements of the Thompson
groups, we realised that the branches of tree pair representation of the group
elements keep track of the support of the groups and detect the changes in the
word length when having the product of the group elements. We will mainly use
the tree pair representations and the branches to find the desired path for
some later arguments.

###  2.4. The estimate of the word lengths via tree pair representations

In the current section, we investigate the word lengths of the elements in the
Brown-Thompson groups and it turns out that the word length relies largely on
the number of the carets in the tree pair representation.

The word length of a group element $g$ with respect to some finite generating
set of this group is the length of the shortest word representing $g$ in this
finite generating set denoted by $|\cdot|$, or $|\cdot|_{\Sigma}$ when the
finite generating set $\Sigma$ needs to be specified.

Since the tree pair representations are usually not unique for a group element
in Thompson’s groups and their generalisations, we take the unique reduced
tree pair representative for each group element [8], define the following form
with respect to the infinite generating set.

######  Definition 2.8 ([11, 26]).

Let the reduced labelled tree pair
$(\mathrm{T}_{n(+)},\sigma,\mathrm{T}_{n(-)})$ represent an element $g$ in
$T_{n}$ or in $V_{n}$ and $\mathrm{T}_{n(+)}$ and $\mathrm{T}_{n(-)}$ each
have $i$ carets. Let $\mathrm{R}$ be the all right tree with $i$ carets, i.e.
a tree constructed by attaching carets consequently to the rightmost leaf of
the previously attached caret. We write $g$ as a product
$\textbf{p}\sigma\textbf{q},$ where:

  1. (1)

p, a positive word in the infinite generating set of $F_{n}$ and of the form
$\textbf{p}=x_{i_{1}}^{r_{1}}x_{i_{2}}^{r_{2}}\cdots x_{i_{y}}^{r_{y}}$ where
$i_{1}<i_{2}\cdots<i_{y}$) with tree pair
$(\mathrm{T}_{n(+)},\mathrm{id},\mathrm{R}).$

  2. (2)

     * •

For $g\in F_{n},$ $\sigma$ is just the identity $\rm id,$

     * •

For $g\in T_{n}$, $\sigma$ is $c_{i-1}^{j},$ the $j$th power of torsion
element $c_{i-1}$ with $1\leq j<i(n-1)+1$, which can be represented by a tree
pair with two $\mathrm{R}$’s with $i$ $n$-carets, i. e.
$(\mathrm{R},c_{i-1}^{j},\mathrm{R}).$

     * •

For $g\in V_{n}$, $\sigma$ is a permutation in the symmetric group
$\mathfrak{S}_{i(n-1)+1},$ which permutes the leaves of the tree pair, we have
$(\mathrm{R},\sigma,\mathrm{R})$ as the factor in the middle,

and

  3. (3)

q, a negative word, is the of the form $x_{j_{z}}^{-s_{z}}\cdots
x_{j_{2}}^{-s_{2}}x_{j_{1}}^{-s_{1}}$ where $j_{1}<j_{2}<\cdots<j_{z}$) for an
element in $F_{n}$ represented by
$(\mathrm{R},\mathrm{id},\mathrm{T}_{n(-)}).$

We call such a product $\textbf{p}\sigma\textbf{q}$, a generalised pcq
factorisation. An example is illustrated in Figure 2.

Figure 2. pcq factorisation

Note that a word in the form of $x_{i_{1}}^{r_{1}}x_{i_{2}}^{r_{2}}\cdots
x_{i_{y}}^{r_{y}}c_{i-1}^{j}x_{j_{z}}^{-s_{z}}\cdots
x_{j_{2}}^{-s_{2}}x_{j_{1}}^{-s_{1}}$ in $T_{n}$, where
$i_{1}<i_{2}\cdots<i_{y}$ and $j_{z}>\cdots>j_{2}>j_{1}$, $r_{k}$’s and
$s_{l}$’s are all positive and $1\leq j\leq i(n-1)+1$ for the cyclic part
$c_{i-1}^{j}$ in the middle. $x_{i_{1}}^{r_{1}}x_{i_{2}}^{r_{2}}\cdots
x_{i_{y}}^{r_{y}}$ and $x_{j_{z}}^{-s_{z}}\cdots
x_{j_{2}}^{-s_{2}}x_{j_{1}}^{-s_{1}}$ are words represented in the standard
infinite generating set of $F_{n}$ Figure 1).

####  2.4.1. The estimate of the word length of elements in $F_{n}$, $T_{n}$

There is a connection between the word length and the uniquely reduced tree
pair representation of a group element by comparing the number of the building
blocks of each $n$-nary trees in the tree pair, which is a rooted $n$-ary tree
with one vertex representing the root and $n$ edges attached to it, called the
$n$-caret, and the number of the leaves in each tree.

As is proved in Burillo et. al [8, 11, 26], these two notion are
interchangeable and the word length of the group elements in $F$, $F_{n}$,
$T$, $T_{n}$ can be estimated by the number of carets of the trees in the tree
pair representation as follows.

######  Lemma 2.9 ([11]).

Let $\omega\in T_{n},$ Let $N_{n}(\omega)$ denote the number of carets in each
of the trees in the unique tree pair representing $\omega$, let $|\omega|_{n}$
denote the word length of $\omega$ with respect to the standard finite
generating set $\\{x_{0},x_{1},\cdots,x_{n-1},c_{0}\\}$ of the group $T_{n},$
then there exists a constant $C^{\prime}$ such that the following estimate is
satisfied for any $\omega$,

$\frac{N_{n}(\omega)}{C^{\prime}}\leq|\omega|_{n}\leq C^{\prime}N_{n}(\omega)$

Thus the word length can also be estimated by the number of leaves in each
tree of the unique tree pair representation.

####  2.4.2. The definition of $V_{n}$ from a logic viewpoint

For group $V,$ the word length of the group elements are proved to have a
lower bound comparable with the number of leaves and the number of carets in
Birget [1] in a slightly more combinatorial description. We follow the method
from [1] and generalise the argument to $V_{n}$ which requires the definition
of the Thompson groups through coding theory and this, in fact, coincides with
the original definition of Thompson’s groups.

We briefly set up the foundation as follows. Let $A$ be a finite alphabet,
given any two elements $u,v$ in the set $A^{\ast}$ of the words, we denote by
$u\cdot v$ the concatenation and say that $u$ is a prefix of $v$ if and only
if $v=ux$ for some $x\in A^{\ast}.$ This provides a partial order for the
words. A prefix code over $A$ is defined to be a subset of $A^{\ast}$ such
that no elements of this set is the strict prefix of any other elements of the
set. For $A,$ we define the right ideal of $A^{\ast}$ to be $R\subseteq
A^{\ast}$ such that $R\cdot A^{\ast}\subseteq R.$ A set $\Gamma\subseteq R$ is
called a right ideal generator of $R$ if and only if $R=\Gamma\cdot A^{\ast}$.
$R$ is called essential if and only if $R$ has a nonempty intersection with
every other right ideal of $A^{\ast}.$ A right ideal homomorphism of
$A^{\ast}$ is a function $\phi:R_{1}\rightarrow R_{2}$ such that $R_{1}$ and
$R_{2}$ are nonempty right ideals of $A^{\ast}$ and such that for all $u\in
R_{1}$ and for all $x\in A^{\ast}:\phi(u)\cdot x=\phi(ux)$ and it is a
bijective right ideal homomorphism. An extension of a right ideal homomorphism
$\phi:R_{1}\rightarrow R_{2}$ is a right ideal homomorphism
$\Phi:J_{1}\rightarrow J_{2}$ where $J_{1},J_{2}$ are right ideals such that
$R_{i}\subseteq J_{i}$ for $i\in\\{1,2\\}$ and $\Phi$ agrees with $\phi$ on
$R_{1}.$ A right ideal homomorphism is said to be maximal if and only if it
has no strict extension in $A^{\ast}.$ Then we have all the basic ingredients
for defining $V$ and it’s generalisation alternatively.

######  Definition 2.10 ([1]).

$V$ is the partial action group on the set of words $\\{a,b\\}^{\ast}$
consisting of all maximal isomorphisms between the finitely generated
essential right ideals of $\\{a,b\\}^{\ast}.$ The group multiplication of two
elements of $V$ is the maximum extension of the composition of two elements of
the group.

Similarly, we define $V_{n}$.

######  Definition 2.11 (The generalised Thompson group $V_{n}$).

$V_{n}$ is defined as the partial action group on the set of words
$\\{a_{0},a_{1},\cdots,a_{n-1}\\}^{\ast}$ consisting of all maximal
isomorphisms between the finitely generated essential right ideals of
$\\{a_{0},a_{1},\cdots,a_{n-1}\\}^{\ast}.$ The group multiplication of two
elements of $V_{n}$ is defined similarly as the maximum extension of the
composition of two elements of the group.

Need to mention that $F_{n}$ can be defined as the subgroup of $V_{n}$
consists of all right ideal isomorphism of
$\\{a_{0},a_{1},\cdots,a_{n-1}\\}^{\ast}$ that preserves the dictionary order
as for $F$ in [2].

######  Definition 2.12 ([1]).

For a right-ideal isomorphism $\phi:P_{1}A^{\ast}\rightarrow P_{2}A^{\ast}$ (a
bijection), where $P_{1}$ and $P_{2}$ are finite maximal prefix codes, the
restriction $P_{1}\rightarrow P_{2}$ of $\phi$ is called the table of $\phi.$
Define the table size $\lVert\phi\rVert$ to be $|P_{1}|$ $(=|P_{2}|).$ For an
element $g\in V$, the table size $\lVert g\rVert$ of $g$ is defined to be the
table size of the maximally extended right-ideal isomorphism that represents
$g$.

####  2.4.3. An estimate of the word length of elements in $V_{n}$ from the
logic viewpoint

The following lemma indicates that the table size and the word length of an
element of $V$ are comparable.

######  Lemma 2.13 ( [1]).

The table size and the word size of an element $g\in V$ are related as
follows:

  1. (1)

There are $c_{\Delta},c^{\prime}_{\Delta}>0$ (depending on the choice of
finite generating set $\Delta$) such that for all $g\in V-\\{1\\}$:
$c^{\prime}_{\Delta}\lVert g\rVert\leq|g|_{\Delta}\leq c_{\Delta}\lVert
g\rVert\cdot\log_{2}\lVert g\rVert.$

  2. (2)

For almost all $g\in V$, $|g|_{\Delta}>\lVert
g\rVert\cdot\log_{2|\Delta|}\lVert g\rVert.$

“Almost all” means that in the set $\\{g\in V:\lVert g\rVert=n\\},$ the subset
that does not satisfy the above inequality has a proportion that tends to $0$
exponentially fast as $n\mapsto\infty.$

The analogue for $V_{n}$ is as follows.

######  Lemma 2.14 (The word length of elements in $V_{n}$).

The table size and the word size of an element $g\in V_{n}$ are related as
follows:

There are $c_{\Delta},c^{\prime}_{\Delta}>0$ (depending on the choice of
finite generating set $\Delta$) such that for all $g\in V-\\{1\\}$:
$c^{\prime}_{\Delta}\lVert g\rVert\leq|g|_{\Delta}\leq c_{\Delta}\lVert
g\rVert\cdot\log_{2}\lVert g\rVert.$

###### Proof of Lemma 2.14.

We extend [1] to find a canonical factorisation for elements in $V_{n}$ to a
right-ideal automorphism and two elements of $F_{n}$, then prove the latter to
have bounded word length.

The ideal comes naturally from Thompson’s orginal definition. When fixing the
maximal prefix code $S_{\lVert g\rVert}$ of some cardinality $l(n-1)+1,$ for
an group element $g$ of $V_{n}$, there exists a unique decomposition such that
$g=\beta_{g}\pi_{g}\alpha_{g}$ where $\alpha_{g},\beta_{g}\in F_{n}$ and
$\pi_{g}$ is a automorphism whose table is a permutation of the $S_{\lVert
g\rVert}$. This is because, by considering the group element as a maximal
right-ideal isomorphism $\phi$, $\alpha_{g}$ and $\beta_{g}$ can be uniquely
defined as order-preserving bijective, hence being uniquely determined by the
maximal prefixed codes. This factorisation coincides with the pcq
factorisation.

Next, we look at the word length of $\alpha_{g}$ and $\beta_{g}$. The linear
boundedness of the word length of the group elements in $F_{n}$ can either be
deduced from [8] or from a similar argument as in [1]. They are essentially
the same estimate.

Finally, we focus only on $\pi_{g}$, which can be regarded as a table of some
automorphisms from the finite maximal prefix code $S_{\lVert g\rVert}$ to
itself and hence is a permutation with bounded number of transpositions. The
transpositions defined in a generalised version of the one in [1] give exactly
the process of the permutation of the labels on the two identical trees in the
tree pair representation. ∎

By the above argument, we also have a lower bound for estimating the word
length of the elements in $V_{n}$.

##  3\. Divergence functions of the generalised Thompson’s groups (the Brown-
Thompson groups)

###  3.1. The Divergence function

For metric spaces, there are several different types of divergence
functions[19, 16, 20, 28]. The intuitive idea of divergence property is to
understand to what extend two geodesics starting from the same point in a
metric space go away from each other. Here we are taking the convention from
[16].

The divergence of the group is narrowed down from the divergence of the metric
spaces by considering the divergence function of the Cayley graph of the
groups with respect to the standard finite generating set defined below in
Subsection 3.4. From now on, we use $|\cdot|$ as the notation for the word
length of the group elements.

######  Definition 3.1 ([16]).

Define $div(a,b,c;\delta)$ as the infimum of the length of the paths for a
geodesic metric space $(X,d)$ connecting $a$, $b$ and avoiding the ball
$B(c,\delta r)$ centered at $c$ with radius $r$, where $a$, $b$, $c$ are
points taken arbitrarily in the metric space $X,$ and $r$ is the minimum of
the distances between $c$ and $a$ or between $b$ and $c.$

######  Definition 3.2 (Divergence function [16]).

The divergence function $Div(m;\delta)$ is the maximum of all
$div(a,b,c;\delta)$ such that the pairs $(a,b)$ are within distance $m$, i.e.
$dist(a,b)\leq m.$

The goal here is to find the shortest path between two points at distance $n$
from the identity in the Cayley graph of the generalised Thompson groups and
compare this path to some functions.

###  3.2. Divergence property of $F_{n},T_{n},V_{n}$

As is in the Subsection 2.3, we denote by $\mathcal{G}$ any of the groups
$F_{n}$, $T_{n}$, $V_{n}.$ For an element $g\in\mathcal{G}$, we denote by
$\mathcal{N}_{\mathcal{G}}(g)$ the number of leaves in each of the trees in
the reduced tree pairs representing $g$ in group $\mathcal{G}$ or
$\mathcal{N}(\cdot)$ when the group is clear.

######  Theorem 3.3.

There exist constants $\delta,D>0$ such that the following holds. Let
$g_{1},g_{2}\in\mathcal{G}$ be two elements with $\mathcal{N}(\cdot)\geq
3n-2.$ Then there is a path of length at most $D(|g_{1}|+|g_{2}|)$ in the
Cayley graph $\Gamma=Cay(\mathcal{G},X)$ which avoids a
$(\delta\min\\{|g_{1}|,|g_{2}|\\})$-neighbourhood of the identity and which
has initial vertex $g_{1}$ and terminal vertex $g_{2}$.

We will prove Theorem 3.3 over the course of this section.

######  Corollary 3.4.

The generalised Thompson’s groups $F_{n}$, $T_{n}$ and $V_{n}$ have linear
divergence.

###### Proof.

This is a direct consequence of Theorem 3.3. ∎

The next Proposition which can be regarded as an analogue of [20, Prop2.1] can
either be proved by the estimate on the number of carets as in [20, 11, 26]or
using the estimate by computation in [1] and the analogues of [1] in the
discussion above.

Let $\mathcal{A}_{n}=\\{x_{0},x_{1},\cdots,x_{n-1}\\}$,
$\mathcal{B}_{n}=\\{x_{0},x_{1},\cdots,x_{n-1},c_{0}\\}$,
$\mathcal{C}_{n}=\\{x_{0},x_{1},\cdots,x_{n-1},c_{0},\pi_{0}\\}$ be the
standard generating sets of $F_{n}$, $T_{n}$, $V_{n}$, respectively,
$\mathcal{N}_{\mathcal{G}}(g)$ be the number of leaves as above.

######  Proposition 3.5.

There exist constants $0<c<1$ and $C>1$ such that the following holds

  1. (1)

For every $g\in F_{n}$ we have
$c\mathcal{N}_{F_{n}}(g)\leq|g|_{\mathcal{A}}\leq C\mathcal{N}_{F_{n}}(g)$.

  2. (2)

For every $g\in T_{n}$ we have
$c\mathcal{N}_{T_{n}}(g)\leq|g|_{\mathcal{B}}\leq C\mathcal{N}_{T_{n}}(g)$.

  3. (3)

For every $g\in V_{n}$ we have $c\mathcal{N}_{V_{n}}(g)\leq|g|_{\mathcal{C}}$.

###### Proof of Proposition 3.5.

Since the number of carets and the number of the leaves of the unique tree
pair representing an element in Brown-Thompson groups $F_{n}$ and $T_{n}$ are
interchangeable [8, 26], together with Lemma 2.9, the first two statements for
the groups $F_{n}$ and $T_{n}$ are satisfied.

A lower bound for the word length of the group elements of $V_{n}$ can be
given by a similar argument as in [1] in the proof of Lemma 2.14. By defining
the table size for $V_{n},$ we could prove that the table size and the word
length of an element in $V_{n}$ are related, and hence find a bound. ∎

The next step is to find out how concatenating one extra generator may change
the number of the carets.

######  Lemma 3.6.

For $g\in\mathcal{G}$ being represented by some reduced tree pair
representation $(\mathrm{G}_{n(+)},\sigma_{g},\mathrm{G}_{n(-)})$ with at
least three carets in each tree, let $\bar{\ell}_{\gamma}(\mathrm{G}_{n(-)})$
denote the length of the branch of the target tree $\mathrm{G}_{n(-)},$ where
$\gamma$ is an $n$-ary word corresponding a path from the root in
$\mathrm{G}_{n(-)}.$ Denote by
$(\mathrm{GX_{i}}_{n(+)},\sigma_{gx_{i}},\mathrm{GX_{i}}_{n(-)})$ the reduced
tree pair representation of $gx_{i}\in\mathcal{G}.$ We then have

$\mathcal{N}(g)-(n-1)\leq\mathcal{N}(gx_{i}^{\pm 1})\leq\mathcal{N}(g)+(n-1)$

where $i\in\\{0,\cdots,n-2\\}$.

  1. (1)

If $\bar{\ell}_{0}(\mathrm{G}_{n(-)})=1,$ then
$\mathcal{N}(gx_{i}^{+1})=\mathcal{N}(g)+(n-1),$ and
$\bar{\ell}_{0}(GX_{i}^{+1})=1.$

  2. (2)

If $\bar{\ell}_{0}(\mathrm{G}_{n(-)})\neq 1,$ then
$\mathcal{N}(gx_{i}^{+1})=\mathcal{N}(g),$ or
$\mathcal{N}(gx_{i}^{+1})=\mathcal{N}(g)-(n-1),$ Moreover,
$\bar{\ell}_{i}(\mathrm{GX_{i}}_{n(-)})=\bar{\ell}_{i}(\mathrm{G}_{n(-)})-1.$

  3. (3)

If $\bar{\ell}_{i}(\mathrm{G}_{n(-)})=1$, then
$\mathcal{N}(gx_{i}^{-1})=\mathcal{N}(g)+(n-1),$
$\bar{\ell}_{j}(\mathrm{GX_{i}}_{n(-)})=1$ for $j\in\\{1,\cdots,n-2\\}.$

  4. (4)

If $\bar{\ell}_{i}(\mathrm{G}_{n(-)})\neq 1$ then
$\mathcal{N}(gx_{i}^{-1})=\mathcal{N}(g)$ or $\mathcal{N}(gx_{i}^{\pm
1})=\mathcal{N}(g)-(n-1),$
$\bar{\ell}_{n}(\mathrm{GX_{i}}_{n(-)})=\bar{\ell}_{n}(\mathrm{G}_{n(-)}).$

###### Proof.

The proof is generalised from the arguments in [20]. The elements
$x_{i}\in\mathcal{G}$ where $i\in\\{0,\cdots,n-2\\}$ are the generators in
$F_{n}$ in the finitely generating set with only two carets in each tree in
the tree pair representation in Figure 1, i.e. $\mathcal{N}(x_{i})=2(n-1)+1$
with the same target tree. The source tree of the unique tree pair
representing $x_{i}$s has the second caret attached to leaves labelled
$\\{0,\cdots,n-1\\}$, and has the branch $ik$ where $k\in\\{0,\cdots,n-1\\}$
in the source tree, respectively. The inverses $x_{i}^{-1}$s are represented
by the tree pairs that reverse the source tree and the target tree in the
pair.

The composition $gx_{i}$ can be represented by the multiplication of the
unique tree pair representation of $g$ and $x_{i}.$ If
$\bar{\ell}_{0}(\mathrm{G}_{n(-)})=1,$ the target tree of $g$ has a path with
only one edge attached to the first leaf. Hence the product add one more
$n$-caret to both the source and the target tree of the tree pair representing
$g.$ Alternatively, this is indicated by the branch
$\ell_{00}(G_{n(+)}(x_{i}))\mapsto\ell_{0i}(G_{n(-)}(x_{i})).$ Since the
number of carets in $gx_{i}$ is larger than the ones in $g,$ the equation in
$(1)$ follows. The branch $\bar{\ell}_{0}(GX_{i{n(-)}})=1$, since there is no
extra $n$-carets added to the $i$th leaf on target tree of the product. This
proves $(1),$ the arguments for the rest three cases are similar. ∎

######  Corollary 3.7.

Let $g\in\mathcal{G}$ be any element that can be represented by a unique tree
pair representation with at least three carets in each tree. Then
$\mathcal{N}(g)<\mathcal{N}(gx_{i}^{+m})\leq\mathcal{N}(g)+m(n-1)-1$ when
$\bar{\ell}_{0}(G_{n(-)}^{+1})=1$ and
$\bar{\ell}_{0^{m}}(GX_{in(-)}^{+m})=m+1;$
$\mathcal{N}(g)<\mathcal{N}(gx_{i}^{-m})\leq\mathcal{N}(g)+m(n-1)-1$ when
$\bar{\ell}_{i}(G_{n(-)}^{+1})=1.$ and
$\bar{\ell}_{i0^{m-1}}(GX_{in(-)}^{-m})$ becomes $m+1.$

###### Proof.

This is a direct consequence of the preceeding lemma. ∎

######  Lemma 3.8.

Let $g\in\mathcal{G}$ be an element with the tree pair representation
$(\mathrm{G}_{n(+)},\sigma,\mathrm{G}_{n(-)}).$ Let
$\ell_{u}(\mathrm{G}_{n(+)})\mapsto\ell_{v}(\mathrm{G}_{n(-)})$ be a branch of
$g$ and let $h$ be an element of $F_{n}$. Let $h^{\prime}=(h)_{[v]}.$ Then

$\mathcal{N}(gh^{\prime})=\mathcal{N}(g)+\mathcal{N}(h)-1.$

###### Proof.

The argument follows from the ones in the interpretation of the branches.
According to Subsection 2.7 the unique tree pair representation of
$h^{\prime}=(h)_{[v]}$ has branches
$\ell_{v\omega_{s}}{(\mathrm{T}_{n(+)}(h^{\prime}))}\mapsto\ell_{v\omega_{t}}{(\mathrm{T}_{n(-)}(h^{\prime}))},$
where $\omega_{s}$ and $\omega_{t}$ are the words corresponding to the paths
in the source tree and the target tree of the tree pair representing $h$
respectively. The branch $\ell_{u}{(g)}\mapsto\ell_{v}{(g)}$ in the product
$gh^{\prime}$ takes the source tree attached to vertex labelled $v$ in the
tree pair representation of $h^{\prime}$ back to the vertex labelled $u$ in
the source tree pf the tree pair represents $g.$ Then we have branches
$\ell_{u\omega_{s}}{(\mathrm{T}_{n(+)}(gh^{\prime}))}\mapsto\ell_{v\omega_{t}}{(\mathrm{T}_{n(-)}(gh^{\prime}))},$
in the tree pair representation of $gh^{\prime}.$ We here give a more
geometric interpretation of this process in Figure 3.

Figure 3. $h^{\prime}=(h)_{[v]}$ in tree pair representation

∎

######  Proposition 3.9.

There exist constants $\delta,D>0$ and a positive integer $Q$ such that the
following holds. Let $g\in\mathcal{G}$ be an element with $\mathcal{N}(g)\geq
2(n-1)+n=3n-2$. Then there is a path represented by $\omega$ of length at most
$D|g|$ in the Cayley graph $\Gamma=Cay(\mathcal{G},X)$ which avoids a
$\delta|g|$-neighbourhood of the identity and which has initial vertex $g$ and
terminal vertex $g\omega$.

Similar to the strategy in the proof of [20], we want to find a path
connecting two points on two geodesics staring from the same point, for
instance, the identity $id,$ with some bounded distance and avoid a
neighbouhood of $id.$ The construction works for $F_{n}$, $T_{n}$ and $V_{n}$.

In the general case, we associate an element $g_{k}$ to some $k\in\mathbb{N}$
which is roughly the length of some group element $h$ in $F_{n}$, and then
find a path from $h$ to $g_{k}$ of which the length is linear to the word
length of the element $h$. Finally, take two such elements so that the path
between them which is linear to some of the positive numbers associate them as
well as avoiding a ball at center $id.$

###### Proof of Proposition 3.9.

Let $C$ and $c$ be the constants from Proposition 3.5. Pick an element $h$
randomly, and associate to the word length of $h$ a group element $g_{|h|}$ so
that the path $\omega$ between these two group elements in the Cayley graph is
linear to the word length $|h|$ with respect to the standard finite generating
set.

For some group element $h$ whose number of carets is at least three and hence
the number of leaves is at least $2(n-1)+n,$ denote by
$(\mathrm{H}_{n(+)},\sigma_{h},\mathrm{H}_{n(-)})$ the tree pair
representation of $h.$ We want the element to be supported on some interval
$[0,a_{|h|}).$ Hence we concatenate the subpath
$\omega_{1}=x_{0}^{2}x_{n-1}^{-1}x_{0}^{-1}$ to obtain $h_{1}=h\omega_{1}$ if
$l_{0}(h)=1$, otherwise we take $\omega_{1}$ to be an empty word.

Figure 4. Adding the first subpath

Next, we want to add on two subpaths to ensure that the newly created word
represent an element in $F_{n}.$ We concatenate the subpath $\omega_{2}$ and
$\omega_{3}$ to $h_{1}$ and the construction of the two subpaths are as
follows: The subpath
$\omega_{2}=x_{0}^{-M\mathcal{N}(h_{1})}x_{n-1}x_{0}^{M\mathcal{N}(h_{1})}$
which is an element in $F_{n}$ and we take $M\geq 10\frac{C}{c(n-1)}.$

The subpath $\omega_{3}$ is a word representing an cyclic element
$h_{\omega_{3}}$ in $T_{n}<V_{n}$ such that the tree pair representation of
the element $h_{\omega_{3}}$ has the tree pair with the source tree being the
target tree of the tree pair of $h$ and the target tree being the source tree
of the tree pair of $h,$ i.e.
$(\mathrm{H}_{n(-)},\sigma_{\omega_{3}},\mathrm{H}_{n(+)})$
$\sigma_{\omega_{3}}$ permutes the leaves from $\mathrm{H}_{n(-)}$ to
$\mathrm{H}_{n(+)}$ in the a cyclic order and with the permutation
$\sigma_{\omega_{3}}(\sigma_{(}h_{1}))(\ell_{0}(h_{\omega_{3}}))=\ell_{0}(h_{\omega_{3}}),$
in other words, $h_{\omega_{3}}$ has the branch
$\ell_{(\sigma_{\omega_{3}}(0))}\mapsto\ell_{0}.$ Since $h_{\omega_{3}}\in
T_{n},$ its word length is bounded linearly by its number of carets and hence
the number of leaves, i.e. $|h_{\omega_{3}}|\leq
C\mathcal{N}(h_{\omega_{3}}).$

By concatenating the paths, we have $h_{3}=h\omega_{1}\omega_{2}\omega_{3}.$
From the construction, we have that
$\mathcal{N}(h_{1})\leq\mathcal{N}(h_{1})+(m-1)(n-1)-1\leq\mathcal{N}(h_{2})\leq\mathcal{N}(h_{3})\leq
2\mathcal{N}(h_{1})+(m-1)(n-1)-2.$ First two in equalities are from Lemma 3.8,
the last ones is from considering the branches with prefix having $0$s and
Corollary 3.7.

The subpath $\omega_{4}$ represents an element in $F_{n}$ with the
construction $x_{0}^{Q|h|}x_{n-1}^{-1}x_{0}^{-Q|h|+1}$ where we take $Q\geq
10\frac{M}{c^{2}(n-1)}.$ This subpath ensures
$\mathcal{N}(h_{3})\leq\mathcal{N}(h_{3}\omega_{4})$ and commutes with
$h_{3}.$ This is because $h_{3}$ is supported on $[0,\ell_{0}(h_{3}))$ while
$\omega_{4}$ from the definition fixes the same interval. We take the last
subpath $\omega_{5}$ to be the word representing $h_{3}^{-1}$, where
$h_{3}=h\omega_{1}\omega_{2}\omega_{3}.$

Now, we can compute the upper bound of the length of the path $\omega$ as
follows:

$\displaystyle\lVert\omega\rVert\leq\lVert
h_{3}\rVert+\lVert\omega_{4}\rVert+\lVert\omega_{5}\rVert=2\lVert
h_{3}\rVert+\lVert\omega_{4}\rVert$ $\displaystyle\leq
2(2M\mathcal{N}(h_{1})+2\mathcal{N}(h_{1}))+(2Q\mathcal{N}(h_{1})+2)$
$\displaystyle\leq 4M\mathcal{N}(h_{1})+(2Q+2)\mathcal{N}(h_{1})+2$
$\displaystyle\leq(4M+2Q+2)\mathcal{N}(h_{1})+2$
$\displaystyle=\Big{(}\frac{4M+2Q+2}{c}\Big{)}|h|+2$

We could take $\delta=\frac{c}{8M}$ since $\lVert\omega\rVert\leq
4M\mathcal{N}(h)$ and $D=\Big{(}\frac{4M+2Q+4}{c}\Big{)}|h|,$ so that
$\delta|g_{|h|}|\leq\lVert g_{|h|}\omega^{\prime}\rVert\leq D|g_{|h|}|$ which
proves the linearity of the length of the path from $h$ to $g_{|h|}.$ ∎

######  Lemma 3.10.

For a prefix $\omega^{\prime}$ of $\omega$, $|h\omega^{\prime}|\geq\delta|h|.$

###### Proof.

By considering the changes in the number of carets, there is no reduction on
the number of carets while adding on the first subpath, since when adding on
the first subpath $\omega_{1},$ if $\bar{\ell}_{0}(H_{-})=1$,
$\mathcal{N}(h\omega_{1})>\mathcal{N}(h)$ by Corollary 3.7, otherwise we add
an empty word and the length is unchanged. For $\omega_{2},$ Lemma 3.6 ensures
the increase in the number of carets and hence the number of word length. The
tree pair representation of $h_{\omega_{3}}$ have the same number of carets as
in $h_{1}$ in the pcq form. Besides $\omega_{3}$ has a cyclic permutation,
hence by [26] whose word length is linear to the number of its carets and
together with Lemma 3.8, we have
$\mathcal{N}(h_{1}\omega_{2})\leq\mathcal{N}(h_{1}\omega_{2}\omega_{3})=\mathcal{N}(h_{3}).$
Since the tree pair representation of $h_{\omega_{3}}$ has the branch
$\ell_{(\sigma_{\omega_{3}}(0))}\mapsto\ell_{0},$ $h_{3}$ is supported on some
$[0,a_{n}).$ The path $h_{2}$ after adding on $\omega_{2}$ ends with powers of
$x_{0}^{-1},$ we choose $\omega_{3}$ to be the shortest word representing the
element $h_{\omega_{3}}$ with the tree pair symmetric to the tree pair
representing $h.$ Since no reduction occurs when concatenating powers of
$x_{0}$s to $h,$ there are also no reductions occurring when concatenating
$\omega_{3}$ to $x_{0}^{-1}$s which is just reflecting the picture of the
operating of the product of the tree pairs.

For the subpath represented by word $\omega_{4},$ the argument is the same as
the ones for the case when adding the subpath representing $\omega_{2}.$ As
for adding on the subpath $\omega_{5},$ it is just the inverse of path
$h\omega_{1}\omega_{2}\omega_{3}.$ ∎

With the preceding Proposition, we could prove Theorem 3.3.

###### Proof of Theorem 3.3.

From Proposition 3.9, we could pick two elements $g_{1}$ and $g_{2}$ randomly,
for each $g_{i}$ where $i\in\\{1,2\\}$, there is a path $\omega_{i}$ avoiding
some ball centered at identity with radius $D(|g_{i}|)$ from element $g_{i}$
to $x_{0}^{Q|g_{i}|}x_{n-1}^{-1}x_{0}^{-Q|g_{i}|+1}.$ Assume without loss of
generality, we have $|g_{1}|\leq|g_{2}|$ and we add a path

$p=x_{0}^{Q|g_{1}|-1}x_{n-1}x_{0}^{-Q|g_{1}-g_{2}|+1}x_{n-1}^{-1}x_{0}^{-Q|g_{2}|+1}.$

Then by the previous lemma and the estimate from of the word length, we have

$\delta\min\\{|g_{1}|,|g_{2}|\\}<cQ\rVert
g_{1}\lVert\leq|x_{0}^{Q|g_{1}|}x_{n-1}^{-1}x_{0}^{-Q|g_{1}\omega^{\prime\prime}|}|.$

∎

For some undistorted subgroups of Thompson’s groups, the method of finding out
the comparison paths for the divergence functions is similar, since the
estimate of the word length replies only on the subgroups $F_{n}$.

Alternatively, we could rely on the quasi-isometric embeddings. Since, the
Brown-Thompson groups $F_{n}$, $T_{n}$ can be embedded quasi-isometrically
into Thompson’s groups $F$, $T,$ respectively, we could consider the quasi-
isometric embedding $\phi^{\ast}$ from $T_{n}$ to $T$ defined in [26], and
pick the shortest path between the image of two points outside the ball
centred at the identity with radius of the element with larger word length in
$T,$ then the paths that are quasi-isometric to the preimage of this linear
path [26] taken back by $\phi^{\ast}$ may also be an estimate.

##  4\. Divergence functions for the braided Thompson’s group

In [20], Golan and Sapir posed the question: if the divergence property of the
braided version of Thompson’s groups can also be found using the same method.
In this section, we would like to investigate this generalised version of
Thompson’s group, the braided Thompson groups. The origin of these groups can
be traced back to Brin [6] and Dehornoy [15] of which the metric properties
are being investigated in [4, 10] where the estimate of the word length is
slightly different.

###  4.1. The braided Thompson groups

The braided version of the Thompson groups can be regarded as some
“Artinification” construction of the original Thompson’s groups. Intuitively,
when considering the tree pair representation of the elements of Thompson’s
groups, instead of just permuting the labelings of the leaves, we add “braids”
to the leaves between the pair. One replaces the permutations by the braid
relations. The group $BV,$ the larger group $\widehat{BV}$ and the braided
analogues of $F,$ namely, $BF$ and $\widehat{BF}$ will be considered in the
rest of this paper.

We recall dyadically subdivided intervals defined in the previous section. For
two dyadically subdivided intervals with the same number of subintervals $I$
and $J,$ we associate them to an element in the braid group $B_{m}$ where $m$
is the number of the subintervals. In order to navigate the map between
subintervals in $I$ and $J,$ we identify each subinterval with the thickened
strands or a elastic band in the braids.

Figure 5.

Figure 5 illustrates one of the group elements of the braided Thompson group
$BV.$ Adding and subtracting the dyadic breakpoints to the subintervals to the
corresponding strands do not change the map. When there are two such maps,
i.e. a map from $I_{1}$ to $J_{1}$ and a map from $I_{2}$ to $J_{2}$, where
$I_{1}$, $J_{1}$, $I_{2}$, $J_{2}$ are dyadically subdivided unit intervals,
and $I_{1}$, $J_{1}$ are the intervals with the same number of subdivisions,
$I_{2}$, $J_{2}$ are the intervals with the same number of subdivisions,
respectively. We compose $J_{1}$ and $I_{2}$ as composing two piecewise-linear
functions and then extend the composition to $I_{1}$ and $J_{2}$ as in Figure
6, the resulted map consists again a pair of dyadically subdivided unit
intervals associated to an element in the braid group.

Figure 6.

######  Definition 4.1 ($BV$).

The braided Thompson group $BV$ is the group of the piecewise-linear maps from
the set of the dyadically subdivided intervals to itself associating pairs of
intervals with intervals having the same number of breakpoints, such that the
subintervals in the source unit interval are associated to those in the target
as follows,

  * •

all subintervals in the source and the corresponding ones in the target are
bijectively connected to each other by gluing a rectangular flat elastic band
(or a ribbon) to each pair of them such that the top of the band is glued to
the source subintervals and the bottom is glued to the the target
subintervals;

  * •

the band is not allowed to flip or twist, but can be stretched;

  * •

the set of the bands can be identified with the thickened strands of a braid
with $m$ strands, $m\in\mathbb{N}$ and can be represented by an element $b$ in
the braid group $B_{m}$ where $m$ is the number of the subintervals in both
the source and the target unit intervals.

The group operation is defined by: First, compositing the target interval in
the band-connected interval pair representing the former element and the
source interval in the pair representing the latter element. Then by extending
the composition through the bands back to the source interval in the pair
representing the former element and also to the target interval in the pair of
the latter one.

This group can also be regarded as the extension of Thompson’s group $V$ by
the “stable braid groups ” with trivial center which includes all braid groups
[18] by $V$. Similar to the case in $F$, $T$, $V$ and the Brown-Thompson
groups, there is a more combinatorial interpretation using the tree pair
representation and the braid diagrams for the braided Thompson groups.

These are called tree-braid-tree diagram or the braided version of the pcq
factorisation where the middle component c comes from the elements of the
braid groups instead of the symmetric groups. Since for $F$, $T$, $V$ as well
as for $F_{n}$, $T_{n}$ and $V_{n},$ we have a symbol with three tuples
indicating the tree pair representation for a group element, analogously, we
have a similar symbol $(\mathrm{T}_{+},\beta,\mathrm{T}_{-})$ for elements in
$BV$ where $\beta$ is a braid induced by an element in the braid group
$B_{m+1}$ with $m$ strand.

Figure 7.  $\tau_{1}$, $\sigma_{1}$, $\phi\in BV$, $\psi\in BF.$

Following [12], we allow the tree-braid-tree diagrams to change to each other
through the following equivalent movements in Figure 8 which form an
equivalence class and hence define the composition of the diagrams.

Figure 8.

When we want to compose two tree-braid-tree diagrams, we lay the two diagrams
vertically, stacking the first diagram pair on top of the second diagram pair
and merge the target tree of the top diagram and the source tree of the bottom
tree using the relations on the left side on Figure 8, then using the
relations on the right side to drag the carets up or down to obtain a new
tree-braid-tree diagram. This operation coincides with the multiplication of
the group elements.

Similarly, we defined the braided version for the group $F$.

######  Definition 4.2 ($BF$).

The braided Thompson group $BF$ is the group of the orientation-preserving
piecewise-linear isotopies from the set of the dyadically subdivided intervals
to itself associating pairs of intervals with intervals having the same number
of breakpoints, such that the subintervals in the source unit interval are
associated to the ones in the target as follows,

  * •

all subintervals in the source and the corresponding ones in the target are
connected together by gluing a rectangular flat elastic band to each pair of
them such that the top of the band is glued to the source subintervals and the
bottom is glued to the corresponding target subintervals;

  * •

the band is not allowed to flip or twist, but can be stretched;

  * •

the set of the bands can be identified with a thickened pure braid with $m$
strands, $m\in\mathbb{N}$ and can be represented by an element $p$ in the pure
braid group $PB_{m}.$ In addition, $m$ is the number of the subintervals in
both the source and the target unit intervals.

The group operation is defined in the same manner as the one in the Definition
4.1.

Both $BF$ and $BV$ are finitely presented [4] and $BV$ has a finite generating
set $x_{0}$, $x_{1}$, $\sigma_{1}$ and $\tau_{1}$ where $x_{0}$ and $x_{1}$
come from the standard generating set of $F$ and the other two are induced by
the braid relations and are illustrated on the left half in Figure 7. These
two groups also have series of infinitely generating sets $x_{i}^{\pm 1}$,
$\sigma_{i}^{\pm 1}$,$\tau_{i}^{\pm 1}$ where $x_{i}^{\pm 1}$ are the infinite
order generators from the original Thompson’s groups and $\sigma_{i}^{\pm
1}$,$\tau_{i}^{\pm 1}$ are the infinite sequence of generators induced from
the braid groups [10, 29] which can be represented by tree-braid-tree diagrams
with a pair of all right trees and a positive crossing at the strands on
$(i-1)$th and $i$th leaves for the diagram pair representing $\tau_{i}^{\pm},$
a positive crossing at the strands on $(i-2)$th and $(i-1)$th leaves for
diagram pair $\sigma_{i}^{\pm 1}.$

Similar as in the non-braided version of Thompson’s groups, a set of the tree-
braid-tree diagrams under some equivalence relation have a unique reduced
diagram pair which corresponds to a standard form. A more precise description
of the equivalent relation of the diagrams and the minimal size
representatives can be referred to [8]. Since the estimate on the word length
later on will depend largely on this standard form, we give a brief
introduction on this combinatorial description below.

######  Definition 4.3 (The braided version of pcq form).

Let the reduced labelled tree pair $(\mathrm{T}_{(+)},\beta,\mathrm{T}_{(-)})$
represent an element $g\in BV$ and $\mathrm{T}_{(+)}$ and $\mathrm{T}_{(-)}$
each have $i$ carets. Let $\mathrm{R}$ be the all right tree with $i$ carets.
We write $g$ as a product $\textbf{p}\beta\textbf{q}$, where:

  1. (1)

p, a positive word in the infinite generating set of $F$ and with the form
$\textbf{p}=x_{i_{1}}^{r_{1}}x_{i_{2}}^{r_{2}}\cdots x_{i_{y}}^{r_{y}}$ where
$i_{1}<i_{2}\cdots<i_{y}$ with tree pair
$(\mathrm{T}_{(+)},\rm{id},\mathrm{R}).$

  2. (2)

$\beta$ part is induced by an element in the braided group of $m$-strands,
which is represented by a $(\mathrm{R},\beta,\mathrm{R}),$ and is a word in
the infinite generating sets $\\{\sigma_{i}\\}$ and $\\{\tau_{i}\\}$ which are
induced by the elements in $B_{m}$ where
$m=\max\\{\mathcal{N}(\mathrm{T}_{(+)}),\mathcal{N}(\mathrm{T}_{(-)})\\}.$

  3. (3)

q, a negative word, is the in the form $x_{j_{z}}^{-s_{z}}\cdots
x_{j_{2}}^{-s_{2}}x_{j_{1}}^{-s_{1}}$ where $j_{1}<j_{2}<\cdots<j_{z}$ for an
element in $F$ represented by $(\mathrm{R},\rm{id},\mathrm{T}_{(-)}).$

We call such a product $\textbf{p}\beta\textbf{q}$, a pcq factorization for
the braided Thompson groups.

Note that a word with the form $x_{i_{1}}^{r_{1}}x_{i_{2}}^{r_{2}}\cdots
x_{i_{y}}^{r_{y}}\mathrm{id}x_{j_{z}}^{-s_{z}}\cdots
x_{j_{2}}^{-s_{2}}x_{j_{1}}^{-s_{1}}$ in $BV,$ where
$i_{1}<i_{2}\cdots<i_{y}\neq j_{z}>\cdots>j_{2}>j_{1}$, $r_{k}$’s and
$s_{l}$’s are all positive and $1\leq j\leq i(n-1)+1$, is a word with respect
to the infinite generating set of $F.$

In [4], there is a silmilar notion, block. By applying reductions to the words
in the pcq form we may naturally obtain a block.

This is another standard form of the group elements when taking the word
$\beta$ to be the right-greedy form [13].

######  Definition 4.4 (Block[4]).

A word in the infinite generators $x_{i}^{\pm 1}$, $\sigma_{i}^{\pm 1}$,
$\tau_{i}^{\pm 1}$ is called a block, if it is of the form
$\omega_{1}\omega_{2}\omega_{3}^{-1}$ where

  1. (1)

$\omega_{1}$ is a positive word as the p part in the pcq representation.

  2. (2)

Let $N=\max{\\{N(\omega_{1}),N(\omega_{3})\\}},$ then there exists an integer
$m\geq N+1$ such that $\omega_{2}$ is a word in the generators
$\\{\sigma_{1},\sigma_{2},\cdots,\sigma_{m-2},\tau_{m-1}\\}.$

  3. (3)

$\omega_{3}^{-1}$ is a negative word as the q part in the pcq representation.

Notice that when the c part of the pcq form of some group element is a word in
the generators induced from some braid group then the word can easily be
transformed to a block.

######  Lemma 4.5.

Every group element in the pcq form can be transformed to a block and
transform back by finitely many steps of moving around letters.

###### Proof.

The p and the q are elements in $F$, they are equivalent to the first block
and the third block in the block form and can possibly be reduced to shorter
words. The middle part of the pcq form denoted by $\beta$, is a word in
$\sigma_{i}^{\pm 1}$,$\tau_{i}^{\pm 1}$ induced by elements in $B_{m}$ for
some positive integer $m$, hence, more precisely, a word in
$\\{\tau_{1},\cdots,\tau_{m-1}\\}$ and $\\{\sigma_{1},\cdots,\sigma_{m}\\}.$
Leave the letters in $\\{\sigma_{1},\cdots,\sigma_{m}\\}$ where they are and
rewrite the letters from $\\{\tau_{2},\cdots,\tau_{m-1}\\}$ by reducing the
subsuffix of the generators using the relation $D_{2}$ and $D_{1}$ from [4] to
move around the generators $x_{i}.$ Since $\beta$ has only finite letters, the
process takes only finitely many steps. ∎

In the original definition [6], there is a “larger” version of the braided
Thompson group which contains the group $BV$ and also sits inside $BV$ which
has the last strand unbraided and we define this group formally below:

######  Definition 4.6 ($\widehat{BV}$).

The braided Thompson group $\widehat{BV}$ is defined as the group of the
piecewise-linear maps from the set of the infinite dyadically subdivided
intervals to itself such that the first $k$ subintervals in the source
interval are associated to the first $k$ subintervals in the target as an
element in $BV$ in the description of Definition 4.1, where $k\in\mathbb{N}.$
The rest of the subintervals in the source and the target are associated to
each other one by one from the left in order.

Similarly, we could the define the larger version of the braided Thompson
groups containing $F.$

######  Definition 4.7 ($\widehat{BF}$).

The braided Thompson group $\widehat{BF}$ is the group of the orientation-
preserving piecewise-linear isotopies from the set of the infinite dyadically
subdivided intervals to itself such that the first $k$ subintervals in the
source interval are associated to the first $k$ subintervals in the target as
an element in $BF$ in the description of Definition 4.2, where
$k\in\mathbb{N}.$ The rest of the subintervals in the source and the target
are associated to each other one by one from the left in order.

Since the braided version of Thompson’s groups involve both Thompson’s groups
and braid groups, we expect the estimate of the word length to depend on the
estimate of the word length of both of Thompson’s groups and braid groups.

###  4.2. Divergence property of the braided Thompson groups $BF$, $BV$

By considering the group elements of the braided Thompson groups as tree-
braid-tree diagrams, we estimate the word length from a purely combinatorial
aspect.

For Thompson’s group $F$ and the generalisations $F_{n}$ without the torsion
elements, the word length is proportional to the number of carets in the tree
pair representation, whereas for the braid groups, the word length with
respect to the standard generators is closely related to the number of the
crossings. Here we denote by $\mathcal{K}(\cdot)$, the number of crossing of
the tree-braid-tree diagram representing some group element. In [10], Burillo
et. al. have considered the Garside elements of the braid groups inside the
$BV$ in order to give the following estimate.

######  Lemma 4.8 ([10]).

For an element $\omega$ in $BV$ which has a reduced tree-pair-tree diagram
representative with $n$ leaves and $k$ total crossings, and with the maximum
number of crossings of a pair of strands is $s$, then there exist constants
$C_{1}$, $C_{2}$, $C_{3}$, $C_{4}$ such that the length $|\omega|$ satisfy the
following:

$C_{1}\max\\{n,\sqrt[3]{k}\\}\leq C_{2}\max\\{n,s\\}\leq|\omega|\leq
C_{3}(n+nk)\leq C_{4}(n+n^{3}s).$

Garside elements which provide the largest number of crossings for the tree-
braid-tree representation of the braid elements in Thompson’s group are the
ones that we want to avoid in the estimate of the word length.

######  Lemma 4.9.

For $g\in BV$ being represented by some reduced tree pair representation
$(\mathrm{G}_{(+)},\sigma_{g},\mathrm{G}_{(-)})$ with at least three carets in
the each tree, and $gx_{0}$ be represented by
$(\mathrm{GX_{0}}_{(+)},\sigma_{gx_{0}},\mathrm{GX_{0}}_{(-)}),$ let
$\bar{\ell}_{o}(\cdot)$ denote length of the branch of a tree, then we have

$\mathcal{N}(g)-1\leq\mathcal{N}(gx_{0}^{\pm 1})\leq\mathcal{N}(g)+1$

In addition,

  1. (1)

If $\bar{\ell}_{0}(\mathrm{G}_{(-)})=1,$ then
$\mathcal{N}(gx_{0})=\mathcal{N}(g)+1$ and
$\bar{\ell}_{0}(\mathrm{GX_{0}}_{(-)})=1.$ The number of the crossing is at
most twice the original number.

  2. (2)

If $\bar{\ell}_{0}(\mathrm{G}_{(-)})\neq 1,$ then
$\mathcal{N}(gx_{0})=\mathcal{N}(g)$ or
$\mathcal{N}(gx_{0})=\mathcal{N}(g)-1.$ Moreover,
$\mathcal{N}(gx_{0})=\mathcal{N}(g)$ only when $1$ and $01$ are strict
prefixes are prefixes of the target tree $G_{(-)}$ in the pair representing
element $g.$ No changes in the number of crossings.

  3. (3)

If $\bar{\ell}_{1}(\mathrm{G}_{(-)})=1,$ then
$\mathcal{N}(gx_{0}^{-1})=\mathcal{N}(g)+1$ and
$\bar{\ell}_{1}(\mathrm{GX_{0}^{-1}}_{(-)})=1.$ The number of the crossing is
at most twice the original number.

  4. (4)

If $\bar{\ell}_{1}(\mathrm{G}_{(-)})\neq 1,$ then
$\mathcal{N}(gx_{0}^{-1})=\mathcal{N}(g)$ or
$\mathcal{N}(gx_{0}^{-1})=\mathcal{N}(g)-1.$ Moreover,
$\mathcal{N}(gx_{0})=\mathcal{N}(g)$ only when $10$ and $11$ are strict
prefixes are prefixes of the target tree $G_{(-)}$ in the pair representing
element $g.$ No changes in the number of crossings.

###### Proof.

The proof is a generalised argument of the proof of Lemma 3.6 and the estimate
of the number of crossing is purely by graphical computation. Take $g$ such
that $\bar{\ell}_{0}(\mathrm{G}_{(-)})=1,$ the first statement follows from
the changes in the carets. The change is the number of crossings depending
changes in the strands under the extra carets added to the product. Since we
have one more caret in each tree in the tree-braid tree diagram of $gx_{0},$
we then have at most twice the number of the original number of crossings. The
rest three cases are by similar arguments. ∎

We hence know that the generators in the braided Thompson groups that are
coming from the torsion free part of Thompson’s groups have affects mainly on
the number of the carets of the tree-braid-tree diagrams.

######  Lemma 4.10.

For $g\in BV$ being represented by some reduced tree pair representation
$(\mathrm{G}_{(+)},\sigma_{g},\mathrm{G}_{(-)})$ with at least three carets in
the each tree, let $\ell_{\gamma}(\cdot)$ denote the branch of an element such
that $\gamma$ is a branch of the source tree in the reduced tree pair
representing $g$ as before, $\bar{\ell}_{\gamma}(\cdot)$ denote the length of
the branch of some tree and let $\mathcal{K}(\cdot)$ be the number of
crossings in the reduced tree pair representation (or tree-braid-tree
diagrams) of an group element.

$\mathcal{N}(g)\leq\mathcal{N}(g\sigma_{1}^{\pm 1})\leq\mathcal{N}(g)+1$
$\mathcal{N}(g\tau_{1}^{\pm 1})=\mathcal{N}(g)$

In addition,

  1. (1)

If $\bar{\ell}_{0}(G_{(-)})=1,$ then $\bar{\ell}_{1}(G_{(-)})\neq 1,$
$\mathcal{N}(g\sigma_{1}^{\pm 1})=\mathcal{N}(g)$ and

    1. (a)

$\bar{\ell}_{0}(\mathrm{GX_{0}}_{(-)})=1$ when
$\ell_{10}(\mathrm{GX_{0}}_{(-)})$ is a branch, $\mathcal{K}(g\sigma_{1}^{\pm
1})\leq\mathcal{K}(g)+1;$

    2. (b)

$\bar{\ell}_{0}(\mathrm{GX_{0}}_{(-)})\neq 1$ when
$\ell_{10}(\mathrm{GX_{0}}_{(-)})$ is not a branch,
$\mathcal{K}(g\sigma_{1}^{\pm 1})\leq 2(\mathcal{K}(g)-2);$

  2. (2)

If $\bar{\ell}_{1}(G_{(-)})=1,$ then $\bar{\ell}_{0}(G_{(-)})\neq 1,$
$\mathcal{N}(g\sigma_{1}^{\pm 1})=\mathcal{N}(g)+1,$ and
$\bar{\ell}_{0}(\mathrm{GX_{0}}_{(-)})\neq 1.$ Moreover,
$\mathcal{K}(g\sigma_{1}^{\pm 1})\leq\mathcal{K}(g)+2;$

  3. (3)

The crossing number of $g\tau_{1}^{\pm},$
$\mathcal{K}(g\tau_{1}^{\pm})\leq\frac{\mathcal{K}(g)^{2}}{4}.$

###### Proof.

When $\bar{\ell}_{0}(G_{(-)})=1,$ $\bar{\ell}_{1}(G_{(-)})\neq 1$ follows from
the assumption, $\sigma_{1}^{\pm 1}$ takes $0$ to $10,$ takes $10$ to $0$ and
$11*$ to $11*.$ Then number of carets is hence unchanged through the
computation. Since there is one crossing between the first and second strand
in $\sigma_{1}^{\pm 1},$ i.e. strand on leaves labelled $0$ and $10,$ for the
number of crossings of $g\sigma_{1}^{\pm 1},$ we only need to consider how
these two strands are interacting with other strands. For $(a),$ the product
does not add extra caret and hence the number of crossings just adds by one
from $\sigma_{1}^{\pm}.$ Similar arguments work for the case $(b).$ For $(2)$
and $(3),$ we do the similar analyse. ∎

######  Example 4.11.

Some elementary cases of the multiplication of the generators induced from
braid relations $\tau_{1}$, $\sigma_{1}$ and the Thompson group generators
$x_{0}$ are illustrated in Figure 9a and Figure 9b.

(a) data The products of $\tau_{1}$ and $x_{0}$, $x_{1}$

(b) data $\sigma_{1}$ and $x_{0}$, $x_{1}$

######  Lemma 4.12.

Let $g\in BV$ be an element with the tree pair representation
$(\mathrm{G}_{(+)},\sigma,\mathrm{G}_{(-)}).$ Let
$\ell_{u}(\mathrm{G}_{(+)})\mapsto\ell_{v}(\mathrm{G}_{(-)})$ be a branch of
$g$ and let $h$ be an element of $F$ and $\mathcal{K}(\cdot)$ be the number of
crossings of some reduced tree pair. Let $h^{\prime}=(h)_{[v]}.$ Then

$\mathcal{K}(g)\leq\mathcal{K}(gh^{\prime})\leq\min{\\{(\mathcal{N}(g)-1)\mathcal{N}(h)+\mathcal{K}(g),\mathcal{N}(h)\mathcal{K}(g)\\}}.$

###### Proof.

The former equality can be obtained from the same argument in Lemma 3.6 and
the latter one is given by multiplication of tree-braid-tree diagrams. ∎

From a few of the above consequence, we can see that for elements coming from
the braid groups inside $BV,$ they do increase the number of carets when
multiplying the torsion-free elements, and they also increase on the number of
crossings, hence, the changes in the number of crossings are in fact very hard
to control, when concatenating elements, while the changes in the number of
carets when adding on elements is partially inherited from those in the
original Thompson’s groups $F$ inside $BV.$ Thus, the “counting arguments” for
constructing our paths will still rely on the “non-braided” part.

###  4.3. Divergence property of the geodesics

For braided Thompson groups, we would like to find a similar path from one
geodesic to another avoiding a ball centered at the identity in their Cayley
graphs with respect to the finite generating sets, and we first want to find
an element $g_{\mid h\mid}$ relying on elements in $F$ which corresponds to
the element $h$ in $BV$ such that there is a path connecting the two elements
and avoids the ball having radius which is the same as the word length of $h.$

######  Proposition 4.13.

There exist constants $\delta,D>0$ and a positive integer $Q$ such that the
following holds. Let $g\in BV$ be an element with $\mathcal{N}(g)\geq 4$. Then
there is a path represented by $\omega$ of length at most $D|g|^{4}$ in the
Cayley graph $\Gamma=Cay(\mathcal{G},X)$ which avoids a
$\delta|g|$-neighbourhood of the identity and which has initial vertex $g$ and
terminal vertex $g\omega.$

###### Proof.

Let the estimate be as in Lemma 4.8. Let $h\in BV$ whose number of leaves is
at least four and we want to first find a subpath $\omega_{1}\omega_{2}$ which
makes $h_{2}=h\omega_{1}\omega_{2}$ supported on some interval $[0,a_{n})$.
Next, we concatenate $h_{2}$ by another subpath $\omega_{3}$ to obtain
$h_{3}=h_{2}\omega_{3}$ with the condition that $h_{3}$ commutes with
$\omega_{4}$. This ensures
$\mathcal{N}(h_{3}\omega_{4})\geq\mathcal{N}(h_{3})$. We finally obtain
$g_{|h|}=h_{4}(h_{3})^{-1}$ by adding another subpath $\omega_{5}=h_{3}^{-1}$
which by definition can be reduced to be $\omega_{4}$ and represents an
element in $F$ inside $BV$.

Let $h$ be represented as a unique tree-braid-tree diagram with the notation
$(\mathrm{T}_{(+)},b,\mathrm{T}_{(-)})$. By definition, $h$ will either have
$\bar{\ell}_{(0)}(h)=1$ or $\bar{\ell}_{(0)}(h)>1$. When
$\bar{\ell}_{(0)}(h)=1$, we add a subpath
$\omega_{1}=x_{0}^{2}x_{1}^{-1}x_{0}^{-1}$ depending on the branches in $h$,
so that $l_{0}(h\omega_{1})>1,$ otherwise, we keep the original element $h.$

Then we add the second subpath

$\omega_{2}=x_{0}^{-M\mathcal{N}(h_{1})}x_{1}x_{0}^{M\mathcal{N}(h_{1})}$

to $h$ or $h\omega_{1}$ which is a path in $F$ linear to the length of $h$,
where $M\geq 10\frac{C_{4}}{C_{1}}$.

The estimate of the length of $\omega_{3}$ is the key part in case of the
braided Thompson group. The subpath corresponding to $\omega_{3}$ is taken to
be the shortest word representing an element with the following tree pair
representation $(\mathrm{T}_{(-)},b^{\prime},\mathrm{T}_{(+)})$ such that
$b^{\prime}$ is taking the strand $b(0)$ to $0,$ i.e. $b^{\prime}(b(0))=id,$
$b^{\prime}$ is not necessarily the inverse of $b.$ If $h\in F$, then the
braid part of $\omega_{3}$ just represents $id.$

The path will satisfy the following properties:

  1. (1)

$\rVert\omega_{3}\lVert\leq C_{3}\mathcal{N}(h_{1})(1+\mathcal{K}(h_{1}))$
which follows from Lemma 4.8.

  2. (2)

$(M-1)\mathcal{N}(h)\leq(M-1)\mathcal{N}(h_{1}\omega_{2})\leq\mathcal{N}(h\omega_{1}\omega_{2}\omega_{3}).$
Moreover,
$(M-1)\mathcal{K}(h)\leq(M-1)\mathcal{K}(h_{1}\omega_{2})\leq\mathcal{K}(h\omega_{1}\omega_{2}\omega_{3}).$
Both sequences of inequalities can be deduced from visualising the products
from graphical interpretation of the group elements. For the increase in the
number of crossings, every time we concatenate a prefix in $\omega_{2}$ which
is a word purely from generators in $F$ from the left, the resulted tree-
braid-tree diagram becomes larger extending below to the right side. Then both
the number of carets an the number of crossings increase linearly which
ensures that “Garside element” (elements with far larger number of crossings
than the number of carets [10]) will not appear.

  3. (3)

$\ell_{0}(h_{1})$ or $0^{\ell_{0}(h_{1})}\mapsto 0^{\ell_{0}(h_{1})}$ is a
branch of $h\omega_{1}\omega_{2}\omega_{3},$ which follows again from the
products of tree-braid-tree diagrams.

For $h_{1}$ supported on $[0,\ell_{0}(h_{1})))$, we add on the fourth subpath
$\omega_{4}=x_{0}^{Q|h|}x_{1}^{-1}x_{0}^{-Q|h|+1}$ where $Q\geq
10\frac{M}{C_{1}^{2}}$ and then obtain $h_{4}=h_{3}\omega_{4}$. The
construction by the tree-braid-tree diagrams could be interpreted as follows:
Noting that $h_{3}$ and $\omega_{4}$ commute on $[0,\ell_{0}(h_{1}))$, since
$h_{3}$ is supported on $[0,\ell_{0}(h_{1})))$ while $\omega_{4}$ from the
definition fixes the braids in the same interval. The difference in $BV$ is
that the number of crossings keep adding up linearly along the path. Finally,
we add on the last subpath $\omega_{5}=h_{3}^{-1}$. Since $h_{4}$ and
$\omega_{3}$ commute,
$g_{|h|}=h_{4}\omega_{5}=h_{3}\omega_{5}h_{3}^{-1}=\omega_{4}$.

We could take $\delta=8M\mathcal{N}(h)+3Q$ by purely considering the changes
in the number of carets, so we have
$\delta|g_{|h|}|\leq|g_{|h|}\omega^{\prime}|.$ However, the upper bound is
given by the product of $\mathcal{N}(g_{|h|})$ and $\mathcal{K}(g_{|h|})$.
Hence, the path from $h$ to $|g_{|h|}|$ has a linear lower bound and at most
polynomial for the upper bound. ∎

######  Lemma 4.14.

For a prefix $\omega^{\prime}$ of $\omega$ representing an element in $BV,$
$|h\omega^{\prime}|\geq\delta|h|.$

###### Proof.

In the case of $BV$, we consider both the changes in the number of carets and
the number of crossings. When adding on the first subpath $\omega_{1}$, when
$\bar{\ell}_{0}(h)\neq 1$, $\mathcal{N}(h)<\mathcal{N}(h\omega_{1})$ and
number of crossings $\mathcal{K}(h\omega_{1})\leq\mathcal{K}(h)$ by Lemma 4.9,
otherwise, we add an empty word and the length is unchanged. For $\omega_{2},$
Lemma 4.12 ensures the increase in the number of carets and hence the number
of word length. The word $\omega_{3}$ represents a subpath with the same
number of carets as $h$ in the pcq form. Besides we choose $\omega_{3}$ to be
the shortest word with a non trivial braids with word length linear to the
ones of $h.$ We have
$\mathcal{N}(h_{1}\omega_{2})\leq\mathcal{N}(h_{1}\omega_{2}\omega_{3})=\mathcal{N}(h_{3}).$
Since the element represents $\omega_{3}$ having the branch
$\ell_{(\sigma_{\omega_{3}}(0))}\mapsto\ell_{0},$ $h_{3}$ is supported on some
$[0,a_{|h|}).$ For the $\omega_{4}$, the argument is the same as the ones for
the case when add $\omega_{3}.$ As for subpath $\omega_{5}$, it is just the
inverse of path $h\omega_{1}\omega_{2}\omega_{3}.$ ∎

######  Theorem 4.15.

There exists constant $\delta>0$ and a polynomial function $p(\cdot)$ such
that the following holds. Let $g_{1},g_{2}\in\mathcal{G}$ be two elements with
$\mathcal{N}(g)\geq 4.$ Then there is a path of length at most
$p(|g_{1}|+|g_{2}|)$ in the Cayley graph $\Gamma=Cay(\mathcal{G},X)$ which
avoids a $\delta\min{\\{|g_{1}|,|g_{2}|\\}}$-neighbourhood of the identity and
which has initial vertex $g_{1}$ and terminal vertex $g_{2}.$

###### Proof.

For element with small carets number the linear divergence of the geodesics
can be deduced directly from the computation. Again by taking two paths we
described in Proposition 4.13 and connect them as in the Brown-Thompson group
case, no reductions will occur. ∎

Taking the above estimate, we could only say that the group has linear
functions as lower bounds for the divergence functions for $BV.$ In fact, a
stronger result has been proved in [24] indicating that the divergence
function of $BV$ is also bounded above by linear functions.

######  Theorem 4.16 (Kodama).

$BV$ has linear divergence function.

Now we consider the group $BF<BV,$ which is a finitely generated group with
finite generating set

$\Sigma_{BF}=\\{x_{0},x_{1},\alpha_{12},\alpha_{13},\alpha_{23},\alpha_{24},\beta_{12},\beta_{13},\beta_{23},\beta_{24}\\}$

where $\alpha_{i,j}$s and $\beta_{i,j}$s are in fact deduced from a sequence
of infinite generating set of the pure braid relations in the braid groups
[4]. These two sequences of infinite generating sets can be written in the
infinite generating sets of $BV$, $\tau_{i}$s and $\sigma_{i}$s, by the
following [29],

$\alpha_{i,j}=\sigma_{i}\sigma_{i+1}\cdots\sigma_{j-2}\sigma_{j-1}^{2}\sigma_{j-2}^{-1}\cdots\sigma_{i}^{-1}$
$\beta_{i,j}=\sigma_{i}\sigma_{i+1}\cdots\sigma_{j-2}\tau_{j-1}^{2}\sigma_{j-2}^{-1}\cdots\sigma_{i}^{-1}.$

We take this finite generating set as the standard generating set when we
consider the word length of $BF.$

######  Lemma 4.17.

Let $g\in BF$ be an element with the tree-braid tree diagram pairs
$(\mathrm{T}_{n(+)},\sigma,\mathrm{T}_{n(-)})$ and let $u\mapsto v$ be a
branch of $g,$ $h$ be an element of $F<BF.$ Let $h^{\prime}=(h)_{[v]}.$ Then

$\mathcal{N}(gh^{\prime})=\mathcal{N}(g)+\mathcal{N}(h)-1.$

###### Proof.

The proof is similar to the one for Lemma 3.8. ∎

######  Proposition 4.18.

There exist constants $\delta,D>0$ such that the following holds. Let $g\in
BF$ be an element with $\mathcal{N}(g)\geq 4$. Then there is a path which can
be interpreted by a word $\omega$ of length at most $D|g|$ in the Cayley graph
$\Gamma=Cay(\mathcal{G},X)$ which avoids a $\delta|g|$-neighbourhood of the
identity and which has initial vertex $g$ and terminal vertex $\omega$.

###### Proof.

Let $h\in BF$ be an element whose reduced tree-braid-tree diagram pair
$(\mathrm{H}_{+},\sigma_{h},\mathrm{H}_{-})$ has at least three carets in each
tree. Again when $\bar{\ell}_{(0)}(h)=1$, we add an extra path
$\omega_{1}=x_{0}^{2}x_{1}^{-1}x_{0}^{-1}$ and then add path
$\omega_{2}=x_{0}^{-M\mathcal{N}(h_{1})}x_{1}x_{0}^{M\mathcal{N}(h_{1})}$
followed by the shortest word $\omega_{3}$ which has the reduced tree-braid-
tree diagram pair $(\mathrm{H}_{-},id,\mathrm{H}_{+}).$ This works because
$BF$ is torsion free, our paths only rely on the changes in the number of
carets. Next we take $\omega_{4}=x_{0}^{Q|h|}x_{1}^{-1}x_{0}^{-Q|h|+1},$ where
$Q\geq 10\frac{M}{C_{1}^{2}}.$ Since the path we are adding on represents an
element in $F,$ the number of the crossings stays the same, thus as long as we
take $Q$ large enough, the there will not be the influence of Garside elements
[10] on the distance. This can also be seen from the Lemma 4.17. The estimate
of the length of the path only relies on the generators $x_{0}$ and $x_{1}$
and hence on the number of carets. ∎

######  Theorem 4.19.

$BF$ has linear divergence.

###### Proof.

Provided the constant $Q$ taken is large enough, the concatenation of the two
subpaths between the two words representing two elements in $BV$ that reduces
the number of crossings are not going to affect the length of the linear path.
∎

######  Remark 4.20.

According to [4, 5], $\widehat{BF}$ and $\widehat{BV}$ are also finitely
generated, where $\widehat{BV}$ is generated by standard generating set in $F$
and the $\sigma$ generating set, $\widehat{BF}$ only contains the standard
generating set of $F$ and the $\alpha$-generators induced from pure braid
relation. In the case of $\widehat{BV}$, the estimate can be made similarly by
taking five subpaths and concatenating them. The middle subpath corresponding
to $\omega_{3}$ taken, however, should be modified since $\tau$-generators are
not in the generating set of $\widehat{BV}.$ In $\widehat{BF},$ the word
length can be estimated similarly as in the case of $BF,$ the path only
consists of generators from the subgroup isomorphic to $F$ and hence the
geodesics all have linear divergence.

######  Corollary 4.21.

$\widehat{BF}$ and $\widehat{BV}$ have linear divergence.

###### Proof.

The arguments here are the analogues of the proof for $BF$ and $BV$ by
focusing on the “braided” part of the tree-braid-tree diagrams pair. ∎

######  Remark 4.22.

In the previous sections, we considered the divergence property of the
geodesics in the Cayley graph of braided Thompson groups. We could also
consider more generally the divergence property of the subgroups such as the
the normal subgroup of the braided Thompson groups for, on one hand, many of
the normal subgroups of the braided Thompson group are being investigated in
[29], and they are still mysterious; On the other hand, the connection between
the distortion of the finitely generated normal subgroups of a finitely
generated group and the divergence of both groups has been indicated in [28],
so we can think of the analogue in braided Thompson groups.

In Thompson’s group $F,$ the proper quotient subgroups of $F$ are abelian
which somehow coincides with the linear divergence property of $F,$ whereas,
in the case of the braided Thompson group, the surjective map from $BF$ to $F$
provide an exact sequence, where the kernel corresponding to the pure braid
part of the groups provide natural subgroups of $BF.$ However, the kernel is
not finitely generated and there are also normal subgroups containing the
commutator subgroups $[BF,BF]$ of which the presentation may be extremely
complicated [29].

Here divergence property gives some indication on the subgroup distortion of
the normal subgroups inside the $BF$ and $BV.$

######  Remark 4.23.

At the point of finishing this work, I was told that Kodama also has a result
on the divergence function of the group $BV.$

## Acknowledgments

I am very grateful for the help and support from my supervisor Takuya Sakasai
during while I am preparing for this work and during this hard time and I
would also like to thank Tomohiro Fukaya and his student Yuya Kodama for
helpful discussions. Last but not the least, I would also like to express my
gratitude to Sadayoshi Kojima for helpful discussion and suggestions and for
anonymous reviewers for carefully going through the manuscript and pointing
out mistakes and inappropriate descriptions.

## Conflict of interests

The authors declare that they have no conflict of interest.

## Data Avaliability statement

Data sharing not applicable to this article as no datasets were generated or
analysed during the current study.

## References

  * [1] Jean-Camille Birget.  The groups of Richard Thompson and complexity.  volume 14, pages 569–626. 2004.  International Conference on Semigroups and Groups in honor of the 65th birthday of Prof. John Rhodes.
  * [2] Jean-Camille Birget.  The groups of Richard Thompson and complexity.  International Journal of Algebra and Computation, 14(05n06):569–626, 2004.
  * [3] Collin Bleak and Olga Salazar-Díaz.  Free products in R. Thompson’s group $V$.  Trans. Amer. Math. Soc., 365(11):5967–5997, 2013.
  * [4] Tom Brady, José Burillo, Sean Cleary, and Melanie Stein.  Pure braid subgroups of braided Thompson’s groups.  Publ. Mat., 52(1):57–89, 2008.
  * [5] Matthew G. Brin.  The Algebra of strand splitting II: A presentation for the braid group on one strand.  International Journal of Algebra and Computation, 16(01):203–219, feb 2006.
  * [6] Matthew G. Brin.  The Algebra of strand splitting. I. A braided version of Thompson’s group $V$.  J. Group Theory, 10(6):757–788, 2007.
  * [7] Kenneth S. Brown.  Finiteness properties of groups.  In Proceedings of the Northwestern conference on cohomology of groups (Evanston, Ill., 1985), volume 44, pages 45–75, 1987.
  * [8] J. Burillo, S. Cleary, and M. I. Stein.  Metrics and embeddings of generalizations of Thompson’s group $F$.  Trans. Amer. Math. Soc., 353(4):1677–1689, 2001.
  * [9] José Burillo.  Quasi-isometrically embedded subgroups of Thompson’s group $F$.  J. Algebra, 212(1):65–78, 1999.
  * [10] José Burillo and Sean Cleary.  Metric properties of braided Thompson’s groups.  Indiana Univ. Math. J., 58(2):605–615, 2009.
  * [11] José Burillo, Sean Cleary, Melanie Stein, and Jennifer Taback.  Combinatorial and metric properties of Thompson’s group $T$.  Trans. Amer. Math. Soc., 361(2):631–652, 2009.
  * [12] Kai-Uwe Bux, Martin G. Fluch, Marco Marschler, Stefan Witzel, and Matthew C. B. Zaremsky.  The braided Thompson’s groups are of type $\rm F_{\infty}$.  J. Reine Angew. Math., 718:59–101, 2016.  With an appendix by Zaremsky.
  * [13] Kai-Uwe Bux and Dmitriy Sonkin.  Some remarks on the braided thompson group BV.  arXiv preprint math:0807.0061.
  * [14] J. W. Cannon, W. J. Floyd, and W. R. Parry.  Introductory notes on Richard Thompson’s groups.  Enseign. Math. (2), 42(3-4):215–256, 1996.
  * [15] Patrick Dehornoy.  Braid-based cryptography.  In Group theory, statistics, and cryptography, volume 360 of Contemp. Math., pages 5–33. Amer. Math. Soc., Providence, RI, 2004.
  * [16] Cornelia Druţu, Shahar Mozes, and Mark Sapir.  Divergence in lattices in semisimple Lie groups and graphs of groups.  Transactions of the American Mathematical Society, 362(5):2451–2505, dec 2009.
  * [17] Daniel Farley.  The action of Thompson’s group on a $\rm CAT(0)$ boundary.  Groups Geom. Dyn., 2(2):185–222, 2008.
  * [18] Louis Funar, Christophe Kapoudjian, and Vlad Sergiescu.  Asymptotically rigid mapping class groups and Thompson’s groups.  In Handbook of Teichmüller theory. Volume III, volume 17 of IRMA Lect. Math. Theor. Phys., pages 595–664. Eur. Math. Soc., Zürich, 2012.
  * [19] Stephen. M. Gersten.  Divergence in $3$-manifold groups.  Geom. Funct. Anal., 4(6):633–647, 1994.
  * [20] Gili Golan and Mark Sapir.  Divergence functions of Thompson groups.  Geom. Dedicata, 201:227–242, 2019.
  * [21] M. Gromov.  Asymptotic invariants of infinite groups.  In Geometric group theory, Vol. 2 (Sussex, 1991), volume 182 of London Math. Soc. Lecture Note Ser., pages 1–295. Cambridge Univ. Press, Cambridge, 1993.
  * [22] Victor S. Guba.  Polynomial isoperimetric inequalities for Richard Thompson’s groups $F$, $T$, and $V$.  In Algorithmic problems in groups and semigroups (Lincoln, NE, 1998), Trends Math., pages 91–120. Birkhäuser Boston, Boston, MA, 2000\.
  * [23] Victor S. Guba.  On the properties of the Cayley graph of Richard Thompson’s group $F$.  volume 14, pages 677–702. 2004.  International Conference on Semigroups and Groups in honor of the 65th birthday of Prof. John Rhodes.
  * [24] Yuya Kodama.  Divergence function of the braided Thompson group.  arXiv preprint math/2012.03785., 12 2020.
  * [25] Ralph McKenzie and Richard J. Thompson.  An elementary construction of unsolvable word problems in group theory.  In Word problems: decision problems and the Burnside problem in group theory (Conf., Univ. California, Irvine, Calif. 1969; dedicated to Hanna Neumann), volume 71 of Studies in Logic and the Foundations of Math., pages 457–478. 1973.
  * [26] Xiaobing Sheng.  Quasi-isometric embedding from the generalised Thompson’s group $T_{n}$ to $T$.  To appear., 03 2018.
  * [27] Melanie Stein.  Groups of piecewise linear homeomorphisms.  Trans. Amer. Math. Soc., 332(2):477–514, 1992.
  * [28] Hung C. Tran.  Relative divergence, subgroup distortion, and geodesic divergence.  ProQuest LLC, Ann Arbor, MI, 2015.  Thesis (Ph.D.)–The University of Wisconsin - Milwaukee.
  * [29] Matthew C. B. Zaremsky.  On normal subgroups of the braided Thompson groups.  Groups Geom. Dyn., 12(1):65–92, 2018.