# Homeomorphism types of Floyd boundaries of infinite-ended groups

Subhajit Chakraborty and Ravi Tomar<EMAIL_ADDRESS>Department of
Mathematical Sciences, Indian Institute of Science Education and Research
Bhopal, Bhopal Bypass Road, Bhauri 462066 Madhya Pradesh, India
<EMAIL_ADDRESS>
(Date: August 28, 2024)

###### Abstract.

Suppose $G$ is a finitely generated infinite group and $\mathcal{G}$ is a
graph of groups decomposition of $G$ such that the edge groups are finite.
This paper establishes that the topology of the Floyd boundary of $G$ is
uniquely determined by the topology of the Floyd boundary of each vertex group
of $\mathcal{G}$.

###### Key words and phrases:

Bass–Serre tree, Floyd boundary, free products

###### 2020 Mathematics Subject Classification:

20F65, 20F67

##  1\. Introduction

One of the first compactification of groups was introduced by W. Floyd in
[Flo80]. It is now known as Floyd boundary. In general, the Floyd boundary
depends on a scaling function known as Floyd function. This compactification
exhibits a strong connection to the theory of hyperbolic and relatively
hyperbolic groups. Given a hyperbolic group $G$, there exists a Floyd function
$f$ such that Floyd boundary of $G$ with respect to $f$ is homeomorphic to
Gromov boundary of $G$ (see [Gro85]). Similarly, when $G$ is a finitely
generated group hyperbolic relative to a collection of subgroups, Gerasimov in
[Ger12] shows that there exists a Floyd function $f$ and a continuous
$G$-equivariant map from Floyd boundary of $G$ with respect to $f$ to Bowditch
boundary of $G$. Floyd boundary also has connection to convergence actions
(see [Kar03]).

In [MS15], Martin–S̀wiatkowski proved that the topology of Gromov boundary of
a free product of hyperbolic groups is uniquely determined by the topology of
the Gromov bounary of each free factor. In this paper, we drop the hypothesis
that each free factor is hyperbolic and prove a similar result for Floyd
boundary. Throughout the paper, all groups are assumed to be finitely
generated. The following is the main result of this paper:

Theorem 1. (Theorem 6.1) Suppose $G$ and $H$ are two infinite groups. Suppose
$\mathcal{G}$ and $\mathcal{H}$ are graphs of groups decomposition of $G$ and
$H$, respectively such that the edge groups of $\mathcal{G}$ and $\mathcal{H}$
are finite. Then, we have the following:

  1. (1)

If each vertex group of $\mathcal{G}$ is elementary and $G$ has infinitely
many ends then the Floyd boundary of $G$ is homeomorphic to the Cantor set.

  2. (2)

Suppose at least one vertex group of either $\mathcal{G}$ or $\mathcal{H}$ is
non-elementary. Let $h(\mathcal{G})$ and $h(\mathcal{H})$ denote the set of
homeomorphism types of Floyd boundaries of non-elementary vertex groups of
$\mathcal{G}$ and $\mathcal{H}$, respectively. If
$h(\mathcal{G})=h(\mathcal{H})$ then the Floyd boundary of $G$ is homeomorphic
to Floyd boundary of $H$.

A non-elementary group is one that is not virtually cyclic. One interesting
consequence of this theorem is that, for any Floyd function, the Floyd
boundary of a virtually free group is homeomorphic to the Cantor set (see
Corollary 7.5). We need the following result to prove the previous theorem,
which tells us the homeomorphism type of the Floyd boundary of a free product.

Theorem 2. (Theorem 6.2) For $n,m\geq 2$, let $G_{1}=A_{1}\ast\dots\ast A_{n}$
and $G_{2}=B_{1}\ast\dots\ast B_{m}$ be two free products where at least one
free factor of either $G_{1}$ or $G_{2}$ is non-elementary. Let $h(G_{1})$ and
$h(G_{2})$ denote the set of homeomorphism types of Floyd boundaries of non-
elementary free factors of $G_{1}$ and $G_{2}$, respectively. If
$h(G_{1})=h(G_{2})$ then the Floyd boundary of $G_{1}$ is homeomorphic to the
Floyd boundary of $G_{2}$.

Remark. In the above theorem, if each free factor of $G_{1}$ is elementary
then either Floyd boundary of $G_{1}$ contains $2$ points or it is
homeomorphic to the Cantor set (see Corollary 5.6).

Next, we look at the connected components of the Floyd boundary of an
infinite-ended group and obtain a partial converse to Theorem 1.

Theorem 3. (Theorem 7.3) Suppose $G$ and $H$ are two finitely generated
infinite-ended groups. Suppose $\mathcal{G}$ and $\mathcal{H}$ are graphs of
groups decomposition of $G$ and $H$, respectively such that the edge groups of
$\mathcal{G}$ and $\mathcal{H}$ are finite, and each non-elementary vertex
group of $\mathcal{G}$ and $\mathcal{H}$ is $1$-ended. Also, assume that at
least one vertex group of $\mathcal{G}$ and $\mathcal{H}$ is non-elementary.
Then, we have the following:

Let $h(\mathcal{G})$ and $h(\mathcal{H})$ denote the set of homeomorphism
types of Floyd boundaries of non-elementary vertex groups of $\mathcal{G}$ and
$\mathcal{H}$, respectively. If Floyd boundary of $G$ is homeomorphic to Floyd
boundary of $H$ then $h(\mathcal{G})=h(\mathcal{H})$.

Organization of the paper: In Section 2, we recall some basic facts about
Floyd boundary and exlain Cayley graphs of amalgamted free products and HNN
extensions. Section 3 contains a construction of Floyd boundaries of the
amalgamated free products and HNN extensions over finite groups. In Section 4,
we discuss the homeomorphism type of Floyd boundaries of free products. Then,
in Section 5, we discuss homeomorphism types of Floyd boundaries of
amalgamated free product and HNN extension. There, we deduce a number of
interesting corollaries too. In Section 6, we give a proof of Theorem 6.1.
Finally, in Section 7, we finish the paper by discussing the connected
components of Floyd boundary of a graph of groups with finite edge groups.

##  2\. Preliminaries

###  2.1. Coarse geometric notion

Let $X$ be a metric space and let $x,y\in X$. A geodesic joining $x$ and $y$
is an isometric embedding $\alpha:[a,b]\subset\mathbb{R}\to X$ such that
$\alpha(a)=x$ and $\alpha(b)=y$. If any two points of $X$ can be joined by a
geodesic then $X$ is called a geodesic metric space. A geodesic joining $x$
and $y$ will be denoted by $[x,y]$. Throughout the paper, all graphs are
assumed to be connected. For $z\in X$ and $A\subset X$, we shall denote by
$d(z,A)$ the quantity $\inf\\{d(z,a):a\in A\\}$.

Suppose $(X,d_{X})$ and $(Y,d_{Y})$ are two metric spaces. Given $L\geq 0$, an
$L$-bi-Lipschitz map $\phi:X\to Y$ is one such that
$\dfrac{1}{L}d_{X}(x,x^{\prime})\leq d_{Y}(\phi(x),\phi(x^{\prime}))\leq
Ld_{X}(x,x^{\prime}).$ The map $\phi$ is said to be bi-Lipschitz if it is
$L$-bi-Lipschitz for some $L\geq 0$.

###  2.2. Floyd completion of groups

The notion of Floyd boundary of a group was introduced by W. Floyd in [Flo80].
Let $G$ be a group generated by a finite set, say, $S$. Let $\Gamma_{G,S}$ be
the Cayley graph of $G$ with respect to $S$. By assigning each edge a unit
length, we see that $\Gamma_{G,S}$ is naturally a complete geodesic metric
space. We denote this metric by $d$. Let $f:\mathbb{N}\to\mathbb{R}_{>0}$ be a
function satisfying the following two conditions:

(1) $\Sigma_{n=1}^{\infty}f(n)<\infty$.

(2) There exists $\lambda>0$ such that $f(n)\geq f(n+1)\geq\lambda f(n)$ for
all $n\in\mathbb{N}$.

We call $f$ a Floyd function. To simplify the construction of the Floyd
boundary, for any Floyd function f, we deﬁne $f(0):=f(1).$ An example of a
Floyd function is $f(n)={1}/{n^{2}}$. We define a new metric on $\Gamma_{G,S}$
by scaling the length of edges. Let $e_{G}$ denote the identity element of
$G$. For an edge $e$ in $\Gamma_{G,S}$ connecting the vertices $g,h$, we
define the new edge length $l_{f}(e)$ of $e$ to be $f(d(e_{G},\\{g,h\\}))$. If
$\alpha$ is a path in $\Gamma_{G,S}$, given by the consecutive edges
$e_{1},e_{2},\dots,e_{n}$, then the Floyd length of $\alpha$ is defined as
$\Sigma_{i=1}^{n}l_{f}(e_{i})$. For $v,w\in\Gamma_{G,S}$, define

$d_{f}(v,w)=\inf\\{l_{f}(\alpha):\alpha\text{ is a path in $\Gamma_{G,S}$
joining $v$ and $w$}\\}.$

It is easy to see that $d_{f}$ is a metric and it is called the Floyd metric
on $\Gamma_{G,S}$.

######  Definition 2.1 (Floyd boundary).

The Cauchy completion $\overline{\Gamma}_{G,S}$ of the metric space
$(\Gamma_{G,S},d_{f})$ is called the Floyd completion. The subspace
$\overline{\Gamma}_{G,S}\setminus\\{\Gamma_{G,S}\\}$ is called Floyd boundary
of $G$. It is denoted by $\partial_{f}(G)$.

This definition does not depend on the choice of finite generating set $S$ of
$G$ (see [Flo80, Lemma 2]). Thus, we suppress the dependence on the generating
set and denote the completion by $\overline{\Gamma}_{G}$. Note that (1)
$\Gamma_{G}$ has finite $d_{f}$-diameter as $f$ is summable. (2)
$\overline{\Gamma}_{G}$ is compact. Since the group $G$ acts by isometries on
$\Gamma_{G}$, this action extends to an action of $G$ on
$\overline{\Gamma}_{G}$ by homeomorphism (see [Kar03] for details). Similarly,
one can define the Floyd boundary of a locally finite graph.

######  Definition 2.2 (Shortest sequence).

A sequence $\\{w_{n}\\}\subset\Gamma_{G}$ is said to be a shortest sequence
if, for all $i$, $d(e_{G},w_{i})=i$ and $d(w_{i},w_{i+1})=1$.

######  Lemma 2.3.

If $\\{w_{n}\\}$ is a shortest sequence in $\Gamma_{G}$ then $\\{w_{n}\\}$ is
a Cauchy sequence in $(\Gamma_{G},d_{f})$.

###### Proof.

Since $\Sigma_{r=1}^{\infty}f(r)<\infty$, given $\epsilon>0$, there exists
$N\in\mathbb{N}$ such that $\Sigma_{r=N}^{\infty}f(r)\leq\epsilon$. As
$\\{w_{n}\\}$ is shortest, for all $i,j\geq N$,

$d_{f}(w_{i},w_{j})\leq\Sigma_{r=N}^{\infty}f(r)\leq\epsilon.$

Hence the lemma. ∎

###  2.3. A tree of Cayley graphs corresponding to amalgamated free product

Suppose $A$ and $B$ are two finitely generated groups, and $H$ is a finite
group. Let $G=A\ast_{H}B$ be an amalgamated free product with monomorphisms
$i_{A}:H\rightarrow A$ and $i_{B}:H\rightarrow B$. Choose finite generating
sets $S_{A}$ and $S_{B}$ of $A$ and $B$ such that $S_{A}\supset
i_{A}(H-\\{1\\})$ and $S_{B}\supset i_{B}(H-\\{1\\})$. Let $\Gamma$ be the
Cayley graph of $G$ with respect to the generating set $S:=S_{A}\cup S_{B}$.
We will denote the associated word metric on $\Gamma$ by $d$. We start this
subsection by defining the Bass–Serre tree ([Ser03]) corresponding to $G$.

Let $\tau$ be a unit interval with vertices $v_{A}$ and $v_{B}$. We define a
tree $T$, called the Bass–Serre tree of $G$, as $G\times\tau$ divided by the
transitive closure of the relation $\sim$ defined as follows:

$(g_{1},v_{A})\sim(g_{2},v_{A})\text{ if }g_{1}^{-1}g_{2}\in A,$
$(g_{1},v_{B})\sim(g_{2},v_{B})\text{ if }g_{1}^{-1}g_{2}\in B,$
$(g_{1},t)\sim(g_{2},t)\text{ if }g_{1}^{-1}g_{2}\in H\text{ and }t\in\tau.$

The tree $T$ comes with a natural action of $G$. For each vertex $v\in T$, we
denote by $G_{v}$ the stabilizer of $v$ in $G$.

Let $\Gamma_{A}$ and $\Gamma_{B}$ be the Cayley graphs of $A$ and $B$ with
respect to $S_{A}$ and $S_{B}$, respectively. Let $\Gamma_{i_{A}(H)}$ and
$\Gamma_{i_{B}(H)}$ be the subgraphs of $\Gamma_{A}$ and $\Gamma_{B}$ having
vertices $i_{A}(H)$ and $i_{B}(H)$, respectively. Since $S_{A}\supset
i_{A}(H-\\{1\\})$ and $S_{B}\supset i_{B}(H-\\{1\\})$, the subgraphs
$\Gamma_{i_{A}(H)}$ and $\Gamma_{i_{B}(H)}$ are subgraphs of diameter $1$.
Define a graph $Y$ as the union of $\Gamma_{A}$ and $\Gamma_{B}$ where
$i_{A}(h)$ is identified with $i_{B}(h)$ for all $h\in H$. Thus, in $Y$, we
obtain a subgraph which is union of $\Gamma_{i_{A}(H)}$ and
$\Gamma_{i_{B}(H)}$ where $i_{A}(h)$ is identified with $i_{B}(h)$ for all
$h\in H$. We will denote this subgraph by $\Gamma_{H}$. Finally, we define the
graph $\Gamma$ as an equivalence relation on $G\times Y$ induced by

$(g_{1},y_{1})\sim(g_{2},y_{2})\text{ if }y_{1},y_{2}\in\Gamma_{A}\text{ and
}g_{2}^{-1}g_{1}y_{1}=y_{2},$ $(g_{1},y_{1})\sim(g_{2},y_{2})\text{ if
}y_{1},y_{2}\in\Gamma_{B}\text{ and }g_{2}^{-1}g_{1}y_{1}=y_{2}.$

Note that $\Gamma$ is the Cayley graph of $G$ with respect to $S$. Define a
map $p:\Gamma\to T$ by sending $(g,\Gamma_{A})$ to $(g,v_{A})$ and
$(g,\Gamma_{B})$ to $(g,v_{B})$ for $g\in G$. As $(g,\Gamma_{A})$ and
$(g,\Gamma_{B}$) both contain the subgraph $(g,\Gamma_{H})$, $p$ sends
$(g,\Gamma_{H})$ to $(g,v_{A})$ as well as to $(g,v_{B})$. Since $(g,v_{A})$
and $(g,v_{B})$ are $1$-distance apart in $T$, $p$ is coarsely well-defined.

Notation: We will write the edge $(g,\tau)$ of $T$ as $g\tau$. If, for $g\in
G$, either $v=(g,v_{A})$ or $v=(g,v_{B})$ is a vertex of $T$ then the Cayley
graph of $G_{v}$ is isometric to either $(g,\Gamma_{A})$ or $(g,\Gamma_{B})$.
For $(g,v_{A})\in T$, we denote by $\Gamma_{gA}$ the preimage of $(g,v_{A})$
under $p$ which is nothing but $(g,\Gamma_{A})$. Similarly, for $(g,v_{B})\in
T$, we denote by $\Gamma_{gB}$ the preimage of $(g,v_{B})$ under $p$. For a
vertex $v\in T$, we denote by $\Gamma_{v}$ the preimage of $v$ under $p$. We
will denote the subgraphs $(g,\Gamma_{H})$ by $\Gamma_{gH}$. After this point,
while using the notation $g\tau,\Gamma_{gA}$ and $gA$, we will assume $g$ to
be the shortest possible coset representative. Similarly, we will assume the
same for cosets of $B$ and $H$.

###  2.4. A tree of Cayley graphs corresponding to HNN extension

Let $A$ be a finitely generated group with finite subgroups $F$ and
$F^{\prime}$ such that $F\simeq F^{\prime}$ as groups. Let $G=A*_{F\simeq
F^{\prime}}$ be a HNN extension of $A$. Let $\phi:F\to F^{\prime}$ be the
isomorphism. Then, the presentation of $G$ is $\langle
S_{A},t:t^{-1}ft=\phi(f),f\in F\rangle$. Let $S_{A}$ be a finite generating
set of $A$, which contains $F\setminus\\{1\\}$ and
$F^{\prime}\setminus\\{1\\}$. Let $\Gamma$ be the Cayley graph of $G$ with
respect to $S:=S_{A}\cup\\{t\\}$, where $t$ is the stable letter. We will
denote the word metric on $\Gamma$ by $d$.

Firstly, we give a description of the Bass–Serre tree, say, $T$ of $G$. The
vertex set of $T$, say, $V(T)$ is the set of left cosets of $A$ in $G$. The
set of edges of $T$, say, $E(T)$ is the set of left cosets of $F$ in $G$. For
all $g\in G$, vertices $gA$ and $gtA$ are connected by the edge $gF$.

Let $\Gamma_{A}$ be the Cayley graph of $A$ with respect to $S_{A}$. Consider
the graph $Y$ induced by the following equivalence relation on
$G\times\Gamma_{A}$ :

$(g_{1},y_{1})\sim(g_{2},y_{2})\text{ if }y_{1},y_{2}\in\Gamma_{A}\text{ and
}g_{2}^{-1}g_{1}y_{1}=y_{2}.$

Denote the equivalence class $[g,x]$ by $gx$ and $\Gamma_{gA}$ be the subgraph
spanned by the vertices in $\\{gx\in Y:x\in\Gamma_{A}\\}$. These are the
connected components of $Y$, which are isometric to $\Gamma_{A}$ with the word
metric. Now, the Cayley graph of $G$ can defined by using the graph $Y$ as
follows:

For all $g\in G$, the vertices $gA$ and $gtA$ are connected in the Bass-Serre
tree $T$. Similarly, join the vertices $gf\in\Gamma_{gA}$ and
$gft(=gt\phi(f))\in\Gamma_{gtA}$, for all $f\in F$. Denote the subgraph
spanned by the vertices in $\\{gf,gt\phi(f):f\in F\\}$ by $\Gamma_{gF}$. From
this discussion, we can say that the subgraphs $\Gamma_{gA}$ and
$\Gamma_{gtA}$ are connected through the subgraph $\Gamma_{gF}$. Hence, we can
define the natural projection map, $p:\Gamma\to T$, such that
$p(\Gamma_{gA}):=gA$, which in this case is well-defined.

##  3\. A Construction of Floyd boundaries of an amalgamated free product and
HNN extension

Amalgamated free product case. Let $G=A\ast_{H}B$ be as in Subsection 2.3. The
next subsection is devoted to a construction of Floyd boundary of $G$. When
$A$ and $B$ are hyperbolic and $H$ is trivial, Martin–S̀wiatkowski gave a
construction of Gromov boundary of $G$ in [MS15]. We closely follow that
construction. Let $f$ be a Floyd function.

Notation: We retain the notations used in the Subsection 2.3. We will denote
the scaled Cayley graph of $G$ with respect to the generating set $S$ by
$\Gamma_{f}$ with the Floyd metric $d_{f}$. For the sake of ease, we will
denote the metric subspaces $(\Gamma_{gA},d_{f})$, $(\Gamma_{gB},d_{f})$ and
$(\Gamma_{gH},d_{f})$ by $\Gamma_{gA}^{f}$, $\Gamma_{gB}^{f}$ and
$\Gamma_{gH}^{f}$ respectively. Similarly, when $v\in T$, we will denote the
metric subspace $(\Gamma_{v},d_{f})$ by $\Gamma_{v}^{f}$. Define
$\partial_{f}(\Gamma_{v}^{f}):=\overline{\Gamma}_{v}^{f}-\Gamma_{v}^{f}$,
where $\overline{\Gamma}_{v}^{f}$ is the metric completion of
$\Gamma_{v}^{f}$. Let $\partial_{f}(A)$ and $\partial_{f}(B)$ be the Floyd
boundary of $A$ and $B$, respectively.

###  3.1. Boundaries of the stabilizers.

Let $\delta_{Stab}(\Gamma_{f})$ be the set
$G\times(\partial_{f}A\sqcup\partial_{f}B)$ divided by the equivalence
relation induced by

$(g_{1},\xi_{1})\sim(g_{2},\xi_{2})\text{ if
}\xi_{1},\xi_{2}\in\partial_{f}A,\hskip 1.9919ptg_{2}^{-1}g_{1}\in A\text{ and
}g_{2}^{-1}g_{1}\xi_{1}=\xi_{2},$ $(g_{1},\xi_{1})\sim(g_{2},\xi_{2})\text{ if
}\xi_{1},\xi_{2}\in\partial_{f}B,\hskip 1.9919ptg_{2}^{-1}g_{1}\in A\text{ and
}g_{2}^{-1}g_{1}\xi_{1}=\xi_{2}.$

The equivalence class of an element $(g,\xi)$ is denoted by $[g,\xi]$. The set
$\delta_{Stab}(\Gamma_{f})$ comes with a natural action of $G$ on the left.
This also comes with a natural projection onto the set of vertices of $T$. The
preimage of each vertex $v\in T$ is denoted by $\partial_{f}(G_{v})$ which is
identified with $\partial_{f}(\Gamma_{v}^{f})$ by Lemma 3.1.

Let $\partial T$ denotes the Gromov boundary of $T$. Then, as a set, we denote
the boundary of $\Gamma_{f}$ as

$\delta(\Gamma_{f}):=\delta_{Stab}(\Gamma_{f})\sqcup\partial T.$

Also, as a set, we define the compactification of $\Gamma_{f}$ as

$\overline{\Gamma}_{f}:=\Gamma_{f}\cup\delta(\Gamma_{f}).$

This set comes with a natural action of $G$ and with a natural map
$p:\overline{\Gamma}_{f}\to\overline{T}$, which is coarsely well-defined. The
preimage of a vertex $v\in T$ is $\Gamma_{v}^{f}\cup\partial_{f}(G_{v})$ that
is identified, as a set, with $\overline{\Gamma}_{v}^{f}$ which is the metric
completion of $\Gamma_{v}^{f}$.

###  3.2. Two topologies on $\overline{\Gamma}_{f}$

There are two ways to put a topology on $\overline{\Gamma}_{f}$. Here, we give
a description of these topologies, and we prove that these two are equivalent.

Topology using neighborhood system. For a point $x\in\Gamma_{f}$, we set a
basis of open neighborhoods of $x$ in $\Gamma_{f}$ to be a basis of open
neighborhoods of $x$ in $\overline{\Gamma}_{f}$. Now we define a basis of open
neighborhoods for points of $\delta(\Gamma_{f})$. Fix a vertex $v_{0}$ of $T$.

  1. (1)

Let $\xi\in\delta_{Stab}(\Gamma_{f})$. Suppose $v$ is the vertex of $T$ such
that $\xi\in\partial_{f}(G_{v})$. Let $U$ be an open neighborhood of $\xi$ in
$\overline{\Gamma}_{v}^{f}$. Define $V_{U}$ to be the set of all
$z\in\overline{\Gamma}_{f}\setminus\overline{\Gamma}_{v}^{f}$ such that the
first edge of the geodesic $[v,p(z)]$ or the geodesic ray $[v,p(z))$ is
$g\tau$ where $g$ is a vertex in $U$. Then we set

$V_{U}(\xi):=U\cup V_{U}.$

A neighborhood basis of $\xi$ in $\overline{\Gamma}_{f}$ is a collection of
set $V_{U}(\xi)$ where $U$ runs over a neighborhood basis of $\xi$ in
$\overline{\Gamma}_{v}^{f}$.

  2. (2)

Let $\eta\in\partial T$. Let $T_{n}(\eta)$ be the subtree of $T$ consisting of
those elements $x$ of $T$ for which the first $n$ edges of $[v_{0},x]$ and
$[v_{0},\eta)$ are the same. Suppose $u_{n}(\eta)$ is the vertex on
$[v_{0},\eta)$ at the distance $n$ from $v_{0}$. Let $\partial(T_{n}{(\eta)})$
denotes the Gromov boundary of $T_{n}{(\eta)}$ and let
$\overline{T_{n}(\eta)}=T_{n}(\eta)\cup\partial(T_{n}(\eta))$. We define

$V_{n}(\eta)=p^{-1}(\overline{T_{n}(\eta)}\setminus\\{u_{n}(\eta)\\}).$

We take the collection $\\{V_{n}(\eta):n\geq 1\\}$ as a basis of open
neighborhoods of $\eta$ in $\overline{\Gamma}_{f}$.

We skip a straightforward verification that the above collections of sets
satisfy the axioms for the basis of open neighborhoods.

Metric on $\overline{\Gamma}_{f}.$ Recall that $d_{f}$ is the Floyd metric on
$\Gamma_{f}$ and $\Gamma_{gA}^{f}$, $\Gamma_{gB}^{f}$ are the notations for
the metric subspaces as indicated in the first paragraph of this section.
Before defining the metric on $\overline{\Gamma}_{f}$, we record the following
observations which follows from [Kar03, p. 5366]:

######  Lemma 3.1.

For $g\in G$, the natural translation map from $\Gamma_{A}^{f}$ to
$\Gamma_{gA}^{f}$ is bi-Lipschitz and hence induces a homeomorphism from
$\partial_{f}(A)$ to $\partial_{f}(\Gamma_{gA}^{f})$. Similarly,
$\partial_{f}(B)$ is homeomorphic to $\partial_{f}(\Gamma_{gB}^{f})$. ∎

Let $\eta\in\partial T$. Fix the vertex $v_{0}=(e_{G},v_{A})$ and let
$\\{v_{n}\\}$ be the sequence of vertices that lies on the geodesic ray
joining $v_{0}$ to $\eta$ in $T$. Suppose for all $i\geq 0$, $v_{i}$ and
$v_{i+1}$ are connected by the edge $g_{i}\tau$. Choose
$x_{i}\in\Gamma_{g_{i}H}$. It is straightforward to show that $\\{x_{i}\\}$ is
a Cauchy sequence in $\Gamma_{f}$ and any other choice $\\{y_{i}\\}$ is Cauchy
equivalent to $\\{x_{i}\\}.$ We call such a sequence $\\{x_{i}\\}$, a Cauchy
representative of $\eta$.

Definition of metric $\bar{d_{f}}$ on $\overline{\Gamma}_{f}$. We define a
metric $\bar{d_{f}}$ in the following manner:

  1. (1)

For $x,y\in\Gamma_{f}$, $\bar{d_{f}}(x,y):=d_{f}(x,y)$.

  2. (2)

Suppose $x\in\Gamma_{f}$ and $\xi\in\partial_{f}(G_{v})$ for some vertex $v\in
T$. Let $\\{x_{n}\\}$ be a Cauchy sequence in $\Gamma_{v}^{f}$ representing
$\xi$. Then, $\bar{d_{f}}(x,\xi):=\lim_{n\to\infty}d_{f}(x,x_{n})$. This limit
exists as $\\{x_{n}\\}$ is Cauchy, and it is independent of the chosen
sequence $\\{x_{n}\\}$.

  3. (3)

Suppose $\xi_{1}\in\partial_{f}(G_{v})$ and $\xi_{2}\in\partial_{f}(G_{w})$
for some $v,w\in T$. Let $\\{x_{n}\\}$ and $\\{y_{n}\\}$ be Cauchy sequences
in $\Gamma_{v}^{f}$ and $\Gamma_{w}^{f}$ representing $\xi_{1}$ and $\xi_{2}$,
respectively. Then, $\bar{d_{f}}:=\lim_{n\to\infty}d_{f}(x_{n},y_{n})$. Since
$\\{x_{n}\\}$ and $\\{y_{n}\\}$ are Cauchy, this limit exists, and it is
independent of chosen sequences.

  4. (4)

Let $\eta,\eta_{1},\eta_{2}$ be three distinct points in $\partial T$ and
$\xi\in\delta_{Stab}(\Gamma_{f})$. Let $\\{x_{n}\\},\\{y_{n}\\},\\{z_{n}\\}$
be Cauchy representatives of $\eta,\eta_{1},\eta_{2}$, respectively, in
$\Gamma_{f}$. The above discussion shows that the following definition does
not depend on the chosen sequences.

    1. (a)

$\bar{d}_{f}(\eta,z):=\lim_{n\to\infty}d_{f}(x_{n},z)$, where
$z\in\Gamma_{f}$.

    2. (b)

$\bar{d}_{f}(\eta,\xi):=\lim_{n\to\infty}d_{f}(x_{n},u_{n})$, where
$\xi\in\partial_{f}(G_{v})$ for some $v\in T$ and $\\{u_{n}\\}$ is Cauchy
sequence in $\Gamma_{v}^{f}$ representing $\xi$.

    3. (c)

$\bar{d}_{f}(\eta_{1},\eta_{2}):=\lim_{n\to\infty}d_{f}(y_{n},z_{n})$.

######  Remark 3.2.

From the definition of the metric $\bar{d}_{f}$, we can directly conclude that
for each vertex $v\in T$, the restriction of $\bar{d_{f}}$ on
$\overline{\Gamma}_{v}^{f}$ is the same as the completion metric of $d_{f}$ on
$\Gamma_{v}^{f}$.

######  Proposition 3.3.

$\overline{\Gamma}_{f}$ is a metric space.

###### Proof.

From the definition, it follows that $\bar{d}_{f}$ is a pseudo metric. It
remains to check the positivity of $\bar{d}_{f}$. If $x\in\Gamma_{f}$ and
$\xi\in\delta_{Stab}(\Gamma_{f})$ then $\bar{d}_{f}(x,\xi)>0$. Thus, the
following three cases are remaining to consider:

Case 1. Suppose $\xi_{1}\neq\xi_{2}\in\delta_{Stab}(\Gamma_{f})$. If
$\xi_{1},\xi_{2}\in\partial_{f}(G_{v})$ for some $v\in T$ then, by Remark 3.2,
$\bar{d}_{f}(\xi_{1},\xi_{2})>0$. Suppose $v\neq w\in T$ such that
$\xi_{1}\in\partial_{f}(G_{v})$ and $\xi_{2}\in\partial_{f}(G_{w})$. Let
$e_{1}=[v,v_{1}]$ and $e_{2}=[v_{n},w]$ be the first and last edge on the
geodesic $[v,w]$ in $T$. Suppose $\Gamma_{v}^{f}$ and $\Gamma_{v_{1}}^{f}$ are
joined through $\Gamma_{g_{1}H}^{f}$. Similarly, suppose $\Gamma_{v_{n}}^{f}$
and $\Gamma_{w}^{f}$ are joined through $\Gamma_{g_{n}H}^{f}$. Then, we have
the following inequality:

(1) $d_{f}(x,y)\geq
d_{f}(x,\Gamma_{g_{1}H}^{f})+d_{f}(\Gamma_{g_{n}H}^{f},y).$

Suppose $\\{x_{n}\\},$ $\\{y_{n}\\}$ are Cauchy sequences in $\Gamma_{v}^{f}$
and $\Gamma_{w}^{f}$ representing $\xi_{1}$ and $\xi_{2}$, respectively. Then,
putting $x=x_{n}$ and $y=y_{n}$ in Equation 1 and applying limit on both
sides, we get

$\bar{d}_{f}(\xi_{1},\xi_{2})\geq\bar{d}_{f}(\xi_{1},\Gamma_{g_{1}H}^{f})+\bar{d}_{f}(\Gamma_{g_{n}H}^{f},\xi_{2}).$

By Remark 3.2 and the fact that $\partial(\Gamma_{v}^{f})$ and
$\Gamma_{g_{1}H}^{f}$ are compact subset of $\overline{\Gamma}_{v}^{f}$, we
have

$\bar{d}_{f}(\xi_{1},g_{1})\geq\bar{d}_{f}(\Gamma_{g_{1}H}^{f},\partial(\Gamma_{v}^{f}))>0.$

Hence, we conclude that $\bar{d}_{f}(\xi_{1},\xi_{2})>0$.

Case 2. Suppose that $\xi\in\delta_{Stab}(\Gamma_{f})$, $\eta\in\partial T$.
Let $v\in T$ such that $\xi\in\partial_{f}(G_{v})$ and let $w_{n}$ be a Cauchy
representative of $\eta$. Let $e_{1}$ be the first edge on the geodesic ray
$[v,\eta)$. Suppose $\Gamma_{v}^{f}$ and $\Gamma_{v_{1}}^{f}$ are joined
through $\Gamma_{g_{1}H}^{f}$. Then, as in Case 1, we get

$d_{f}(x,w_{n})\geq d_{f}(x,\Gamma_{g_{1}H}^{f})\hskip 7.11317pt\text{ for all
}x\in\Gamma_{v}\text{ and sufficiently large }n.$

Let $\\{x_{n}\\}$ be a Cauchy representative of $\xi$ in $\Gamma_{v}^{f}$.
Putting $x=x_{n}$ in the last equation and taking the limit on both sides, we
get

$\bar{d}_{f}(\xi,\eta)\geq\bar{d}_{f}(\xi,\Gamma_{g_{1}H}^{f})>0.$

Case 3. Suppose $\eta_{1},\eta_{2}\in\partial T$ and $\\{x_{n}\\}$ and
$\\{y_{n}\\}$ be their respective Cauchy representatives. Let $\gamma_{1}$ and
$\gamma_{2}$ be the geodesics from $v_{A}$ to $\eta_{1}$ and $\eta_{2}$,
respectively. Then there exists a unique vertex $v_{0}\in T$ such that
$\gamma_{1}\cap\gamma_{2}=[v_{A},v_{0}].$ Let $e_{1}=[v_{0},v_{1}]$ and
$e_{2}=[v_{0},v_{2}]$ be the first edges on $\gamma_{1}$ and $\gamma_{2}$ just
after $[v_{A},v_{0}]$. Suppose $\Gamma_{v_{0}}^{f}$, $\Gamma_{v_{1}}^{f}$ are
joined through $\Gamma_{g_{1}H}^{f}$ and $\Gamma_{v_{0}}^{f}$,
$\Gamma_{v_{2}}^{f}$ are joined through $\Gamma_{g_{2}H}^{f}$. Then, for all
sufficiently large $n$, each geodesic joining $x_{n}$ and $y_{n}$ in
$\Gamma_{f}$ passes through $\Gamma_{g_{1}H}^{f}$ and $\Gamma_{g_{2}H}^{f}$
which implies that $d_{f}(x_{n},y_{n})\geq
d_{f}(\Gamma_{g_{1}H}^{f},\Gamma_{g_{2}H}^{f})>0.$ Hence, we conclude that
$\bar{d}_{f}(\eta_{1},\eta_{2})>0.$ ∎

Now, we are ready to prove the following:

######  Proposition 3.4.

The Floyd boundary $\partial_{f}(G)$ of $G$ is isometric to
$\delta(\Gamma_{f})$.

###### Proof.

Recall that for every $z\in\Gamma$, $p(z)$ has two possible. For this proof,
we will choose the value closest to $v_{A}$.

Let $i:(\Gamma_{f},d_{f})\rightarrow(\overline{\Gamma}_{f},\bar{d}_{f})$ be
the natural inclusion map, which is an isometric embedding. Hence it extends
to an isometric embedding
$\bar{i}:\Gamma_{f}\cup\partial_{f}(G)\to\overline{\Gamma_{f}\cup\delta(\Gamma_{f})}$,
where $\overline{\Gamma_{f}\cup\delta(\Gamma_{f})}$ denotes the Cauchy
completion of $\Gamma_{f}\cup\delta(\Gamma_{f})$. We show that $\bar{i}$ is in
fact an isometry. Note that
$\bar{i}(\Gamma_{f}\cup\partial_{f}(G))=\overline{i(\Gamma_{f})}$ (which is
nothing but the Cauchy completion of $\Gamma_{f}$). Let $\\{w_{n}\\}\in\Gamma$
be a shortest sequence. Then, by Lemma 2.3, $\\{w_{n}\\}$ is a Cauchy sequence
in $(\Gamma_{f},d_{f})$. Also, the distance between $v_{A}$ and $p(w_{n})$ is
non-decreasing. Hence, we have the following two cases:

Case 1. There exists $N\in\mathbb{N}$ such that for all sufficiently large n,
the distance in $T$ between $v_{A}$ and $p(w_{n})$ is $N$. Since $\\{w_{n}\\}$
is a shortest sequence in $\Gamma$, there exists a vertex $v_{N}\in T$ such
that $d_{T}(v_{A},v_{N})=N$ and $w_{n}\in\Gamma_{v_{N}}^{f}$ for all large
$n$. Hence, $\\{w_{n}\\}$ converges to a point
$\xi\in\partial_{f}(\Gamma_{v_{N}}^{f})\subset\delta(\Gamma_{f})$.

Case 2. The distance between $v_{A}$ and $p(w_{n})$ goes to $\infty$ as
$n\rightarrow\infty.$ Since $\\{w_{n}\\}$ is shortest, there exists a geodesic
ray, say, $\gamma$ in $T$ such that $\gamma(0)=v_{A}$ and $p(w_{n})\in\gamma$.
Let $\eta\in\partial T$ be the point represented by $\gamma$. Then, a
subsequence $\\{w_{n_{k}}\\}$ becomes a Cauchy representative of $\eta$.
Hence, $\\{w_{n}\\}$ converges to $\eta$ in
$(\overline{\Gamma}_{f},\bar{d}_{f})$.

Let $z\in\partial_{f}(G)$. Then there exists a shortest sequence
$\\{w_{n}\\}\in\Gamma_{f}$ such that $w_{n}\rightarrow z$ in
$\Gamma_{f}\cup\partial_{f}(G)$. From the above two cases, we know that there
exists $z^{\prime}\in\delta(\Gamma_{f})$ such that $w_{n}\rightarrow
z^{\prime}$ in $\Gamma_{f}\cup\delta(\Gamma_{f})$. Hence, by continuity of
$\bar{i}$, we have that $\bar{i}(z)=z^{\prime}$. This implies that
$\overline{i}(\partial_{f}(G))\subset\delta(\Gamma_{f})$. Hence, we conclude
that
$\overline{i(\Gamma_{f})}=\bar{i}(\Gamma_{f}\cup\partial_{f}(G))\subset\Gamma_{f}\cup\delta\Gamma_{f}.$
Since, by construction, $i(\Gamma_{f})$ is dense in
$\Gamma_{f}\cup\delta\Gamma_{f}$, we have that
$\overline{\Gamma_{f}\cup\delta\Gamma_{f}}=\overline{i(\Gamma_{f})}\subset\Gamma_{f}\cup\delta\Gamma_{f}$.
This in turn implies that
$\overline{\Gamma_{f}\cup\delta\Gamma_{f}}=\Gamma_{f}\cup\delta\Gamma_{f}=\overline{i(\Gamma_{f})}$.
Hence, $\bar{i}$ is an isometry between $\Gamma_{f}\cup\partial_{f}G$ and
$\Gamma_{f}\cup\delta(\Gamma_{f}),$ that gives us the desired result. ∎

###  3.3. Equivalence of topologies on $\overline{\Gamma}_{f}$

Let $\mathcal{T}_{o}$ be the topology defined by the neighborhood basis on
$\overline{\Gamma}_{f}$ and let $\mathcal{T}_{m}$ be the topology on
$\overline{\Gamma}_{f}$ induced by the metric $\bar{d}_{f}$. In this
subsection, we show that $\mathcal{T}_{o}$ is the same as $\mathcal{T}_{m}$.

######  Proposition 3.5.

$\mathcal{T}_{o}=\mathcal{T}_{m}$.

###### Proof.

$\mathcal{T}_{o}$ is finer than $\mathcal{T}_{m}$: Let
$\xi\in\delta_{Stab}(\Gamma_{f})$. Let $v\in T$ represents coset $g_{0}A$ with
$g_{0}$ being a reduced representative such that $\xi\in\partial_{f}(G_{v})$.
Let $B_{r}$ be a $r$-radius ball about $\xi$ in
$(\overline{\Gamma}_{f},\bar{d}_{f})$. We show that there exists a
neighborhood $U$ of $\xi$ in $\overline{\Gamma}_{v}^{f}$ such that
$V_{U}(\xi)\subset B_{r}(\xi)$. Choose $N\in\mathbb{N}$ such that
$\sum_{j=N}^{\infty}f(j)<\frac{r}{3}$ and $d(e_{G},\Gamma_{g_{0}A})<N-1$. Let

$W:=\\{x\in\Gamma\hskip 4.26773pt|\hskip 4.26773ptd(x,e_{G})\leq N\\}.$

Note that $W$ is a compact subset of $(\overline{\Gamma}_{f},\bar{d}_{f})$.
Define $U:=(B_{r/3}-W)\cap\Gamma_{v}^{f}$. It is evident that $U$ is an open
neighborhood of $\xi$ in $\overline{\Gamma}_{v}^{f}.$ Let $z$ be an element in
$V_{U}(\xi)$ and let $e_{1}=[v,v_{1}]$ be the first edge either on the
geodesic $[v,p(z)]$ or on the geodesic ray $[v,p(z))$. Assume that
$\Gamma_{v}^{f}$ and $\Gamma_{v_{1}}^{f}$ are joined through
$\Gamma_{gH}^{f}$. Since every path in $\Gamma_{f}$ joining $e_{G}$ and $z$
passes through $\Gamma_{gH}^{f}$, we have

(2)
$\bar{d}_{f}(e,\Gamma_{gH}^{f})+\bar{d}_{f}(\Gamma_{gH}^{f},z)\leq\bar{d}_{f}(e,z).$

Since a representative $g_{1}$ (not necessarily reduced) of $gH$ is contained
in $U$, we get $d(e_{G},\Gamma_{gH})\geq N$. Thus, we have
$\bar{d}_{f}(\Gamma_{gH}^{f},e)\geq\sum_{i=0}^{N-1}f(i),$ and
$\bar{d}_{f}(e,z)\leq\sum_{i=0}^{\infty}f(i).$ By using inequality 2, we get
$\bar{d}_{f}(\Gamma_{gH}^{f},z)\leq\sum_{i=N}^{\infty}f(i)<\frac{r}{3}.$ By
using triangle inequality, we get

$\bar{d}_{f}(\xi,z)\leq\bar{d}_{f}(\xi,\Gamma_{gH}^{f})+\bar{d}_{f}(\Gamma_{gH}^{f},z)+f(N)<\frac{r}{3}+\frac{r}{3}+\frac{r}{3}=r.$

Hence, we conclude that $B_{r}\supset V_{U}(\xi).$

Now, let $\eta\in\partial T$ and let $U_{s}$ be an open ball of radius $s\geq
0$ about $\eta$ in $(\overline{\Gamma}_{f},\bar{d}_{f})$. Choose $N$ such that
$\Sigma_{j=N}^{\infty}f(j)<\frac{s}{3}$. We have a unique sequence $v_{i}$ in
$T$, with $v_{0}=v_{A}$, which along the unique geodesic converges to $\eta$.
Let $g_{i}\tau$ be the edge between $v_{i}$ and $v_{i+1}$ with $g_{i}$ being
the reduced representative of the coset associated to vertex $v_{i+1}$. Then
$\\{g_{i}\\}$ is a Cauchy representative of $\eta$. Thus, $\\{g_{i}\\}$
converges to $\eta$. Choose $N_{0}>N$ such that
$\bar{d}_{f}(\Gamma_{g_{N_{0}}H}^{f},\eta)<\frac{s}{3}.$ It is easy to see
that $d(\Gamma_{g_{N_{0}}H},e_{G})\geq N_{0}.$ For $z\in V_{N_{0}}(\eta)$,
every path between $e_{G}$ and $z$ passes through $\Gamma_{g_{N_{0}}H}$.
Hence, by using inequality 2 and triangle inequality, we get

$\bar{d}_{f}(z,\eta)\leq\bar{d}_{f}(z,\Gamma_{g_{N_{0}}H}^{f})+\bar{d}_{f}(\eta,\Gamma_{g_{N_{0}}H}^{f})+f(N_{0})<\frac{s}{3}+\frac{s}{3}+\frac{s}{3}=r.$

Hence $V_{N_{0}}(\eta)\subset U_{s}$.

$\mathcal{T}_{m}$ is finer than $\mathcal{T}_{o}$: Let
$\xi\in\partial_{f}(\Gamma_{v}^{f})\subset\delta_{Stab}({\Gamma_{f}})$ for
some vertex $v\in T$. Let $U$ be a neighborhood of $\xi$ in
$\overline{\Gamma}_{v}^{f}$ such that $V_{U}(\xi)$ is a neighborhood of $\xi$
in $\overline{\Gamma}_{f}$. Since $U$ is open in $\overline{\Gamma}_{v}^{f}$,
there exists $r\geq 0$ such that the $\overline{\Gamma}_{v}^{f}$-ball of
radius $r$ contained in $U$. Let $B_{r}(\xi)$ is the $r$-radius ball in
$(\overline{\Gamma}_{f},\bar{d}_{f})$. Then, it follows from the definition of
neighborhoods that $B_{r}(\xi)\subset V_{U}(\xi)$. Let $\eta\in\partial T$ and
let $V_{M}(\eta)$ be a neighborhood in $\overline{\Gamma}_{f}$. Choose $t\geq
0$ such that $\Sigma_{j=M}^{\infty}f(j)<\frac{t}{3}$. Let $B_{t}(\eta)$ is
ball of radius $t$ in $(\overline{\Gamma}_{f},\bar{d}_{f})$. Then, again, it
is easy to see that $V_{M}(\eta)\subset B_{t}(\eta)$. ∎

###  3.4. HNN extension case

In this subsection, we retain the notations of Subsection 2.4. Let $f$ be a
Floyd function. We will denote the scaled Cayley graph of $G$ with respect to
the generating set $S$ by $\Gamma_{f}$ with the Floyd metric $d_{f}$. For the
sake of ease, we will denote the metric subspaces $(\Gamma_{gA},d_{f})$ by
$\Gamma_{gA}^{f}$. Similarly, when $v\in T$, we will denote the metric
subspace $(\Gamma_{v},d_{f})$ by $\Gamma_{v}^{f}$. Define
$\partial_{f}(\Gamma_{v}):=\overline{\Gamma}_{v}^{f}-\Gamma_{v}^{f}$, where
$\overline{\Gamma}_{v}^{f}$ is the metric completion of $\Gamma_{v}^{f}$.

Boundaries of the stabilizers. Let $\delta_{Stab}(\Gamma_{f})$ be the set
$G\times\partial_{f}(A)$ divided by the equivalence relation induced by

$(g_{1},\xi_{1})\sim(g_{2},\xi_{2})\text{ if
}\xi_{1},\xi_{2}\in\partial_{f}A,\hskip 1.9919ptg_{2}^{-1}g_{1}\in A\text{ and
}g_{2}^{-1}g_{1}\xi_{1}=\xi_{2}.$

The equivalence class of an element $(g,\xi)$ is denoted by $[g,\xi]$. The set
$\delta_{Stab}(\Gamma_{f})$ comes with a natural action of $G$ on the left.
This also comes with a natural projection onto the set of vertices of $T$. A
result similar to Lemma 3.1 holds for $G$, which implies that
$\partial_{f}(A)$ is homeomorphic to $\partial_{f}(\Gamma_{gA}^{f})$ for all
$g\in G$. Hence, for every vertex $v\in T$, $\partial_{f}(G_{v})$ can be
identified with $\partial_{f}(\Gamma_{v}^{f})$.

Let $\partial T$ denotes the Gromov boundary of $T$. Then, as a set, we define
the boundary of $\Gamma$ as

$\delta(\Gamma_{f}):=\delta_{Stab}(\Gamma_{f})\sqcup\partial T.$

Also, as a set, we define the compactification of $\Gamma_{f}$ as

$\overline{\Gamma}_{f}:=\Gamma_{f}\cup\delta(\Gamma_{f}).$

This set comes with a natural action of $G$ and with a natural map
$p:\overline{\Gamma}_{f}\to\overline{T}$, which in this case is well-defined.
The preimage of a vertex $v\in T$ is $\Gamma_{v}^{f}\cup\partial_{f}(G_{v})$
that is identified, as a set, with $\overline{\Gamma}_{v}^{f}$ which is the
metric completion of $\Gamma_{v}^{f}$. As in Subsection 3.2, we can similarly
define topology $\mathcal{T}_{o}$ and metric $\bar{d}_{f}$ on $\Gamma_{f}$.
Following the steps employed in previous subsections, one can easily show that
$\overline{\Gamma}_{f}$ with the topology $\mathcal{T}_{o}$ and the topology
induced by the metric $\bar{d}_{f}$ are homeomorphic as well as
$\delta(\Gamma_{f})$ with the metric $\bar{d}_{f}$ is isometric to
$\partial_{f}(G)$. Hence, $\delta(\Gamma_{f})$ with the subspace topology
induced by $\mathcal{T}_{o}$ is homeomorphic to $\partial_{f}(G)$.

##  4\. Homeomorphism types of Floyd boundaries of free products

The main goal of this section is to prove the following theorem. In this
section, assume all the groups to be infinite unless mentioned otherwise.

######  Theorem 4.1.

Suppose $G_{1}=A_{1}\ast B_{1}$ and $G_{2}=A_{2}\ast B_{2}$ are two free
products of infinite groups. Let $f_{1}$ and $f_{2}$ are two Floyd functions
such that $\partial_{f_{1}}(A_{1})\simeq\partial_{f_{2}}(A_{2})$ and
$\partial_{f_{1}}(B_{1})\simeq\partial_{f_{2}}(B_{2})$ then there is a
homeomorphism from $\partial_{f_{1}}(G_{1})$ to $\partial_{f_{2}}(G_{2})$.

An easy application of induction gives the following:

######  Corollary 4.2.

For $k\geq 2$, suppose $G_{1}=A_{1}\ast\dots\ast A_{k}$ and
$G_{2}=B_{1}\ast\dots\ast B_{k}$ are two free products of infinite groups. Let
$f_{1}$ and $f_{2}$ are two Floyd functions such that
$\partial_{f_{1}}(A_{i})\simeq\partial_{f_{2}}(B_{i})$ for $1\leq i\leq k$.
Then, $\partial_{f_{1}}(G_{1})\simeq\partial_{f_{2}}(G_{2})$. ∎

Notation: Let $h_{A}:\partial_{f_{1}}(A_{1})\to\partial_{f_{2}}(A_{2})$ and
$h_{B}:\partial_{f_{1}}(B_{1})\to\partial_{f_{2}}(B_{2})$ be the fixed
homeomorphism of Floyd boundaries. We denote by
$q:\partial_{f_{1}}(A_{1})\cup\partial_{f_{1}}(B_{1})\to\partial_{f_{2}}(A_{2})\cup\partial_{f_{2}}(B_{2})$
the homeomorphism induced by $h_{A}$ and $h_{B}$. Let $T_{1}$ and $T_{2}$ be
the Bass–Serre trees of $A_{1}\ast B_{1}$ and $A_{2}\ast B_{2}$ respectively.
Finally, we denote by $\Gamma_{f_{1}}$ and $\Gamma_{f_{2}}$ the Cayley graphs
of $G_{1}$ and $G_{2}$, respectively.

######  Remark 4.3.

The construction of a Cayley graph of a free product is same as that of
$\Gamma_{f}$ as in Section 2.3 by taking $H$ to be trivial.

Suppose $G$ and $H$ are two finitely generated groups, and $f$, $f^{\prime}$
are two Floyd functions for $G$ and $H$, respectively. Let $\Gamma_{G}$ and
$\Gamma_{H}$ be the Cayley graphs of $G$ and $H$ respectively with the Floyd
metrics $d_{f,G}$ and $d_{f^{\prime},H}$. Then,
$\overline{\Gamma}_{G}=\Gamma_{G}\cup\partial_{f}(G),\overline{\Gamma}_{H}=\Gamma_{H}\cup\partial_{f^{\prime}}(H)$
are compact metrizable spaces. Then one has metric subspaces
$G\cup\partial_{f}(G)$ and $H\cup\partial_{f^{\prime}}(H)$ with metric
$d_{f,G}$ and $d_{f^{\prime},H}$, respectively.. The following lemma is an
analog of [MS15, Lemma 4.2] in the context of Floyd boundaries.

######  Lemma 4.4.

Let $\psi:\partial_{f}(G)\to\partial_{f^{\prime}}(H)$ be a homeomorphism. Then
there is a bijection $b:G\to H$ such that $b(1)=1$ and
$b\cup\psi:G\cup\partial_{f}(G)\to H\cup\partial_{f^{\prime}}(H)$ is a
homeomorphism.

###### Proof.

Define a map $\pi:G\to\partial_{f}(G)$ which takes $g\in G$ to
$\xi\in\partial_{f}(G)$ such that $d_{f,G}(g,\partial_{f}(G))=d_{f,G}(g,\xi)$
(such a $\xi$ exists as $\partial_{f}(G)$ is compact). An easy application of
triangle inequality shows that $\pi(G)$ is dense in $\partial_{f}(G)$. Order
the elements of $G\setminus\\{1\\}$ and $H\setminus\\{1\\}$ into sequences
$\\{g_{k}\\}_{k\in\mathbb{N}}$ and $\\{h_{k}\\}_{k\in\mathbb{N}}.$ Define
$b(1)=1$. To choose the required $b$, iterate the following two steps
alternatively.

Step 1. Suppose $k$ is the smallest number for which $b(g_{k})$ is not yet
defined. Since $H$ is dense in $H\cup\partial_{f^{\prime}}(H)$, choose some
$l\in\mathbb{N}$ such that $h_{l}$ is not an image of any $g_{i}$ under the
map $b$ and

$d_{f^{\prime},H}(h_{l},\psi(\pi(g_{k})))<\dfrac{1}{k}.$

Then, define $b(g_{k})=h_{l}$.

Step 2. Suppose $k$ is the smallest number for which $h_{k}$ is not chosen as
the image of any $g\in G$ under $b$. Since $\pi(G)$ is dense in
$\partial_{f}(G)$, hence $\psi(\pi(G))$ is dense in
$\partial_{f^{\prime}}(H)$. Choose $g\in G\setminus\\{1\\}$ such that $b$ has
not yet been defined on $g$ and

$d_{f^{\prime},H}(h_{k},\psi(\pi(g)))<d_{f^{\prime},H}(h_{k},\partial_{f^{\prime}}(H))+\dfrac{1}{k}.$

Then, define $b(g)=h_{k}$.

By performing the above two steps alternatively, we see that $b$ is a
bijection. Since $G\cup\partial_{f}(G)$ and $H\cup\partial_{f^{\prime}}(H)$
are compact metrizable, to prove that $b\cup\psi$ is a homeomorphism, it is
sufficient to prove that $b\cup\psi$ is continuous.

Consider a sequence $\\{x_{i}\\}$ of $G$ that converges to
$\xi\in\partial_{f}(G)$. Then, $\lim_{i}d_{f,G}(x_{i},\xi)=0$ and thus
$\lim_{i}d_{f,G}(x_{i},\partial_{f}(G))\to 0$. This implies that
$\lim_{i}d_{f,G}(x_{i},\pi(x_{i}))=0$ and , by using triangle inequality,
therefore $\pi(x_{i})$ converges to $\xi$. Hence, by continuity of $\psi$,
$\psi(\pi(x_{i}))$ converges to $\psi(\xi)$. From the definition of $b$, it
follows that $\lim_{i}d_{f^{\prime},H}(b(x_{i}),\psi(\pi(x_{i})))=0$. Thus,
$b(x_{i})$ converges to $\psi(\pi(\xi))$. Hence, $b\cup\psi$ is continuous. ∎

We immediately have the following:

######  Corollary 4.5.

There exists a bijection $b:G\to H$ such that the following holds:

Let $\xi\in\partial_{f^{\prime}}(H)$ and let $U_{2}$ be an open neighborhood
of $\xi$ in $\overline{\Gamma}_{H}$. Then there exists a neighborhood $U_{1}$
of $f^{-1}(\xi)$ in $\overline{\Gamma}_{G}$ such that $b(G\cap U_{1})\subset
U_{2}$.

###### Proof.

Let $b$ be the bijection as in Lemma 4.4. Thus, $\bar{b}:=b\cup
f:G\cup\partial_{f}(G)\to H\cup\partial_{f^{\prime}}(H)$ is a homeomorphism.
We prove that $b$ is the required bijection. Since $U_{2}$ is an open
neighborhood of $\xi\in\partial_{f^{\prime}}(H)$, viewing
$H\cup\partial_{f}^{\prime}(H)$ as a subspace of $\overline{\Gamma}_{H}$,
$\bar{b}^{-1}(U_{2})$ is an open neighborhood of $f^{-1}(\xi)$ in
$G\cup\partial_{f}(G)$. Since
$K=(G\cup\partial_{f}(G))\setminus\bar{b}^{-1}(U_{2})$ is compact and
$f^{-1}(\xi)\in\overline{\Gamma}_{G}\setminus K$, there exists an open
neighborhood $U_{1}$ of $f^{-1}(\xi)$ in $\overline{\Gamma}_{G}$ such that
$U_{1}\subset\overline{\Gamma}_{G}\setminus K$. Now it is clear that $b(G\cap
U_{1})\subset U_{2}$. ∎

Using the arguments in the Lemma 4.4, we get the following topological result
as a corollary, which we will use in the coming sections.

######  Corollary 4.6.

Let $M,$ $N$ be compact metric spaces with countable dense subspaces $X\subset
M$, $Y\subset N$ such that every element of $X$ and $Y$ are isolated in $M$
and $N,$ respectively. Denote $\partial M:=M-X$ and $\partial N:=N-Y$. Assume
that there exists a homeomorphism $\psi:\partial M\rightarrow\partial N$. Then
there exists a bijection $b:X\rightarrow Y$ such that the map
$b\cup\psi:M\rightarrow N$ is a homeomorphism. ∎

Using Corollary 4.6, we prove the following topological lemma, which turns out
to be useful later. In the following lemma, the closure of a subset $U$ of a
metric space $X$ shall be denoted by $\overline{U}$.

######  Lemma 4.7.

Let $(X,d)$ be a compact metric space and $Y$ be a countable dense subset of
$X$ containing all isolated points of $X$. Define $\partial X:=X\setminus Y.$
Then, there exist metric subspaces $Y_{1}$, $Y_{2}$ of $Y$ such that

  1. (1)

$Y=Y_{1}\sqcup Y_{2}$.

  2. (2)

There exist bijections $\varphi_{1}:Y\rightarrow Y_{1}$ and
$\varphi_{2}:Y\rightarrow Y_{2}$, which induce homeomorphisms
$\bar{\varphi_{1}}:\overline{Y}=X\rightarrow Y_{1}\cup\partial X$ and
$\bar{\varphi_{2}}:\overline{Y}=X\rightarrow Y_{2}\cup\partial X$ such that
$\bar{\varphi_{i}}\bigr{|}_{\partial X}$ is the identity map.

###### Proof.

Let $(X^{\prime},d^{\prime})$ be a metric space such that $\psi:X\rightarrow
X^{\prime}$ is an isometric embedding. Let $Y^{\prime}=\psi(Y)$ and let
$\partial X^{\prime}=X^{\prime}\setminus Y^{\prime}=\psi(\partial X)$. Let
$q:X\sqcup X^{\prime}\to(X\sqcup X^{\prime})/_{\sim}$ be a quotient map where
$x\sim\psi(x)$ for all $x\in\partial X$. Note that , as a set, $(X\sqcup
X^{\prime})/_{\sim}$ is same as $Z:=\partial X\sqcup Y\sqcup Y^{\prime}$. Now,
by [BH99, Lemma 5.24, I.5], we define a metric $d_{q}$ on $Z$ as follows:

  * •

$d_{q}(x,y):=d(x,y)$ if $x,y\in Y\cup\partial X$

  * •

$d_{q}(x,y):=d^{\prime}(x,y)$ if $x,y\in Y^{\prime}$.

  * •

$d_{q}(x,y):=d^{\prime}(x,\psi(y))$ if $x\in Y^{\prime}$, $y\in\partial X$.

  * •

$d_{q}(x,y):=\inf_{z\in\partial
X}\left\\{d(x,z)+d^{\prime}(y,\psi(z))\right\\}$ if $x\in Y$, $y\in
Y^{\prime}$.

The metric $d_{q}$ is compatible with the topology defined on $Z$ by the
quotient map $q$. It is clear from the definition of $d_{q}$ that the
restriction of $q$ on either $X$ or on $X^{\prime}$ is an isometric embedding.
Also, $Y\sqcup Y^{\prime}$ is a countable dense subset of $Z$, which contains
all isolated points of $Z$. Now, we define

$\partial Z:=Z\setminus(Y\sqcup Y^{\prime}).$

Note that the restriction of $q$ on $\partial X$ is an isometry onto $\partial
Z$, which is nothing but the identity map. Since $Z$ is compact, by applying
Corollary 4.6, we get a bijection $\varphi:Y\sqcup Y^{\prime}\rightarrow Y$
and it induces a homeomorphim $\bar{\varphi}:Z\rightarrow X.$ Let
$\varphi(Y)=Y_{1}$ and $\varphi(Y^{\prime})=Y_{2}$. Then,
$\bar{\varphi}(\overline{Y})=Y_{1}\cup\partial X$ and
$\bar{\varphi}(Y^{\prime}\cup\partial X)=Y_{2}\cup\partial X$. We take
$\varphi_{1}$ to be the restriction of $\varphi$ to $Y$ and $\varphi_{2}$ to
be the composition $\varphi\bigr{|}_{Y^{\prime}}\circ q\circ\psi$ restricted
to $Y$. Hence the lemma. ∎

######  Remark 4.8.

(1) The previous lemma can be generalized to $n$ many $Y_{i}$’s with the
properties as in the lemma.

(2) Using the notations used in Lemma 4.7, assume there $n$ many $Y_{i}$’s
with the properties mentioned in the lemma. Let $\xi\in\partial X$ and $U_{2}$
be a neighborhood of $x$ in $X$. It is straightforward to check that there
exists a neighborhood $U_{1}$ of $x$ in $X$ such that
$\bar{\varphi}_{i}(U_{1})\subset U_{2}$.

Isomorphism between $T_{1}$ and $T_{2}$. Corresponding to homeomorphisms
$h_{A}$ and $h_{B}$, let $\alpha$ and $\beta$ be the bijections by Lemma 4.4.
For $i=1,2$, let $\tau_{i}=[v_{i},u_{i}]$ be the edge of $T_{i}$ such that
$v_{i}$ and $u_{i}$ are stabilized by $A_{i}$ and $B_{i}$ respectively. Recall
that each non-trivial element $g\in G_{1}$ can be expressed uniquely, in
reduced form, as $g=a_{1}b_{1},\dots,b_{n}$, with $a_{j}\in
A_{1}\setminus\\{1\\},b_{j}\in B_{1}\setminus\\{1\\}$, allowing also that
$a_{1}=1$ and that $b_{n}=1$. Define a map $\iota:T_{1}\to T_{2}$ that maps
any edge $a_{1}b_{1}\dots a_{n}b_{n}\tau_{1}$ to the edge
$\alpha(a_{1})\beta(b_{1})\dots\alpha(a_{n})\beta(b_{n})\tau_{2}$. It is easy
to check that $\iota$ is an isomorphism such that $\iota(\tau_{1})=\tau_{2}$.

Now, we are ready to prove that $\partial_{f_{1}}(G_{1})$ is homeomorphic to
$\partial_{f_{2}}(G_{2})$. For that it is sufficient to prove that
$\delta(\Gamma_{f_{1}})$ is homeomorphic to $\delta(\Gamma_{f_{2}})$. Note
that, if $\xi\in\partial_{f_{1}}(A_{1})$ and $g\in G_{1}$ then the set of all
representatives of $[g,\xi]$ are of the form $(ga,a^{-1}\xi)$ where $a\in
A_{1}$. Similarly, if $\xi\in\partial_{f_{1}}(B_{1})$ and $g\in G$ then the
set of all representatives are of the form $(gb,b^{-1}\xi)$ for $b\in B_{1}$.
When $\xi\in\partial_{f_{1}}(A_{1})$, we choose a unique
$ga=a_{1}b_{1},\dots,a_{n}b_{n}$ for which $n$ is smallest possible (in this
case we have $b_{n}\neq 1$). When $\xi\in\partial_{f_{1}}(B_{1})$, we choose a
unique $gb=a_{1}b_{1},\dots,a_{n}b_{n}$ for which $b_{n}=1$. These
representatives of an element $[g,\xi]\in\delta_{Stab}(\Gamma_{f_{1}})$ are
called reduced represenatives. Finally, we define a map
$F:\delta(\Gamma_{f_{1}})\to\delta(\Gamma_{f_{2}})$ in the following manner:

  1. (1)

Let $[g,\xi]\in\delta_{Stab}(\Gamma_{f_{1}})$ and let $(g,\xi)$ be its reduced
representative. Define

$F([a_{1}b_{1},\dots,a_{n}b_{n},\xi])=[\alpha(a_{1})\beta(b_{1}),\dots,\alpha(a_{n})\beta(b_{n}),q(\xi)].$

  2. (2)

Let $\eta\in\partial T_{1}$. We can represent it as an infinite word
$\eta=a_{1}b_{1},\dots$ such that for each $n$, the subword consisting of its
first $n$ letters corresponds to the $n$-th edge of the geodesic from $v_{1}$
to $\eta$ via the correspondence $g\to g.\tau_{1}$. Define

$F((a_{1}b_{1},\dots))=\alpha(a_{1})\beta(b_{1}),\dots$

where the infinite word on the right gives a geodesic ray in $T_{2}$ starting
from $v_{2}$.

Note that the restriction of $F$ to $\partial(T_{1})$ is same as the map
$\partial(T_{1})\to\partial(T_{2})$ induced from the isomorphism
$\iota:T_{1}\to T_{2}$.

######  Lemma 4.9.

The map $F$ is a homeomorphism.

###### Proof.

From the definition of $F$, it follows that $F$ is a bijection. To show that
$F$ is a homeomorphism, it is sufficient to prove that $F$ is continuous.
There are two cases to be considered:

Case 1. Let $\xi\in\delta_{Stab}(\Gamma_{f_{1}})$ and let $v$ be the vertex of
$T_{1}$ such that $\xi\in\partial_{f_{1}}(G_{v})$. Let $U_{2}$ be an open
neighborhood of $F(\xi)$ in $\overline{\Gamma}_{\iota{(v)}}^{f_{2}}$. By
Corollary 4.5, we have an open neighborhood
$U_{1}\subset\overline{\Gamma}_{v}^{f_{1}}$ of $\xi$. Then, from the
definition of neighborhoods in $\overline{\Gamma}_{f_{1}}$ and in
$\overline{\Gamma}_{f_{2}}$, it follows that
$F(V_{U_{1}}\cap\delta(\Gamma_{f_{1}}))\subset
V_{U_{2}}(F(\xi))\cap\delta(\Gamma_{f_{2}})$.

Case 2. Let $\eta\in\partial(T_{1})$. For an integer $n\geq 1$, consider the
subtree $(T_{2})_{n}(F(\eta))\subset T_{2}$, defined with respect to $v_{2}$.
Let $(T_{1})_{n}(\eta)=\iota^{-1}((T_{2})_{n}(F(\eta)))$, which is subtree of
$T_{1}$ with respect to the base vertex $v_{1}$. Again, from definition of
neighborhoods, it follows that
$F(V_{n}(\eta)\cap\delta(\Gamma_{f_{1}}))=V_{n}(F(\eta))\cap\delta(\Gamma_{f_{2}})$.
∎

This completes the proof of Theorem 4.1. We end this section by proving the
following interesting case. Let $f$ be a Floyd function.

######  Proposition 4.10.

Let $G=A*A,K=A*F$, where $A$ is an infinite group and $F$ is a finite group.
Then $\partial_{f}(G)$ is homeomorphic to $\partial_{f}(K)$.

###### Proof.

Assume that $F=\\{e,h_{1},h_{2},\dots h_{n}\\}$. From Lemma 4.7, we have
$A_{i}$ such that $A=\sqcup_{i=1}^{n}A_{i}$, where $1\in A_{1}$ and, for
$1\leq i\leq n$, there exists a bijection $\varphi_{i}:A\rightarrow A_{i}$
which extends to a homeomorphism
$\bar{\varphi}_{i}:\overline{A}\rightarrow\overline{A}_{i}$ such that
$\varphi_{1}(1)=1$. For the sake of distinction, let $G=A*A^{\prime}$ where
$A^{\prime}=A$. Let $\varphi_{i}^{\prime}:A\rightarrow A_{i}^{\prime}$ which
extends to homeomorphisms
$\bar{\varphi^{\prime}}_{i}:\overline{A}\rightarrow\overline{A^{\prime}}_{i}$.
We denote the scaled Cayley graphs of $K$ and $G$ by $\Gamma_{f}$ and
$\Gamma^{\prime}_{f}$ and Bass–Serre trees by $T_{1}$ and $T_{2}$,
respectively. Let $V$ be the set of vertices of $T_{1}$ that represents the
cosets of $A$ and $V(T_{2})$ is the vertex set of $T_{2}$. From discussions in
Section 3, we know that it is sufficient to produce a homeomorphism between
$\delta(\Gamma_{f})$ and $\delta(\Gamma_{f}^{\prime})$.

Since $F$ is a finite group, $\partial_{f}(F)$ is empty. This implies that for
every $v\in T_{1}$ representing a coset of $F$, $\partial_{f}(\Gamma_{v}^{f})$
is empty. Progressing along the lines of the proof of Theorem 4.1, we will
define a homeomorphism $\phi:V\cup\partial T_{1}\rightarrow
V(T_{2})\cup\partial T_{2}$, where the domain and the codomain have the
subspace topology induced from $\overline{T}_{1}$ and $\overline{T}_{2}$. Let
$v\in V$ represents $gA$. Then $g$ has a unique reduced form as discussed in
Section 4 such that $g=a_{1}h_{i_{1}}a_{2}h_{i_{2}}\dots a_{n}h_{i_{k}}$. Let
$m$ denotes a natural number. Define a map $\iota:V\rightarrow V(T_{2})$ as
follows:

$\iota(gA):=\begin{cases}\varphi_{i_{1}}(a_{1})\varphi^{\prime}_{i_{2}}(a_{2})\dots\varphi^{\prime}_{i_{2m-2}}(a_{2m-2})\varphi_{i_{2m-1}}(a_{2m-1})A^{\prime}&\text{if
}g\notin A,\hskip 4.26773ptk=2m-1,\\\
\varphi_{i_{1}}(a_{1})\varphi^{\prime}_{i_{2}}(a_{2})\dots\varphi_{i_{2m-1}}(a_{2m-1})\varphi^{\prime}_{i_{2m}}(a_{2m})A&\text{if
}g\notin A,\hskip 4.26773ptk=2m,\\\ A&\text{ if }g=e_{K}.\end{cases}$

This map induces the required homeomorphism $\phi$. As discussed in the proof
of Theorem 4.1, for every element in $\delta_{Stab}(\Gamma_{f})$, there exists
a unique reduced representative $[g,\xi]$ where
$g=a_{1}h_{i_{1}}a_{2}h_{i_{2}}\dots a_{n}h_{i_{k}}$.

Define the map $F:\partial_{f}(A*F)\to\partial_{f}(A*F)$ in the following
manner:

  1. (1)

$F([g,\xi]):=\begin{cases}[\varphi_{i_{1}}(a_{1})\varphi^{\prime}_{i_{2}}(a_{2})\dots\varphi_{i_{2m-1}}(a_{2m-1}),\xi]&\text{
if }g\notin A,\hskip 4.26773ptk=2m-1,\\\
[\varphi_{i_{1}}(a_{1})\varphi^{\prime}_{i_{2}}(a_{2})\dots\varphi_{i_{2m-1}}(a_{2m-1})\varphi^{\prime}_{i_{2m}}(a_{2m}),\xi]&\text{
if }g\notin A,\hskip 4.26773ptk=2m,\\\ [e_{K},\xi]&\text{ if
}g=e_{K}.\end{cases}$

  2. (2)

$F(\eta):=\phi(\eta)\text{ where }\eta\in\partial K.$

From Remark 4.8(2) and Corollary 4.5, we can deduce that $F$ is continuous at
every $\xi\in\delta_{Stab}(\Gamma_{f})$. It is straightforward to check that
$F$ is also continuous at every point in $\partial T_{1}$. Hence, we conclude
that $F$ is continuous. Since $\overline{\Gamma}_{f}$ is compact and $F$ is a
bijection, $F$ is also a homeomorphism. Hence, we conclude the proof. ∎

##  5\. Homeomorphism types of Floyd boundaries of amalgamated free product
and HNN extension

So far, we have proved that homeomorphism types of Floyd boundaries of free
products of infinite groups depends only on homeomorphism types of Floyd
boundary of free factors. Now, we proceed to prove similar results for groups
admitting a decomposition as amalgamted free products and HNN extension over
finite groups. We fix a Floyd function $f$ for this section.

###  5.1. Amalgam case

The main result of this subsection is the following:

######  Theorem 5.1.

Suppose $A$ is an infinite group and $K=A*_{H}B$, $G=A*B$. Then, we have the
following:

  1. (1)

$\partial_{f}(K)\simeq\partial_{f}(G),\text{ if }B\text{ is also infinite},$

  2. (2)

$\partial_{f}(K)\simeq\partial_{f}(A*A),\text{ if }B\text{ is finite and
}B\neq H,$

  3. (3)

$\partial_{f}(B*_{H}C)$ is homeomorphic to the Cantor set if $B,C$ are non-
trivial finite groups such that $B\ast_{H}C$ has infinitely many ends.

###### Proof.

Denote the scaled Cayley graph of $G$ and $K$ by $\Gamma_{f}$ and
$\Gamma^{\prime}_{f}$, and Bass–Serre trees by $T$ and $T^{\prime}$,
respectively. Let $\tau$ and $\tau^{\prime}$ be the edges joining $A$ and $B$
in $T$ and $T^{\prime}$, respectively. Let $A_{1}\subset A$, $B_{1}\subset B$
be subsets that contain exactly one coset representative for each coset of $H$
in $A$ and $B$, respectively. We also assume that identity is contained in
both of them. For each edge $g\tau^{\prime}\in T^{\prime}$, we have the unique
reduced form of $g$ such that
$g=a^{\prime}_{1}b^{\prime}_{1}a^{\prime}_{2}b^{\prime}_{2}\dots
a^{\prime}_{n}b^{\prime}_{n}$, where $a^{\prime}_{j}\in A_{1}-\\{1\\}$ and
$b^{\prime}_{j}\in B_{1}-\\{1\\}$, allowing $a^{\prime}_{1}=1$ and
$b^{\prime}_{n}=1$.

Case 1. If $B$ is infinite, then $\partial_{f}B$ is not empty. It is easy to
check that $A_{1}$ and $B_{1}$ are countable, dense subsets of
$A\cup\partial_{f}A$ and $B\cup\partial_{f}B$, respectively. By applying
Corollary 4.6, we get bijections $\varphi:A\rightarrow A_{1}$ and
$\psi:B\rightarrow B_{1}$ with $\varphi(1)=1$ and $\psi(1)=1$, that extend to
homeomorphisms $\varphi\cup\text{Id}:A\cup\partial_{f}A\rightarrow
A_{1}\cup\partial_{f}A$ and $\psi\cup\text{Id}:B\cup\partial_{f}B\rightarrow
B_{1}\cup\partial_{f}B$.

Define $\iota:T\rightarrow T^{\prime}$, such that

$\iota(a_{1}b_{1}a_{2}b_{2}\dots
a_{n}b_{n}\tau)=\varphi(a_{1})\psi(b_{1})\varphi(a_{2})\psi(_{2})\dots\varphi(a_{n})\psi(b_{n})\tau^{\prime},$

where $a_{1}b_{1}a_{2}b_{2}\dots a_{n}b_{n}$ is the reduced form of an element
in $G$. The map $\iota$ is a graph isomorphism inducing homeomorphism
$\bar{\iota}:\overline{T}\rightarrow\overline{T^{\prime}}$. We will define a
map $F:\delta(\Gamma_{f})\rightarrow\delta(\Gamma^{\prime}_{f})$ as follows:

  1. (1)

Following the discussion in the proof of Theorem 4.1, for every element in
$\delta_{Stab}(\Gamma_{f})$, we choose the reduced representative $[g,\xi]$
with $g$ having the reduced form $a_{1}b_{1}\dots a_{n}b_{n}$. Define

$F([g,\xi]):=[\varphi(a_{1})\psi(b_{1})\varphi(a_{2})\psi(b_{2})\dots\varphi(a_{n})\psi(b_{n}),\xi].$

  2. (2)

For $\eta\in\partial T$, define $F(\eta):=\bar{\iota}(\eta).$

Assume that $H=\\{e,h_{1},h_{2},\dots,h_{l}\\}$. For $1\leq k\leq l$, let
$\varphi_{k}:A\rightarrow A_{1}h_{k}$ be the maps that take $a$ to
$\varphi(a)h_{k}.$ Since $A=A_{1}\sqcup A_{1}h_{k}$, by Remark 4.8(2), we can
show that $F$ is continuous at every point of $\partial_{f}(G_{v})$, for each
vertex $v\in T$ that represents a coset of A. Applying a similar argument for
$B$ gives the conclusion that $F$ is continuous at every point of
$\delta_{Stab}(\Gamma_{f})$. Since $\overline{\iota}$ is a homeomorphism, it
is easy to see that $F$ is also continuous at every point of $\partial T$.
Since $F$ is a continuous bijection and $\overline{\Gamma}_{f}$ is compact,
$F$ is a homeomorphism.

Case 2. If $B$ is a finite group then $\partial_{f}B$ is empty. Let $m$ be the
index of $H$ in $B$. We show that
$\partial_{f}(A*\mathbb{Z}_{m})\simeq\partial_{f}(K).$ We will use the map
$\varphi$ defined in the last case. Let $\psi_{1}:\mathbb{Z}_{m}\rightarrow
B_{1}$ be a bijection with $\psi_{1}(1)=1$. Let $\tau_{1}$ be the edge joining
$A$ and $\mathbb{Z}_{m}$ in the Bass–Serre tree $T_{1}$ of $A*\mathbb{Z}_{m}$.
Define $\iota_{1}:T_{1}\rightarrow T^{\prime}$, that takes the edge
$g\tau_{1}$ to
$\varphi(a_{1})\psi_{1}(b_{1})\varphi(a_{2})\psi_{1}(b_{2})\dots\varphi(a_{n})\psi_{1}(b_{n})\tau^{\prime}$,
where $a_{1}b_{1}a_{2}b_{2}\dots a_{n}b_{n}$ is the reduced form of $g\in
A*\mathbb{Z}_{m}$. Note that $\iota_{1}$ is an isomorphism and hence it
induces a homeomorphism
$\overline{\iota}_{1}:\overline{T}_{1}\rightarrow\overline{T^{\prime}}$. Now,
$\bar{\iota}_{1}$ can be used to define a homeomorphism between
$\partial_{f}(K)$ and $\partial_{f}(A*\mathbb{Z}_{m})$ as we did in the last
case. Finally, by Proposition 4.10, we get that

$\partial_{f}{K}\simeq\partial_{f}(A*\mathbb{Z}_{m})\simeq\partial_{f}(A*A),$

which completes the proof.

Case 3. Let $L:=B*_{H}C$. We denote the Bass–Serre tree and Cayley Graph of
$L$ by $T^{\prime\prime}$ and $\Gamma_{f}^{\prime\prime}$, respectively. From
Section 3, we know that
$\partial_{f}(L)\simeq\delta_{Stab}\Gamma_{f}^{\prime\prime}\cup\partial
T^{\prime\prime}$. Since, $B$ and $C$ are finite groups, $\partial_{f}(B)$ and
$\partial_{f}(C)$ are empty, which implies that
$\delta_{Stab}(\Gamma_{f}^{\prime\prime})$ is empty. Hence,
$\partial_{f}(L)\simeq\partial T^{\prime\prime}$. Since the degree of the
vertices of $T^{\prime\prime}$ are bounded above by a fixed natural number,
$\partial T^{\prime\prime}$ is homeomorphic to the Cantor set. ∎

######  Remark 5.2.

If $B\ast_{H}C$, in Theorem 5.1(3), is $2$-ended then $G$ is virtually cyclic
and hence $|\partial_{f}(G)|=2$ (see [Kar03, Proposition 7]).

We immediately have the following corollary whose proof is an easy application
of induction:

######  Corollary 5.3.

Let $Y$ be a tree with $k>1$ vertices. Let $({\mathcal{G}},Y)$ be a graph of
groups over $Y$ such that each edge group is finite. Suppose $G$ is the
fundamental group of $({\mathcal{G}},Y)$ and $G_{1},G_{2},\dots,G_{k}$ are the
vertex groups. Then, we have the following:

  1. (1)

If each $G_{i}$ is infinite then
$\partial_{f}(G)\simeq\partial_{f}(G_{1}\ast\dots\ast G_{k})$.

  2. (2)

If $G_{j}$ is infinite for some $j$ and, for each $i\neq j$, $G_{i}$ is
finite. Then, $\partial_{f}(G)\simeq\partial_{f}(G_{j}\ast G_{j})$.

  3. (3)

By reindexing (if required), suppose there exists $2\leq k^{\prime}\leq k$
such that $G_{i}$ is infinite for $1\leq i\leq k^{\prime}$ and $G_{i}$ is
finite for $i>k^{\prime}$. Then,
$\partial_{f}(G)\simeq\partial_{f}(G_{1}\ast\dots\ast G_{k^{\prime}})$.

  4. (4)

If $G_{i}$’s are finite and $G$ is infinite-ended then $\partial_{f}(G)$ is
homeomorphic to the Canter set.

∎

###  5.2. HNN extension Case

This subsection is devoted to prove the following theorem:

######  Theorem 5.4.

Let $G=A*_{F\simeq F^{\prime}}$, where $F$ and $F^{\prime}$ are finite
subgroups. Then, we have the following:

  1. (1)

If $A$ is an infinite group then $\partial_{f}(G)\simeq\partial_{f}(A*A)$.

  2. (2)

If $A$ is finite and either $F$ or $F^{\prime}$ is not equal to $A$ then
$\partial_{f}(G)$ is homeomorphic to the Cantor set.

###### Proof.

(1) For the sake of distinction, let $K=A*A^{\prime}$ where $A^{\prime}$ is an
isomorphic copy of $A$. Let $S_{A}$ be a generating set for $A$. We denote the
Bass–Serre tree and Cayley graph with respect to generating set $S_{A}\cup
S_{A}^{\prime}$ of $K$ by $T^{\prime}$ and $\Gamma^{\prime}_{f}$,
respectively. From the discussion in Subsection 3.4, it is sufficient to
produce a homeomorphism between $\delta(\Gamma_{f})$ and
$\delta(\Gamma^{\prime}_{f})$. Let $A_{1}$ and $A_{2}$ be the set of left
coset representatives of $F$ and $F^{\prime}$ in $A$, respectively, such that
$1\in A_{1}\cap A_{2}$. For every coset of $A$ in $G$, there exists a unique
reduced representative $g$ of that coset of the form
$g=a_{0}t^{\varepsilon_{0}}a_{1}t^{\varepsilon_{1}}\dots
a_{n}t^{\varepsilon_{n}}$, $\varepsilon_{i}=\pm 1$, $a_{i}\in A_{k}$ when
$\varepsilon_{i}=(-1)^{(k+1)}$, where $k\in\\{1,2\\}$, and for any $i\geq 1$,
$\varepsilon_{i-1}=-\varepsilon_{i}$, $a_{i}=1$ is not possible.

For this proof, we will denote the identity map on $\partial_{f}(A)$ by Id.
From Lemma 4.7, we have disjoint subsets $B_{1},B_{2}\subset A$ such that
$A=B_{1}\sqcup B_{2}$, with bijections $\psi_{i}:A\rightarrow B_{i}$ such that
$\psi_{i}\cup\text{Id}$ are homeomorphisms. Since $A_{i}$’s are dense in
$A_{i}\cup\partial_{f}(A)$, there exist bijections
$\alpha_{i}:A_{i}\rightarrow A$ such that $\alpha_{i}\cup\text{Id}$ are
homeomorphisms (see Corollary 4.6). Define a map $\varphi_{i}:A_{i}\rightarrow
B_{i}$ as $\varphi_{i}:=\psi_{i}\circ\alpha_{i}$. Similarly, define the maps
$\varphi^{\prime}_{i}:A_{i}\to B^{\prime}_{i}$, where $B_{i}^{\prime}$’s are
the isometric copy of $B_{i}$ in $A^{\prime}$. By construction, we conclude
that $\varphi_{i}\cup\text{Id}$ and $\varphi_{i}^{\prime}\cup\text{Id}$ are
homeomorphisms.

Firstly, we define a graph isomorphism $\iota:T\to T^{\prime}$ as follows:

$\iota(gA):=\begin{cases}\varphi^{\prime}_{\varepsilon_{0}}(a_{0})\varphi_{\varepsilon_{1}}(a_{1})\dots\varphi^{\prime}_{\varepsilon_{n}}(a_{n})A&\text{
if }n\text{ is even,}\\\
\varphi^{\prime}_{\varepsilon_{0}}(a_{0})\varphi_{\varepsilon_{1}}(a_{1})\dots\varphi_{\varepsilon_{n}}(a_{n})A^{\prime}&\text{
if }n\text{ is odd,}\\\ A&\text{ if }g=e_{G}.\end{cases}$

which induces the homeomorphism
$\bar{\iota}:\overline{T}\to\overline{T^{\prime}}$. Define the map
$F:\delta(\Gamma_{f})\to\delta(\Gamma^{\prime}_{f})$ as follows:

  * •

Let $\xi\in\partial_{f}(\Gamma^{f}_{gA})$, then define

$F([g,\xi]):=\begin{cases}[\varphi^{\prime}_{\varepsilon_{0}}(a_{0})\varphi_{\varepsilon_{1}}(a_{1})\dots\varphi^{\prime}_{\varepsilon_{n}}(a_{n}),\xi]&\text{
if }n\text{ is even,}\\\
[\varphi^{\prime}_{\varepsilon_{0}}(a_{0})\varphi_{\varepsilon_{1}}(a_{1})\dots\varphi_{\varepsilon_{n}}(a_{n}),\xi]&\text{
if }n\text{ is odd}\\\ [e_{G},\xi]&\text{ if }g=e_{G}.\end{cases}$

  * •

If $\eta\in\partial T$, then define $F(\eta):=\bar{\iota}(\eta)$.

Using Remark 4.8(2), we can show that $F$ is continuous. Since
$\delta(\Gamma_{f})$ is compact and $F$ is a bijection, we conclude that $F$
is a homeomorphism.

(2) Since $A$ is finite, $\delta_{Stab}(\Gamma_{f})$ is empty. Hence,
$\partial_{f}(G)\simeq\partial T$. However, as $T$ is $k$-regular where $k\geq
3$, $\partial T$ is homeomorphic to the Cantor set. ∎

######  Remark 5.5.

In Theorem 5.4(2), if $A=F$ then $A=F^{\prime}$ too as $A$ is finite. In this
case, $G$ is virtually cyclic and hence $|\partial_{f}(G)|=2$.

######  Corollary 5.6.

For $n\geq 2$, let $G=A_{1}\ast\dots\ast A_{n}$ be a free product of
elementary groups such that at least one free factor is infinite. Then, the
Floyd boundary of $G$ is homeomorphic to the Cantor set.

###### Proof.

There are two cases to be consider:

Case 1. By reindexing, if required, assume that $A_{1}$ is infinite and
$A_{i}$ is finite for $2\leq i\leq n$. Then, by Corollary 5.3(2),
$\partial_{f}(G)\simeq\partial_{f}(A_{1}\ast A_{1})$. Since $A_{1}$ is
virtually cyclic, $\partial_{f}(A_{1})\simeq\partial_{f}(\mathbb{Z})$. Thus,
by Theorem 4.1, $\partial_{f}(A\ast
A)\simeq\partial_{f}(\mathbb{Z}\ast\mathbb{Z})$. Let $B$ be a non-trivial
finite group. Then, by Proposition 4.10,
$\partial_{f}(\mathbb{Z}\ast\mathbb{Z})\simeq\partial_{f}(\mathbb{Z}\ast B)$.
Since $B_{\ast_{\\{1\\}}}$ is isomorphic to $B\ast\mathbb{Z}$, their Floyd
boundaries are homeomorphic. Now, by Theorem 5.4(2),
$\partial_{f}(B\ast\mathbb{Z})$ is homeomorphic to the Cantor set.

Case 2. By reindexing, if necessary, suppose there exists $2\leq
k^{\prime}\leq n$ such that $A_{i}$ is infinite for $1\leq i\leq k^{\prime}$
and $A_{j}$ is finite for $j>k^{\prime}$. Then, by Corollary 5.3(3),
$\partial_{f}(G)\simeq\partial_{f}(A_{1}\ast\dots\ast A_{k^{\prime}})$. As
$A_{i}$’s are infinite virtually cyclic for $1\leq i\leq k^{\prime}$, by
induction, it is easy to prove that $\partial_{f}(G)$ is homeomorphic to the
Cantor set. ∎

We end this subsection by proving the following which remove the dependence of
Floyd boundary of free products on virtually cyclic groups.

######  Corollary 5.7.

For $k\geq 2$, let $G=B_{1}\ast\dots\ast B_{k}$ be a free product of infinite
groups such that, for $2\leq i\leq k$, $B_{i}$ is virtually cyclic. Then,
$\partial_{f}(G)$ is homeomorphic to $\partial_{f}(B_{1}\ast B_{1})$.

###### Proof.

We prove it by induction on $k$. Let $k=2$. Then,
$\partial_{f}(G)\simeq\partial_{f}(B_{1}\ast\mathbb{Z})$. Since
$B_{1}\ast\mathbb{Z}$ is isomorphic to $B_{\ast_{\\{1\\}}}$,
$\partial_{f}(G)\simeq\partial_{f}(B_{\ast_{\\{1\\}}})$ which is homeomorphic
to $\partial_{f}(B_{1}\ast B_{1})$ by Theorem 5.4(1). Suppose
$\partial_{f}(B_{1}\ast\dots\ast B_{k-1})\simeq\partial_{f}(B_{1}\ast B_{1})$.
Now, by Theorem 4.1, $\partial_{f}(G)\simeq\partial_{f}(B_{1}\ast B_{1}\ast
B_{k})$ which is homeomorphic to $\partial_{f}(B_{1}\ast B_{k})$ by Theorem
6.3. Now the corollary follows from the case when $k=2$. ∎

By combining Corollary 5.3(3) and Corollary 5.7, we have the following:

######  Corollary 5.8.

For $k,l\in\mathbb{N}$, let $G=A_{1}\ast\dots\ast A_{k}\ast B_{1}\ast\dots\ast
B_{l}$ where $A_{i}$’s are non-elementary and $B_{j}$’s are elementary groups.
Then, we have the following:

(1) If $k\geq 2,l\geq 1$ then the Floyd boundary of $G$ is homeomorphic to
$\partial_{f}(A_{1}\ast\dots\ast A_{k})$.

(2) If $k=1$ and $l\geq 1$ then $\partial_{f}(G)\simeq\partial_{f}(A_{1}\ast
A_{1})$. ∎

##  6\. Main Theorem

Let $f$ and $f^{\prime}$ be two Floyd functions which is fixed for this and
the next section. Here, the main goal is to prove the following:

######  Theorem 6.1.

Suppose $G$ and $H$ are two finitely generated infinite groups. Suppose
$\mathcal{G}$ and $\mathcal{H}$ are graphs of groups decomposition of $G$ and
$H$, respectively such that the edge groups of $\mathcal{G}$ and $\mathcal{H}$
are finite. Then, we have the following:

  1. (1)

If each vertex group of $\mathcal{G}$ is elementary and $G$ has infinitely
many ends then $\partial_{f}(G)$ is homeomorphic to the Cantor set.

  2. (2)

Suppose at least one vertex group of either $\mathcal{G}$ or $\mathcal{H}$ is
non-elementary. Let $h(\mathcal{G})$ and $h(\mathcal{H})$ denote the set of
homeomorphism types of Floyd boundaries of non-elementary vertex groups of
$\mathcal{G}$ and $\mathcal{H}$ with respect to $f$ and $f^{\prime}$,
respectively. If $h(\mathcal{G})=h(\mathcal{H})$ then $\partial_{f}(G)$ is
homeomorphic to $\partial_{f^{\prime}}(H)$.

To prove the above theorem, we use the following result:

######  Theorem 6.2.

For $n,m\geq 2$, let $G_{1}=A_{1}\ast\dots\ast A_{n}$ and
$G_{2}=B_{1}\ast\dots\ast B_{m}$ be two free products where at least one free
factor of either $G_{1}$ or $G_{2}$ is non-elementary. Let $h(G_{1})$ and
$h(G_{2})$ denote the set of homeomorphism types of Floyd boundaries of non-
elementary free factors of $G_{1}$ and $G_{2}$ with respect to $f$ and
$f^{\prime}$, respectively. If $h(G_{1})=h(G_{2})$ then the Floyd boundary of
$G_{1}$ is homeomorphic to the Floyd boundary of $G_{2}$.

Before getting into the proof of Theorem 6.2, we prove the following result
which turns out to be useful in proving the above theorem:

######  Theorem 6.3.

Suppose that $G=A\ast B\ast C$ is a free product of infinite groups. If
$\partial_{f}(A)$ is homeomorphic to $\partial_{f}(C)$ then $\partial_{f}(G)$
is homeomorphic to $\partial_{f}(A\ast B)$.

We immediately have the following corollaries:

######  Corollary 6.4.

Suppose that $G=A\ast B\ast C\ast D$ is a free product of infinite groups. If
$\partial_{f}(A)\simeq\partial_{f}(C)$ and
$\partial_{f}(B)\simeq\partial_{f}(D)$ then
$\partial_{f}(G)\simeq\partial_{f}(A\ast B)$.

###### Proof.

By applying Theorem 6.3, we see that $\partial_{f}(G)\simeq\partial_{f}(A\ast
B\ast D)$ which is again homeomorphic to $\partial_{f}(A\ast B)$ by Theorem
6.3. ∎

######  Corollary 6.5.

For $n\geq 2$, suppose $A=A_{1}\ast\dots\ast A_{n}$ is a free product of
infinite groups. Suppose $2\leq i\leq n$, $\partial_{f}(A_{i})$ is
homeomorphic to $\partial A_{1}$. Then, $\partial_{f}(A)$ is homeomorphic to
$\partial_{f}(A_{1}\ast A_{1})$.

###### Proof.

We prove it by induction on $n$. From Theorem 6.3, it follows that
$\partial_{f}(A_{1}\ast A_{2}\ast A_{3})\simeq\partial_{f}(A_{1}\ast A_{1}).$
Suppose $\partial_{f}(A_{1}\ast\dots\ast A_{n-1})\simeq\partial_{f}(A_{1}\ast
A_{1})$. Then, by Theorem 4.1, $\partial_{f}(A)\simeq\partial_{f}(A_{1}\ast
A_{1}\ast A_{1})$ which is homeomorphic to $\partial_{f}(A_{1}\ast A_{1})$. ∎

Using Theorem 6.3 and Proposition 4.10, we immediately have the following
corollary:

######  Corollary 6.6.

Let $G=A_{1}*B_{1}$ and $H=A_{2}*B_{2}$, where $A_{1}$ and $A_{2}$ are
infinite groups while $B_{1}$ and $B_{2}$ can be finite such that
$\partial_{f}(A_{1})\simeq\partial_{f}(A_{2})$ and
$\partial_{f}(B_{1})\simeq\partial_{f}(B_{2})$. Then
$\partial_{f}(G)\simeq\partial_{f}(H).$ ∎

######  Corollary 6.7.

Let $G=A\ast F_{1}\dots\ast F_{n}$ be a free product of groups such that $A$
is an infinite group and $F_{i}$’s are finite groups. Then
$\partial_{f}(G)\simeq\partial_{f}(A*A)$.

###### Proof.

We prove it by induction on $n$. For $n=1$, it is Proposition 4.10. Let
$H=A\ast F_{1}\ast\dots*F_{n-1}$. Assume that
$\partial_{f}(H)\simeq\partial_{f}(A\ast A)$. Now, by Proposition 4.10, we see
that $\partial_{f}(G)\simeq\partial_{f}(H\ast H)$. By inductive hypothesis, we
have that $\partial_{f}(H)\simeq\partial_{f}(A\ast A)$. Hence, by Theorem 4.1,
$\partial_{f}(G)\simeq\partial_{f}(A\ast A\ast A\ast A)$. Finally, by
Corollary 6.4, we have that $\partial_{f}(G)\simeq\partial_{f}(A\ast A)$. ∎

###  6.1. Proof of Theorem 6.3

Let $G$ be as in Theorem 6.3 and let $T$ be its Bass–Serre tree. Suppose
$G^{\prime}=A\ast B$ and let $T^{\prime}$ be its Bass–Serre tree. In this
subsection, first of all, we prove that there is an isomorphism between $T$
and $T^{\prime}$. Using this isomorphism, we define a map from
$\partial_{f}(G)$ to $\partial_{f}(G^{\prime})$ and prove that this is in fact
a homeomorphism.

Isomorphism between $T$ and $T^{\prime}$: Let $\Gamma_{B}$ be the Cayley graph
of $B$ with respect to a finite generating set. Then,
$\overline{\Gamma}_{B}=\Gamma_{B}\cup\partial_{f}(B)$ is a compact metrizable
space. With restricted metric, $B\cup\partial_{f}(B)$ is compact, $B$ is dense
in $\overline{\Gamma}_{B}$ and $B$ contains all isolated points of
$\overline{\Gamma}_{B}.$ Then, by Lemma 4.7, there exist $B_{1},B_{2}$ such
that $B\setminus\\{e_{B}\\}=B_{1}\sqcup B_{2}$ and there exist bijections
$\beta_{1}:B\to B_{1},\beta_{2}:B\to B_{2}$ that induce homeomorphisms
$\bar{\beta_{1}}:B\cup\partial_{f}(B)\to B_{1}\cup\partial_{f}(B)$ and
$\bar{\beta_{2}}:B\cup\partial_{f}(B)\to B_{2}\cup\partial_{f}(B)$,
respectively. Since $\partial_{f}(A)$ is homeomorphic to $\partial_{f}(C)$, by
Lemma 4.4, there exists a bijection $\gamma:C\to A$ such that $\gamma(1)=1$
and it induces a homeomorphism from $A\cup\partial_{f}(A)$ to
$C\cup\partial_{f}(C)$. Let $\alpha$ be the identity map from $A$ to $A$. Any
element $g\in A*B*C$ can be written as $g=a_{1}b_{1}d_{1}b_{2}d_{2}\dots
b_{n}d_{n}$, where $a_{1}\in A$, $b_{i}\in B$, and $d_{i}\in
A\setminus\\{e_{A}\\}\cup C\setminus\\{e_{C}\\}$, and $a_{1},b_{i},d_{n}$ can
be the identity in their respective groups. We shall call this the standard
form of $g$.

Let $\tau$ be the edge between $A$ and $B$ in $T^{\prime}$, and $\tau_{1}$,
$\tau_{2}$ be the edges between $A$ and $B$, $B$ and $C$, respectively in $T$.
Note that any edge in $T^{\prime}$ is a $G^{\prime}$-translate of $\tau$.
Similarly, any edge of $T$ is either a $G$-translate of $\tau_{1}$ or a
$G$-translate of $\tau_{2}$. Define a map $\phi:T\to T^{\prime}$ in the
following manner:

Case 1. Let $g\tau_{1}$ be an edge in $T$ for some $g\in G$ and
$a_{1}b_{1}d_{1},\dots,b_{n}d_{n}$ be the standard form for $g$. There are two
subcases:

Subcase 1(a). Suppose $d_{n}=1$. Then, we define

$h:=a_{1}\beta_{i_{1}}(b_{1})\delta_{1}(d_{1}),\dots,\beta_{i_{n}}(b_{n})$

where for all $1\leq k<n$, $i_{k}=1$ if $d_{k}\in A\setminus\\{e_{A}\\}$
otherwise $i_{k}=2$, and $\delta_{k}=\alpha$ if $d_{k}\in
A\setminus\\{e_{A}\\}$ otherwise $\delta_{k}=\gamma$, and $i_{n}=1$. In this
case, we define $\phi(g\tau_{1})=h\tau$.

Subcase 1(b). Suppose $d_{n}\neq 1$. Then, $d_{n}\in A$ and we define

$h:=a_{1}\beta_{i_{1}}(b_{1})\delta_{1}(d_{1}),\dots,\beta_{i_{n}}(b_{n})\delta_{n}(d_{n})$

where, for $1\leq k\leq n$, $i_{k}=1$ if $d_{k}\in A\setminus\\{e_{A}\\}$
otherwise $i_{k}=2$, and $\delta_{k}=\alpha$ if $d_{k}\in
A\setminus\\{e_{A}\\}$ otherwise $\delta_{k}=\gamma$. In this case, we define
$\phi(g\tau_{1})=h\tau$.

Case 2. Let $g\tau_{2}$ be an edge in $T$ for some $g\in G$ and
$a_{1}b_{1}d_{1},\dots,b_{n}d_{n}$ be the standard form for $g$. Again there
are two subcases:

Subcase 2(a). Suppose $d_{n}\neq 1$. Then, we define

$h:=a_{1}\beta_{i_{1}}(b_{1})\delta_{1}(d_{1}),\dots,\beta_{i_{n}}(b_{n})\delta_{n}(d_{n})$

where, for $1\leq k\leq n$, $i_{k}=1$ if $d_{k}\in A\setminus\\{e_{A}\\}$
otherwise $i_{k}=2$, and $\delta_{k}=\alpha$ if $d_{k}\in
A\setminus\\{e_{A}\\}$ otherwise $\delta_{k}=\gamma$. In this case, we define
$\phi(g\tau_{2})=h\tau$.

Subcase 2(b). Suppose $d_{n}=1$. Then, we define

$h:=a_{1}\beta_{i_{1}}(b_{1})\delta_{1}(d_{1}),\dots,\beta_{i_{n}}(b_{n})$

where for all $1\leq k<n$, $i_{k}=1$ if $d_{k}\in A\setminus\\{e_{A}\\}$
otherwise $e_{k}=2$, and $\delta_{k}=\alpha$ if $d_{k}\in
A\setminus\\{e_{A}\\}$ otherwise $\delta_{k}=\gamma$, and $i_{n}=2$. In this
case, we define $\phi(g\tau_{2})=h\tau$.

From the construction of $\phi$, it follows that $\phi$ is an isomorphism.

Homeomorphism from $\partial_{f}(G)$ to $\partial_{f}(G^{\prime})$: Let
$\Gamma_{f}$ and $\Gamma_{f}^{\prime}$ be the scaled Cayley graphs of $G$ and
$G^{\prime}$, respectively and let $\Gamma_{f}$ and $\Gamma^{\prime}_{f}$ be
compact metrizable spaces as constructed in Section 3. From Proposition 3.4,
it follows that $\partial_{f}(G)$ is homeomorphic to $\delta(\Gamma_{f})$ and
$\partial_{f}(G^{\prime})$ is homeomorphic to $\delta(\Gamma^{\prime}_{f})$.
Let $h_{A}:\partial_{f}(A)\rightarrow\partial_{f}(A),$
$h_{B}:\partial_{f}(B)\rightarrow\partial_{f}(B)$ be the identity
homeomorphisms, and let $h_{C}:\partial_{f}(C)\rightarrow\partial_{f}(A)$ be a
homeomorphism such that $\gamma\cup h_{C}:C\cup\partial_{f}(C)\rightarrow
A\cup\partial_{f}(A)$ is a homeomorphism. Since $\phi:T\rightarrow T^{\prime}$
is an isomorphism, it induces a homeomorphism $\partial\phi:\partial
T\rightarrow\partial T^{\prime}$.

We define a map $F:\delta(\Gamma_{f})\rightarrow\delta(\Gamma^{\prime}_{f})$
in the following manner:

  1. (1)

Suppose $\xi\in\delta_{Stab}(\Gamma_{f})$. Let $v$ be a vertex in $T$ such
that $\xi\in\partial_{f}(G_{v})$. As discussed in the proof of Theorem 4.1,
for $\xi$, there exists a unique reduced representative $[g,\xi]$. Define

$F([g,\xi]):=\begin{cases}[\phi(g),\xi]&\text{ if }v=gA,\\\
[\phi(g),\xi]&\text{ if }v=gB,\\\ [\phi(g),h_{C}(\xi)]&\text{ if
}v=gC,\end{cases}$

  2. (2)

Suppose $\eta\in\partial T$. Then, define $F(\eta)=\partial\phi(\eta)$.

From Remark 4.8(2) and Corollary 4.5, the continuity of $F$ on every point of
$\delta_{Stab}(\Gamma_{f})$ follows. From the definition of neighborhoods, it
is straightforward to check the continuity of $F$ at every point of $\partial
T$. Since $\delta(\Gamma_{f})$ is compact and $F$ is a bijection, we conclude
that $F$ is a homeomorphism. Hence, we complete the proof of Theorem 6.3. ∎

###  6.2. Proof of Theorem 6.2

Let $n_{1}<n_{2}<\dots<n_{k}<n_{k+1}$ be the natural numbers such that
$n=n_{1}+\dots+n_{k}+n_{k+1}$. By reindexing the groups, if required, we have
that $G_{1}=(A_{1}\ast\dots\ast
A_{n_{1}})\ast\dots\ast(A_{n_{k-1}+1}\ast\dots\ast
A_{n_{k}})\ast\dots\ast(A_{n_{k}+1}\ast\dots\ast A_{n_{k+1}})$ such that
$\partial_{f}(A_{s_{j}})\simeq\partial_{f}(A_{n_{j}})$, where $1\leq j\leq k$
and $1+n_{j-1}\leq s_{j}\leq n_{j}$ with $n_{0}=0$, and $A_{i}$’s are
elementary groups for $n_{k+1}+1\leq i\leq n_{k+2}$. Now, the following two
cases can occur:

Case 1. Suppose $k=1$. If $n_{1}=1$ too then, by Corollary 5.8(2),
$\partial_{f}(G_{1})\simeq\partial_{f}(A_{1}\ast A_{1})$. Since
$h(G_{1})=h(G_{2})$, by reindexing if necessary,
$\partial_{f}(G_{2})\simeq\partial_{f^{\prime}}(B_{1}\ast B_{1})$. Then, by
Theorem 4.1, we are done. If $n_{1}\geq 2$ then, by Corollary 5.8(1),
$\partial_{f}(G_{1})\simeq\partial_{f}(A_{1}\ast\dots\ast A_{n_{1}})$. By
Corollary 6.5, $\partial_{f}(G_{1})\simeq\partial_{f}(A_{n_{1}}\ast
A_{n_{1}})$. As $h(G_{1})=h(G_{2})$, there exists $m_{1}$ such that
$\partial_{f^{\prime}}(G_{2})\simeq\partial_{f^{\prime}}(B_{m_{1}}\ast\dots
B_{m_{1}})$. Again, we are done by Theorem 4.1.

Case 2. Suppose $k\geq 2$. By Corollary 5.8(1), we get
$\partial_{f}(G_{1})\simeq\partial_{f}(A_{1}*A_{2}*\dots A_{n_{k}})$. Using
Corollary 6.5 and Corollary 4.2, we see that
$\partial_{f}(G_{1})\simeq\partial_{f}(A_{n_{1}}\ast A_{n_{1}}\ast
A_{n_{2}}\ast A_{n_{2}}\ast\dots\ast A_{n_{k}}\ast A_{n_{k}})$. An easy
application Corollary 6.4 and induction on $k$ gives that
$\partial_{f}(G_{1})\simeq\partial_{f}(A_{n_{1}}\ast\dots\ast A_{n_{k}})$.
Since $h(G_{1})=h(G_{2})$, by following the same process as above, there exist
$m_{1},m_{2},\dots,m_{k}$ such that
$\partial_{f^{\prime}}(G_{2})\simeq\partial_{f^{\prime}}(B_{m_{1}}\ast\dots\ast
B_{m_{k}})$. Finally, by Corollary 4.2, we have that
$\partial_{f}(G_{1})\simeq\partial_{f^{\prime}}(G_{2})$. ∎

###  6.3. Proof of Theorem 6.1

Let $Y_{G}$ and $Y_{H}$ be the undeline graphs of $\mathcal{G}$ and
$\mathcal{H}$. Let $T_{G}$ and $T_{H}$ be the maximal trees in $Y_{G}$ and
$Y_{H}$, respectively. Then, the restriction of $\mathcal{G}$ and
$\mathcal{H}$ to $T_{G}$ and $T_{H}$ are trees of groups with finite edge
groups whose fundamental groups are, say, $G^{\prime}$ and $H^{\prime}$.

(1) If $G^{\prime}$ is infinite then, by Theorem 5.4,
$\partial_{f}(G)\simeq\partial_{f}(G^{\prime}\ast G^{\prime})$ which is
homeomorphic to the Cantor set by Corollary 5.3(3,4) and Corollary 5.6.
Suppose $G^{\prime}$ is finite. Since $G$ is repeated HNN extension of
$G^{\prime}$ and $G$ has infinitely many ends, at least one edge outside
$T_{G}$ gives an HNN extension, say, $G^{\prime\prime}$ such that it is an
infinite group. Then, by repeated application of Theorem 5.4(1), we see that
$\partial_{f}(G)\simeq\partial_{f}(G^{\prime\prime}\ast G^{\prime\prime})$.
Again, by Theorem 5.4(2), $\partial_{f}(G^{\prime\prime})$ is homeomorphic to
the Cantor set. Now, Theorem 4.1 and Corollary 5.6 give that
$\partial_{f}(G)\simeq\partial_{f}(\mathbb{Z}\ast\mathbb{Z}\ast\mathbb{Z}\ast\mathbb{Z})$
which is again homeomorphic to $\partial_{f}(\mathbb{Z}\ast\mathbb{Z})$ by
Corollary 6.4. Hence, the Floyd boundary of $G$ is homeomorphic to the Cantor
set. Similarly, by doing the same analysis, one can show that
$\partial_{f^{\prime}}(H)$ is homeomorphic to the Cantor set.

(2) Let $G_{1},\dots,G_{k}$ and $H_{1},\dots,H_{l}$ be the vertex groups in
$\mathcal{G}$ and $\mathcal{H}$, respectively. There are two cases to be
consider:

Case 1. By reindexing (if required), we assume that $G_{1}$ is non-elementary
and, for $i\neq 1$, $G_{i}$ is elementary. Then,
$\partial_{f}(G^{\prime})\simeq\partial_{f}(G_{1}\ast G_{1})$ by Corollary
5.8. Since $h(\mathcal{G})=h(\mathcal{H})$, by reindexing (if required), we
assume that $H_{1}$ is non-elementary and, for $j\neq 1$, $H_{j}$ is
elementary. Thus,
$\partial_{f^{\prime}}(H^{\prime})\simeq\partial_{f^{\prime}}(H_{1}\ast
H_{1})$. Now, Theorem 5.4(1) and Corollary 6.4 give us that
$\partial_{f}(G)\simeq\partial_{f}(G_{1}\ast G_{1})$. Similarly,
$\partial_{f^{\prime}}(H)\simeq\partial_{f^{\prime}}(H_{1}\ast H_{1})$. Hence,
$\partial_{f}(G)\simeq\partial_{f}(H)$ by Theorem 4.1.

Case 2. By reindexing (if required), we assume that there exists $2\leq
k^{\prime}\leq k$ such that $G_{i}$ is non-elementary for $1\leq i\leq
k^{\prime}$ and $G_{i}$ is elementary $k^{\prime}+1\leq i\leq k$. Similarly,
we assume that there exists $2\leq l^{\prime}\leq l$ such that $H_{j}$ is non-
elementary for $1\leq j\leq l^{\prime}$ and $H_{j}$ is elementary for
$l^{\prime}+1\leq j\leq l$. Thus, by combining Corollary 5.3(3) and Corollary
5.7, we have $\partial_{f}(G^{\prime})\simeq\partial_{f}(G_{1}\ast\dots\ast
G_{k^{\prime}})$ and
$\partial_{f^{\prime}}(H)\simeq\partial_{f^{\prime}}(H_{1}\ast\dots\ast
H_{l^{\prime}})$. Now, Theorem 5.4(1) gives that
$\partial_{f}(G)\simeq\partial_{f}((G_{1}\ast\dots\ast
G_{k^{\prime}})\ast(G_{1}\ast\dots\ast G_{k^{\prime}}))$ and
$\partial_{f^{\prime}}(H)\simeq\partial_{f^{\prime}}((H_{1}\ast\dots\ast
H_{l^{\prime}})\ast(H_{1}\ast\dots\ast H_{l^{\prime}}))$. Since
$h(\mathcal{G})=h(\mathcal{H})$, Theorem 6.2 gives us that
$\partial_{f}(G)\simeq\partial_{f^{\prime}}(H)$. This complete the proof of
the theorem. ∎

##  7\. Components of Floyd boundary of infinite-ended groups

It is well known that Gromov boundary of a hyperbolic group is connected if
and only if it is $1$-ended. In this section, first of all, we prove an analog
of this fact in the context of Floyd boundary. Let $G$ be a $1$-ended group
and $f$ be a Floyd function. We denote the scaled Cayley graph of $G$ with
respect to a finite generating set by $\Gamma_{f}$ with the Floyd metric
$d_{f}$, and $d$ denotes the metric on the Cayley graph. Assume that
$\overline{\Gamma}_{f}$ is the metric completion with the metric
$\bar{d}_{f}$.

######  Proposition 7.1.

$\partial_{f}(G)$ is connected.

###### Proof.

Define a map $\pi:G\to\partial_{f}(G)$ such that
$\bar{d}_{f}(g,\pi(g))=\bar{d}_{f}(g,\partial_{f}(G))$. For the sake of
contradiction, assume that $\partial_{f}(G)=V_{1}\sqcup V_{2}$, where
$V_{i}$’s are non-empty, disjoint open subsets of $\partial_{f}(G)$. Let
$B_{i}:=\pi^{-1}(V_{i})$ for $i=1,2$. Denote the closure of $B_{i}$ in
$\overline{\Gamma}_{f}$ by $\overline{B}_{1}$. From definition of $\pi$, it
follows that if $\\{g_{i}\\}$ is a sequence converging to a point
$\xi\in\partial_{f}(G)$ then $\pi(g_{i})$ also converges to $\xi$. Thus, it is
see to check that $\overline{B}_{i}\subset B_{i}\cup V_{i}$.

Let $\xi\in V_{1}$ and let $\\{g_{n}\\}$ be a sequence in $\Gamma_{f}$ such
that $g_{n}\to\xi$. Suppose there exists an infinite subsequence of
$\\{g_{n}\\}$ which is contained in $B_{2}$. Then $\xi\in\overline{B}_{2}$
which in turn implies that $\xi\in V_{2}$. This gives a contradiction as
$V_{1}$ and $V_{2}$ are disjoint. Hence, $\\{g_{n}\\}$ is eventually contained
in $B_{1}$. Applying a similar argument to $\xi^{\prime}\in V_{2}$ leads us to
the following conclusion:

$\overline{B}_{i}:=B_{i}\cup V_{i}.$

Since $V_{1}$ and $V_{2}$ are non-empty, there exists $\xi_{1}\in V_{1},$
$\xi_{2}\in V_{2}$. Assume that $\\{x_{n}\\}$ and $\\{y_{n}\\}$ are sequences
in $B_{1}$ and $B_{2}$, respectively, such that $x_{n}\to\xi_{1}$ and
$y_{n}\to\xi_{2}$. For every $m\in\mathbb{N}$, denote $K_{m}$ the closed
$d$-metric ball of radius $m$ about $e_{G}$. Since $G$ is $1$-ended, there
exist subsequences $\\{x_{n_{m}}\\}$ and $\\{y_{n_{m}}\\}$ and paths
$\gamma_{m}$ joining $x_{n_{m}}$ and $y_{n_{m}}$ such that $\gamma_{m}$ is
contained in $\Gamma_{f}\setminus K_{m}$. Let $e_{m}=[a_{m},b_{m}]$ be an edge
such that $a_{m}\in B_{1}$ and $b_{m}\in B_{2}$. Since $\Gamma_{f}$ is
compact, there exists subsequence $\\{a_{m_{k}}\\}$ that converges to $\eta\in
V_{1}$. Since $d(a_{m_{k}},b_{m_{k}})=1$, it is easy to see that
$b_{m_{k}}\to\eta$ in $\overline{\Gamma}_{f}$ too. This implies that $\eta\in
V_{2}$, which gives us a contradiction. Hence, we have the desired result. ∎

Now, we record the following fact whose proof is same as the proof of [MS15,
Proposition 6.3,6.4].

######  Proposition 7.2.

Let $G$ be a group which splits as a graph of groups $\mathcal{G}$ such that
the edge groups are finite and $G$ is infinite-ended. Suppose each non-
elementary vertex group is $1$-ended. Then, each connected component is either
homeomorphic to the Floyd boundary of a non-elementary vertex group or it is a
singleton set. ∎

Now, we are ready to prove the following which gives a partial converse of
Theorem 6.1.

######  Theorem 7.3.

Suppose $G$ and $H$ are two finitely generated infinite-ended groups. Suppose
$\mathcal{G}$ and $\mathcal{H}$ are graphs of groups decomposition of $G$ and
$H$, respectively such that the edge groups of $\mathcal{G}$ and $\mathcal{H}$
are finite, and each non-elementary vertex group of $\mathcal{G}$ and
$\mathcal{H}$ is $1$-ended. Also, assume that at least one vertex group of
$\mathcal{G}$ and $\mathcal{H}$ is non-elementary. Then, we have the
following:

Let $h(\mathcal{G})$ and $h(\mathcal{H})$ denote the set of homeomorphism
types of Floyd boundaries of non-elementary vertex groups of $\mathcal{G}$ and
$\mathcal{H}$, respectively. If $\partial_{f}(G)$ is homeomorphic to
$\partial_{f^{\prime}}(H)$ then $h(\mathcal{G})=h(\mathcal{H})$.

###### Proof.

Let $G_{1},\dots,G_{k}$ and $H_{1},\dots,H_{l}$ be the non-elementary vertex
groups of $\mathcal{G}$ and $\mathcal{H}$, respectively. There are two cases
to be consider:

Case 1. Suppose $k=1$ and $l=1$. Then,
$\partial_{f}(G)\simeq\partial_{f}(G_{1}\ast G_{1})$ and
$\partial_{f^{\prime}}(H)\simeq\partial_{f^{\prime}}(H_{1}\ast H_{1})$. Thus,
$\partial_{f}(G_{1}\ast G_{1})\simeq\partial_{f^{\prime}}(H_{1}\ast H_{1})$.
Using Proposition 7.2 and the fact that a homeomorphism takes connnected
components to connected components, we see that
$\partial_{f}(G_{1})\sim\partial_{f^{\prime}}(H_{1})$. Suppose $k=1$ and
$l\geq 2$. Then, $\partial_{f}(G)\simeq\partial_{f}(G_{1}\ast G_{1})$ and
$\partial_{f^{\prime}}(H)\simeq\partial_{f^{\prime}}(H_{1}\ast\dots\ast
H_{l})$. Again, by the same logic, we see that
$\partial_{f}(G_{1})\simeq\partial_{f^{\prime}}(H_{i})$ for $1\leq i\leq l$. A
similar analysis can be done when $l=1$ and $k$ is any natural number.

Case 2. Suppose $k,l\geq 2$. In this case, we have that
$\partial_{f}(G)\simeq\partial_{f}(G_{1}\ast\dots\ast G_{k})$ and
$\partial_{f^{\prime}}(H)=\partial_{f^{\prime}}(H_{1}\ast\dots H_{l})$. Thus,
$\partial_{f}(G_{1}\ast\dots\ast
G_{k})\simeq\partial_{f^{\prime}}(H_{1}\ast\dots\ast H_{l})$. Now, using
Proposition 7.2 and the fact that homeomorphism takes connected components to
connected components, we see that $h(\mathcal{G})=h(\mathcal{H})$. ∎

In [Dun85]. M. Dunwoody proved that each finitely presented group $G$ has a
terminal splitting over finite subgroups, i.e. it is isomorphic to the
fundamental group of a graph of groups whose vertex groups are $1$-ended or
finite and whose edge groups are finite. Thus, by combining Theorem 6.1 and
Theorem 7.3, we have the following:

######  Corollary 7.4.

Suppose $G$ and $H$ are two finitely presented infinite-ended groups. Suppose
$\mathcal{G}$ and $\mathcal{H}$ are terminal splittings of $G$ and $H$,
respectively such that at least one vertex of $\mathcal{G}$ and $\mathcal{H}$
is non-elementary. Then, we have the following:

Let $h(\mathcal{G})$ and $h(\mathcal{H})$ denote the set of homeomorphism
types of Floyd boundaries of non-elementary vertex groups of $\mathcal{G}$ and
$\mathcal{H}$, respectively. Then, $\partial_{f}(G)$ is homeomorphic to
$\partial_{f^{\prime}}(H)$ if and only if $h(\mathcal{G})=h(\mathcal{H})$. ∎

We end the paper by noting the following:

######  Corollary 7.5.

Let $G$ be a virtually free group. Then, for any Floyd function $f$,
$\partial_{f}(G)$ is homeomorphic to the Cantor set. Conversely, if, for some
Floyd function $f$, $\partial_{f}(G)$ is homeomorphic to the Cantor set then
$G$ is virtually free.

###### Proof.

Since virtually free groups admit a decomposition as graphs of finite groups,
by Theorem 6.1(1), $\partial_{f}(G)$ is homeomorphic to the Cantor set.
Conversely, suppose that $\partial_{f}(G)$ is the Cantor set. This implies
that $G$ is infinite-ended. Since virtually free groups are finitely
presented, $G$ admits a terminal splitting where each vertex group is finite.
This implies that $G$ is virtually free (see [Ser03]). ∎

## References

  * [BH99] M. Bridson and A Haefliger, _Metric spaces of nonpositive curvature_ , Grundlehren der mathematischen Wissenchaften, Vol 319, Springer-Verlag (1999).
  * [Dun85] M. J. Dunwoody, _The accessibility of finitely presented groups_ , Invent. Math. 81 (1985), no. 3, 449–457.
  * [Flo80] W. J. Floyd, _Group Completions and Limit Sets of Kleinian Groups_ , Invent. Math. vol.57 (1980), 205–218.
  * [Ger12] Victor Gerasimov, _Floyd maps for relatively hyperbolic groups_ , Geom. Funct. Anal. 22 (2012), no. 5, 1361–1399.
  * [Gro85] M. Gromov, _Hyperbolic Groups_ , in Essays in Group Theory, ed. Gersten, MSRI Publ.,vol.8, Springer Verlag (1985), 75–263.
  * [Kar03] Anders Karlsson, _Free subgroups of groups with nontrivial Floyd boundary_ , Comm. Algebra 31 (2003), no. 11, 5361–5376.
  * [MS15] Alexandre Martin and Jacek Światkowski, _Infinitely-ended hyperbolic groups with homeomorphic Gromov boundaries_ , J. Group Theory 18 (2015), no. 2, 273–289.
  * [Ser03] J.-P. Serre, _Trees_ , Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003, Translated from the French original by John Stillwell, Corrected 2nd printing of the 1980 English translation. MR 1954121