On the connectedness of the boundary of hierarchically hyperbolic spaces

TL;DR

This paper proves that under mild conditions, the boundary of a one-ended hierarchically hyperbolic space is connected, and explores the implications for group boundaries and their homeomorphisms.

Contribution

It establishes the connectedness of boundaries in hierarchically hyperbolic spaces and groups, linking boundary connectedness to one-endedness, and characterizes boundary homeomorphisms for free products.

Findings

Boundaries of one-ended hierarchically hyperbolic spaces are connected.

Boundary connectedness is equivalent to one-endedness of the group.

Homeomorphism of boundaries in free products depends on component boundaries.

Abstract

We prove that, under a mild assumption, any metrizable compactification of a one-ended proper geodesic metric space is connected. As a consequence, we deduce that the boundary, introduced by Durham--Hagen--Sisto, of a one-ended hierarchically hyperbolic space is connected. Moreover, we prove that the connectedness of the boundary of a hierarchically hyperbolic group is equivalent to the one-endedness of the group. As an application, we show that if, for $n\geq 2$, $G_1=A_1\ast\dots\ast A_n$ and $G_2=B_1\ast\dots\ast B_n$ are free products of one-ended hierarchically hyperbolic groups, then the boundary of $G_1$ is homeomorphic to the boundary of $G_2$ if and only if the boundary of $A_i$ is homeomorphic to the boundary of $B_i$ for $1\leq i\leq n$.

Equations

$$B_{\{U_S\},ϵ}(p):={\mathcal{B}}_{\{U_S\},ϵ}^{rem}(p)\cup {\mathcal{B}}_{\{U_S\},ϵ}^{non}(p)\cup {\mathcal{B}}_{\{U_S\},ϵ}^{int}(p).$$

$$\overline{G}:=\Gamma \cup \partial (G,\mathfrak{S})$$

$$\pi (g)=\xi ,\text{ where }\xi \in \partial (G,\mathfrak{S})\text{ such that }{d}_{△}(g,\xi )=d_{△}(g,\partial (G,\mathfrak{S}))$$

$$\overline{\pi }(x)=\xi ,\text{ where }\xi \in \partial X\text{ such that }\overline{d}(x,\xi )=\overline{d}(x,\partial X).$$

$$\partial X=V_1\bigsqcup V_2$$

$$(g_1,v_A)\sim (g_2,v_A)\text{ if }{g}_1^{-1}{g}_2\in A,$$

$$(g_1,v_B)\sim (g_2,v_B)\text{ if }{g}_1^{-1}{g}_2\in B.$$

$$(g_1,y_1)\sim (g_2,y_2)\text{ if }{y}_1,y_2\in {\Gamma }_A\text{ and }{g}_2^{-1}{g}_1{y}_1=y_2,$$

$$(g_1,y_1)\sim (g_2,y_2)\text{ if }{y}_1,y_2\in {\Gamma }_B\text{ and }{g}_2^{-1}{g}_1{y}_1=y_2.$$

$$\mathfrak{S}:=\{\hat{\Gamma }\}\cup (\underset{v\in T}{⨆}{\mathfrak{S}}_v).$$

$$(g_1,{\xi }_1)\sim (g_2,{\xi }_2)\text{ if }{\xi }_1,{\xi }_2\in \partial A,g_2^{-1}{g}_1\in A\text{ and }{g}_2^{-1}{g}_1{\xi }_1={\xi }_2,$$

$$(g_1,{\xi }_1)\sim (g_2,{\xi }_2)\text{ if }{\xi }_1,{\xi }_2\in \partial B,g_2^{-1}{g}_1\in B\text{ and }{g}_2^{-1}{g}_1{\xi }_1={\xi }_2.$$

$$\delta (\Gamma ):={\delta }_{Stab}(\Gamma )\bigsqcup \partial T.$$

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

$$V_U(\xi ):=U\cup V_U.$$

$$V_n(\eta )={\pi }_T^{-1}(\overline{T_n(\eta )}\setminus \{u_n(\eta )\}).$$

$$\partial (G,\mathfrak{S})=(\underset{v\in T}{⨆}\partial (G_v,{\mathfrak{S}}_v))\cup \partial \hat{\Gamma }.$$

$$\psi (x)=\{\begin{array}{ll}x& \text{if }x\in \Gamma \cup ({⨆}_{v\in T}\partial G_v),\\ \bar{\varphi }(x)& \text{if }x\in \partial \hat{\Gamma }\end{array}$$

$${\mathcal{B}}_U^{rem}(\xi )=\{{\xi }^{\prime }\in \underset{v\in T}{⨆}\partial G_v:\text{ if ξ′∈∂Gv then the cone-point corresponding to Γv is in }U\}.$$

$${\mathcal{B}}_U^{non}(\xi )=\{{\xi }^{\prime }\in \partial \hat{\Gamma }:{\xi }^{\prime }\in U\}.$$

$$d_{{△}_2}(h_l,\partial f(\pi (g_k)))<\frac{1}{k}.$$

$$d_{{△}_2}(h_k,\partial f(\pi (g)))<d_{{△}_2}(h_k,\partial G_2)+\frac{1}{k}.$$

$$F([a_1{b}_1,\dots ,a_n{b}_n,\xi ])=[f_1(a_1)f_2(b_1),\dots ,f_1(a_n)f_2(b_n),q(\xi )].$$

$$F((a_1{b}_1,\dots ))=f_1(a_1)f_2(b_1),\dots$$

Full text

1. Introduction

Motivated by the seminal work of Masur–Minsky [17, 16], Behrstock–Hagen–Sisto introduced the notion of a hierarchically hyperbolic space and group [6]. This provides a common framework to study mapping class groups and cubical groups. In [10], Durham–Hagen–Sisto introduced a boundary of a hierarchically hyperbolic group that coincides with the Gromov boundary when the group is hyperbolic. This boundary also gives a compactification of the group, and is called the hierarchical boundary of a hierarchically hyperbolic group. In [3], Abbott–Behrstock–Russell showed that if a hierarchically hyperbolic group $G$ is hyperbolic relative to a collection of subgroups, then the Bowditch boundary of $G$ is a quotient of the hierarchical boundary of $G$. This fact is also crucial to prove our main result.

It is well known that the Gromov boundary of a hyperbolic group is connected if and only if it is one-ended. The same is true for the Bowditch boundary of a relatively hyperbolic group [7, Theorem 10.1]. So it is natural to look for the relationship between the hierarchical boundary of a hierarchically hyperbolic group and its ends. In [2], Abbott–Behrstock–Durham introduced a ‘maximized hierarchical structure’ for a given hierarchically hyperbolic space. Using this maximalization, Abbott–Behrstock–Russell [3, Corollary 5.6] proved that if the hyperbolic space associated to the maximal nested element in the maximized hierarchical structure of a hierarchically hyperbolic group $G$ is one-ended, then the hierarchical boundary with respect to any hierarchical structure on $G$ is connected. One of our main aims in this note is to directly prove that the hierarchical boundary of a one-ended hierarchically hyperbolic space is connected. However, from the proof and under a mild assumption, we see that this holds for any compactification of a proper geodesic metric space. Let $X$ be a proper geodesic metric space and $\partial X$ be a set such that $\overline{X}:=X\cup\partial X$ is a compactification of $X$, i.e. the inclusion $X\to\overline{X}$ is a topological embedding such that its image is a dense and open subset of $\overline{X}$. We further assume that the topology on $\overline{X}$ is metrizable.

Definition 1.1**.**

We call $\partial X$ a weakly visible boundary of $X$ if the following holds:

Suppose $\{x_{n}\}$ and $\{y_{n}\}$ are uniformly bounded sequences in $X$. If $\{x_{n}\}$ is converging to $\xi\in\partial X$ then $\{y_{n}\}$ is converging to $\xi.$

The Gromov boundary of a proper hyperbolic space, the Bowditch boundary of a proper relatively hyperbolic space, and the hierarchical boundary of a proper hierarchically hyperbolic space are weakly visible. In this note, the following is our first main result.

Theorem 1 (Theorem 3.1)*A weakly visible boundary of a one-ended proper geodesic metric space is connected. *

In [13], Hamenstädt introduced a $Z$-boundary of the mapping class group of a surface of finite type. In Lemma 3.2, we show that this $Z$-boundary is weakly visible, and hence by Theorem 1, it is connected (Proposition 3.3). As an application of Theorem 1 and [4, Theorem 1.3], we obtain the following, and answer a question that appears in [1]:

Theorem 2 (Theorem 3.6). Let $G$ be a hierarchically hyperbolic group. Then, the hierarchical boundary of $G$ is connected if and only if $G$ is one-ended.

For the definition of a hierarchically hyperbolic group and its boundary, one is referred to Section 2. In [10, p. 3672], the authors conjectured Theorem 2. Here, we prove their conjecture. An application of Theorem 2 implies that the hierarchical boundary of the mapping class group of a connected orientable surface of finite type is connected, see Corollary 3.7 for the precise statement.

In [15], the authors proved that the topology of the Gromov boundary of a free product of hyperbolic groups is uniquely determined by the topology of the Gromov boundaries of the free factors. Zbinden [23], Tomar [22], and Chakraborty–Tomar [9] proved similar results for the Morse boundary, Bowditch boundary, and Floyd boundary of a free product of groups, respectively. We also prove a result of the same flavour for hierarchical boundary (Theorem 5.4). As an application of this result, we prove the following:

Theorem 3 (Theorem 6.3.) For $n\geq 2$, suppose $G_{1}=A_{1}\ast\dots\ast A_{n}$ and $G_{2}=B_{1}\ast\dots\ast B_{n}$, where $A_{i}$ and $B_{i}$ are one-ended hierarchically hyperbolic groups for all $i$. Suppose $G_{1}$ and $G_{2}$ have hierarchical structures as described in Subsection 4.2. Then, the hierarchical boundary of $G_{1}$ is homeomorphic to the hierarchical boundary of $G_{2}$ if and only if the hierarchical boundary of $A_{i}$ is homeomorphic to the hierarchical boundary of $B_{i}$ for all $1\leq i\leq n$.

Remark 1.2**.**

It is possible to give more than one hierarchical structure on a given group. However, it remains an open question whether different hierarchical structures on the same group yield homeomorphic hierarchical boundaries [10, Question 1]. In this paper, for one-ended groups, we give equivalent conditions for this question using free products (see Corollary 6.4). Throughout the paper, for free products of hierarchically hyperbolic groups, we take the hierarchical structure as described in Subsection 4.2.

2. Background

In this section, we collect the necessary definitions and results. The definition of hierarchically hyperbolic space (HHS) is rather technical, and we refer the reader to [5, Definition 1.2] for a complete account. Roughly, an HHS is an $E$-quasigeodesic metric space with an index set $\mathfrak{S}$, with some extra data. We include some axioms for being an HHS that are relevant to us. Let $E>0$ and $\mathcal{X}$ be a $E$-quasigeodesic metric space. Let $\{\mathcal{C}W:W\in\mathfrak{S}\}$ be a collection of $E$-hyperbolic spaces.

Projections. For each $U\in\mathfrak{S}$, there exists an $E$-coarsely Lipschitz $E$-coarse map $\pi_{U}:\mathcal{X}\to\mathcal{C}U$ such that $\pi_{U}(\mathcal{X})$ is $E$-quasiconvex in $\mathcal{C}U$.

Nesting. If $\mathfrak{S}\neq\phi$, then $\mathfrak{S}$ is equipped with a partial order $\sqsubseteq$ and it has a unique $\sqsubseteq$-maximal element. If $U,V\in\mathfrak{S}$ and $U\sqsubseteq V$, then we say that $U$ is nested in $V$. Moreover, for all $U,V\in\mathfrak{S}$ with $U\sqsubsetneq V$ there is a specified subset $\rho_{V}^{U}\subset\mathcal{C}V$ such that Diam${}_{\mathcal{C}V}(\rho_{V}^{U})\leq E$. Also, there is a projection $\rho_{U}^{V}:\mathcal{C}V\to\mathcal{C}U$

Orthogonality. $\mathfrak{S}$ has a symmetric and antireflexive relation called orthogonality. We write $U\perp V$ when $U$ and $V$ are orthogonal.

Transversality. If $U,V\in\mathfrak{S}$ are not orthogonal and neither is nested in the other, then we say $U$ and $V$ are transverse, denoted $U\pitchfork V$. Moreover, for all $U,V\in\mathfrak{S}$ with $U\pitchfork V$, there are non-empty sets $\rho_{U}^{V}\subseteq\mathcal{C}U$ and $\rho_{V}^{U}\subseteq\mathcal{C}V$ each of diameter at most $E$.

Bounded geodesic image. For all $U,V\in\mathfrak{S}$ such that $U\sqsubsetneq V$, and all geodesic $\alpha$ in $\mathcal{C}V$, either Diam${}_{\mathcal{C}U}(\rho_{U}^{V}(\alpha))\leq E$ or $N_{E}(\rho_{V}^{U})\cap\alpha\neq\emptyset$.

We use $\mathfrak{S}$ to denote the hierarchically hyperbolic space structure, including the index set $\mathfrak{S}$, spaces $\{\mathcal{C}W:W\in\mathfrak{S}\}$, projections $\{\pi_{W}:W\in\mathfrak{S}\}$, and relations $\sqsubseteq,\perp,\pitchfork$. A quasigeodesic metric space $\mathcal{X}$ is said to be hierarchically hyperbolic space with constant $E$ if there exists a hierarchically hyperbolic structure on $\mathcal{X}$ with constant $E$. The pair $(\mathcal{X},\mathfrak{S})$ denotes a hierarchically hyperbolic space equipped with the specific HHS structure $\mathfrak{S}$.

A hierarchically hyperbolic group (HHG) is a finitely generated group $G$ that acts on an HHS $(\mathcal{X},\mathfrak{S})$ such that

1. (1)

the action of $G$ on $\mathcal{X}$ is geometric, 2. (2)

$G$ acts on $\mathfrak{S}$ by a $\sqsubseteq$-,$\perp$-, and $\pitchfork$-preserving bijection, and $\mathfrak{S}$ has finitely many $G$-orbits, 3. (3)

the action is compatible with the HHS structure on $\mathcal{X}$ [18, p. 483].

For a precise definition, see [5, Definition 1.21]. In this case, we say that $\mathfrak{S}$ is an HHG structure for the group $G$ and use the pair $(G,\mathfrak{S})$ to denote the hierarchically hyperbolic group $G$ equipped with the specific hierarchically hyperbolic group structure $\mathfrak{S}$. From condition (1) in the definition of an HHG, it follows that $G$ is quasiisometric to $X$. From here, one can give a hierarchical structure on the Cayley graph of $G$, and the left action of $G$ on the Cayley graph also satisfies all the conditions for being an HHG. Thus, to define an HHG, one can use a Cayley graph of $G$ itself. Also, it is easy to show that the definition of HHG does not depend on the choice of the Cayley graph.

Hierarchical boundary. In [10], the authors introduced the notion of a boundary of an HHS. From [10, Section 2], we recall the definition of the hierarchical boundary and its topology. For $S\in\mathfrak{S}$, $\partial\mathcal{C}S$ denote the Gromov boundary [11] of $\mathcal{C}S$.

Definition 2.1**.**

Let $(\mathcal{X},\mathfrak{S})$ be an HHS. A subset $\overline{S}\subset\mathfrak{S}$ is said to be a support set if $S_{i}\perp S_{j}$ for all $S_{i},S_{j}\in\overline{S}$. Given a support set $\overline{S}$, the boundary point with support $\overline{S}$ is a formal sum $p=\sum_{S\in\overline{S}}a_{S}^{p}p_{S}$, where $p_{S}\in\partial\mathcal{C}S$, $a_{S}>0$, and $\sum_{S\in\overline{S}}a_{S}^{p}=1$. By [5, Lemma 2.1], such sums are necessarily finite. We denote the support of the boundary point $p$ by Supp($p$). The hierarchical boundary $\partial(\mathcal{X},\mathfrak{S})$ of $(\mathcal{X},\mathfrak{S})$ is the set of all boundary points.

When the specific HHS structure is clear, we write $\partial\mathcal{X}$ instead of $\partial(\mathcal{X},\mathfrak{S})$. For an HHG $(G,\mathfrak{S})$, let $(\mathcal{X},\mathfrak{S})$ be a corresponding HHS. Then, the hierarchical boundary $\partial(G,\mathfrak{S})$ of $(G,\mathfrak{S})$ is defined to the hierarchical boundary of $(\mathcal{X},\mathfrak{S})$.

Topology on $\partial\mathcal{X}$. Before defining the topology, we need the notion of a remote point and boundary projection.

Definition 2.2** (Remote point).**

A point $q\in\partial\mathcal{X}$ is called a remote point with respect to a support set $\overline{S}$ if Supp($q$)$\cap\overline{S}=\emptyset$ and, for all $S\in\overline{S}$, there exists $T_{S}\in$ Supp($q$) such that $S\notperp T_{S}$. The set of all remote points of $\partial\mathcal{X}$ with respect to $\overline{S}$ is denoted by $\partial^{rem}_{\overline{S}}(\mathcal{X})$.

For a support set $\overline{S}$, we denote $\overline{S}$⟂ the set of all $U\in\mathfrak{S}$ such that $U\perp V$ for all $V\in\overline{S}$. Given a support set $\overline{S}$ and $q\in\partial^{rem}_{\overline{S}}\mathcal{X}$, let $\overline{S}_{q}$ denote the union of $\overline{S}$ and the set of all $U\in\overline{S}$⟂ such that $U$ is not orthogonal to some $T_{U}\in$ Supp($q$).

Definition 2.3** (Boundary projection).**

Define a boundary projection $\partial\pi_{\overline{S}}(q)\in\prod_{S\in\overline{S}_{q}}\partial\mathcal{C}S$ as follows. Let $q=\sum_{T\in\text{Supp}(q)}a_{T}^{q}q_{T}$. For each $S\in\overline{S}_{q}$ , let $T_{S}\in\text{Supp}(q)$ be chosen so that $S$ and $T_{S}$ are not orthogonal. Define the $S$-coordinate $(\partial\pi_{\overline{S}}(q))_{S}$ of $\partial\pi_{\overline{S}}(q)$ as follows:

1. (1)

If $T_{S}\sqsubseteq S$ or $T_{S}\pitchfork S$, then $(\partial\pi_{\overline{S}}(q))_{S}=\rho_{S}^{T_{S}}$. 2. (2)

Otherwise, $S\sqsubseteq T_{S}$. Choose a $(1,20E)$–quasigeodesic ray $\gamma$ in $\mathcal{C}T_{S}$ joining $\rho_{T_{S}}^{S}$ to $q_{T_{S}}$. By the bounded geodesic image axiom, there exists $x\in\gamma$ such that $\rho_{S}^{T_{S}}$ is coarsely constant on the subray of $\gamma$ beginning at $x$. Let $(\partial\pi_{\overline{S}}(q))_{S}=\rho_{S}^{T_{S}}(x)$.

The map $\partial\pi_{\overline{S}}$ is coarsely independent of the choice of $\{T_{S}\}_{S\in\overline{S}}$ (see [10, Lemma 2.1]). Now, we are ready to define the topology on $\partial\mathcal{X}$.

Fix a base point $x_{0}\in\mathcal{X}$. We define a neighborhood basis for each point $p=\sum_{S\in\overline{S}}a_{S}^{p}p_{S}$, where $p_{S}\in\partial\mathcal{C}S$ for each $S\in\text{Supp}(p)=\overline{S}$. For each $S\in\mathfrak{S}$, choose a neighborhood $U_{S}$ of $p_{S}$ in $\mathcal{C}S\cup\partial\mathcal{C}S$, and choose $\epsilon>0$. Now, we define the following three subsets of $\partial\mathcal{X}$ which contribute in the definition of a neighborhood around $p$.

Definition 2.4** (Remote part).**

The remote part $\mathcal{B}^{rem}_{\{U_{S}\},\epsilon}(p)$ is the set of all $q\in\partial\mathcal{X}$ such that the following hold:

1. (1)

For all $S\in\overline{S}$, $(\partial\pi_{\overline{S}}(q))_{S}\in U_{S}$. 2. (2)

$\sum_{T\in\overline{S}^{\perp}}a_{T}^{q}<\epsilon$. 3. (3)

For all $S\in\overline{S}_{q}$ and $S^{\prime}\in\overline{S}$, \bigg{|}\dfrac{d_{S}(x_{0},(\partial\pi_{\overline{S}}(q))_{S})}{d_{S^{\prime}}(x_{0},(\partial\pi_{\overline{S}}(q))_{S^{\prime}})}-\dfrac{a_{S}^{p}}{a_{S^{\prime}}^{p}}\bigg{|}<\epsilon.

Definition 2.5** (Non-remote part).**

The non-remote part $\mathcal{B}^{non}_{\{U_{S}\},\epsilon}(p)$ is the set of points $q=\sum_{T\in\text{Supp}(q)}a_{T}^{q}q_{T}\in\partial\mathcal{X}-\partial^{rem}_{\overline{S}}\mathcal{X}$ such that the following hold, where $A=\overline{S}\cap\text{Supp}(q)$:

1. (1)

For all $T\in A$, $q_{T}\in U_{T}$. 2. (2)

For all $T\in A$, $|a_{T}^{p}-a_{T}^{q}|<\epsilon.$ 3. (3)

$\sum_{V\in\text{Supp}(q)-A}a_{V}^{q}<\epsilon$.

Definition 2.6** (Interior part).**

The interior part $\mathcal{B}^{int}_{\{U_{S}\},\epsilon}(p)$ is the points $x\in\mathcal{X}$ such that the following conditions are satisfied:

1. (1)

For all $S\in\overline{S}$, $\pi_{S}(x)\in U_{S}$. 2. (2)

For all $S,S^{\prime}\in\overline{S}$, \bigg{|}\dfrac{d_{S}(x_{0},\pi_{S}(x))}{d_{S^{\prime}}(x_{0},\pi_{S^{\prime}}(x)}-\dfrac{a_{S}^{p}}{a_{S^{\prime}}^{p}}\bigg{|}<\epsilon. 3. (3)

For all $S\in\overline{S}$ and $T\in\overline{S}^{\perp}$, $\dfrac{d_{T}(x_{0},x)}{d_{S}(x_{0},x)}<\epsilon$.

Definition 2.7** (Topology on $\mathcal{X}\cup\partial\mathcal{X}$).**

For each $p\in\partial\mathcal{X}$ with support $\overline{S}$, and $\{U_{S}\}_{S\in\overline{S}}$, $\epsilon>0$ as above, let

[TABLE]

We declare the set of all such $B_{\{U_{S}\},\epsilon}(p)$ to form a neighborhood basis around $p$. Also, we include the open subsets of $\mathcal{X}$ in the topology of $\mathcal{X}\cup\partial\mathcal{X}$.

In [12, Remark 1.3], Hagen clarified why this indeed forms a valid neighborhood basis. This topology does not depend on the choice of the base point $x_{0}$.

Theorem 2.8**.**

([10, Theorem 3.4],[12])* If $\mathcal{X}$ is proper, then $\overline{\mathcal{X}}:=\mathcal{X}\cup\partial X$ is a compact metrizable space. Moreover, $\mathcal{X}$ is dense in $\overline{\mathcal{X}}$.*

Let $(G,\mathfrak{S})$ be an HHG, and let $\partial(G,\mathfrak{S})$ be its hierarchical boundary. Define

[TABLE]

where $\Gamma$ denotes a Cayley graph of $G$ with respect to a finite generating set. By the previous theorem, $\overline{G}$ is a compact metrizable space. We denote this metric by $d_{\triangle}$. Define a map $\pi:G\to\partial(G,\mathfrak{S})$ as

[TABLE]

Note that for (1), such a $\xi$ exists as $\partial(G,\mathfrak{S})$ is compact, but (2) $\xi$ may not be unique; however, uniqueness is not required for our purposes.

Lemma 2.9**.**

$\pi(G)$* is dense in $\partial(G,\mathfrak{S})$.*

Proof.

Let $\xi\in\partial G$. Since $G$ is dense in $\overline{G}$, there exists a sequence $\{g_{i}\}\subset G$ such that $\lim_{i\to\infty}d_{\triangle}(g_{i},\xi)=0$. Thus, by (1), $\lim_{i\to\infty}d_{\triangle}(\pi(g_{i}),\xi)=0$. Hence, $\pi(G)$ is dense in $\partial(G,\mathfrak{S})$. ∎

We finish this section by recording the following fact that is relevant to us. For relative hyperbolicity and Bowditch boundary, one is referred to [7].

Theorem 2.10**.**

([3, Theorem 1.3])* Let $G$ be an HHG that is hyperbolic relative to a finite collection of subgroups $\mathcal{P}$. Then, the Bowditch boundary of $G$ with respect to $\mathcal{P}$ is the quotient of hierarchical boundary of $G$ obtained by collapsing the limit set of each coset of a parabolic subgroup to a point.*

3. Proof of Theorem 3.1

This section is devoted to proving our main result.

Theorem 3.1**.**

A weakly visible boundary of a one-ended proper geodesic metric space is connected.

Proof.

Let $X$ be a proper geodesic metric space with weakly visible boundary $\partial X$, and $\overline{X}:=X\cup\partial X$ be a compactification of $X$. Let $d$ and $\overline{d}$ denote the metric on $X$ and $\overline{X}$, respectively. Define a map $\overline{\pi}:X\to\partial X$ as

[TABLE]

Note that such a $\xi$ exists as $\partial X$ is compact, but $\xi$ may not be unique; however, uniqueness is not required for our purposes. We prove the theorem by contradiction. Let if possible

[TABLE]

where $V_{1}$ and $V_{2}$ are non-empty disjoint open subsets of $\partial X$. Let $B_{i}=\overline{\pi}^{-1}(V_{i})$ for $i=1,2$. Denote the closure of $B_{i}$ in $\overline{X}$ by cl$(B_{i})$ (with respect to the topology induced by $\overline{d}$).

Claim: $B_{1}$ and $B_{2}$ are non-empty, disjoint, and satisfy cl$(B_{i})=B_{i}\cup V_{i}$ for $i=1,2$.

Let $\xi\in$ cl$(B_{i})\setminus\{B_{i}\}$. Then, there exists a sequence $\{b_{n}\}\subset B_{i}$ such that $b_{n}\to\xi$ as $n\to\infty$. An easy application of triangle inequality shows that $\overline{\pi}(b_{n})\to\xi$ as $n\to\infty$. This implies that cl$(B_{i})\subseteq B_{i}\cup V_{i}$. For the converse, let $\xi\in V_{1}$. Since $X$ is dense in $\overline{X}$, let $\{x_{n}\}$ be a sequence in $X$ such that $x_{n}\to\xi$ (in the original topology of $\overline{X}$). As the topology induced by $\overline{d}$ on $\partial X$ is same as the original topology on $\partial X$, $\{x_{n}\}\to\xi$ with respect to the metric $\overline{d}$. Suppose there exists a subsequence of $\{x_{n}\}$ contained in $B_{2}$. Then, $\xi\in$ cl$(B_{2})$, which in turn implies that $\xi\in V_{2}$. This gives a contradiction as $V_{1}$ and $V_{2}$ are disjoint. Hence, $\{x_{n}\}$ is eventually contained in $B_{1}$ and therefore, $\xi\in$ cl$(B_{1})$. Similarly, one can show that $V_{2}\subseteq$ cl$(B_{2})$. This also shows that $B_{1}$ and $B_{2}$ are non-empty disjoint subsets of $X$. Hence, the claim.

Since $V_{1}$ and $V_{2}$ are non-empty, let $\xi_{1}\in V_{1}$ and $\xi_{2}\in V_{2}$. As in the proof of the claim, there exist sequences $\{x_{n}\}$ and $\{y_{n}\}$ in $B_{1}$ and $B_{2}$, respectively, such that $x_{n}\to\xi_{1}$ and $y_{n}\to\xi_{2}$. Since closed $d$-balls in $X$ are compact, up to passing to subsequences, we can assume that $\{x_{n}\}$ and $\{y_{n}\}$ are unbounded in $X$. For every $m\in\mathbb{N}$, denote $K_{m}$ the closed $d$-metric ball of radius $m$ about a fixed base point $x_{0}\in X$. Since $X$ is one-ended, there exist subsequences $\{x_{n_{m}}\}$ and $\{y_{n_{m}}\}$ and a sequence of geodesics $\{\gamma_{m}\}$ joining $x_{n_{m}}$ and $y_{n_{m}}$ such that $\gamma_{m}$ is contained in $X\setminus K_{m}$. Let $[a_{m},b_{m}]$ be a subpath of $\gamma_{m}$ such that $a_{m}\in B_{1}$ and $b_{m}\in B_{2}$ and $d(a_{m},b_{m})\leq 1$ (such a subpath always exists as $B_{1}\cup B_{2}$ cover the whole $X$). Note that the sequence $\{a_{m}\}$ is unbounded in $X$. Since $\overline{X}$ is compact, there exists a subsequence $\{a_{m_{k}}\}$ that converges to $\eta\in\partial X$. Since $d(a_{m_{k}},b_{m_{k}})\leq 1$, it follows $b_{m_{k}}\to\eta$ in $\overline{X}$ as $\partial X$ is weakly visible that also $\{a_{m_{k}}\}\to\eta$ in the topology induced by the metric $\overline{d}$ ( since $\overline{d}$ induces the original topology on $\partial X$). Thus, $\eta\in V_{1}$. By the same logic, $\eta\in V_{2}$. This gives us a contradiction as $V_{1}$ and $V_{2}$ are disjoint. Hence, we have the desired result. ∎

Let $S_{g}^{n}$ be the connected orientable surface of genus $g$ and punctures $n$ such that $3g+n-3\geq 2$. Let Mod$(S_{g}^{n})$ and $\mathcal{T}(S_{g}^{n})$ denote the mapping class group and the Teichmüller space of $S_{g}^{n}$, respectively. For $\epsilon<\epsilon_{0}$, $\mathcal{T}_{\epsilon}(S_{g}^{n})$ denote a subset of $\mathcal{T}(S_{g}^{n})$ containing all those elements whose systole is at least $\epsilon$ (for constant $\epsilon$ and $\epsilon_{0}$, see [13, p.1]). By [13, Theorem 1], $\mathcal{T}_{\epsilon}(S_{g}^{n})$ is a manifold, and Mod$(S_{g}^{n})$ acts properly and cocompactly on $\mathcal{T}_{\epsilon}(S_{g}^{n})$. Recently, Hamenstädt introduced a $Z$-structure for every torsion free finite index subgroup of Mod$(S_{g}^{n})$ [13, Theorem 4]. In fact, this gives a compactification $\overline{\mathcal{T}(S_{g}^{n})}:=\mathcal{T}_{\epsilon}(S_{g}^{n})\cup X(S_{g}^{n})$ of $\mathcal{T}_{\epsilon}(S_{g}^{n})$. For our purpose, we do not need the full definition of the topology on $\overline{\mathcal{T}(S_{g}^{n})}$. Rather, we just need to know how a sequence of interior points converges to a point of $X(S_{g}^{n})$. This is Definition 4.2 in [13]. Now, we are ready to prove the following:

Lemma 3.2**.**

The $Z$-boundary $X(S_{g}^{n})$ is weakly visible.

Proof.

Let $d_{\mathcal{T}}$ denote the Teichmüller metric on $\mathcal{T}_{\epsilon}(S_{g}^{n})$. Let $\{X_{j}\}$ and $\{Y_{j}\}$ be two sequences in $\mathcal{T}_{\epsilon}(S_{g}^{n})$ such that $d_{\mathcal{T}}(X_{j},Y_{j})\leq K$ for some $K\geq 0$. Suppose $X_{j}\to\xi\in X(S_{g}^{n})$ as $j\to\infty$. We need to show that $Y_{j}\to\xi$ as $j\to\infty.$ For that, we check conditions $(1)$, $(2)$, and $(3)$ of [13, Definition 4.2]. The first condition is clear. For conditions (2) and (3), we use Lemma 3.19 and the idea of the proof of Lemma 3.20 of [4]. Using the distance formula [19, Theorem 6.1] and the fact that projections to subsurfaces are coarsely Lipschitz, we see that the distance between the projection of $X_{j}$ and $Y_{j}$ in the curve graphs of the subsurfaces is uniformly bounded. Then, the lemma follows from [4, Lemma 3.19]. ∎

Now, Theorem 3.1 implies the following:

Proposition 3.3**.**

The $Z$-boundary $X(S_{g}^{n})$ is connected. ∎

Since the hierarchical boundary of a proper HHS is weakly visible [4, Lemma 3.20], we immediately have the following:

Corollary 3.4**.**

The hierarchical boundary of a one-ended proper hierarchically hyperbolic space is connected. ∎

It is known that $\mathcal{T}(S_{g}^{n})$ with respect to either Teichmüller metric or Weil–Petersson metric is an HHS [6, Theorem G]. Since $\mathcal{T}(S_{g}^{n})$ is one-ended, we have the following:

Corollary 3.5**.**

The HHS boundary of $\mathcal{T}(S_{g}^{n})$ is connected. ∎

For hierarchically hyperbolic groups, we prove the converse of Corollary 3.4.

Theorem 3.6**.**

Let $(G,\mathfrak{S})$ be an HHS. The hierarchical boundary $\partial(G,\mathfrak{S})$ is connected if and only if $G$ is one-ended.

Proof.

Suppose $G$ is one-ended. Then, by Corollary 3.4, $\partial(G,\mathfrak{S})$ is connected. Conversely, suppose $\partial(G,\mathfrak{S})$ is connected. Suppose, if possible, $G$ is not one-ended. Then, there are the following two cases:

Case 1. Suppose $G$ is two-ended. Then, $G$ is virtually cyclic. Therefore, $G$ is hyperbolic and by [10, Theorem 4.3] $\partial(G,\mathfrak{S})$ has only two elements. Thus, this case is not possible.

Case 2. Suppose $G$ has infinitely many ends. Then, by [8, Chapter I, Theorem 8.32(5)], $G$ splits as a graph of groups over finite edge groups. If all the vertex groups are finite, then $G$ is virtually a non-Abelian free group, and hence $\partial(G,\mathfrak{S})$ is homeomorphic to the Cantor set. Since the Cantor set is not a connected space, at least one vertex group has to be infinite. Hence, $G$ is hyperbolic relative to infinite vertex groups [7]. Note that the Bowditch boundary of $G$ is disconnected as the coned-off Cayley graph of $G$ with respect to the infinite vertex groups is quasiisometric to the Bass–Serre tree of the splitting of $G$. Now, by Theorem 2.10, the Bowditch boundary of $G$ is a quotient of $\partial(G,\mathfrak{S})$. Thus, $\partial(G,\mathfrak{S})$ is disconnected. This gives us a contradiction. Hence, this case is also not possible. Since a finitely generated group, either one-ended, two-ended, or infinite-ended, by the above two cases, we see that $G$ is one-ended. ∎

It is well known that Mod$(S_{g}^{n})$ is a hierarchically hyperbolic group [5, Theorem 11.1]. The following is immediate from the above theorem.

Corollary 3.7**.**

For $3g+n-3\geq 2$, the HHG boundary of Mod$(S_{g}^{n})$ is connected.

Proof.

Since $3g+n-3\geq 2$, Mod$(S_{g}^{n})$ is neither a hyperbolic nor a relatively hyperbolic group. Hence, Mod$(S_{g}^{n})$ is one-ended. Thus, by Theorem 3.6, we are done. ∎

We end this section with the following remark.

Remark 3.8**.**

Let $G$ be a finitely generated group that is hyperbolic relative to a finite collection of subgroups $\mathcal{P}$. Let $\partial_{\text{rel}}(G)$ denote the Bowditch boundary of $G$ with respect to $\mathcal{P}$. Suppose $G$ is one-ended. Then, using the same idea of the proof of Theorem 3.6, one can show that $\partial_{\text{rel}}(G)$ is connected. This recovers the result of Bowditch, which says that $\partial_{\text{rel}}(G)$ is connected if $G$ does not split non-trivially over finite groups relative to parabolic subgroups.

4. Hierarchical boundaries of free products of HHGs

Throughout this section, we fix a free product $G=A\ast B$. When $A$ and $B$ are HHGs, this section aims to give a hierarchical structure of $G$. Using this structure, we then give a description of the hierarchical boundary of $G$. This description is crucial for proving Theorem 3 in the following sections.

4.1. A model space for a free product of HHGs

In this subsection, we associate a graph to the splitting of $G$ that is naturally quasiisometric to a Cayley graph of $G$. This construction can be extended in a straightforward way to free products of finitely many groups.

The definition of the Bass–Serre tree of a graph of groups is classical [21]. For completeness, we recall it for free products of groups. Let $\tau$ be a unit interval with vertices $v_{A}$ and $v_{B}$. We define a tree $T$, called the Bass–Serre tree of the splitting of $G$, as $G\times\tau$ divided by the transitive closure of the following relation $\sim$

[TABLE]

[TABLE]

Let $S_{A}$ and $S_{B}$ be generating sets of $A$ and $B$, respectively. Let $\Gamma_{A}$ and $\Gamma_{B}$ be the Cayley graphs of $A$ and $B$ with respect to $S_{A}$ and $S_{B}$, respectively. Define a graph $Y$ as the union of $\Gamma_{A},\Gamma_{B}$ and $\tau$, where $v_{A}$ is identified with the identity of $A$ and $v_{B}$ is identified with the identity of $B$. Define a graph $\Gamma$ as $G\times Y$ modulo an equivalence relation induced by

[TABLE]

[TABLE]

Thus, we obtain a tree of Cayley graphs $\Gamma\to T$, where the preimage of each vertex $v\in T$, called the vertex space corresponding to $v$, is isometric to the Cayley graph of the stabilizer $G_{v}$ of $v$ in $G$ ([15, Subsection 2.1]). For $v\in T$, we denote the vertex space corresponding to $v$ by $\Gamma_{v}$. The following are a few observations about $\Gamma$.

(1) There is a bijection between all the edges of $\Gamma$ connecting different Cayley graphs and the edges of $T$. We call them lifts of the corresponding edges of $T$.

(2) By collapsing lifts of all the edges of $T$ to points, we get a natural quotient map from $\Gamma$ to the Cayley graph of $G$ with respect to $S_{A}\cup S_{B}$ which is a $G$-equivariant quasiisometry. Also, the natural left action of $G$ on $\Gamma$ is geometric.

(3) Since $G$ is hyperbolic relative to $\{A,B\}$ [7], the graph $\Gamma$ is hyperbolic relative to $\{\Gamma_{v}:v\in T\}$.

4.2. HHG structure on $G$

Let $\mathfrak{S}_{A}$ and $\mathfrak{S}_{B}$ be HHG structures on $A$ and $B$, respectively. For $g\in G$, let $g\mathfrak{S}_{A}$ be a copy of $\mathfrak{S}_{A}$ with its associated hyperbolic spaces and projections in such a way that there is a hieromorphism (see [5]) $A\to gA$ equivariant with respect to the conjugation isomorphism $A\to A^{g}$. Similarly, one can put a hierarchical structure on the cosets of $B$ in $G$. For each vertex $v\in T$, we denote the hierarchical structure on $\Gamma_{v}$ by $\mathfrak{S}_{v}$. Since $\Gamma$ is hyperbolic relative to $\{\Gamma_{v}\}$ and $(\Gamma_{v},\mathfrak{S}_{v})$’s are hierarchically hyperbolic, by [5, Theorem 9.1], $\Gamma$ is an HHS. We denote this HHS structure on $\Gamma$ by $\mathfrak{S}$. This implies that $(G,\mathfrak{S})$ is an HHG. Now, we briefly explain the HHG structure $\mathfrak{S}$ on $\Gamma$.

Indexing set. Let $\hat{\Gamma}$ denote the graph obtained by coning-off each subspace $\Gamma_{v}$. Define

[TABLE]

Hyperbolic spaces. The hyperbolic space $\mathcal{C}\hat{\Gamma}$ for $\hat{\Gamma}$ is $\hat{\Gamma}$ itself, while the hyperbolic space for $U\in\mathfrak{S}_{v}$, for some $v$, was defined above.

Relations. The nesting, orthogonality, transversality relations on each $\mathfrak{S}_{v}$ are as above. If $U,V\in\mathfrak{S}_{v},\mathfrak{S}_{w}$, and $v\neq w$, then declare $U\pitchfork V$. Finally, for all $U\in\mathfrak{S}$, let $U\sqsubseteq\hat{\Gamma}$.

Projections. For $\hat{\Gamma}$, $\pi_{\hat{\Gamma}}:\Gamma\to\hat{\Gamma}$ is the inclusion which is coarsely surjective. For each $v\in T$, let $\pi_{v}$ denotes the nearest point projection of $\Gamma$ onto $\Gamma_{v}$. Then, for $U\in\mathfrak{S}_{v}$, define $\pi_{U}:=\pi_{U,v}\circ\pi_{v}$, where $\pi_{U,v}:\Gamma_{v}\to\mathcal{C}U$ is the projection in $(\Gamma,\mathfrak{S}_{v})$.

Relative projections. For $U\in\mathfrak{S}_{v}$, $\rho_{\hat{\Gamma}}^{U}$ is the cone-point corresponding to $\Gamma_{v}$. If $U,V\in\mathfrak{S}_{v}$ then the coarse maps $\rho_{U}^{V}$ and $\rho_{V}^{U}$ were already defined. If $U,V\in\mathfrak{S}_{u},\mathfrak{S}_{v}$ and $u\neq v$, then $\rho_{V}^{U}:=\pi_{V}(\pi_{v}(\Gamma_{u}))$ and $\rho_{U}^{V}:=\pi_{U}(\pi_{u}(\Gamma_{v}))$. Finally, if, for $U\in\mathfrak{S}_{v}$, $U\sqsubsetneq\hat{\Gamma}$ then $\rho_{U}^{\hat{\Gamma}}:\hat{\Gamma}\to\mathcal{C}U$ is defined as follows:

(i) If $x\in\Gamma$, then $\rho_{U}^{\hat{\Gamma}}(x):=\pi_{U}(x)$.

(ii) If $x$ is the cone-point over $\Gamma_{w}$ and $v\neq w$. Then, $\rho_{U}^{\hat{\Gamma}}(x):=\rho_{U}^{S_{w}}$, where $S_{w}$ is $\sqsubseteq$-maximal element of $\mathfrak{S}_{w}$. The cone-point over $\Gamma_{v}$ may be sent anywhere in $\mathcal{C}U$.

For $v\in T$, let $\hat{\Gamma_{v}}$ denote the coned-off graph obtained by coning-off $\Gamma_{v}$. We already have observed a one-one correspondence between the edges of $T$ and edges in $\Gamma$ connecting different Cayley graphs. We conclude this subsection by noting the following, whose proof is clear, and thus we omit it.

Lemma 4.1**.**

Let $\phi:\hat{\Gamma}\to T$ be a map that sends $\hat{\Gamma}_{v}$ to $v$, and the edges connecting different Cayley graphs to the corresponding edges of $T$. Then, $\phi$ is a continuous quasiisometry. ∎

4.3. Construction of the compactification

Suppose $(A,\mathfrak{S}_{A})$ and $(B,\mathfrak{S}_{B})$ are HHGs. Let $\mathfrak{S}$ be the hierarchical structure as described in Subsection 4.2. Here, we construct a compact metrizable space which turns out to be the hierarchical boundary of $(G,\mathfrak{S})$. For this, we follow the construction of Martin–Świ\polhkatkowski [15, Subsection 2.2], the only difference is that we are taking the hierarchical boundary in place of the Gromov boundary.

Boundaries of the stabilizers. Let $\delta_{Stab}(\Gamma)$ be the set $G\times(\partial(A,\mathfrak{S}_{A})\cup\partial(B,\mathfrak{S}_{B}))$ divided by the equivalence relation induced by

[TABLE]

[TABLE]

The equivalence class of an element $(g,\xi)$ is denoted by $[g,\xi]$. The set $\delta_{Stab}(\Gamma)$ comes with a natural action of $G$ on the left. This also comes with a natural projection onto the set of vertices of $T$, which sends the boundary of each vertex stabilizer to the vertex. The preimage of each vertex $v\in T$ is denoted by $\partial(\Gamma_{v},\mathfrak{S}_{v})$.

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

[TABLE]

Also, we define a set $\overline{\Gamma}$ (which will be called the compactification of $\Gamma$) as

[TABLE]

This set has a natural action of $G$ and a natural map $\pi_{T}:\overline{\Gamma}\to T\cup\partial T$, which sends $\Gamma\cup\delta_{Stab}(\Gamma)$ to $T$. The preimage of a vertex $v\in T$ is $\Gamma_{v}\cup\partial G_{v}$ that is identified as a set with $\overline{\Gamma_{v}}.$

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

(1) Let $\xi\in\delta_{Stab}(\Gamma)$. Suppose $v$ is the vertex of $T$ such that $\xi\in\partial(G_{v})$. Let $U$ be an open neighborhood of $\xi$ in $\overline{\Gamma_{v}}$. Define $V_{U}$ to be the set of all $z\in\overline{\Gamma}$ such that $\pi_{T}(z)\neq v$ and the first edge of the geodesic in $T$ from $v$ to $\pi_{T}(z)$ lifts to an edge of $\Gamma$ that is glued to a point of $U$. Then we set

[TABLE]

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

(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 $\partial(T_{n}{(\eta)})$ denote the Gromov boundary of $T_{n}{(\eta)}$, and let $\overline{T_{n}(\eta)}=T_{n}(\eta)\cup\partial(T_{n}(\eta))$. We define

[TABLE]

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

We skip a verification that the above collections of sets satisfy the axioms for the basis of open neighborhoods, for an idea of proof, one is referred to [14, Theorem 6.17]. We denote this topology by $\tau$ on $\delta(\Gamma)$. A proof of the following lemma is the same as [15, Lemma 3.3].

Lemma 4.2**.**

The space $(\delta(\Gamma),\tau)$ is Hausdorff.

4.4. Equivalence of two topologies on $(G,\mathfrak{S})$

From the hierarchical structure of $G$, as a set, we see that

[TABLE]

Let $\phi$ be the map as in Lemma 4.1. Hence, it induces a homeomorphism from $\partial\hat{\Gamma}\to\partial T$. Thus, we have a continuous map $\bar{\phi}:\hat{\Gamma}\cup\partial\hat{\Gamma}\to T\cup\partial T$. Let $\mathcal{T}$ be the topology $\overline{G}:=\Gamma\cup\partial(G,\mathfrak{S})$ as defined in Section 2. Define a natural map $\psi:\overline{G}\to\Gamma\cup\delta(\Gamma)$ in the following manner:

[TABLE]

Proposition 4.3**.**

$\psi$* is a homeomorphism.*

Proof.

Clearly, $\psi$ is a bijection. Since $\overline{G},\overline{\Gamma}$ are compact Hausdorff spaces, to show that $\psi$ is a homeomorphism, it is sufficient to show that $\psi$ is continuous. It is continuous on the points of $\Gamma$. Thus, we have the following two cases to consider:

Case 1. Let $p\in\partial G_{v}$ for some $v\in T$. Let $p=\sum_{S\in\overline{S}}a_{S}^{p}p_{S}$, where $\overline{S}$ is the support set of $p$ in $\mathfrak{S}_{v}$. Let $\epsilon>0$ and $U_{S}$ be a neighborhood of $p_{S}$ in $\partial\mathcal{C}S$ such that $\mathcal{B}_{\{U_{S}\},\epsilon}^{v}(p)$ is a neighborhood of $p$ in $\overline{\Gamma_{v}}$. For $U:=\mathcal{B}_{\{U_{S}\},\epsilon}^{v}(p)$, let $V_{U}(p)=U\cup V_{U}$ be a neighborhood of $p$ in $\overline{\Gamma}$. We show that the neighborhood $\mathcal{B}_{\{U_{S}\},\epsilon}(p)$ of $p$ in $\overline{G}$ satisfies $\psi(\mathcal{B}_{\{U_{S}\}}(p))=V_{U}(p)$.

Remote part. Note that $\mathcal{B}_{\{U_{S}\},\epsilon}^{rem,v}(p)\subset\mathcal{B}_{\{U_{S}\},\epsilon}^{rem}(p)$. Every point in $(\bigsqcup_{v\neq w\in T}\partial G_{w})\cup\partial\hat{\Gamma}$ is remote with respect to $\overline{S}$. From the definition of neighborhoods and the hierarchical structure of $\Gamma$, it follows that, for $w\neq v$, a point $q\in\partial G_{w}$ belongs to $\mathcal{B}_{\{U_{s}\},\epsilon}^{rem}(p)$ if and only the vertex of the lift of $e$ attached to $\Gamma_{v}$ belongs to $\mathcal{B}_{\{U_{S}\},\epsilon}^{int,v}(p)\subset U$, where $e$ is the first edge of the geodesic segment $[v,w]\subset T$. Hence, such $q\in\mathcal{B}_{\{U_{s}\},\epsilon}^{rem}(p)$ if and only $q\in V_{U}$. Similarly, a point $\xi\in\hat{\Gamma}$ is in $\mathcal{B}_{\{U_{s}\},\epsilon}^{rem}(p)$ if and only if the vertex of the lift of $e$ attached to $\Gamma_{v}$ belongs to $\mathcal{B}_{\{U_{S}\},\epsilon}^{int,v}(p)\subset U$, where $e$ is the first edge of the geodesic ray $[v,\phi(\xi))\subset T$. Hence, such $\xi\in\mathcal{B}_{\{U_{S}\},\epsilon}^{rem}(p)$ if and only $\phi(\xi)\in V_{U}$.

Non-remote part. Since non-remote points are in $\partial G_{v}$, it is clear that $\mathcal{B}_{\{U_{S}\},\epsilon}^{non}(p)=\mathcal{B}_{\{U_{S}\},\epsilon}^{non,v}(p)$.

Interior part. The interior part of $\mathcal{B}_{\{U_{S}\},\epsilon}^{v}(p)$ lies in the interior part of $\mathcal{B}_{\{U_{S}\},\epsilon}(p)$. Again, from the definition of neighborhoods and the hierarchical structure of $\Gamma$, it follows that, for $w\neq v$, a point $q\in\Gamma_{w}$ belongs to $\mathcal{B}_{\{U_{s}\},\epsilon}^{int}(p)$ if and only the vertex of the lift of $e$ attached to $\Gamma_{v}$ belongs to $\mathcal{B}_{\{U_{S}\},\epsilon}^{int,v}(p)\subset U$, where $e$ is the first edge of the geodesic segment $[v,w]\subset T$. Hence, such $q\in\mathcal{B}_{\{U_{s}\},\epsilon}^{int}(p)$ if and only $q\in V_{U}$.

From the above discussion, it follows that $\psi(\mathcal{B}_{\{U_{S}\},\epsilon}(p))=V_{U}(p)$ and hence $\psi$ is continuous at $p$.

Case 2. Let $\xi\in\partial\hat{\Gamma}$ and let $\bar{\phi}(\xi)=\eta\in\partial T$. Let $V_{n}(\eta)$ be a neighborhood of $\eta$ in $\overline{\Gamma}$. Choose a neighborhood $U\subset\hat{\Gamma}\cup\partial\hat{\Gamma}$ such that $\bar{\phi}(U)\subseteq T_{n}(\eta)$. Note that Supp$(\xi)=\{\hat{\Gamma}\}$ and each point in $\bigsqcup_{v\in T}\partial G_{v}$ is a remote point with respect to $\{\hat{\Gamma}\}$. We use $U$ to construct the required neighborhood of $\xi$ in $(\overline{G},\mathcal{T})$. In each of the following parts, the conditions involving $\epsilon$ are vacuous. Thus, we remove the dependency of the neighborhood on $\epsilon$.

Remote part. Note that, for each $v\in T$, each domain in $\mathfrak{S}_{v}$ is nested in $\hat{\Gamma}$ and each element in $\partial(G_{v},\mathfrak{S}_{v})$ is remote with respect to $\{\hat{\Gamma}\}$. Also, for $V\in\mathfrak{S}_{v}$, $\rho_{\hat{\Gamma}}^{V}$ is the cone-point corresponding to $\Gamma_{v}$. Thus, the remote part

[TABLE]

Non-remote part. Elements in $\partial\hat{\Gamma}$ are the only non-remote points in $\partial(G,\mathfrak{S})$. Thus, the non-remote part

[TABLE]

Interior part. $\mathcal{B}_{U}^{int}(\xi)=\{x\in\Gamma:x\in U\}.$

Let $\mathcal{B}_{U}(\xi)=\mathcal{B}_{U}^{rem}(\xi)\cup\mathcal{B}_{U}^{non}(\xi)\cup\mathcal{B}_{U}^{int}(\xi)$. Then, from the definition of the neighborhood in $(\overline{\Gamma},\tau)$, we see that $\psi(\mathcal{B}_{U}(\xi))\subseteq V_{n}(\eta)$. Thus, $\psi$ is continuous at $\xi$. ∎

5. Homeomorphism types of hierarchical boundaries of free products

Suppose $(G_{1},\mathfrak{S}_{1}),(G_{2},\mathfrak{S}_{2})$ are two HHGs, and $\Gamma_{1},\Gamma_{2}$ are Cayley graphs of $G_{1},G_{2}$, respectively. By Theorem 2.8, $\overline{\Gamma_{1}}=\Gamma_{1}\cup\partial(G_{1},\mathfrak{S}_{1}),\overline{\Gamma_{2}}=\Gamma_{2}\cup\partial(G_{2},\mathfrak{S}_{2})$ are compact metrizable. Then, one has induced metrics $d_{\triangle_{1}}$ and $d_{\triangle_{2}}$ on $G_{1}\cup\partial(G_{1},\mathfrak{S}_{1})$ and $G_{2}\cup\partial(G_{2},\mathfrak{S}_{2})$, respectively. The following lemma is an analogue of [15, Lemma 4.2] in the context of HHGs, which helps to prove Theorem 5.3. In particular, this lemma helps us to define an isomorphism between Bass–Serre trees of free products given in Theorem 5.3.

Lemma 5.1**.**

Let $\partial f:\partial(G_{1},\mathfrak{S}_{1})\to\partial(G_{2},\mathfrak{S}_{2})$ be a homeomorphism. Then, there is a bijection (need not be a homomorphism) $f:G_{1}\to G_{2}$ such that $f(1)=1$ and $f\cup\partial f:G_{1}\cup\partial(G_{1},\mathfrak{S}_{1})\to G_{2}\cup\partial(G_{2},\mathfrak{S}_{2})$ is a homeomorphism.

Proof.

For $i=1,2$, we write $\partial G_{i}$ in place of $\partial(G_{i},\mathfrak{S}_{i})$. Let $\pi:G_{1}\to\partial G_{1}$ as defined in Equation (1). Order the elements of $G_{1}\setminus\{1\}$ and $G_{2}\setminus\{1\}$ into sequences $\{g_{k}\}_{k\in\mathbb{N}}$ and $\{h_{k}\}_{k\in\mathbb{N}}.$ Define $f(1)=1$. To get the required $f$, iterate the following two steps alternatively.

Step 1. Suppose $k$ is the smallest number for which $f(g_{k})$ is not yet defined. Since $G_{2}$ is dense in $G_{2}\cup\partial G_{2}$, choose some $l\in\mathbb{N}$ such that $h_{l}$ is not an image of any $g_{i}$ under the map $f$ and

[TABLE]

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

Step 2. Suppose $k$ is the smallest number for which $h_{k}$ is not chosen as the image of any $g\in G_{1}$ under $f$. Since, by Lemma 2.9, $\pi(G_{1})$ is dense in $\partial G_{1}$, $\partial f(\pi(G_{1}))$ is dense in $\partial(G_{2})$. Choose $g\in G_{1}\setminus\{1\}$ such that $f$ has not yet been defined on $g$ and

[TABLE]

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

By performing the above two steps alternatively, we see that $f$ is a bijection. Since $G_{1}\cup\partial G_{1}$ and $G_{2}\cup\partial G_{2}$ are compact and Hausdorff, to prove that $f\cup\partial f$ is a homeomorphism, it is sufficient to prove that $f\cup\partial f$ is continuous.

Clearly, $f\cup\partial f$ is continuous at the points of $G_{1}$. Now, consider a sequence $\{g_{k}\}$ in $G_{1}$ that converges to $\xi\in\partial G_{1}$ (in the original topology on $\partial G_{1}$). Since the topology induced by $d_{\triangle_{1}}$ on $\partial G_{1}$ is same as the original topology on $\partial G_{1}$, $\lim_{k\to\infty}d_{\triangle_{1}}(g_{k},\xi)=0$. This, in turn, implies that $\lim_{k\to\infty}d_{\triangle_{1}}(g_{k},\partial G_{1})=0$. Thus, $\lim_{k\to\infty}d_{\triangle_{1}}(g_{k},\pi(g_{k}))=0$ by the definition of $\pi$. Using triangle inequality, we see that $\pi(g_{k})$ converges to $\xi$. Thus, by continuity of $\partial f$, $\partial f(\pi(g_{k}))$ converges to $f(\xi)$. From the definition of $f$, it follows that $\lim_{k\to\infty}d_{\triangle_{2}}(\partial f(\pi(g_{k})),f(g_{k}))=0$. This implies that $f(g_{k})$ converges to $f(\xi)$. Hence, $f\cup\partial f$ is continuous. ∎

We immediately have the following:

Corollary 5.2**.**

Let $\partial f:\partial G_{1}\to\partial G_{2}$ be a homeomorphism. Then, there exists a bijection $f:G_{1}\to G_{2}$ such that the following holds:

Let $\xi\in\partial G_{2}$ and $U_{2}$ be an open neighborhood of $\xi$ in $\overline{\Gamma_{2}}$. Then, there exists a neighborhood $U_{1}$ of $(\partial f)^{-1}(\xi)$ in $\overline{\Gamma_{1}}$ such that $f(G\cap U_{1})\subset U_{2}$.

Proof.

Let $f$ be the bijection as in Lemma 5.1 such that $\bar{f}:=f\cup\partial f:G_{1}\cup\partial G_{1}\to G_{1}\cup\partial G_{2}$ is a homeomorphism. We show that $f$ is the required bijection. Since $U_{2}$ is an open neighborhood of $\xi\in\partial G_{2}$, $V:=\bar{f}^{-1}(U_{2})$ is an open neighborhood of $(\partial f)^{-1}(\xi)$ in $G_{1}\cup\partial G_{1}$. Thus, $(G_{1}\cup\partial G_{1})\setminus V$ is closed in $G_{1}\cup\partial G_{1}$ and $G_{1}\cup\partial G_{1}$ is closed in $\overline{\Gamma_{1}}$, $K:=\overline{\Gamma_{1}}\setminus V$ is compact in $\overline{\Gamma_{1}}$. Since $(\partial f)^{-1}(\xi)\in\overline{\Gamma_{1}}\setminus K$, there exists an open neighborhood $U_{1}$ of $f^{-1}(\xi)$ in $\overline{\Gamma_{1}}$ such that $U_{1}$ is disjoint from $K$, and hence $U_{1}\subset\bar{f}^{-1}(U_{2})$. Now, it is clear that $f(G_{1}\cap U_{1})\subset U_{2}$. ∎

The remainder of this section is devoted to proving the following theorem.

Theorem 5.3**.**

Suppose $G_{1}=A_{1}\ast B_{1}$ and $G_{2}=A_{2}\ast B_{2}$ are two free products of HHGs. For $i=1,2$, let $\mathfrak{S}_{A_{i}}$ and $\mathfrak{S}_{B_{i}}$ be HHG structures on $A_{i}$ and $B_{i}$, respectively. Let $\mathfrak{S}_{1}$ and $\mathfrak{S}_{2}$ be HHG structures on $G_{1}$ and $G_{2}$ as described in Subsection 4.2. If $\partial(A_{1},\mathfrak{S}_{A_{1}})$ is homeomorphic to $\partial(A_{2},\mathfrak{S}_{B_{2}})$ and $\partial(B_{1},\mathfrak{S}_{B_{1}})$ is homeomorphic to $\partial(B_{2},\mathfrak{S}_{B_{2}})$, then $\partial(G_{1},\mathfrak{S}_{1})$ is homeomorphic to $\partial(G_{2},\mathfrak{S}_{2})$.

Notation: Let $\partial f_{1}:\partial A_{1}\to\partial A_{1}$ and $\partial f_{2}:\partial B_{1}\to\partial B_{2}$ be the fixed homeomorphism of the hierarchical boundaries. We denote by $q:\partial A_{1}\cup\partial B_{1}\to\partial A_{2}\cup\partial B_{2}$ the homeomorphism induced by $\partial f_{1}$ and $\partial f_{2}$. Let $f_{1}$ and $f_{2}$ be the bijections provided by Lemma 5.1. Let $T_{1}$ and $T_{2}$ be the Bass–Serre trees of $A_{1}\ast B_{1}$ and $A_{2}\ast B_{2}$, respectively. We denote by $\Gamma_{1}$ and $\Gamma_{2}$ the trees of Cayley graphs of $G_{1}$ and $G_{2}$, respectively as described in Subsection 4.1. Finally, $\delta(\Gamma_{1})$ and $\delta(\Gamma_{2})$ denote the spaces as constructed in Subsection 4.3.

Isomorphism between $T_{1}$ and $T_{2}$. 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 $f_{1}(a_{1})f_{2}(b_{1}),\dots,f_{1}(a_{n})f_{2}(b_{n})\tau_{2}$. It is easy to check that $\iota$ is an isomorphism such that $\iota(\tau_{1})=\tau_{2}$.

Note that if $\xi\in\partial A_{1}$ and $g\in G_{1}$, then the set of all representatives of $[g,\xi]\in\delta_{Stab}(\Gamma_{1})$ is of the form $(ga,a^{-1}\xi)$ where $a\in A_{1}$. Similarly, if $\xi\in\partial B_{1}$ and $g\in G$ then the set of all representatives are of the form $(gb,b^{-1}\xi)$ for $b\in B_{1}$. When $\xi\in\partial A_{1}$, we choose a unique $ga=a_{1}b_{1},...,a_{n}b_{n}$ for which $n$ is the smallest possible (in this case we have $b_{n}\neq 1$). When $\xi\in\partial B_{1}$, we choose a unique $gb=a_{1}b_{1},...,a_{n}b_{n}$ for which $b_{n}=1$. These representatives of an element $[g,\xi]\in\delta_{Stab}(\Gamma_{1})$ are called reduced representatives.

Proof of Theorem 5.3. To prove that $\partial G_{1}$ is homeomorphic to $\partial G_{2}$, it is sufficient to prove that $\delta(\Gamma_{1})$ is homeomorphic to $\delta(\Gamma_{2})$. We define a map $F:\delta(\Gamma_{1})\to\delta(\Gamma_{2})$ in the following manner:

(1) Let $[g,\xi]\in\delta_{Stab}(\Gamma_{1})$ and let $(g,\xi)$ be its reduced representative. Write $g=a_{1}b_{1},\dots,a_{n}b_{n}$. Define

[TABLE]

(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

[TABLE]

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

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

Claim: The map $F$ is a homeomorphism.

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 $\xi\in\delta_{Stab}(\Gamma_{1})$ and let $v$ be the vertex of $T_{1}$ such that $\xi\in\partial G_{v}$. Let $U_{2}$ be an open neighborhood of $F(\xi)$ in $\overline{\Gamma_{\iota{(v)}}}$. By Corollary 5.2, we have an open neighborhood $U_{1}\subset\overline{\Gamma_{v}}$ of $\xi$. Then, from the definition of neighborhoods in $\overline{\Gamma_{1}}$ and in $\overline{\Gamma_{2}}$, it follows that $F(V_{U_{1}}(\xi)\cap\delta(\Gamma_{1}))\subset V_{U_{2}}(F(\xi))\cap\delta(\Gamma_{2})$.

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 a subtree of $T_{1}$ with respect to the base vertex $v_{1}$. Again, from the definition of neighborhoods, it follows that $F(V_{n}(\eta)\cap\delta(\Gamma_{1}))=V_{n}(F(\eta))\cap\delta(\Gamma_{2})$.

This completes the proof of the claim, and hence the theorem. ∎

A straightforward generalization of Theorem 5.3 to the free product of finitely many HHGs gives the following:

Theorem 5.4**.**

For $n\geq 2$, let $G_{1}=A_{1}\ast,\dots,\ast A_{n}$ and $G_{2}=B_{1}\ast,\dots,\ast B_{n}$ be free products of HHGs. For $1\leq i\leq n$, let $\mathfrak{S}_{A_{i}}$ and $\mathfrak{S}_{B_{i}}$ be HHG structures on $A_{i}$ and $B_{i}$, respectively. Suppose $\mathfrak{S}_{1}$ and $\mathfrak{S}_{2}$ are HHG structures on $G_{1}$ and $G_{2}$ as described in Subsection 4.2. If, for $1\leq i\leq n$, $\partial(A_{i},\mathfrak{S}_{A_{i}})$ is homeomorphic to $\partial(B_{i},\mathfrak{S}_{B_{i}})$, then $\partial(G_{1},\mathfrak{S}_{1})$ is homeomorphic to $\partial(G_{2},\mathfrak{S}_{2})$.

6. Applications

Suppose $G=A\ast B$ where $(A,\mathfrak{S}_{A})$ and $(B,\mathfrak{S}_{B})$ are HHGs. Then, $(G,\mathfrak{S})$ is an HHG with the hierarchical structure $\mathfrak{S}$ described in Subsection 4.2. Corresponding to $G$, let $\delta(\Gamma)$ be the space constructed in Subsection 4.3. Using Theorem 3.6, the following proposition describes the connected components of $\delta(\Gamma)$ whose proof is the same as the proof of Proposition 6.3 and Proposition 6.4 from [15]. Hence, we skip its proof.

Proposition 6.1**.**

We have the following:

1. (1)

Let $T$ be the Bass–Serre tree of $G$. Then, a point $\eta\in\partial T$ is its own connected component in $\delta(\Gamma)$. 2. (2)

Suppose $A$ and $B$ are one-ended groups. Then, for each vertex $v\in T$, $\partial G_{v}$ is a connected component of $\delta(\Gamma)$.

Now, we are ready to prove the following:

Theorem 6.2**.**

Suppose $G_{1}$ and $G_{2}$ satisfy the hypotheses of Theorem 5.3. Additionally, assume that $A_{i}$ and $B_{i}$ are one-ended groups for $i=1,2$. Then, $\partial(G_{1},\mathfrak{S}_{1})$ is homeomorphic to $\partial(G_{2},\mathfrak{S}_{2})$ if and only if $\partial(A_{1},\mathfrak{S}_{A_{1}})$ is homeomorphic to $\partial(B_{1},\mathfrak{S}_{B_{1}})$ and $\partial(A_{2},\mathfrak{S}_{A_{2}})$ is homeomorphic to $\partial(B_{2},\mathfrak{S}_{B_{2}})$.

Proof.

Let $\delta(\Gamma_{1})$ and $\delta(\Gamma_{2})$ be the spaces as constructed in Subsection 4.3. Suppose $\partial(G_{1},\mathfrak{S}_{1})$ is homeomorphic to $\partial(G_{2},\mathfrak{S}_{2})$. This implies that $\delta(\Gamma_{1})$ is homeomorphic to $\delta(\Gamma_{2})$. As a homeomorphism maps connected components to connected components, we see that $\partial(A_{1},\mathfrak{S}_{A_{1}})$ is homeomorphic to $\partial(B_{1},\mathfrak{S}_{B_{1}})$ and $\partial(A_{2},\mathfrak{S}_{A_{2}})$ is homeomorphic to $\partial(B_{2},\mathfrak{S}_{B_{2}})$. The converse is the content of Theorem 5.3. ∎

Theorem 6.2 can be generalised in a straightforward manner for free products of finitely many HHGs.

Theorem 6.3**.**

For $n\geq 2$, suppose $G_{1}=A_{1}\ast\dots\ast A_{n}$ and $G_{2}=B_{1}\ast\dots\ast B_{n}$, where $A_{i}$ and $B_{i}$ are one-ended hierarchically hyperbolic groups for all $i$. Suppose $G_{1}$ and $G_{2}$ have hierarchical structures as described in Subsection 4.2. Then, the hierarchical boundary of $G_{1}$ is homeomorphic to the hierarchical boundary of $G_{2}$ if and only if the hierarchical boundary of $A_{i}$ is homeomorphic to the hierarchical boundary of $B_{i}$ for all $1\leq i\leq n$.

By combining Theorem 5.3 and Theorem 6.2, we immediately have the following corollary:

Corollary 6.4**.**

Let $G$ be a one-ended group. Suppose $\mathfrak{S}_{1}$ and $\mathfrak{S}_{2}$ are two hierarchical structures on $G$. Let $\mathfrak{S}$ and $\mathfrak{S}^{\prime}$ be the hierarchical structures on $(G,\mathfrak{S}_{1})\ast(G,\mathfrak{S}_{1})$ and $(G,\mathfrak{S}_{2})\ast(G,\mathfrak{S}_{2})$, respectively, as described in Subsection 4.2. Then, $\partial(G,\mathfrak{S}_{1})$ is homeomorphic to $\partial(G,\mathfrak{S}_{2})$ if and only if $\partial(G\ast G,\mathfrak{S})$ is homeomorphic to $\partial(G\ast G,\mathfrak{S}^{\prime})$.

6.1. Locally quasiconvex HHGs

In [5], the authors introduce the notion of hierarchical quasiconvexity in HHSs.

Definition 6.5**.**

([5, Definition 5.1]) Let $(\mathcal{X},\mathfrak{S})$ be an HHS, and $k:[0,\infty)\to[0,\infty)$ be a map. A subset $\mathcal{Y}$ of $\mathcal{X}$ is said to be $k$-hierarchically quasiconvex if the following hold:

1. (1)

For each $U\in\mathfrak{S}$, $\pi_{U}(\mathcal{Y})$ is $k(0)$-quasiconvex subset of $\mathcal{C}U$. 2. (2)

If $x\in\mathcal{X}$ satisfies $d_{U}(x,\mathcal{Y})\leq r$ for each $U\in\mathfrak{S}$, then $d_{\mathcal{X}}(x,\mathcal{Y})\leq k(r)$.

The subspace $\mathcal{Y}$ is said to be hierarchically quasiconvex (HQC) if it is $k$-hierarchically quasiconvex for some $k:[0,\infty)\to[0,\infty)$. A subgroup $H$ of an HHG $(G,\mathfrak{S})$ is hierarchically quasiconvex if $H$ is a hierarchically quasiconvex subset of $G$ equipped with a finitely generated word metric.

The definition of an HQC subgroup does not depend on the choice of a finite generating set of the ambient group [20, Proposition 5.7]. Also, an HQC subgroup of an HHG $(G,\mathfrak{S})$ is finitely generated and undistorted [3, Lemma 2.10].

Definition 6.6**.**

Let $(G,\mathfrak{S})$ be an HHG. We say that $G$ is locally hierarchically quasiconvex if every finitely generated subgroup of $G$ is hierarchically quasiconvex.

We suspect that, under natural conditions on the HHG structure, locally HQC HHGs are hyperbolic and locally quasiconvex. We plan to explore this in future work.

In this subsection, we prove a combination theorem for a free product of locally HQC.

Theorem 6.7**.**

Let $G=A\ast B$, where $(A,\mathfrak{S}_{A})$ and $(B,\mathfrak{S}_{B})$ are locally hierarchically quasiconvex HHGs. Then, with respect to the HHG structure described in Subsection 4.2, $G$ is locally HQC.

Proof.

Let $\Gamma$ be the graph constructed in Subsection 4.1 and $H$ be a finitely generated subgroup of $G$. Then, $H$ has an induced graph of groups structure whose vertex groups are the intersection of $H$ and the conjugates of $A$ or $B$ in $G$ (some vertex groups may be trivial), and the edge groups are trivial. We continue to use the same notation as in Section 4. Since $\Gamma_{v}$’s are isometric to the Cayley graphs of $G_{v}$’s, we can realize $H$ as a subset of $\Gamma$ in the following manner:

Let $T_{H}\subset T$ be the Bass–Serre tree of the graph of groups decomposition of $H$. Note that, for each vertex $v\in T_{H}$, the stabilizer $H_{v}$ of $v$ in $H$ is $H\cap G_{v}$. Thus, we can see $H_{v}$ as a subset of $\Gamma_{v}$. In this way, we realize $H$ as a subset of $\Gamma$.

We show that $H$ is an HQC subset of $\Gamma$ which shows that $H$ is an HQC subgroup of $G$. Since $H$ is finitely generated, $H_{v}$ is finitely generated for each vertex $v\in T_{H}$. Hence, $H_{v}$’s are HQC subsets of $(\Gamma_{v},\mathfrak{S}_{v})$’s. As there are finitely many vertex groups in the graph of groups decomposition of $H$, we can assume that $H_{v}$’s are $k$-HQC for some $k:[0,\infty)\to[0,\infty)$.

Condition (1) of Definition 6.5. From the hierarchical structure on $\Gamma$, it follows that, for $v\in T_{H}$ and $U\in\mathfrak{S}_{v}$, $\pi_{U}(H)=\pi_{U}(H_{v})$. Since $H_{v}$ is $k$-HQC, we see that $\pi_{U}(H)$ is $k(0)$-quasiconvex subset of $\mathcal{C}U$ for each $v\in T_{H}$ and all $U\in\mathfrak{S}_{v}$. If $v\in T\setminus T_{H}$, then, for all $U\in\mathfrak{S}_{v}$, $\pi_{U}(H)$ is a subset of diameter $E$ in $\mathcal{C}U$, where $E$ is a constant in the HHS structure on $\Gamma$. Clearly, $\pi_{\hat{\Gamma}}(H)$ is $C$-quasiconvex in $\hat{\Gamma}$ for some $C\geq 0$. By redefining the function $k$ at [math] as max$\{k(0),E,C\}$, we see that for all $U\in\mathfrak{S}$, $\pi_{U}(H)$ is $k(0)$-quasiconvex in $\mathcal{C}U$.

Condition (2) of Definition 6.5. Let $x\in\Gamma$ such that $d_{U}(x,H)\leq r$ for each $U\in\mathfrak{S}$. If $x\in H$, then there is nothing to prove. We consider the following two cases:

Case 1. Suppose $x\notin H$ but $x\in\Gamma_{w}$ for some vertex $w\in T_{H}$. By our assumption, $d_{U}(x,H_{w})=d_{U}(x,H)\leq r$ for all $U\in\mathfrak{S}_{w}$. However, $H_{w}$ is $k$-HQC in $G_{w}$. This implies that $d_{\Gamma_{w}}(x,H_{w})\leq k(r)$. Since $\Gamma_{w}$ is isometrically embedded in $\Gamma$, $d_{\Gamma}(x,H_{w})=d_{\Gamma}(x,H)\leq k(r)$.

Case 2. Suppose $x\notin\Gamma_{w}$ for all $w\in T_{H}$. Suppose $x\in\Gamma_{u}$ for some vertex $u\in T\setminus T_{H}$. Let $w\in T_{H}$ be the closest vertex to $u$ and let $y$ be the point of $\Gamma_{w}$ to which the first edge of the geodesic $[w,u]\subset T$ is attached. Then, from the HHS structure on $\Gamma$, it follows that $d_{U}(x,H)=d_{U}(y,H_{w})$ for all $U\in\mathfrak{S}_{w}$. Since $H_{w}$ is $k$-HQC in $G_{w}$, $d_{\Gamma_{w}}(y,H_{w})\leq k(r)$ and hence $d_{\Gamma}(y,H_{w})\leq k(r)$. Observer that, for all $u\neq v\in T$ and $U\in\mathfrak{S}_{v}$, $d_{U}(x,G_{w})=0$. For $U\in\mathfrak{S}_{u}$, $d_{U}(x,G_{w})=d_{U}(x,H)\leq r$. Similarly, $d_{\hat{\Gamma}}(x,G_{w})=d_{\hat{\Gamma}}(x,H)\leq r$. Since $G_{w}$ is $k_{1}$-HQC in $G$ for some $k_{1}:[0,\infty)\to[0,\infty)$ [20, Theorem 1.2], $d_{\Gamma}(x,G_{w})\leq k_{1}(r)$. However, $d_{\Gamma}(x,G_{w})=d_{\Gamma}(x,y)$. Thus, using triangle inequality, we see that $d_{\Gamma}(x,H)=d_{\Gamma}(x,H_{w})\leq k(r)+k_{1}(r)$. By redefining the function $k$ at $r$ as $k(r)+k_{1}(r)$, we are done. ∎

By induction and using Theorem 6.7, we have the following general version of the previous theorem.

Theorem 6.8**.**

For $n\geq 2$, let $G=A_{1}\ast\dots\ast A_{n}$ be a free product of locally hierarchically quasiconvex HHGs. Then, with respect to the HHG structure described in Subsection 4.2, $G$ is locally HQC. ∎

Acknowledgements. I want to express my deepest gratitude to Mark Hagen for reviewing the draft version of this note and pointing out an error. I am also sincerely thankful to both Jason Behrstock and Mark Hagen for their encouragement and support during the writing of this note. Finally, I would like to thank the referee for encouraging me to write down the main result for the compactification of one-ended proper geodesic metric spaces.