[columns=3, title=Alphabetical Index, intoc]

# Sublinearly Morse Boundary of $\operatorname{CAT}(0)$ admissible groups

Hoang Thanh Nguyen  Department of Mathematics
The University of Danang - University of Science and Education
459 Ton Duc Thang, Da Nang, Vietnam<EMAIL_ADDRESS>and  Yulan Qing
Shanghai Center for Mathematical Sciences, Fudan University, Songhu Road 2005,
Shanghai, China<EMAIL_ADDRESS>
###### Abstract.

We show that if $G$ is an admissible group acting geometrically on a
$\operatorname{CAT}(0)$ space $X$, then $G$ is a hierarchically hyperbolic
space and its $\kappa$-Morse boundary $(\partial_{\kappa}G,\nu)$ is a model
for the Poisson boundary of $(G,\mu)$ where $\nu$ is the hitting measure
associated to the random walk driven by $\mu$( with mild assumptions of
$\mu$).

##  1\. Introduction

Sublinearly Morse boundaries are recently constructed for all finitely
generated groups [QRT19, QRT20]. It is a metrizable topological space that is
a group invariant. Similar Gromov boundary of hyperbolic spaces, sublinearly
Morse boundaries are particularly illuminating in revealing features of groups
that contains hyperbolic-like features [IZ21, MQZ20, QZ21, Zal20]. One of
these feature centers around asymptotic behavior of random walk on the
associated groups. In several important classes of groups, such right-angled
Artin groups, relative hyperbolic groups, and the mapping class groups of
surfaces of finite type, an appropriately chosen sublinear function yields a
sublinearly Morse boundary that serves as a topological model for the Poisson
boundaries(with mild assumptions) of the group.

In this paper we continue to prove the connection between a geometric boundary
and random walk boundary for a new class of groups: $\operatorname{CAT}(0)$
admissible groups.

$\operatorname{CAT}(0)$ admissible groups is a particular type of graph of
groups that generalizes graph manifolds. A compact, orientable, irreducible
3-manifold with empty or tori boundary is called a graph manifold if it is
obtained by gluing finitely many Seifert manifolds with orientable hyperbolic
base orbifolds where the gluing maps between the Seifert components do not
identify (unoriented) Seifert fibers up to homotopy. $\operatorname{CAT}(0)$
admissible group was first introduced by Croke-Kleiner [CK02] as key
counterexamples to the nice properties of Gromov boundary when the
hyperbolicity assumption is dropped. This property is further studied in
[Qin16].

Roughly speaking, a $\operatorname{CAT}(0)$ admissible group models the JSJ
structure of nonpositively curved graph manifolds where the Seifert fibration
is replaced by the following central extension of a general hyperbolic group:

(1) $1\to Z(G_{v})=\mathbb{Z}\to G_{v}\to H_{v}\to 1$

In this paper, we also refer to $\operatorname{CAT}(0)$ admissible groups as
CKA groups (CKA stands for Croke-Kleiner admissible). This class of groups
serves as a simple algebraic means to produce interesting groups from an
arbitrary finite collection of $n$ $\operatorname{CAT}(0)$ hyperbolic groups.

A basic example. Let $H_{1}$ and $H_{2}$ be two torsion-free hyperbolic groups
which act geometrically on $\operatorname{CAT}(0)$ spaces $X_{1}$ and $X_{2}$
respectively. Then $G_{i}=H_{i}\times\langle t_{i}\rangle$ (with $i=1,2$) acts
geometrically on the $\operatorname{CAT}(0)$ space
$Y_{i}=X_{i}\times\mathbb{R}$. Any primitive hyperbolic element $h_{i}$ in
$H_{i}$ gives a totally geodesic torus $T_{i}$ in the quotient space
$Y_{i}/G_{i}$ with basis $([h_{i}],[t_{i}])$. We rescale $Y_{i}$ so that the
lengths of $[h_{1}]$ and of $[t_{2}]$ are the same, and the lengths of
$[t_{1}]$ and $[h_{2}]$ are the same. Let $f\colon T_{1}\to T_{2}$ be an
isometry that flips the $\mathbb{Z}$-axis with the axis of an loxodromic
element in $H_{i}$. Let $M$ be the space obtained by gluing $Y_{1}$ to $Y_{2}$
by the isometry $f$. One can show that there exists a metric on $M$ such that
with respect to this metric, $M$ is a locally $\operatorname{CAT}(0)$ space
(see e.g. [BH99, Proposition 11.6]). By the Cartan-Hadamard Theorem, then the
universal cover $\widetilde{M}$ with the induced length metric from $M$ is a
$\operatorname{CAT}(0)$ space, and thus $\pi_{1}(M)$ is a
$\operatorname{CAT}(0)$ admissible group and
$\pi_{1}(M)\curvearrowright\widetilde{M}$ is a $\operatorname{CAT}(0)$ action.

######  Theorem A.

Let $G$ be a $\operatorname{CAT}(0)$ admissible group. The following holds.

  1. (1)

$G$ is a hierarchically hyperbolic space with respect to a word metric.

  2. (2)

Let $\mu$ be a finitely supported, non-elementary probability measure on $G$.
Then for $\kappa(t)=\log^{p}(t)$, where $p$ is the complexity of the HHS. Then
almost every sample path $(w_{n})$ converges to a point in
$\partial_{\kappa}G$; and the $\kappa$-Morse boundary
$(\partial_{\kappa}G,\nu)$ is a model for the Poisson boundary of
$(G,\mu)$where $\nu$ is the hitting measure associated to the random walk
driven by $\mu$.

The key step to show Theorem A involves proving that $\operatorname{CAT}(0)$
admissible groups acts geometrically on hierarchically hyperbolic spaces.
Hierarchically hyperbolic spaces are introduced axiomatically, modeling the
Masur–Minsky deconstruction of mapping class groups [MM00]. A hierarchically
hyperbolic space (HHS) consists of a metric space $X$ together with a
partially ordered set of $\delta$-hyperbolic spaces. There are projections
maps from $X$ to each of the hyperbolic spaces and from the hyperbolic spaces
to each other. In some sense, this set of hyperbolic spaces form a tree-like
metric that can be used to measure distances in $X$. Many interesting groups
are quasi-isometric to hierarchically hyperbolic spaces, examples of classes
of groups include but are not limited to:

  * •

hyperbolic groups and relative hyperbolic groups

  * •

Mapping class groups of orientable surfaces of finite type.

  * •

Fundamental groups of compact three-manifolds with no Nil or Sol in their
prime decomposition.

  * •

Groups that act properly and cocompactly on a special $\operatorname{CAT}(0)$
cube complex, and more generally any cubical group which admits a factor
system.

It is established in [BHS19] that all graph manifolds are a hierarchically
hyperbolic space. To show this, we have Kapovich-Leeb [KL98] to show that all
graph manifolds are quasi-isometric to flip graph manifolds, and the
combination theorem of [BHS19] shows that the fundamental group of a flip
graph manifold is HHS. When we lose the manifold structure and instead we look
at graph of groups and in the setting of $\operatorname{CAT}(0)$ admissible
groups, it is unknown whether all fundamental group of graph of hyperbolic-
by-Z groups are quasi-isometric to the ones whose gluing map is a flip
isometry(here flip roughly means the same as in the basic example). In Theorem
A, we drop both the assumption of flip and also the assumption of the space
being a manifold.

Lastly, an intermediate step in the proof of Theorem A depends on the
following fact:

######  Theorem B.

Let $\mu$ be a non-elementary, finitely supported measure on $G$ and assume
that $G$ acts geometrically on an hierarchically hyperbolic space. Then for
$\kappa(r):=\log^{p}(r)$, where $p$ is the complexity of the hierarchy, the
$\kappa$-Morse boundary is a topological model for the Poisson boundary of
$(G,\mu)$.

The claim is proven for all mapping class groups and is remarked in [QRT20].
We include the proof for hierarchically hyperbolic spaces in the appendix of
this paper.

###  1.1. Acknowledgments

Hoang Thanh Nguyen is partially supported by Project ICRTM04_2021.07 of the
International Centre for Research and Postgraduate Training in Mathematics,
VietNam. We thank Mark Hagen and Kasra Rafi for useful conversations.

##  2\. Preliminaries

In this section, we review definitions of $\kappa$-projection, $\kappa$-Morse
boundary in Section 2.1, random walk on groups and Posisson boundaries in
Section 2.2. These two sections (Section 2.1 and Section 2.2) are only used in
Section 4, so the reader could skip these two sections until Section 4 . In
Section 2.3, we recall the definition of hierachically hyperbolic space (HHS)
given in [BHS19]. We then review backgrounds on CAT(0) admissible groups in
Section 2.4.

###  2.1. $\kappa$-projection, $\kappa$-Morse boundary

We refer to and follow the construction in [QRT20] as a complete reference for
$\kappa$-Morse boundaries of proper geodesic metric spaces. Here we briefly
recall the construction.

######  Definition 2.1.

Let $(X,d_{X})$ be a proper geodesic metric space and $Z\subseteq X$ a closed
subset, and let $\kappa$ be a concave sublinear function. A map
$\pi_{Z}\colon\thinspace X\to\mathcal{P}(Z)$ is a $\kappa$-_projection_ if
there exist constants $D_{1},D_{2}$, depending only on $Z$ and $\kappa$, such
that for any points $x\in X$ and $z\in Z$,

$\operatorname{diam}_{X}(\\{z\\}\cup\pi_{Z}(x))\leq D_{1}\cdot
d_{X}(x,z)+D_{2}\cdot\kappa(x).$

A $\kappa$-projection differs from a nearest point projection by a uniform
multiplicative error and a sublinear additive error[QRT20].

######  Lemma 2.2.

[QRT20] Given a closed set $Z$, we have for any $x\in X$

$\operatorname{diam}_{X}(\\{x\\}\cup\pi_{Z}(x))\leq(D_{1}+1)\cdot
d_{X}(x,Z)+D_{2}\cdot\kappa(x).$

######  Definition 2.3.

Let $Z\subseteq X$ be a closed set, and let $\kappa$ be a concave sublinear
function. We say that $Z$ is _$\kappa$ -Morse_ if there exists a proper
function $m_{Z}:\mathbb{R}^{2}\to\mathbb{R}$ such that for any sublinear
function $\kappa^{\prime}$ and for any $r>0$, there exists $R$ such that for
any $(q,Q)$-quasi-geodesic ray $\beta$ with $m_{Z}(q,Q)$ small compared to
$r$, if

$d_{X}(\beta_{R},Z)\leq\kappa^{\prime}(R)\qquad\text{then}\qquad\beta|_{r}\subset\mathcal{N}_{\kappa}\big{(}Z,m_{Z}(q,Q)\big{)}$

The function $m_{Z}$ will be called a Morse gauge of $Z$.

######  Definition 2.4.

Let $X$ be a proper geodesic space with a base-point ${\mathfrak{o}}$. Two
$\kappa$-Morse quasi-geodesic rays $\alpha$ and $\beta$ are equivalent,
written as $\alpha\sim\beta$, if they diverge sublinearly. The set of all
equivalence classes of $\kappa$-Morse geodesic quasi-geodesic rays, together
with the _fellow traveling quasi-geodesic_ topology [QRT20].

######  Theorem 2.5.

[QRT20] Let $X$ be a proper geodesic metric space and let $\kappa$ be a
sublinear function. The associated $\kappa$-Morse boundary
$\partial_{\kappa}X$ is a metrizable topological space and
$\partial_{\kappa}X$ is quasi-isometrically invariant.

###  2.2. Random walk and Poisson boundaries

We now review random walk on groups per the study of this project. Let $G$ be
a locally compact, second countable group, with left Haar measure $m$, and let
$\mu$ be a Borel probability measure on $G$, which we assume to spread-out,
i.e. such that there exists $n$ for which $\mu^{n}$ is not singular w.r.t.
$m$. Given $\mu$, we consider the _step space_
$(G^{\mathbb{N}},\mu^{\mathbb{N}})$, whose elements we denote as $(g_{n})$.
The _random walk driven by_ $\mu$ is the $G$-valued stochastic process
$(w_{n})$, where for each $n$ we define the product

$w_{n}:=g_{1}g_{2}\dots g_{n}.$

We denote as $(\Omega,\mathbb{P})$ the _path space_ , i.e. the space of
sequences $(w_{n})$, where $\mathbb{P}$ is the measure induced by pushing
forward the measure $\mu^{\mathbb{N}}$ from the step space. Elements of
$\Omega$ are called _sample paths_ and will be also denoted as $\omega$. Let
$T:\Omega\to\Omega$ be the left shift on the path space, and let
$(B,\mathcal{A})$ be a measurable space on which $G$ acts by measurable
isomorphisms; a measure $\nu$ on $B$ is $\mu$-_stationary_ if
$\nu=\int_{G}g_{\star}\nu\ d\mu(g)$, and in that case the pair $(B,\nu)$ is
called a $(G,\mu)$-_space_. Recall that a _$\mu$ -boundary_ is a measurable
$(G,\mu)$-space $(B,\nu)$ such that there exists a $T$-invariant, measurable
map $\textbf{bnd}:(\Omega,\mathbb{P})\to(B,\nu)$, called the _boundary map_.

Moreover, a function $f:G\to\mathbb{R}$ is $\mu$-_harmonic_ if
$f(g)=\int_{G}f(gh)\ d\mu(h)$ for any $g\in G$. We denote by
$H^{\infty}(G,\mu)$ the space of bounded, $\mu$-harmonic functions. One says a
$\mu$-boundary is the _Poisson boundary_ of $(G,\mu)$ if the map

$\Phi:H^{\infty}(G,\mu)\to L^{\infty}(B,\nu)$

given by $\Phi(f)(g):=\int_{B}f\ dg_{\star}\nu$ is a bijection. The Poisson
boundary $(B,\nu)$ is the maximal $\mu$-boundary, in the sense that for any
other $\mu$-boundary $(B^{\prime},\nu^{\prime})$ there exists a
$G$-equivariant, measurable map $p:(B,\nu)\to(B^{\prime},\nu^{\prime})$. For
more details, see [Kai00].

### The $\kappa$-Morse boundary realizes the Poisson boundary

The following result is due to Ilya Gekhtman and [QRT20].

######  Theorem 2.6.

Let $G$ be a finitely generated group, and let $(X,d_{X})$ be a Cayley graph
of $G$. Let $\mu$ be a probability measure on $G$ with finite first moment
with respect to $d_{X}$, such that the semigroup generated by the support of
$\mu$ is a non-amenable group. Let $\kappa$ be a concave sublinear function,
and suppose that for almost every sample path $\omega=(w_{n})$, there exists a
$\kappa$-Morse geodesic ray $\gamma_{\omega}$ such that

(2) $\lim_{n\to\infty}\frac{d_{X}(w_{n},\gamma_{\omega})}{n}=0.$

Then almost every sample path converges to a point in $\partial_{\kappa}X$,
and moreover the space $(\partial_{\kappa}X,\nu)$, where $\nu$ is the hitting
measure for the random walk, is a model for the Poisson boundary of $(G,\mu)$.

###  2.3. Hierarchically hyperbolic spaces

We first recall the definition of a hierarchically hyperbolic space (HHS) as
given in [BHS19].

######  Definition 2.7 (Hierarchically hyperbolic spaces).

A $q$–quasigeodesic space $(X,d)$ (i.e, there exists a constant $q$ such that
any two points in $X$ is joining by a $(q,q)$–quasigeodesic segment) is a
hierarchically hyperbolic space if there exists $\delta\geq 0$, an index set
$\Lambda$, a collection of $\delta$–hyperbolic spaces
$\\{\mathcal{C}W:W\in\Lambda\\}$ such that the following conditions are
satisfied:

Projections: There exists a uniform constant $\xi\geq 0$, there is a set

$\\{\pi_{W}\colon X\to 2^{\mathcal{C}W},W\in\Lambda\\}$

of projections such that
$\operatorname{diam}_{\mathcal{C}W}(\pi_{W}(x))\leq\xi$ for all
$x\in\mathcal{C}W$. Moreover, there exists $K$ such that for all
$W\in\Lambda$, $\pi_{W}$ is coarsely Lipschitz and the image $\pi_{1}(W)(X)$
(which is defined to be the union of $\pi_{W}(x)$) is a $K$–quasiconvex set in
$\mathcal{C}W$.

Nesting: $\Lambda$ is equipped with a partial order $\sqsubseteq$ such that
either $\Lambda=\emptyset$ or $\Lambda$ contains a unique
$\sqsubseteq$–maximal element. In this paper the maximal element is always
denoted $S$. When $V\sqsubseteq W$ we say $V$ is nested in $W$. For each
$W\in\Lambda$, define $\Lambda_{W}=\\{V\in\Lambda:V\sqsubseteq W\\}$. If
$V,W\in\Lambda$ such that $V$ is properly nested in $W$ then there is a
specific subset $\rho_{W}^{V}\subset\mathcal{C}W$ such that
$\operatorname{diam}(\rho_{W}^{V})\leq\xi$. Also, there is a projection
$\rho_{V}^{W}$ from $\mathcal{C}W$ to subsets of $\mathcal{C}V$.

Orthogonality: $\Lambda$ has a symmetric and anti-reflexive relation called
orthogonality: (we use notation $\perp$). Also, whenever $V\sqsubseteq W$ and
$W\perp U$ then we require that $V\perp U$. We require that for each
$S\in\Lambda$, for each $U\in\Lambda_{S}$ such that the set
$\\{V\in\Lambda_{S}:V\perp U\\}\neq\emptyset$ then there exists
$W\in\Lambda_{S}$, $W\neq S$ so that whenever $V\perp U$, $V\sqsubseteq S$
then we have $V\sqsubseteq W$. Finally, if $V\perp W$, then $V,W$ are not
$\sqsubseteq$–comparable.

Transversality and consistency: If $V$ and $W$ are not orthogonal and neither
is nested in the other, then we say that $V$ and $W$ are transverse , denoted
$V\pitchfork W$, then there are subsets $\rho^{V}_{W}\subset\mathcal{C}W$ and
$P^{W}_{V}\subset\mathcal{C}V$ such that
$\operatorname{diam}(\rho^{V}_{W})\leq\xi$,
$\operatorname{diam}(P^{W}_{V})\leq\xi$ and satisfying:

For any $x\in X$ we have

($\clubsuit$)
$\min\\{d_{W}(\pi_{W}(x)),P^{V}_{W})\,,d_{V}(\pi_{V}(x),P^{W}_{V})\\}\leq\kappa_{0}$

For each $V,W\in\Lambda$ such that $V\sqsubseteq W$, and for each $x\in X$ we
have:

($\spadesuit$)
$\min\\{d_{W}(\pi_{W}(x),P^{V}_{W})\,,\operatorname{diam}(\pi_{V}(x)\cup\rho^{W}_{V}(\pi_{W}(x)))\\}\leq\kappa_{0}$

($\diamondsuit$) $\text{If}\qquad U\sqsubseteq
V,\,\qquad\text{then}\qquad\,d_{W}(\mathcal{P}^{U}_{W},\mathcal{P}^{V}_{W})\leq\kappa_{0}$

whenever $W\in\Lambda$ satisfies either $V$ is properly nested in $W$

Finite complexity:  The complexity of $X$ is an integer $n\geq 0$ such that
any set of pairwise $\sqsubseteq$–compatible elements has cardinality at most
$n$.

Large links: There exists $\lambda\geq 1$ and $E\geq\xi,\kappa_{0}$ such that
the following holds. Let $W\in\Lambda$ and let $x,x^{\prime}\in X$. Let
$N=\lambda d_{w}(\pi_{W}(x),\pi_{W}(x^{\prime}))+\lambda$. Then there exists
$A_{1},A_{2},\cdots,A_{\lfloor N\rfloor}\in\Lambda_{W}\backslash\\{W\\}$ such
that for all $A\in\Lambda_{W}\backslash\\{W\\}$ then either $A=T_{i}$ for some
$i$, or $d_{A}(\pi_{A}(x),\pi_{A}(x^{\prime}))<E$. Also,
$d_{W}(\pi_{W}(x),P^{A_{i}}_{W})\leq N$ for each $i$.

Bounded geodesic image:  There exists $E>0$ such that for all $W\in\Lambda$,
all $V\in\Lambda_{W}\backslash\\{W\\}$, and all geodesic $\gamma$ of
$\mathcal{C}W$, either

($\oplus$) $\operatorname{diam}_{\mathcal{C}V}(\rho^{W}_{V}(\gamma))\leq
E\,\,\text{or}\,\,\gamma\cap\mathcal{N}_{E}(\rho^{V}_{W})\neq\emptyset$

Partial Realization: There exists a constant $\alpha$ with the following
property. Let $\\{V_{j}\\}$ be a family of pairwise orthogonal elements of
$\Lambda$, and let $p_{j}\in\pi_{V_{j}}(X)$. Then there exists an element
$x\in X$ such that

  * •

$d_{V_{j}}(x,p_{j})\leq\alpha$ for all $j$

  * •

For each $j$, and each $v\in\Lambda$ with $V_{j}\sqsubseteq V$, we have
$d_{V}(x,\rho^{V_{j}}_{V})\leq\alpha$ and

  * •

If $W$ and $V_{j}$ are transverse for some $j$ then
$d_{W}(x,p^{V_{j}}_{W})\leq\alpha$.

Uniqueness: For each $k\geq 0$, there exists $\Theta=\Theta(k)$ such that for
any $x,y$ in $X$, $d_{X}(x,y)\geq\Theta$ then there exists $V\in\Lambda$ such
that $d_{V}(x,y)\geq k$.

###  2.4. $\operatorname{CAT}(0)$ admissible groups

_Admissible groups_ firstly introduced in [CK02] are a particular class of
graph of groups that includes fundamental groups of $3$–dimensional graph
manifolds. In this section, we review admissible groups and their properties
that will used throughout in this paper.

We often consider oriented edges from $e_{-}$ to $e_{+}$ and denote by
$e=[e_{-},e_{+}]$. Then $\bar{e}=[e_{+},e_{-}]$ denotes the oriented edge with
reversed orientation. Denote by $\mathcal{G}^{0}$ the set of vertices and by
$\mathcal{G}^{1}$ the set of all oriented edges.

######  Definition 2.8.

A graph of group $\mathcal{G}$ is _admissible_ if

  1. (1)

$\mathcal{G}$ is a finite graph with at least one edge.

  2. (2)

Each vertex group ${G}_{v}$ has center $Z({G}_{v})\cong\mathbb{Z}$,
${Q}_{v}\colon={G}_{v}/Z({G}_{v})$ is a non-elementary hyperbolic group, and
every edge subgroup ${G}_{e}$ is isomorphic to $\mathbb{Z}^{2}$.

  3. (3)

Let $e_{1}$ and $e_{2}$ be distinct directed edges entering a vertex $v$, and
for $i=1,2$, let $K_{i}\subset{G}_{v}$ be the image of the edge homomorphism
${G}_{e_{i}}\to{G}_{v}$. Then for every $g\in{G}_{v}$, $gK_{1}g^{-1}$ is not
commensurable with $K_{2}$, and for every $g\in G_{v}{\smallsetminus}K_{i}$,
$gK_{i}g^{-1}$ is not commensurable with $K_{i}$.

  4. (4)

For every edge group ${G}_{e}$, if $\alpha_{i}\colon{G}_{e}\to{G}_{v_{i}}$ are
the edge monomorphism, then the subgroup generated by
$\alpha_{1}^{-1}(Z({G}_{v_{1}}))$ and $\alpha_{2}^{-1}(Z({G}_{v_{1}}))$ has
finite index in ${G}_{e}$.

A group $G$ is _admissible_ if it is the fundamental group of an admissible
graph of groups.

######  Definition 2.9.

We say that an admissible group $G$ is $\operatorname{CAT}(0)$ if it acts
properly discontinuously, cocompactly and by isometries on a complete proper
$\operatorname{CAT}(0)$ space. For the remainder of this paper, the pair
$(G,X)$ will always denote a $\operatorname{CAT}(0)$ admissible group with the
associated $\operatorname{CAT}(0)$ space. For examples of admissible groups
that are not $\operatorname{CAT}(0)$, see [KL96].

Let $G$ be a $\operatorname{CAT}(0)$ admissible group, and let
$G\curvearrowright T$ be the action of $G$ on the associated Bass-Serre tree
$T$ of the graph of group $\mathcal{G}$ (we refer the reader to Section 2.5 in
[CK02] for a brief discussion). Let $V(T)$ and $E(T)$ be the vertex and edge
sets of $T$. For each $\sigma\in V(T)\cup E(T)$, we let $G_{\sigma}\leq G$ be
the stabilizer of $\sigma$.

In the rest of this section, we review facts from Section 3.2 in [CK02] that
will be used thoroughly this paper. We refer the reader to [CK02] for further
explanation. Given $(G,X)$ pair,

  1. (1)

for every vertex $v\in V(T),$ the set $B_{v}:=\cap_{g\in
Z(G_{v})}\operatorname{minset}(g)$ splits as metric product
$H_{v}\times\mathbb{R}$ where $Z(G_{v})$ acts by translation on the
$\mathbb{R}$–factor and the quotient $G_{v}/Z(G_{v})$ acts geometrically on
the Hadamard space $H_{v}$.

  2. (2)

for every edge $e\in E(T)$, the minimal set $B_{e}:=\cap_{g\in
G_{e}}\operatorname{minset}(g)$ splits as
$\overline{B_{e}}\times\mathbb{R}^{2}\subset B_{v}$ where $\overline{B_{e}}$
is a compact $\operatorname{CAT}(0)$ space and $G_{e}=\mathbb{Z}^{2}$ acts co-
compactly on the Euclidean plane $\mathbb{R}^{2}$.

We note that the assignments $v\to B_{v}$ and $e\to B_{e}$ are $G$–equivariant
with respect to the natural $G$ actions in the sense that $gB_{v}=B_{gv}$ and
$gB_{e}=B_{ge}$ for any $g\in G$.

Strips in X: We reviews strips in $X$ that was discussed in the second
paragraph in Section 4.1 in [CK02]). We first choose, in a $G$–equivariant, a
plane $F_{e}\subset B_{e}$ for each edge $e$ in $T$ (i.e, $gF_{e}=F_{ge}$ for
each $g\in G$). We will call $F_{e}$ is a boundary plane of $B_{e_{\pm}}$. For
every pair of adjacent edges $e_{1}$, $e_{2}$. we choose, again equivariantly,
a shortest geodesic $\alpha_{e_{1}e_{2}}$ from $F_{e_{1}}$ to $F_{e_{2}}$
(equivariantly here we mean $\alpha_{(ge)(ge^{\prime})}=g\alpha_{ee^{\prime}}$
for adjacent edges $e$, $e^{\prime}$ ). By the convexity of
$B_{v}=H_{v}\times\mathbb{R}$, $v:=e_{1}\cap e_{2}$, this geodesic
$\alpha_{e_{1}e_{2}}$ determines a Euclidean strip
$\mathcal{S}_{e_{1}e_{2}}:=\gamma_{e_{1}e_{2}}\times\mathbb{R}$ (possibly of
width zero) for some geodesic segment $\gamma_{e_{1}e_{2}}\subset H_{v}$.

######  Remark 2.10.

There exists a $G$–equivariant coarse $L$–Lipschitz map $\pi\colon X\to V(T)$
such that $x\in X_{\rho(x)}$ for all $x\in X$. The map $\pi$ is called a
index map. We refer the reader to Section 3.3 in [CK02] for the existence of
such a map $\rho$.

######  Lemma 2.11.

[CK02, Lemma 2.9] There exists a constant $\epsilon>0$ such that the following
hold. Let $F_{e}$ and $F_{e}^{\prime}$ be two boundary planes in a vertex
space $B_{v}$. Let
$\mathcal{S}_{e,e^{\prime}}=\gamma_{e,e^{\prime}}\times\mathbb{R}$ be the
strip in $B_{v}$. Let $\ell:=F_{e}\cap H_{v}$ and
$\ell^{\prime}=F_{e^{\prime}}\cap H_{v}$. Then the
$\ell\cup\gamma_{e,e^{\prime}}\cup\ell^{\prime}$ is a $\epsilon$–quasiconvex
in the $\operatorname{CAT}(0)$ hyperbolic space $H_{v}$.

######  Definition 2.12 (Boundary lines).

Let $\\{F_{e}\\}$ be the collection of boundary planes of the block $B_{v}$.
We note that the intersection of a boundary plane $F_{e}$ of $B_{v}$ with the
hyperbolic space $H_{v}$ is a line. We define the collection of lines
$\mathbb{L}_{v}$, which we call boundary lines, in the hyperbolic space
$H_{v}$ as the following:

$\mathbb{L}_{v}=\\{\ell_{e}=F_{e}\cap H_{v}\,|\,e_{-}=v\\}$

######  Remark 2.13.

There exists a constant $r>0$ such that for any vertex $v\in T$, for any $x\in
H_{v}$, there exists a boundary line $\ell\in\mathbb{L}_{v}$ such that
$d(x,\ell)\leq r$.

The following lemmas are needed in Section 3 to check the HHS Axioms.

######  Lemma 2.14.

$Q_{v}$ is hyperbolic relative to
$\mathbb{P}:=\\{\bar{K}_{e}:e_{-}=v,e\in\mathcal{G}^{1}\\}$ where
$\bar{K}_{e}$ is the projection of $G_{e}$ under the quotient $G_{v}\to
Q_{v}$.

###### Proof.

Let $K_{e}$ be the image of an edge group $G_{e}$ into $G_{v}$ and
$\bar{K}_{e}$ be its projection in $Q_{v}$ under $G_{v}\to
Q_{v}=G_{v}/Z(G_{v})$. Then
$\mathbb{P}:=\\{\bar{K}_{e}:e_{-}=v,e\in\mathcal{G}^{1}\\}$ is an almost
malnormal collection of virtually cyclic subgroups in $H_{v}$. Indeed, since
$Z(G_{v})\subset K_{e}\cong\mathbb{Z}^{2}$, we have
$\bar{K}_{e}=K_{e}/Z(G_{v})$ is virtually cyclic. The almost malnormality
follows from non-commensurability of $K_{e}$ in $G_{v}$. To see this, assume
that $\bar{K}_{e}\cap h\bar{K}_{e^{\prime}}h^{-1}$ contains an infinite order
element by the hyperbolicity of $Q_{v}$. If $g\in G_{v}$ is sent to $h$, then
$K_{e}\cap gK_{e^{\prime}}g^{-1}$ is sent to $\bar{K}_{e}\cap
h\bar{K}_{e^{\prime}}h^{-1}$. Thus, $K_{e}\cap gK_{e^{\prime}}g^{-1}$ contains
an abelian group of rank 2. The non-commensurability of $K_{e}$ in $G_{v}$
implies that $e=e^{\prime}$ and $g\in K_{e}$. This shows that $\mathbb{P}$ is
almost malnormal. It is well-known that a hyperbolic group is hyperbolic
relative to any almost malnormal collection of quasi-convex subgroups
([Bow12]). Thus $Q_{v}$ is hyperbolic relative to $\mathbb{P}$. ∎

##  3\. $\operatorname{CAT}(0)$ admissible groups are HHS

In this section, we are going to prove the following proposition (that is also
part (1) in Theorem A).

######  Proposition 3.1.

Let $G$ be a $\operatorname{CAT}(0)$ admissible group. Then $G$ is a
hierarchically hyperbolic space with respect to a word metric.

First at all, we will construct an index set $\Lambda$ and a set
$\\{\mathcal{C}W:W\in\Lambda$ } of $\delta$–hyperbolic spaces that satisfies
HHS's axioms (see Definition 2.7). We note that if metric spaces
$\widetilde{X}$ and $\widetilde{Y}$ are quasi-isometric, then $\widetilde{X}$
is a hierarchically hyperbolic space whenever $\widetilde{Y}$ is a
hierarchically hyperbolic space (see Proposition 1.10 in [BHS19]). Hence to
see that $G$ is a hierarchically hyperbolic space with respect to word
metrics, we only need to work on its appropriate finite index subgroups.

Define $V(T):=\mathcal{V}_{1}\cup\mathcal{V}_{2}$ where $\mathcal{V}_{i}$
consists of vertices in $T$ with pairwise even distances. Passing to a finite
index subgroup if necessary, we could assume that $G$ preserves each
$\mathcal{V}_{i}$ (see Lemma 4.6 in [NY20]). We emphasize here that each
vertex group $G_{v}$ in a $\operatorname{CAT}(0)$ admissible groups maybe not
split trivially as a product of a hyperbolic space with $\mathbb{Z}$. We need
to pass to a further finite index subgroup to make our group is in a simpler
form. By Lemma 7.2 in [HNY21], there exists a finite index subgroup of the
$\operatorname{CAT}(0)$ admissible group $G$ such that this subgroup of $G$
preserves $\mathcal{V}_{1}$, $\mathcal{V}_{2}$, and each vertex group of the
subgroup is a product of a nonelementary hyperbolic group with $\mathbb{Z}$.
By abuse of notations, we still denote this subgroup by $G$ (since HHS is
preserved under quasi-isometry [BHS19]).

###  3.1. Constructing an index set

####  3.1.1. Elements in index set $\Lambda$: A collection of coned-off
hyperbolic spaces

Recall from Section 2.4 that for each vertex $v$ in the Bass-Serre tree $T$,
there is associated block $B_{v}=H_{v}\times\mathbb{R}$. For a line $\ell$ and
a constant $r>0$, a hyperbolic $r$–cone, denoted by $cone_{r}(\ell)$ is the
quotient space of $\ell\times[0,r]$ by collapsing $\ell\times 0$ as a point
called apex. A metric is endowed on $cone_{r}(\ell)$ so that it is isometric
to the metric completion of the universal covering of a closed disk of radius
$r$ in the real hyperbolic plane $\mathbb{H}^{2}$ punctured at the center (see
[BH99, Part I, Ch. 5]).

Given a vertex $v$ in the tree $T$. Let $\mathbb{L}_{v}=\\{\ell_{e}=F_{e}\cap
H_{v}\,|\,e_{-}=v\\}$ be the set of boundary lines in the hyperbolic space
$H_{v}$ (see Definition 2.12). We attach a hyperbolic $r$–cone to each
boundary line $\ell_{e}$. The resulted space is denoted by $\hat{H}_{v}$.

Since $\mathbb{L}_{v}$ has the bounded intersection property, it follows from
Corollary 5.39 in [DGO] that for a sufficiently large constant $r$, the space
$\widehat{H}_{v}$ is a hyperbolic space. Since there are finitely many
vertices up to $G$–action, the constant $r$ can be chosen large enough so that
$\widehat{H}_{v}$ is a $\delta(r)$–hyperbolic space for all vertex $v$ in $T$.

The collection of $\delta(r)$–hyperbolic spaces
$\bigl{\\{}\widehat{H}_{v}\,\bigl{|}v\in\mathcal{V}(T)\bigr{\\}}$ is a part in
the index set $\Lambda$ (in Definition 2.7) that we will define later.

####  3.1.2. Elements in the index set $\Lambda$: a collection of quasi-lines

We need to add another collection of hyperbolic spaces into the collection
above as the following. Let $e$ be an oriented edge in $T$ such that
$e_{+}=v$. Let $\mathfrak{f}_{e}$ be a line in the boundary plane $F_{e}$
parallel to the $\mathbb{R}$-direction in $B_{v}=H_{v}\times\mathbb{R}$.
Recall that two lines $\mathfrak{f}_{e}\neq\mathfrak{f}_{\bar{e}}$ in the
plane $F_{e}$ must be intersect. The action of $G_{e}$ on the set of lines
orthogonal to $\mathfrak{f}_{e}$ induces an action of $G_{e}$ on
$\mathfrak{f}_{e}$. Moreover, this action of $G_{e}$ on $\mathfrak{f}_{e}$
extends to the boundary plane $F_{e}$ so that it commutes with the orthogonal
projection to $\mathfrak{f}_{e}$. Note that this new action of $G_{e}$ on
$F_{e}$ preserves $\mathfrak{f}_{e}$ so does not coincide with the original
action on $F_{e}$.

$v$
---
$w$
---
$e$
---
$H_{w}\cap F_{e}$
---
$F_{e}$
---

Figure 1. Dashed-line is the boundary line $\ell_{w}:=H_{w}\cap F_{e}$ of
$H_{w}$. Each fiber line in the collection $\mathbb{K}_{\bar{e}}$ which is
perpendicular to $H_{w}\cap F_{e}$ is coned-off.

We choose a $Z(G_{v})$–invariant set $\mathbb{K}_{\bar{e}}$ of fiber lines of
$B_{w}$ in $F_{e}$ orthogonal to the boundary line $H_{w}\cap F_{e}$ of
$H_{w}$ in $F_{e}$. A hyperbolic cone of radius $r=1$ is then attached to
$F_{e}$ along each line in $\mathbb{K}_{\bar{e}}$. The resulted coned-off
space denoted by $\widehat{F}_{e}$. We note that $\widehat{F}_{e}$ is quasi-
isometric to a line and thus it is a hyperbolic space.

Let $Lk(v)$ and $St(v)$ be the link and the star of $v$ in $T$. Then
$Fl(v):=St(v)\times\mathbb{R}$ is the union of flat strips $S_{e}$'s (of width
1) corresponding to the set of oriented edges $e=[w,v]$ towards $v$.

Figure 2. The picture illustrates the space $\mathcal{R}_{v}$ is obtained by
identifying boundary lines of strips $S_{e}$'s with marked lines
$\mathfrak{f}_{e}$ in $\widehat{F}_{e}$. The set of hyperbolic cones attached
along lines in $\mathbb{K}_{\bar{e}_{1}}$ not shown in the figure. The
resulted space $\mathcal{R}_{v}$ is a quasi-line.

Let $\mathcal{R}_{v}$ be the space obtained from the disjoint union
$Fl(v)\sqcup_{e\in St(v)}\widehat{F}_{e}$ with the boundary lines of every
strip $S_{e}$ in $Fl(v)$ are identified with the marked lines
$\mathfrak{f}_{e}$ in $\widehat{F}_{e}$. Endowed with length metric,
$\mathcal{R}_{v}$ is quasi-isometric a quasi-line, so it is a hyperbolic
space. We refer the reader to Figure 2 for this illustration.

######  Remark 3.2.

By our construction of $\mathcal{R}_{v}$'s, there exists a uniform constant
$r>0$ such that the following holds.

  1. (1)

$\mathcal{R}_{v}$ is a $r$–hyperbolic space for every vertex $v$ in $T$.

  2. (2)

Let $v$ be an arbitrary vertex in $T$, and let $w$ be a vertex that is
adjacent to $v$. Denote the edge joining two vertices $v$ and $w$ by $e$. Let
$F_{e}$ be the boundary plane in $B_{v}$ associated to the edge $e$. Let
$\ell$ be an arbitrary line in the plane $F_{e}$ that is parallel to the
$\mathbb{R}$–factor of the adjace pice $B_{w}=H_{w}\times\mathbb{R}$ of
$B_{v}$. Then the diameter of $\ell$ in
$(\mathcal{R}_{v},d_{\mathcal{R}_{v}})$ is no more than $r$.

  3. (3)

Let $\ell$ and $\ell^{\prime}$ be two lines in the plane $F_{e}$ that is
parallel to the $\mathbb{R}$–factor of the adjacent piece $B_{w}$. These two
lines intersect the boundary line $\ell_{w}:=F_{e}\cap H_{w}$ at two points
$x$ and $y$ respectively. Then
$\bigl{|}\operatorname{diam}_{(\mathcal{R}_{v},d_{\mathcal{R}_{v})}}(\ell\cup\ell^{\prime})-d(x,y)\bigr{|}\leq
2r$.

######  Definition 3.3 (The index set $\Lambda$).

We define the index set $\Lambda$ to be the union of the Bass-Serre tree $T$,
the coned-off spaces $\widehat{H}_{v}$, and the thickened fiber lines
$\mathcal{R}_{v}$, i.e,

$\Lambda:=\\{T\\}\cup\bigl{\\{}\mathcal{R}_{v}:v\in
V(T)\bigr{\\}}\cup\bigl{\\{}\widehat{H}_{v}:v\in V(T)\bigr{\\}}$

###  3.2. Verifying the HHS Axioms

To prove Proposition 3.1, we need to associate to each element in this index
set $\Lambda$ a hyperbolic space, and verify that this index set together with
these hyperbolic spaces satisfy HHS's axioms in Definition 2.7.

Recall that $G\curvearrowright X$ is a $\operatorname{CAT}(0)$ admissible
action. Since HHS preserves under quasi-isometries (see Proposition 1.10 in
[BHS19]), it suffices to show that the orbit space $G(x_{0})\subset X$ is a
HHS for some $x_{0}\in X$. Thus we assume, without loss of generality that
$x_{0}$ belongs to a plane $F_{e_{0}}$ for some edge $e_{0}$ in $T$. By
abusing notation, we still denote $G(x_{0})$ by $X$.

First at all, each element $W$ in the index set $\Lambda$ is associated to a
hyperbolic space as the following: If $W=T$, we let $\mathcal{C}W$ denote the
Bass-Serre tree $T$ equipped with the metric $d_{T}$. If $W=\mathcal{R}_{v}$
for some vertex $v$, we let $\mathcal{C}W$ denote the space $\mathcal{R}_{v}$
equipped with the metric $d_{\mathcal{R}_{v}}$ described early. We recall that
$\bigl{(}\mathcal{R}_{v},d_{\mathcal{R}_{v}}\bigr{)}$ is quasi-isometric to a
line, and thus $\bigl{(}\mathcal{R}_{v},d_{\mathcal{R}_{v}}\bigr{)}$ is a
hyperbolic space. If $W=\widehat{H}_{v}$ for some vertex $v$, we let
$\mathcal{C}W$ denote the space $\widehat{H}_{v}$ equipped with the coned-off
metric $d_{\widehat{H}_{v}}$ described early in previous paragraphs. Note that
these spaces are all hyperbolic spaces, and there are finitely many vertices
up to $G$ action, thus there exists a constant $\delta>0$ such that
$\mathcal{C}W$ is a $\delta$–hyperbolic space for all $W$ in the index set
$\Lambda$.

####  3.2.1.  Projections

In this section we are going to verify the Projection Axiom of Definition 2.7.
Recall that $x_{0}$ belongs to a plane $F_{e_{0}}$ for some edge $e_{0}$ in
$T$. Let $W$ be an element in the index set $\Lambda$, we are going to define
projections $\pi_{W}\colon X\to 2^{CW}$ as follows.

Assume that $W=T$, then we define the $\pi_{W}=\pi_{T}$ to be the index map
$\pi\colon X\to T$ given by Remark 2.10.

Now, assume that $W=\widehat{H}_{v}$ for some vertex $v\in T$. then we define
the projection $\pi_{W}\colon X\to 2^{CW}$ to be the nearest point projection
from $X$ into $H_{v}\subset\widehat{H}_{v}$.

Suppose that $W=\mathcal{R}_{v}$ for some vertex $v\in V(T)$. We are going to
define the projection $\pi_{X}\colon X\to 2^{CW}=2^{\mathcal{R}_{v}}$ as the
following: Let $x$ be an arbitrary element in $X$, then $\pi(x)$ is a vertex
in the tree $T$. We note that $x\in Y_{\pi(x)}$. If $\pi(x)=v$ then we define
$\pi_{W}(x):=x$. If $\pi(x)$ and $v$ are two adjacent vertices then we let
$\bar{x}$ be the projection of $x$ into be base $H_{\pi(x)}$ of
$Y_{\pi(x)}=H_{\pi(x)}\times\mathbb{R}$. Let $\alpha$ be a shortest geodesic
in $H_{\pi(x)}$ joining $\bar{x}$ to the boundary line $\ell_{e}:=F_{e}\cap
H_{\pi(x)}$ where $e:=[v,\pi(x)]$. Let $\mathcal{S}_{\bar{x}}$ be the strip in
$Y_{\pi(x)}$ defined by $\alpha\times\mathbb{R}$. We then define $\pi_{W}(x)$
to be the subset $\mathcal{S}_{\bar{x}}\cap F_{e}$ of
$F_{e}\subset\mathcal{R}_{v}$. If $d_{T}(v,\pi(x))\geq 2$, then we let
$\gamma$ be the geodesic in $T$ connecting $\pi(x)$ to $v$. We write
$\gamma=e_{1}\cdots e_{k-1}e_{k}$ with $(e_{1})_{-}=\pi(x)$ and
$(e_{k})_{+}=v$, then we define $\pi_{W}(x):=\mathcal{S}_{e_{k-1},e_{k}}\cap
F_{e_{k}}\subset\mathcal{R}_{v}$. We remind here that the intersection
$\mathcal{S}_{e_{k-1},e_{k}}\cap F_{e_{k}}$ is a line in the plane
$F_{e_{k}}$, but this line in the coned-off space $\mathcal{R}_{v}$ is no
longer a line, and it is a bounded set in $\mathcal{R}_{v}$ (see Remark 3.2).

To verify the HHS Axioms in Definition 2.7, we will use the following lemma.

######  Lemma 3.4.

There exists a uniform constant $A>0$ such that the following holds. Let $u$,
$v$, and $w$ be vertices in $T$ such that $u,v\neq w$, and $u,v,w\in lk(o)$
for some vertex $o$. Let $e=[w,o]$, $e_{1}=[u,o]$ and $e_{2}=[v,o]$. Let
$\ell_{e}$, $\ell_{e_{1}}$, $\ell_{e_{2}}$ be boundary lines in $H_{v}$
associated to the edges $e$, $e_{1}$, and $e_{2}$ respectively. Suppose that
$x$ and $y$ are two points in $X$ such that $\pi(x)=u$ and $\pi(y)=v$. Then

$\operatorname{diam}\bigl{(}\pi_{\mathcal{R}_{w}}(x,y)\bigr{)}\leq
Ad_{\ell_{e}}(\ell_{e_{1}},\ell_{e_{2}})+A$

where
$\pi_{\mathcal{R}_{w}}(x,y):=\pi_{\mathcal{R}_{w}}(x)\cup\pi_{\mathcal{R}_{w}}(y)$
and
$d_{\ell_{e}}(\ell_{e_{1}},\ell_{e_{2}}):=\operatorname{diam}\pi_{\ell_{e}}(\ell_{e_{1}})\cup\pi_{\ell_{e}}(\ell_{e_{1}})$
with $\pi_{\ell_{e}}(\ell_{e_{i}})$ is the projection from $\ell_{e_{i}}$ into
$\ell_{e}$ in the CAT(0) hyperbolic space $H_{o}$.

###### Proof.

If $u=v$ then the result is vacuously true since both
$\pi_{\mathcal{R}_{w}}(x)$ and $\pi_{\mathcal{R}_{w}}(y)$ are the same. We
thus assume that $u\neq v$. By our definition of projections, we have
$\pi_{\mathcal{R}_{w}}(x)=\mathcal{S}_{e_{1}e}\cap F_{e}$ and
$\pi_{\mathcal{R}_{w}}(y)=\mathcal{S}_{e_{2}e}\cap F_{e}$. Note that
$\pi_{\mathcal{R}_{w}}(x)$ and $\pi_{\mathcal{R}_{w}}(y)$ are parallel lines
in $F_{e}$. These two lines intersect the boundary line $\ell_{e}$ at two
points denoted by $\bar{x}$ and $\bar{y}$. By Remark 3.2, we have that

$\bigl{|}\operatorname{diam}_{(\mathcal{R}_{w},d_{\mathcal{R}_{w}})}\bigl{(}\pi_{\mathcal{R}_{w}}(x,y)\bigr{)}-d(\bar{x},\bar{y})\bigr{|}\leq
2r$

We note that $\bar{x}$ and $\bar{y}$ are within a uniformed bounded distance
from $\pi_{\ell_{e}}(\ell_{e_{1}})$ (projection from $\ell_{e_{1}}$ into
$\ell_{e}$ in the CAT(0) hyperbolic space $H_{o}$) and
$\pi_{\ell_{e}}(\ell_{e_{2}})$ respectively. It follows that there exists a
uniform constant $A>0$ such that the conclusion of the lemma holds.

$u$$\ell_{e_{1}}$$\ell_{e_{2}}$$\ell_{e}$$H_{0}$$e_{1}$$e_{2}$$e$$o$$v$$w$
Figure 3. Two arrows are within a uniform Hausdorff distance from projections
of $\mathcal{S}_{e_{1}e}$ and $\mathcal{S}_{e_{2}e}$ into $H_{o}$

∎

######  Lemma 3.5.

$\bigl{(}X,\Lambda,\\{\pi_{W}\\}\bigr{)}$ satisfies the Projection Axiom.

###### Proof.

By our definition of projection maps in Section 3.2.1, it is obvious that
$\operatorname{diam}_{\mathcal{C}W}\bigl{(}\pi_{W}(x)\bigr{)}\leq\xi$ for some
uniform constant $\xi>0$. Note that each $\pi_{W}$ is coarsely Lipschitz map.
Indeed, this fact is clear when $W\in\Lambda\backslash\\{T\\}$. In the case
$W=T$ then the coarse Lipschitz property of $\pi_{W}$ is followed from Remark
2.10. Also, it is clear from our definition of projection maps that
$\pi_{W}(X)$ is $K$–quasiconvex in $\mathcal{C}W$ for some constant $K$ large
enough. Hence Projection Axiom is verified. ∎

####  3.2.2. Nesting

We equip the index set $\Lambda$ with a partial order $\sqsubseteq$ as
follows: We declare $W\sqsubseteq W$ (i.e, $W$ is nested in itself) for all
$W$ in the index set $\Lambda$. For each $W\in\Lambda$, we declare that
$W\sqsubseteq T$ (i.e, every element in $\Lambda$ is nested in $T$). Let $V,W$
be distinct elements in $\Lambda\backslash\\{T\\}$. We say $V$ is properly
nested in $W$ if $\mathcal{C}W=\widehat{H}_{v}$ for some $v\in V(T)$ and
$\mathcal{C}V=\mathcal{R}_{w}$ for some vertex $w\in T$ such that $v$ and $w$
are adjacent vertices.

######  Lemma 3.6.

$\bigl{(}X,\Lambda,\sqsubseteq,\\{\pi_{W}\\}\bigr{)}$ satisfies the Nesting
Axiom.

###### Proof.

By our definition of $\sqsubseteq$, every element in the index set is nested
in $T$, hence the element $T\in\Lambda$ is the unique $\sqsubseteq$–maximal
element. For each $W$ in the index set $\Lambda$, we denote by $\Lambda_{W}$
the set of $V\in\Lambda$ such that $V\sqsubseteq W$.

Assume that $V,W\in\Lambda$ so that $V$ is properly nested in $W$. We are
going to define a specific subset $\rho_{W}^{V}\subset\mathcal{C}W$ such that
$\operatorname{diam}(\rho_{W}^{V})\leq\xi$. Also, there is a projection
$\rho_{V}^{W}$ from $\mathcal{C}W$ to subsets of $\mathcal{C}V$. Note that in
the Definition 2.7, there is no restriction on $\rho_{V}^{W}$. If
$W=\mathcal{R}_{v}$ then there is no $V$ such that $V$ is properly nested in
$W$. Thus, we only need to consider the following cases:

Case 1: Assume that $W$ is the Bass-Serre tree $T$. In this case, we have that
$\mathcal{C}W=(T,d_{T})$. Since $V$ is properly nested in $W$, it follows that
$V\in\Lambda\backslash\\{T\\}$. Hence $\mathcal{C}V$ is either the coned-off
space $\bigl{(}\widehat{H}_{v},d_{\widehat{H}_{v}}\bigr{)}$ or the thickened
fiber line $\bigl{(}\mathcal{R}_{v},d_{\mathcal{R}_{v}}\bigr{)}$ for some
vertex $v$ in $T$. We thus define $\rho_{W}^{V}$ to be the vertex $v\in T=CW$.
Obviously, $\operatorname{diam}(\rho_{W}^{V})=0<\xi$.

A projection map $\rho_{V}^{W}\colon\mathcal{C}W=(T,d_{T})\to
2^{\mathcal{C}V}$ from $\mathcal{C}W$ to subsets of $\mathcal{C}V$ is defined
as follows. Let $x$ be an arbitrary vertex in the Bass-Serre tree $T$. If
$d_{T}(x,v)\leq 1$, then we define $\rho^{W}_{V}(x)$ to be the entire space
$\mathcal{C}V$.

Now we assume that $d_{T}(x,v)\geq 2$. Let $e_{1}$ and $e_{2}$ be the last two
consecutive oriented edges in the geodesic $[x,v]$ in $T$ such that
$v=(e_{2})_{+}$. Let $u$ be the common vertex of $e_{1}$ and $e_{2}$, and let
$\mathcal{S}_{e_{1}e_{2}}$ be the strip in the space $B_{u}$ (see Subsection
2.4). Recall that $V$ is either $\widehat{H}_{v}$ or $\mathcal{R}_{v}$. When
$V=\mathcal{R}_{v}$, then we define $\rho^{W}_{V}(x)$ to be
$\mathcal{S}_{e_{1}e_{2}}\cap F_{e_{2}}\subset\mathcal{R}_{v}$ . Let $r>0$ be
the constant given by Remark 3.2. Applying Remark 3.2 to
$\ell:=\mathcal{S}_{e_{1}e_{2}}\cap F_{e_{2}}$, we have that the diameter of
$\rho^{W}_{V}(x)=\ell$ in
$\bigl{(}\mathcal{R}_{v},d_{\mathcal{R}_{v}}\bigr{)}$ is bounded above by $r$.
When $V=\widehat{H}_{v}$ then we define $\rho^{W}_{V}(x)$ to be the apex-point
$c_{e_{2}}$ in $\widehat{H}_{v}$.

Case 2: The last case we should consider is that $W=\widehat{H}_{w}$. Since
$V$ is properly nested in $W$, it follows that $V=\mathcal{R}_{v}$ for some
vertex $w$ where $v$ and $w$ are two adjacent vertices in $T$. Let $e$ denote
the edge $[w,v]$. We define $\rho^{V}_{W}$ to be the apex-point $c_{e}$ in
$\widehat{H}_{w}$, and thus $\operatorname{diam}(\rho_{W}^{V})=0<\xi$. Now, we
are going to define $\rho^{W}_{V}\colon\mathcal{C}W\to 2^{\mathcal{C}V}$ as
follows. Let $x$ be a point in
$\mathcal{C}W=\bigl{(}\widehat{H}_{w},d_{\widehat{H}_{w}}\bigr{)}$. If $x$ is
the apex-point $c_{e}$ then we define $\rho^{W}_{V}(x)$ to be
$\mathcal{R}_{v}$. Now we assume that $x$ is not the apex-point $c_{e}$. Let
$\bar{x}$ the projection of $x$ to the subspace $H_{v}$ of $\widehat{H}_{w}$.
Let $\mathcal{S}_{\bar{x}}$ be a strip in $B_{w}$ such that its projection
into $H_{w}$ is a shortest path from $\bar{x}$ to the boundary line
$\ell=F_{e}\cap H_{w}$. We define $\rho^{W}_{V}(x)$ to be the
$\mathcal{S}_{\bar{x}}\cap F_{e}\subset\mathcal{R}_{v}$. ∎

####  3.2.3. Orthogonality

We define orthogonality $\perp$ as the following:

  * •

We declare $\mathcal{R}_{v}\perp\widehat{H}_{v}$ for all vertex $v$ in $T$.

  * •

We declare $\mathcal{R}_{v}\perp\mathcal{R}_{w}$ whenever $v$ and $w$ are
adjacent vertices.

######  Lemma 3.7.

$\bigl{(}X,\Lambda,\perp\bigr{)}$ satisfies Orthogonality Axiom.

###### Proof.

Let $V$, $W$, and $U$ be elements in the index set $\Lambda$ such that
$V\sqsubseteq W$ and $W\perp U$. Note that $W\neq T$ since every element in
$\Lambda$ is nested in $T$. If $V=W$ then it is obvious that $V\perp U$. We
thus assume that $V$ is properly nested in $W$. It follows that
$W=\widehat{H}_{v}$ for some vertex $v$ (as no element in $\Lambda$ is nested
in $\mathcal{R}_{v}$, so $W$ could not be $\mathcal{R}_{v}$). Hence
$V=\mathcal{R}_{w}$ for some adjacent vertex $w$ of $v$. Since
$\mathcal{R}_{v}\perp\mathcal{R}_{w}$ by our definition above, it follows that
$V\perp U$. Also, it is clear from our definition that if $V\perp W$, then
$V,W$ are not $\sqsubseteq$–comparable.

For the rest of the proof, we are going to verify the statement in
Orthogonality Axiom (see Definition 2.7): ``for each $S\in\Lambda$, for each
$U\in\Lambda_{S}$ such that the set $\\{V\in\Lambda_{S}\,|\,V\perp
U\\}\neq\emptyset$ then there exists $W\in\Lambda_{S}$, $W\neq S$ so that
whenever $V\perp U$, $V\sqsubseteq S$ then we have $V\sqsubseteq W$''.

We observe that the element $S$ must be $T$. Indeed, suppose that $S\neq T$.
In this case then $S$ is either $\widehat{H}_{v}$ or $\mathcal{R}_{v}$ for
some vertex $v\in T$. When $S=\mathcal{R}_{v}$ then
$\Lambda_{S}=\\{\mathcal{R}_{v}\\}$. As $U$ and $V$ are in $\Lambda_{S}$, it
follows that the set $\\{V\in\Lambda_{S}\,\bigl{|}\,V\perp U\\}=\emptyset$.
Now we assume that $S=\widehat{H}_{v}$. As $U\in\Lambda_{S}$, we have that $U$
is either $\widehat{H}_{v}$ or $U=\mathcal{R}_{w}$ for some vertex $w$ that is
adjacent to $v$. If $U$ is $\widehat{H}_{v}$ then the set
$\\{V\in\Lambda_{S}:V\perp U\\}=\emptyset$. Now we consider
$U=\mathcal{R}_{w}$. Since $V\in\Lambda_{S}$, it follows that
$V=\widehat{H}_{v}$ or $V=\mathcal{R}_{a}$ where $a$ and $v$ are adjacent.
Note that $\widehat{H}_{v}$ is not orthogonal to $U=\mathcal{R}_{w}$, and
$V=\mathcal{R}_{a}$ is not orthogonal to $U=\mathcal{R}_{w}$ (since $a$ and
$w$ are not adjacent vertices). In other words, the set
$\\{V\in\Lambda_{S}:V\perp U\\}=\emptyset$.

Therefore, the element $S=T$, and hence $\Lambda_{S}=\Lambda$. Since
$U\in\Lambda_{S}$, it follows that $U$ is either $T$, or $\mathcal{R}_{v}$ or
$\widehat{H}_{v}$ for some vertex $v\in T$. Since no element in $\Lambda$ is
orthogonal to $T$, it follows that $U$ could not be $T$, otherwise the set
$\\{V\in\Lambda_{S}:V\perp U\\}=\emptyset$. When $U=\mathcal{R}_{v}$ then we
define $W$ to be $\widehat{H}_{v}$. Note that whenever $V\perp
U=\mathcal{R}_{v}$ and $V\sqsubseteq S=T$ then by our definition of $\perp$,
the element $V$ is either $\widehat{H}_{v}=W\sqsubseteq W$ or
$V=\mathcal{R}_{w}$ for some vertex $w$ that is adjacent to $v$. By definition
of $\sqsubseteq$ in Section 3.2.2, the element $V$ is nested in $W$. When
$U=\widehat{H}_{v}$ then we choose $W$ to be the $\mathcal{R}_{v}$ in the
index set $\Lambda$. It is obvious that if $V\perp U=\widehat{H}_{v}$ and
$V\sqsubseteq S=T$ then $V$ must be $\mathcal{R}_{v}=W\sqsubseteq W$. ∎

####  3.2.4. Transversality and consistency

If $V$ and $W$ are not orthogonal and neither is nested in the other, then we
say that $V$ and $W$ are transverse, denoted $V\pitchfork W$. We have the
following possibilities for $V$ and $W$.

Case 1: $V=\mathcal{R}_{v}$ and $W=\mathcal{R}_{w}$ for some vertices $v$ and
$w$ in $T$. We note that the distance between $v$ and $w$ in $T$ is at least
$2$. Otherwise, $V\perp W$ if $v$ and $w$ are adjacent and $V=W$ if $v=w$.

Case 2: $V=\mathcal{R}_{v}$ and $W=\widehat{H}_{w}$ for some vertices $v,w\in
T$. Again, we note that the distance between $v$ and $w$ is at least $2$.
Indeed, if $v$ and $w$ are adjacent vertices then $\mathcal{R}_{v}$ is nested
in $\widehat{H}_{w}$ by our definition which contradicts to the fact $V$ and
$W$ are transverse. If $v=w$ then $V=W$ that contradicts to the fact $V$ and
$W$ are transverse.

Case 3: $V=\widehat{H}_{v}$ and $W=\widehat{H}_{w}$ for some distinct vertices
$v,w\in T$.

Regarding Case 1: Let $e_{1}e_{2}\dots e_{k-1}e_{k}$ ($k\geq 2$) be the
geodesic edge path in the Bass-Serre tree $T$ connecting $v$ to $w$ with
$v=(e_{1})_{-}$ and $(e_{k})_{+}=w$. Let $F_{e_{k}}$ and $F_{e_{1}}$ be the
boundary planes (see Subsection 2.4) of the blocks $B_{w}$ and $B_{v}$
respectively. Let $\mathcal{S}_{e_{k-1}e_{k}}$ and $\mathcal{S}_{e_{1}e_{2}}$
be the given strips (see Subsection 2.4) in the blocks $B_{(e_{k})_{-}}$ and
$B_{(e_{1})_{+}}$ respectively.

We define

$P^{V}_{W}:=\mathcal{S}_{e_{k-1}e_{k}}\cap F_{e_{k}}$

and

$P^{W}_{V}:=\mathcal{S}_{e_{1}e_{2}}\cap F_{e_{1}}$

######  Lemma 3.8.

$V,W,P^{V}_{W},P^{W}_{V}$ defined in Case 1 above satisfying ($\clubsuit$ ‣
2.7), in Definition 2.7.

###### Proof.

Let $r>0$ be the constant given by Remark 3.2. We have that
$\operatorname{diam}P^{V}_{W}\leq r\leq\xi$ and
$\operatorname{diam}P^{W}_{V}\leq r\leq\xi$. As we have shown early, the
distance between $v$ and $w$ is at least $2$. Let $u=\pi(x)\in V(T)$ where
$\pi\colon X\to T$ be the index map given by Remark 2.10.

Suppose that $u$ does not belong to the geodesic $[v,w]$. If $v$ lies between
$u$ and $w$ then by the definition of $\pi_{W}$ (see Subsection 3.2.1) and
$P^{V}_{W}$, we see that $\pi_{W}(x)$ and $P^{V}_{W}$ are the same subset in
$\mathcal{C}W$. Similarly, if $w$ lies in $[v,u]$ then $\pi_{V}(x)$ and
$P^{W}_{V}$ are the same subset in $\mathcal{C}V$. By Remark 3.2, we have that
($\clubsuit$ ‣ 2.7) holds.

Now, we assume that $u$ belongs to $[v,w]$. Suppose that $d(u,v)\geq 2$ (resp.
$d(u,w)\geq 2$) then we have that $\pi_{V}(x)$ and $P^{W}_{V}$ are the same
subset in $\mathcal{C}V$ (resp, $\pi_{W}(x)$ and $P^{V}_{W}$ are the same
subset in $\mathcal{C}W$). It follows from Remark 3.2 that ($\clubsuit$ ‣ 2.7)
holds whenever $d(u,v)\geq 2$ or $d(u,w)\geq 2$.

The final case we should consider is that $d(v,w)=2$ and $u$ lies between
$[v,w]$. Let $\mathcal{S}:=\mathcal{S}_{[v,u],[u,w]}$ be the strip in $B_{u}$.
Since $x\in X=G(x_{0})$, and $x_{0}$ is in the boundary plane $F_{e_{0}}$, it
follows that $x$ must lie in a boundary plane of $B_{u}$. We denote this
boundary plane by $F_{e^{\prime}}$. If $e^{\prime}=[v,u]$ (the case
$e^{\prime}=[u,w]$ is similar) then $\pi_{W}(x)$ and $\mathcal{S}\cap
F_{[u,w]}$ are identical and their diameter is no more than $r$ in
$\mathcal{C}W$ by Remark 3.2. Now we assume that the edges $e$, $[v,u]$, and
$[u.w]$ are distinct.

We endow each $H_{v}$ with the HHS structure originating from the fact that
$H_{v}$ is hyperbolic relative to the collection of boundary lines
$\mathbb{L}_{v}$ (see Lemma 2.14). As a result, The Transversality Axiom and
the Consistency Axiom is applied to this HHS structure. In particular,

$\min\bigr{\\{}\pi_{\ell_{1}}(\ell,\ell_{2}),\pi_{\ell_{2}}(\ell_{1},\ell)\bigr{\\}}\leq\lambda$

for any boundary line $\ell\in\mathbb{L}_{u}$. It implies that ($\clubsuit$ ‣
2.7) holds. ∎

Regarding Case 2: Assume that $V=\mathcal{R}_{v}$ and $W=\widehat{H}_{w}$ for
some vertices $v,w\in T$. Since $V$ and $W$ are transverse, it follows that
$v\neq w$ (otherwise they are orthogonal). Note that $d(v,w)\geq 2$. Indeed,
if $v$ and $w$ are adjacent vertices then $\mathcal{R}_{v}$ is nested in
$\widehat{H}_{w}$ by our definition which contradicts to the fact $V$ and $W$
are transverse. Now, let $e_{1}e_{2}\dots e_{k-1}e_{k}$ be the geodesic edge
path in the Bass-Serre tree connecting $v$ to $w$ with $v=(e_{1})_{-}$ and
$(e_{k})_{+}=w$. We define

$P^{W}_{V}:=\mathcal{S}_{e_{1}e_{2}}\cap F_{e_{1}}\subset\mathcal{R}_{v}$

and

$P^{V}_{W}:=c_{e_{k}}$

the apex-point in $\widehat{H}_{w}$. Hence, subsets $P^{W}_{V}$ and
$P^{V}_{W}$ are bounded sets in $\mathcal{R}_{v}$ (see Remark 3.2) and
$\widehat{H}_{w}$ respectively. Therefore
$\operatorname{diam}(\rho^{V}_{W})\leq\xi$, and
$\operatorname{diam}(P^{W}_{V})\leq\xi$.

######  Lemma 3.9.

$V,W,P^{V}_{W},P^{W}_{V}$ defined in Case 2 above satisfying ($\clubsuit$ ‣
2.7), in Definition 2.7.

###### Proof.

Let $u=\pi(x)\in V(T)$ where $\pi\colon X\to T$ be the index map given by
Remark 2.10. If $v$ lies between $u$ and $w$ then by the definition of
$\pi_{W}$ in Subsection 3.2.1, we have that $\pi_{W}(x)$ is a point in the
boundary line $H_{w}\cap F_{e}$ where $e$ is the last edge in the geodesic
$[v,w]$. Since $P^{V}_{W}$ is the apex-point $c_{e}$, it follows that the
distance between $\pi_{W}(x)$ and $P^{V}_{W}$ in $\widehat{H}_{w}$ is bounded
above by $r$. Thus ($\clubsuit$ ‣ 2.7) holds. If $w$ lies in $[v,u]$, then
similar arguments as above show that $\pi_{W}(x)$ and $P^{W}_{V}$ are the same
subset in $\mathcal{C}V$, and thus ($\clubsuit$ ‣ 2.7) holds.

Now we assume that $u$ belongs to $[v,w]$. Let $e$ be the last edge in the
geodesic path $[v,w]$. Then $P^{V}_{W}$ is the apex-point $c_{e}$ in
$\widehat{H}_{w}$ and $\pi_{W}(x)$ is a point in the boundary line $H_{w}\cap
F_{e}$. Since the distance between $\pi_{W}(x)$ and $c_{e}$ in
$\widehat{H}_{w}$ is bounded above by $r$. It follows that ($\clubsuit$ ‣ 2.7)
holds. ∎

Regarding Case 3: Finally, suppose that $V=\widehat{H}_{v}$ and
$W=\widehat{H}_{w}$ for some vertices $v,w\in T$. Since $V$ and $W$ are
transverse so $v\neq w$. Let $e_{1}\dots e_{k-1}e_{k}$ be the geodesic edge
path in the Bass-Serre tree connecting $v$ to $w$ with $v=(e_{1})_{-}$ and
$(e_{k})_{+}=w$. We then define $P^{V}_{W}:=c_{e_{k}}$ and
$P^{W}_{V}:=c_{e_{1}}$ to be apex-points in $\widehat{H}_{w}$ and
$\widehat{H}_{v}$ respectively. Therefore
$\operatorname{diam}(\rho^{V}_{W})\leq\xi$,
$\operatorname{diam}(P^{W}_{V})\leq\xi$.

######  Lemma 3.10.

$V,W,P^{V}_{W},P^{W}_{V}$ defined in Case 3 above satisfying ($\clubsuit$ ‣
2.7) in Definition 2.7.

###### Proof.

$V=\widehat{H}_{v}$ and $W=\widehat{H}_{w}$ where $d(v,w)\geq 2$. The proof is
similar as the previous paragraphs, and we leave it to the reader. ∎

######  Lemma 3.11.

$V,W,P^{V}_{W},P^{W}_{V}$ satisfy ($\spadesuit$ ‣ 2.7) in Definition 2.7

###### Proof.

Let $u=\pi(x)$ where $\pi\colon X\to T$ be the index map given by Remark 2.10.

Case 1: Suppose that $W=T$. Since $V$ is nested in $W$, it follows that
$V=\mathcal{R}_{v}$ or $V=\widehat{H}_{v}$ for some vertex $v$ in $T$. We note
that if $d(u,v)\leq 2$ then

$d_{W}(\pi_{W}(x),\rho^{V}_{W})=d_{T}(\pi_{T}(x),\rho^{V}_{T})=d(u,v)\leq 2$

and thus ($\spadesuit$ ‣ 2.7) holds.

Now we assume that $d(u,v)>2$. Let $e$ and $e^{\prime}$ be two consecutive
edges in the geodesic $[u,v]$ such that $(e^{\prime})_{+}=v$,
$(e)_{+}=(e^{\prime})_{-}$. If $V=\mathcal{R}_{v}$ then by our definition of
$\pi_{V}$ (see Subsection 3.2.1), we have
$\pi_{V}(x)=\mathcal{S}_{e,e^{\prime}}\cap F_{e^{\prime}}$ is a bounded subset
in $\mathcal{C}V$ by Remark 3.2, and

$\rho_{V}^{W}(\pi_{W}(x))=\rho_{V}^{T}(\pi(x))=\rho_{V}^{T}(u)=\mathcal{S}_{e,e^{\prime}}\cap
F_{e^{\prime}}$

which is a bounded subset in $\mathcal{C}V$ by Remark 3.2; thus ($\spadesuit$
‣ 2.7) holds. If $V=\widehat{H}_{v}$ then $\pi_{V}(x)$ is a point in the
boundary line $H_{v}\cap F_{e^{\prime}}$ and $\rho^{W}_{V}(\pi_{W}(x))$ is the
apex-point $c_{e^{\prime}}$ in $\widehat{H}_{v}$. Since the distance between
$\pi_{V}(x)$ and $c_{e^{\prime}}$ is bounded above by $r$ in
$\widehat{H}_{v}$, it follows that ($\spadesuit$ ‣ 2.7) holds.

Case 2: $W=\widehat{H}_{w}$ for some vertex $w$ in $T$. Since $V$ is nested in
$W$, it follows that $V=\mathcal{R}_{v}$ for some vertex $v$ that is adjacent
to $w$. We note that $P^{V}_{W}$ is the apex-point $c_{e}$ in
$\widehat{H}_{w}$ where $e=[v,w]$.

When $v\in[u,w]$ then $\pi_{W}(x)$ is a point in the boundary line $H_{w}\cap
F_{e}$ of $H_{w}$. In $\widehat{H}_{w}$, this point and the apex-point $c_{e}$
have a distance that is bounded above by $r$, thus ($\spadesuit$ ‣ 2.7) holds.

When $w\in[v,u)$, let $e^{\prime}$ be the edge in the geodesic $[u,w]$ with
$(e^{\prime})_{+}=w$. We have
$\pi_{V}(x)=\pi_{\mathcal{R}_{v}}(x)=\mathcal{S}_{e^{\prime},e}\cap
F_{e}\subset\mathcal{R}_{v}$. By the definition of $\pi_{W}$, we have that
$\pi_{W}(x)$ is a point in a boundary line $\ell$ of $H_{w}$. Note that this
line is a subset of $F_{e^{\prime}}$. By Lemma 2.11,
$\rho^{W}_{V}(\pi_{W}(x))$ and $\mathcal{S}_{e^{\prime},e}\cap F_{e}$ are
within an uniform finite Hausdorff distance in the plane $F_{e}$, and thus
their distance is uniformly bounded with respect to the metric in
$\mathcal{R}_{v}$. Thus ($\spadesuit$ ‣ 2.7) holds. For the case $u=w$, then
since $x$ lies in the orbit $G(x_{0})$, so $x\in F_{f}$ for some edge $f$ with
$(f)_{+}=v$. The proof in this case is identical as the previous case. ∎

So far, we have the notation $\rho^{W}_{V}$ (a map from $\mathcal{C}W$ to
subsets of $\mathcal{C}V$) whenever $V$ is properly nested in $W$, and the
notation $P^{W}_{V}$ (a bounded subset of $\mathcal{C}V$) whenever
$V\pitchfork W$. For simplicity, we will write $\mathcal{P}^{W}_{V}$ to mean
either $\rho^{W}_{V}$ or $P^{W}_{V}$ when it is clear from the context.

######  Lemma 3.12.

$V,W,\mathcal{P}^{W}_{V},\mathcal{P}^{V}_{W}$ satisfy ($\diamondsuit$ ‣ 2.7)
in Definition 2.7.

###### Proof.

We will assume that $U$ is properly nested in $V$ since if $U=V$ then there is
nothing to show.

Assume that $V=T$, then $U$ is either $\mathcal{R}_{v}$ or $\widehat{H}_{v}$
for some vertex $v\in T$. The set of $W\in\Lambda$ satisfies either $V$ is
properly nested in $W$ or $V\pitchfork W$ and $W$ is not orthogonal to $U$ is
empty. The claim is vacuously true. Assume that $V=\mathcal{R}_{v}$ for some
vertex $v$, then the set of $U\in\Lambda$ that is properly nested in $V$ is
empty. The claim is vacuously true. Assume that $V=\widehat{H}_{v}$ for some
vertex $v$, then since $U$ is properly nested in $V$, it follows that
$U=\mathcal{R}_{w}$ for some vertex $w$ that is adjacent to $v$.

When $V$ is properly nested in $W$ then $W=T$. In this case, we note that
$\rho^{U}_{W}=\rho^{\mathcal{R}_{w}}_{T}=w$, and $\rho^{V}_{W}=v$, thus
$d_{W}(\rho^{U}_{W},\rho^{V}_{W})=d_{T}(w,v)=1\leq\kappa_{0}$. Hence,
($\diamondsuit$ ‣ 2.7) holds. When $\widehat{H}_{v}=V\pitchfork W$ and $W$ is
not orthogonal to $U=\mathcal{R}_{w}$. Then there are two cases we should
consider:

Case 1: $W=\mathcal{R}_{u}$ with $u\neq v,w$, $d(u,w)\geq 2$. Note that
$d(u,v)\geq 2$ (otherwise $d(u,v)=1$ then
$\mathcal{R}_{u}\sqsubseteq\widehat{H}_{v}$, so $W\sqsubseteq V$, a
contradiction). In this case, either $v\in[u,w]$ or $w\in[v,u]$. Let $\gamma$
be the geodesic in $T$ containing $[v,w]$ with the endpoint $u$. Let
$e,e^{\prime}$ be the two edges in $\gamma$ such that
$(e)_{+}=(e^{\prime})_{-}$ and $(e^{\prime})_{+}=u$. We have
$d_{W}(P^{U}_{W},P^{V}_{W})\leq\kappa_{0}$ since both $P^{U}_{W}$ and
$P^{V}_{W}$ are $\mathcal{S}_{e,e^{\prime}}\cap\mathfrak{f}_{e^{\prime}}$ that
is a bounded subset in $\mathcal{C}W$. Hence ($\diamondsuit$ ‣ 2.7) holds.

Case 2: $W=\widehat{H}_{u}$ where $u\neq v,w$ and $d(u,w)\geq 2$. Let $\gamma$
be the geodesic in $T$ containing $[v,w]$ with the endpoint $u$. Let $e$ be
the edge in $\gamma$ with $(e)_{+}=u$. Then
$d_{W}(P^{U}_{W},\rho^{V}_{W})\leq\kappa_{0}$ since $P^{U}_{W}$ and
$P^{V}_{W}$ both are the apex-point $c_{e}$ and the lemma is proven. ∎

####  3.2.5.  Finite complexity

It follows from the construction that the complexity of $X$ is $3$. Indeed, we
have that $V\sqsubseteq W\sqsubseteq T$ whenever $V=\mathcal{R}_{v}$,
$W=\widehat{H}_{w}$ for some adjacent vertices $v$ and $w$ in $T$.

####  3.2.6. Large Links

######  Lemma 3.13.

$(X,\Lambda)$ satisfies the Large Links Axiom.

###### Proof.

We endow each $H_{v}$ with the HHS structure originating from the fact that
$H_{v}$ is hyperbolic relative to the collection of boundary lines
$\mathbb{L}_{v}$ (see Lemma 2.14). As a result, Large Link Axiom is applied to
this HHS structure. Since there are finitely many vertices up to $G$–action,
there are uniform constants $\lambda\geq 1$, $L\geq 1$ such that for each
vertex $v\in T$ the following holds: Let $x$ and $y$ be two points in $H_{v}$,
and let $N^{\prime}=\lambda
d_{\widehat{H}_{v}}(\pi_{\ell}(x),\pi_{\ell}(x^{\prime}))+\lambda$. Then there
exists $\ell_{1},\ell_{2},\cdots,\ell_{\lfloor N^{\prime}\rfloor}$ such that
for all boundary lines $\ell\in\mathbb{L}_{v}$ then either $\ell=\ell_{i}$ for
some $i$, or $d_{\ell}(\pi_{\ell}(x),\pi_{\ell}(x^{\prime}))<L$. Also,
$d_{\widehat{H}_{v}}(\pi_{\widehat{H}_{v}}(x),P^{\ell_{i}}_{\widehat{H}_{v}})\leq
N^{\prime}$ for each $i$.

Let $r>0$ be the constant given by Remark 3.2. We could enlarge $\lambda$ if
necessary and assume that $\lambda>2r$. We are going to verify the following
statement in the Large Links Axiom of Definition 2.7. `` Let $W\in\Lambda$ and
let $x,x^{\prime}\in X$. Let $N=\lambda
d_{w}(\pi_{W}(x),\pi_{W}(x^{\prime}))+\lambda$. Then there exists
$A_{1},A_{2},\cdots,A_{\lfloor N\rfloor}\in\Lambda_{W}\backslash\\{W\\}$ such
that for all $A\in\Lambda_{W}\backslash\\{W\\}$ then either $A=A_{i}$ for some
$i$, or $d_{A}(\pi_{A}(x),\pi_{A}(x^{\prime}))<E$. Also,
$d_{W}(\pi_{W}(x),P^{A_{i}}_{W})\leq N$ for each $i$ ''.

Recall that $x_{0}$ is in the boundary plane $F_{e_{0}}$ for some edge
$e_{0}$. Since $X$ is the orbit $G(x_{0})$ and the collection of boundary
planes is $G$–equivariant, it follows that $x$ and $x^{\prime}$ belong to
boundary planes $F_{e}$ and $F_{e^{\prime}}$ for some edges $e$ and
$e^{\prime}$.

We first observe that if $W=\mathcal{R}_{v}$ for some vertex $v\in V(T)$ then
there is nothing to show since the set
$\Lambda_{W}\backslash\\{W\\}=\emptyset$. Hence, in the rest of the proof, we
only need to consider the cases $W=\widehat{H}_{v}$ and $W=T$.

Case 1: Suppose that $W=\widehat{H}_{v}$ for some vertex $v\in T$. Let $A$ be
an arbitrary element in $\Lambda_{W}\backslash\\{W\\}$. As $A$ is nested in
$W=\widehat{H}_{v}$ and $A\neq W$, it follows that $A=\mathcal{R}_{w}$ for
some vertex $w$ which is adjacent to $v$.

Case 1.1: $v$ is a vertex of both edges $e$ and $e^{\prime}$. Without loss of
generality, we assume that $(e)_{+}v$ and $(e^{\prime})_{+}=v$. In this case,
we note that $F_{e}$ and $F_{e^{\prime}}$ are two boundary planes in the
vertex space $B_{v}=H_{v}\times\mathbb{R}$. Let $\ell_{e}$ and
$\ell_{e^{\prime}}$ be the boundary lines in $H_{v}$ corresponding to edges
$e$ and $e^{\prime}$ respectively. Let $\bar{x}$ and $\bar{x^{\prime}}$ be the
projection point of $x$ and $x^{\prime}$ into the boundary line $\ell_{e}$ and
$\ell_{e^{\prime}}$ respectively.

Each boundary line $\ell_{j}$ with $j\in\\{1,\dots,k\\}$ is associated to an
edge $e_{j}$ with $(e_{j})_{+}=v$. We denote $v_{j}$ by $(e_{j})_{-}$. Let
$A_{j}:=\mathcal{R}_{v_{j}}$ with $j\in\\{1,\dots,k\\}$. If the vertex $w$ is
equal to $v_{j}$ for some $j\in\\{1,\dots,k\\}$ then $A=A_{j}$. Now we assume
that $w\notin\\{v_{1},\dots,v_{k}\\}$. Let $\ell$ be a boundary line of
$H_{v}$ associated to the edge $[v,w]$. We have that
$d_{\ell}(x,x^{\prime})\leq L$, otherwise $w=v_{j}$ for some $j$. Let $A>0$ be
the constant given by Lemma 3.4. We have

$\displaystyle d_{A}(\pi_{A}(x),\pi_{A}(x^{\prime}))$
$\displaystyle=diam\bigl{(}\pi_{\mathcal{R}_{w}}(\mathcal{R}_{(e)_{-}},\mathcal{R}_{(e^{\prime})_{-}})\bigr{)}$
$\displaystyle\leq Ad_{\ell}(\ell_{e},\ell_{e^{\prime}})+A$ $\displaystyle\leq
Ad_{\ell}(x,x^{\prime})+A\leq AL+A\leq E$

It follows from the second paragraph above that with each $i=1,\dots,k$, we
have

$\displaystyle d_{W}(\pi_{W}(x),\rho^{\ell_{i}}_{W})$
$\displaystyle=d_{\widehat{H}_{v}}\bigl{(}\pi_{\widehat{H}_{v}}(x),\rho^{A_{i}}_{\widehat{H}_{v}}\bigr{)}\leq
N^{\prime}\leq N$

Case 1.2: Assume that $(e)_{+}=v$ and $v$ is not a vertex of $e^{\prime}$ (the
case $(e^{\prime})_{+}=v$ and $v$ is not a vertex of $e$ is proved similarly).

When $v$ lies between $(e)_{-}$ and $e^{\prime}$ then the proof is identical
as Case 1.1.

When $(e)_{-}$ lies between $v$ and $e^{\prime}$ then we define
$A_{1}=A_{2}=\dots,=A_{\lfloor N\rfloor}=\mathcal{R}_{(e)_{-}}$. We note that
$d_{W}\bigl{(}\pi_{W}(x),c_{e}\bigr{)}=d_{\widehat{H}_{v}}\bigl{(}\pi_{\widehat{H}_{V}}(x),c_{e}\bigr{)}\leq
r$, and
$d_{\widehat{H}_{v}}\bigl{(}P^{\mathcal{R}_{(e)_{-}}}_{\widehat{H}_{v}},c_{e}\bigr{)}\leq
r$, thus

$d_{W}(\pi_{W}(x),P^{A_{i}}_{W})\leq 2r\leq N$

Let $A$ be an arbitrary element in $\Lambda_{W}\backslash\\{W\\}$. As $A$ is
nested in $W=\widehat{H}_{v}$ and $A\neq W$, it follows that
$A=\mathcal{R}_{w}$ for some vertex $w$ which is adjacent to $v$. If
$A\notin\\{A_{1},\dots,A_{\lfloor N\rfloor}\\}$ then $w\neq(e)_{-}$. Since
$\pi_{A}(x^{\prime})=\pi_{\mathcal{R}_{w}}(x)$ is $\mathcal{S}_{[v,w],e}\cap
F_{[v,w]}\subset\mathcal{R}_{w}$, and $\pi_{A}(x)$ and
$\mathcal{S}_{[v,w],e}\cap F_{[v,w]}$ are close within a uniform Hausdorff
distance in the plane $F_{[v,w]}$, it follows that the distance of two bounded
subsets (with respect to the metric of $\mathcal{R}_{w}$) $\pi_{A}(x)$ and
$\pi_{A}(x^{\prime})$ no more than $E$.

Case 1.3: Suppose that $F_{e}$ and $F_{e^{\prime}}$ are not boundary planes of
$B_{v}$.

Denote the last edge in the geodesic from $e$ to $v$ is $f$, and the last edge
in the geodesic from $e^{\prime}$ to $v$ is $f^{\prime}$. Note that both
$(f)_{+}=(f^{\prime})_{+}=v$. We will consider the collection
$\\{A_{i}\\}_{i=1}^{\lfloor
N\rfloor}=\\{\mathcal{R}_{(f)_{-}},\mathcal{R}_{(f^{\prime})_{-}}\\}$.
Similarly as in Case 1.2, if $A$ is nested in $W$ and $A\neq W$ then
$A=\mathcal{R}_{w}$ for some vertex $w$ that is adjacent to $v$. If
$A\notin\\{A_{1},\dots,A_{\lfloor N\rfloor}\\}$ then
$w\notin\\{(f)_{-},(f^{\prime})_{-}\\}$. It follows from Lemma 3.4 that
$d_{A}\bigl{(}\pi_{A}(x),\pi_{A}(x^{\prime})\bigr{)}<E$.

For each $i$, since $\pi_{W}(x)$ is a point in the boundary line $\ell_{f}$ of
$H_{v}$, and $P^{A_{i}}_{W}$ is the apex-point $c_{f}$, and thus
$d_{W}(\pi_{W}(x),P^{A_{i}}_{W})\leq 2r\leq N$.

Case 2: $W$ is the Bass-Serre tree $T$. Recall that $x\in F_{e}$ and
$x^{\prime}\in F_{e^{\prime}}$.

Let $\alpha$ be the geodesic in $T$ joining the edge $e$ to $e^{\prime}$. Let

$\\{A_{1},A_{2},\cdots,A_{\lfloor
N\rfloor}\\}=\\{\mathcal{R}_{v},\widehat{H}_{v}\,\bigl{|}\,v\,\text{is a
vertex in}\,\alpha\\}$

Since $A$ is properly nested in $W=T$, it follows that $A=\mathcal{R}_{w}$ or
$\widehat{H}_{w}$ for some vertex $w$ in $T$.

If $A\notin\\{A_{1},A_{2},\cdots,A_{\lfloor N\rfloor}\\}$ then
$w\notin\alpha$. Using the same argumenrts as in the previous subcase, we have
that $d_{A}(\pi_{A}(x),\pi_{A}(x^{\prime}))<E$. For each
$i\in\\{1,\cdots,\lfloor N\rfloor\\}$ then $A_{i}$ is either $\mathcal{R}_{v}$
or $\widehat{H}_{v}$ for some vertex $v\in\alpha$. Let $\pi\colon X\to T$ be
the index map given by Remark 2.10. Recall that we define the projection
$\pi_{T}$ given by $\pi_{T}(x)=\pi(x)$ for any vertex $x\in T$. Denote
$\alpha(0)$ and $\alpha(1)$ be the initial and terminal vertices of the
geodesic $\alpha$. Since $\alpha$ is a geodesic joining $e$ to $e^{\prime}$
and $x\in F_{e}$, $x^{\prime}\in F_{e^{\prime}}$, it follows that
$d(\pi(x),\alpha(0))\leq 2$ and $d(\pi(x^{\prime}),\alpha(1))\leq 2$. As
$\rho^{A_{i}}_{W}=\rho^{A_{i}}_{T}=v$, we have

$\displaystyle d_{W}\bigl{(}\pi_{W}(X),\rho^{A_{i}}_{W}\bigr{)}$
$\displaystyle=d_{T}\bigl{(}\pi(x),v\bigr{)}\leq
1+d_{T}\bigl{(}\alpha(0),\alpha(1)\bigr{)}$ $\displaystyle\leq
5+d_{T}\bigl{(}\pi(x),\pi(x^{\prime})\bigr{)}$ $\displaystyle\leq\lambda
d_{T}\bigl{(}\pi(x),\pi(x^{\prime})\bigr{)}+\lambda$ $\displaystyle=\lambda
d_{W}\bigl{(}\pi_{W}(x),\pi_{W^{\prime}}(x^{\prime})\bigr{)}+\lambda=N$

The lemma is proved. ∎

####  3.2.7. Bounded Geodesic Image

######  Lemma 3.14.

$(X,\Lambda)$ satisfies Bounded Geodesic Image Axiom.

###### Proof.

We endow each $H_{v}$ with the HHS structure originating from the fact that
$H_{v}$ is hyperbolic relative to the collection of boundary lines
$\mathbb{L}_{v}$ (see Lemma 2.14). As a result, Bounded Geodesic Image Axiom
is applied to this HHS structure. Since there are finitely many vertices up to
$G$–action, there exists a constant $E>0$ such that properties in Bounded
Geodesic Image Axiom hold for every vertices $v\in T$. Let $r>0$ be the
constant given by Remark 3.2. We could enlarge $E$ so that $E>r$.

We are going to verify ($\oplus$ ‣ 2.7) in the Definition 2.7. Let $\gamma$ be
a geodesic in $\mathcal{C}W$. Let $\gamma_{-}$ and $\gamma_{+}$ be the initial
and terminal points of $\gamma$. We consider the following cases:

Case 1: Suppose that $W=T$. Since $V$ is properly nested in $W$, it follows
that $V$ is either $\mathcal{R}_{v}$ or $\widehat{H}_{v}$ for some vertex $v$
in $T$. There are two subcases we will consider: $d(v,\gamma)\geq 2$ or
$d(v,\gamma)\leq 1$. If $d(v,\gamma)\geq 2$ then the two geodesics
$[\gamma_{-},v]$ and $[\gamma_{+},v]$ share the last common two consecutive
edges, denoted by $e_{1},e_{2}$. When $V=\mathcal{R}_{v}$ then it follows from
the definition of $\rho^{W}_{V}$ that
$\rho^{W}_{V}(x)=\mathcal{S}_{e_{1},e_{2}}\cap
F_{e_{2}}\subset\mathcal{R}_{v}$ for any $x\in\gamma$. Since
$\mathcal{S}_{e_{1},e_{2}}\cap F_{e_{2}}$ is a bounded subset in
$\mathcal{R}_{v}$ with the diameter no more than $r$ (see Remark 3.2), it
follows that $\operatorname{diam}_{\mathcal{C}V}(\rho^{W}_{V}(\gamma))\leq E$.
If $V=\widehat{H}_{v}$ then $\rho^{W}_{V}(x)$ is the apex-point $c_{e_{2}}$
for any $x\in\gamma$. Hence ($\oplus$ ‣ 2.7) is verified since
$\operatorname{diam}_{\mathcal{C}V}(\rho^{W}_{V}(\gamma))\leq E$. If
$d(v,\gamma)\leq 1$ then
$\gamma\cap\mathcal{N}_{2}(\rho^{V}_{W})=\gamma\cap\mathcal{N}_{2}(v)\neq\emptyset$.
Thus ($\oplus$ ‣ 2.7) is verified.

Case 2: Suppose that $W=\mathcal{R}_{v}$ for some vertex $v\in T$. Then there
is nothing to show since there is no element $V\in\Lambda$ such that $V$ is
properly nested in $W$.

Case 3: Suppose that $W=\widehat{H}_{v}$. Since $V$ is properly nested in $W$,
it follows that $V=\mathcal{R}_{w}$ where $w$ and $v$ are adjacent vertices.
Let $e$ denote the edge $[v,w]$, and let $\ell$ be the boundary line in
$H_{v}$ associated to the edge $e$. Since ($\oplus$ ‣ 2.7) holds for
hierarchically hyperbolic structure of $H_{v}$, we have that

$\operatorname{diam}_{\ell}(\rho^{\widehat{H}_{v}}_{\ell}(\gamma))\leq
E\,\,\text{or}\,\,\gamma\cap\mathcal{N}_{E}(\rho^{\ell}_{\widehat{H}_{v}})\neq\emptyset$

Note that $\operatorname{diam}_{\ell}(\rho^{\widehat{H}_{v}}_{\ell}(\gamma))$
and
$\operatorname{diam}_{\mathcal{R}_{w}}(\rho^{\widehat{H}_{v}}_{\ell}(\gamma))$
are within a uniform bounded distance, and $\rho^{\ell}_{\widehat{H}_{v}}$ and
$(\rho^{\mathcal{R}_{W}}_{\widehat{H}_{v}}$ are the apex point $c_{e}$. Thus
it follows that there exists a constant $E^{\prime}$ which does not depend on
$v$, $\ell$, and $\gamma$ such that

$\operatorname{diam}_{\mathcal{R}_{w}}(\rho^{\widehat{H}_{v}}_{\ell}(\gamma))\leq
E^{\prime}\,\,\text{or}\,\,\gamma\cap\mathcal{N}_{E^{\prime}}(\rho^{\mathcal{R}_{W}}_{\widehat{H}_{v}})\neq\emptyset$

Replacing $E$ by $\max\\{E,E^{\prime}\\}$ if necessary, we have that ($\oplus$
‣ 2.7) holds. ∎

####  3.2.8. Partial Realization

######  Lemma 3.15.

$(X,\Lambda)$ satisfies Partial Realization Axiom.

###### Proof.

Since $\\{V_{i}\\}$ be a family of pairwise orthogonal elements of $\Lambda$.
It follows from the definition of orthogonality (see Section 3.2.3) that the
collection $\\{V_{i}\\}$ is either $\\{\mathcal{R}_{v},\widehat{H}_{v}\\}$ for
some vertex $v\in T$ or $\\{V_{i}\\}=\\{\mathcal{R}_{v},\mathcal{R}_{w}\\}$
for some adjacent vertices $v$ and $w$ in $T$. In other words, the collection
$\\{V_{i}\\}$ consists only two elements $\\{V_{1},V_{2}\\}$.

Let $p_{1}$ and $p_{2}$ be two arbitrary elements in $\pi_{V_{1}}(X)$ and
$\pi_{V_{2}}(X)$. We are going to define a point $x\in X$ so that it satisfies
the conditions in Partial Realization Axiom as the following.

Case 1: Assume that $V_{1}=\mathcal{R}_{v}$ and $V_{2}=\widehat{H}_{v}$. Let
$r>0$ be a sufficiently large constant such that it applies to both Remark 3.2
and Remark 2.13. We note that $p_{2}$ is a point in $H_{v}$. By Remark 3.2,
there exists a point $\bar{x}$ in a boundary line $\ell$ of $H_{v}$ such that
$d_{H_{v}}(p_{2},\bar{x})\leq r$. Let $x$ be a point in
$B_{v}=H_{v}\times\mathbb{R}$ such that its projection into $H_{v}$ is
$\bar{x}$ and its $\mathbb{R}$–coordinate is the same as $p_{1}$.

Case 2: Assume that $V_{1}=\mathcal{R}_{v}$ and $V_{2}=\widehat{H}_{w}$ where
$v$ and $w$ are adjacent vertices. Let $e$ denote the edge $[v,w]$. Since
$p_{1}\in\pi_{\mathcal{R}_{v}}(X)$ and $p_{2}\in\pi_{\mathcal{R}_{w}}(X)$, it
follows that they belong to boundary planes of $B_{v}$ and $B_{w}$ (this
follows from the definition of $\pi_{\mathcal{R}_{v}}$ and
$\pi_{\mathcal{R}_{w}}$). Choose $q_{1}$ and $q_{2}$ to be points in the plane
$F_{e}$ such that the $\mathbb{R}$–coordinate of $q_{1}$ in
$B_{v}=H_{v}\times\mathbb{R}$ is the same as of $p_{1}$ and the
$\mathbb{R}$–coordinate of $q_{2}$ in $B_{w}=H_{w}\times\mathbb{R}$ is the
same as of $p_{2}$. Let $\ell$ be the line in the plane $F_{e}$ that is
parallel to $\mathbb{R}$–factor of $B_{v}$ and passes through $q_{1}$. Let
$\ell^{\prime}$ be the line in the plane $F_{e}$ that is parallel to
$\mathbb{R}$–factor of $B_{w}$ and passes through $q_{2}$. These two lines
$\ell$ and $\ell^{\prime}$ intersect at a point, and we will denote this point
by $x$.

Verifying the point $x$ satisfies the conditions in Partial Realization uses
similar arguments as in Section 3.2.4 and is hence omitted to avoid
redundancy. ∎

####  3.2.9. Uniqueness

######  Lemma 3.16.

$(X,\Lambda)$ satisfies Uniqueness Axiom.

###### Proof.

We are going to verify the following statement. For each $k\geq 0$, there
exists $\Theta=\Theta(k)$ such that for any $x,y$ in $X$, $d(x,y)\geq\Theta$
then there exists $V\in\Lambda$ such that $d_{V}(x,y)\geq k$.

We endow $H_{v}$ with the HHS structure originating from the fact that $H_{v}$
is hyperbolic relative to the collection of boundary lines $\mathbb{L}_{v}$
(see Lemma 2.14). As a result, Uniqueness Axiom is applied to this HHS
structure. Since there are finitely many vertices up to $G$–action, there
exists a constant $\xi(k)>0$ such that for each vertex $v$, for any two points
$\bar{x}$ and $\bar{y}$ in $H_{v}$ with $d_{H_{v}}(\bar{x},\bar{y})\geq\xi(k)$
then either $d_{\widehat{H}_{v}}(\bar{x},\bar{y})\geq k$ or
$d_{\ell}(\bar{x},\bar{y})\geq k$ for some boundary line
$\ell\in\mathbb{L}_{v}$.

Let $\mu>1$ be the constant given by Lemma 4.6 in [NY20]. Let $\Theta$ be a
sufficiently large constant such that

$\Theta\geq\mu\bigl{(}\mu+k(4\sqrt{2}(k+3))\bigr{)}+4\sqrt{2}\xi(k)$

Let $\pi\colon X\to T$ be the index map given by Remark 2.10.

Case 1: Two vertices $\pi(x)$ and $\pi(y)$ has distance at least $k$ in $T$.
Recall that we define the projection $\pi_{T}=\pi$. In this case, we will
choose $V$ to be $T$. It follows that
$d_{V}(x,y)=d_{T}(x,y):=d_{T}(\pi_{T}(x),\pi_{T}(y))=d_{T}(\pi(x),\pi(y))\geq
k$.

Case 2: Two vertices $\pi(x)$ and $\pi(y)$ has distance at most $k$ in $T$.

Suppose that $\pi(x)=\pi(y)$. Let $v$ be the vertex $\pi(x)=\pi(y)$. Then we
have that $x\in Y_{\pi(x)}=Y_{v}$ and $y\in Y_{\pi(y)}=Y_{v}$. Let $\gamma$ be
a geodesic in $B_{v}=H_{v}\times\mathbb{R}$ connecting $x$ to $y$. Let
$\gamma_{1}$ and $\gamma_{2}$ be the projections of $\gamma$ into factors
$H_{v}$ and $\mathbb{R}$ of $B_{v}$ respectively. Since $d(x,y)\geq\Theta$, it
follows that

$\operatorname{Len}(\gamma_{1})+\operatorname{Len}(\gamma_{2})\geq\frac{\Theta}{\sqrt{2}}$

Hence either $\operatorname{Len}(\gamma_{1})$ or
$\operatorname{Len}(\gamma_{2})$ is greater than or equal to
$\frac{\Theta}{4\sqrt{2}}$. If
$\operatorname{Len}(\gamma_{2})\geq\frac{\Theta}{4\sqrt{2}}$ then we will
choose $V=\mathcal{R}_{v}$. If

$\operatorname{Len}(\gamma_{1})\geq\frac{\Theta}{4\sqrt{2}}\geq\xi(k),$

i.e, $d_{H_{v}}((\gamma_{1})_{-},(\gamma_{1})_{+})\geq\xi(k)$, then either
$d_{\widehat{H}_{v}}(\bar{x},\bar{y})\geq k$ or $d_{\ell}(\bar{x},\bar{y})\geq
k$ for some boundary line $\ell\in\mathbb{L}_{v}$. In the first case, we will
let $V=\widehat{H}_{v}$. In the later case, let $e$ be the edge in $T$ with
$(e)_{-}=v$ such that $\ell$ is associated to $e$. We then let
$V=\mathcal{R}_{w}$ where $w=(e)_{+}$.

Now we assume that $\pi(x)\neq\pi(y)$. Let $e_{1}\cdots e_{n}$ be the geodesic
edge path connecting $\pi(x)$ to $\pi(y)$ and let
$p_{i}=\mathcal{S}_{e_{i-1}e_{i}}\cap\mathcal{S}_{e_{i}e_{i+1}}$ be the
intersection point of adjacent strips, where $e_{0}:=x$ and $e_{n+1}:=y$.
Hence we have the following path which is a concatenation of geodesics:

$\gamma:=[p_{0},p_{1}][p_{1},p_{2}]\cdots[p_{{n-1}},p_{n}][p_{n},p_{n+1}]$

where $p_{0}:=x$ and $p_{n+1}:=y$. Since we assume that
$d_{T}(\pi(x),\pi(y))\leq k$, it follows that $n\leq k+2$.

By Lemma 4.6 in [NY20], the path $\gamma$ is a $(\mu,\mu)$–quasi-geodesic. It
follows that $d_{X}(x,y)/\mu-\mu\leq\operatorname{Len}(\gamma)$, and hence
$\Theta/\mu-\mu\leq\operatorname{Len}(\gamma)=\sum_{i=0}^{n}\operatorname{Len}([p_{i},p_{i+1}])$
as $d(x,y)\geq\Theta$. It implies that there exists $i_{0}\in\\{0,\cdots,n\\}$
such that

$\frac{\Theta/\mu-\mu}{2(n+1)}\leq\operatorname{Len}([p_{i_{0}},p_{{i_{0}}+1}])$

Let $\alpha$ and $\beta$ be the projections of $[p_{i_{0}},p_{{i_{0}}+1}]$
into the factors $H_{v_{i_{0}}}$ and $\mathbb{R}$ of $B_{v_{i_{0}}}$
respectively (here $v_{i_{0}}:=e_{i_{0}}\cap e_{i_{0}+1}$). The above
inequality implies

$\frac{\Theta/\mu-\mu}{2\sqrt{2}(n+1)}\leq\operatorname{Len}(\alpha)+\operatorname{Len}(\beta)$

Thus either $\operatorname{Len}(\alpha)$ or $\operatorname{Len}(\beta)$ is
greater than or equal to $\frac{\Theta/\mu-\mu}{2\sqrt{2}(n+1)}$. We choose
$V$ similarly as above (case $\pi(x)=\pi(y)$). ∎

###### Proof of Proposition 2.14.

The proof is the combination of the previous lemmas in this section. ∎

##  4\. Appendix: Poisson boundaries of hierarchically hype rbolic spaces

Let $G$ be a group that acts geometrically on a hierarchically hyperbolic
space (Definition 2.7). Such a group we refer to as an HHS group, and note
that conjecturally it is not equivalent to being a hierarchically hyperbolic
group. In this section, we show that for an appropriate choice of $\kappa$
that only depends on the HHS group $G$, the $\kappa$-Morse boundary of $G$
serves as a topological model for the Poisson boundaries of the pair
$(G,\mu)$, where $\mu$ is a finitely-supported , non-elementary, generating
measure.

We need to show that a generic sample path of such a random walk sublinearly
tracks a $\kappa$-Morse quasi-geodesic ray. We will do so by showing that, in
fact, the limiting quasi-geodesic ray is $\kappa$-weakly
contracting((Definition 5.3 in [QRT20]). This proof follows Section 8 of
[QRT20] where the same result is claimed for mapping class groups.

Without loss of generality, equip $X$ with a base-point ${\mathfrak{o}}$ and
we use ${\mathfrak{o}}_{S}$ to denote a point in $\pi_{S}({\mathfrak{o}})$.
Fix ${\mathfrak{o}}_{S}$ once and for all.

### The hierarchy of geodesics

Let $S$ denote the maximal element in $\Lambda$ and recall $\mathcal{C}Y$
denoted the associated hyperbolic spaces for each $Y\in\Lambda$. To every pair
of points $x,y\in G$ one can associate a _hierarchy of geodesics_ , which are
a set of geodesic segments in each $\mathcal{C}Y$ that connects $\pi_{Y}(x)$
to $\pi_{Y}(y)$. Hence we also write $H(x,y)=\\{[x,y]_{Y}\\}$ for specific
given $Y\in\Lambda$. Given $H(x,y)=\\{[x,y]_{Y}\,|Y\in\Lambda\\}$, a
realization of a hierarchy $H(x,y)$ is a uniform quasi-geodesic segment
$\mathcal{G}(x,y)$ in $G$ connecting $x$ to $y$ where, for any element of
$Y\in\Lambda$, the projection of $\mathcal{G}(x,y)$ to $\mathcal{C}Y$ is
contained in a uniformly bounded neighborhood of the geodesic segment
$[x,y]_{Y}$. The set of all realizations we denote $\mathcal{G}(x,y)$.

We can also replace $x$ or $y$ with elements of $G$ whose projection to
$\mathcal{C}S$ are infinite quasi-geodesic rays.

[We start with a (tight) geodesic $[x,\xi)_{S}$ in $\mathcal{C}S$ and build
$H(x,\xi)$, the same as before replacing, for every element $Y\in\Lambda$,
$\pi_{Y}(y)$ with $\pi_{Y}(\xi)$. The realization $\mathcal{G}(x,\xi)$ of
$H(x,\xi)$ is then a uniform quasi-geodesic in $G$ starting from $x$ such that
the _shadow_ of $\mathcal{G}(x,\xi)$ in $\mathcal{C}S$ converges to $\xi$. ]

######  Definition 4.1 (Centers).

Since elements of $\Lambda$ are uniformly hyperbolic spaces, let them all be
$D$–hyperbolic. There exists a constant $\delta(D)$ such that for any three
points $x,y,z$, the intersection of the $\delta(D)$–neighbourhood of the
geodesic segments $[x,y],[y,z]$ and $[x,z]$ are non-empty. Given any
$y\in\Lambda$, we use $ctr_{Y}(x,y,z)$ to denote any point in the said
intersection. It follows from the construction of HHS that there exists a
constant $D^{\prime}$ such that: Given $x,y,z\in G$, for any $Y$ there exists
a point $\eta\in G$ such that, for any $Y\subseteq S$, we have

(3) $d_{Y}\big{(}\eta_{Y},\operatorname{ctr}_{Y}(x,y,z)\big{)}\leq
D^{\prime}.$

We call $\eta$ the _center_ of $x$, $y$ and $z$ and we denote it by
$\operatorname{ctr}(x,y,z)$.

We use the hierarchy paths to show:

######  Proposition 4.2.

Let $p$ be the complexity of the hierarchy of $G$. For any $x,y\in X$, assume
that $d_{Y}(x,y)\leq E$ for all $Y\neq S,Y\in\mathcal{G}$ and some $E>1$. Then
we have

$d_{G}(x,y)\prec d_{S}(x,y)\cdot E^{p}.$

###### Proof.

In view of the Distance Formula, we need to show

$|H(x,y)|\prec d_{S}(x,y)\cdot E^{p}.$

The restriction of $H(x,y)$ to a element of $\Lambda$ is again a hierarchy
which we denote with $H_{Y}(x,y)$. Let $\alpha$ denotes minimal elements in
the poset $\Lambda$, and let We check the statement of Proposition 4.2
inductively. When $p=1$ the statement is vacuously true. Now let $p\geq 2$. By
induction, that for every element of $\Lambda$, the hierarchy $H_{Y}(x,y)$
satisfies $|H_{Y}(x,y)|\prec d_{Y}(x,y)\cdot E^{p-1}$. We have

$\displaystyle|H(x,y)|$
$\displaystyle\prec\sum_{\alpha\in[x,y]_{S}}\left(\big{|}[x,y]_{\alpha}\big{|}+\sum_{\alpha\sqsubset
Y\sqsubset S}|H_{Y}(x,y)|\right)$
$\displaystyle\prec\big{|}[x,y]_{S}\big{|}\cdot(E+2d_{Y}(x,y)\cdot E^{p-1}).$

But $d_{Y}(x,y)\leq E$ and $\big{|}[x,y]_{S}\big{|}\prec d_{S}(x,y)$, thus by
the Distance Formula $|H(x,y)|\prec d_{S}(x,y)\cdot E^{p}$. ∎

Note that, once again we can replace each of $x,y,z$ with an element
$\xi\in\partial\mathcal{C}S$. That is, $\operatorname{ctr}(x,y,\xi)$ is a
well-defined element of $G$. From now on, we will denote as ${\mathfrak{o}}$
the identity element in $G$, which will function as base point.

######  Definition 4.3.

(Projections in HHS) Let $D$ be given from 3, and let $\xi$ be an element of
$G$ whose shadow in $\mathcal{C}S$ is an infinite diameter, quasi-geodesic ray
in $\partial\mathcal{C}S$. We define a _$D$ -cloud of a ray in the direction
of $\xi$_ to be

$\mathcal{Z}({\mathfrak{o}},\xi):=\Big{\\{}z\in G\ \big{|}\
d_{\mathcal{C}Y}\big{(}z_{Y},[{\mathfrak{o}},\xi)_{Y}\big{)}\leq
D\quad\forall\,Y\Big{\\}}.$

By construction, the realization $\mathcal{G}({\mathfrak{o}},\xi)$ of the
hierarchy $H({\mathfrak{o}},\xi)$ is contained in
$\mathcal{Z}({\mathfrak{o}},\xi)$. Fixing $\xi\in\partial\mathcal{C}S$, we
define a projection map to the cloud:

$\displaystyle\Pi_{\xi}\colon\thinspace
G\to\mathcal{Z}({\mathfrak{o}},\xi)\qquad\text{where}\qquad\Pi_{\xi}(x):=\operatorname{ctr}({\mathfrak{o}},x,\xi),\qquad
x\in G.$

We now check that $\Pi_{\xi}$ is a $\kappa$-projection according to Definition
2.1.

######  Lemma 4.4.

Let $\xi\in G$ be an element whose shadow in $\mathcal{C}S$ is an infinite
diameter, quasi-geodesic ray in $\partial\mathcal{C}S$. , the map $\Pi_{\xi}$
is coarsely Lipschitz with respect to $d_{G}$. Furthermore, if
$x\in\mathcal{Z}({\mathfrak{o}},\xi)$, then $d_{G}(x,\Pi_{\xi}(x))$ is
uniformly bounded. As a consequence, $\Pi_{\xi}$ is a $\kappa$-projection.

###### Proof.

Consider points $x,x^{\prime}\in G$ where $d_{w}(x,x^{\prime})\leq 1$. Then
$x(\theta)$ and $x^{\prime}(\theta)$ have a uniformly bounded intersection
number, which implies that there exists a uniform constant $C_{1}>0$ such that

$\forall\,Y,\qquad d_{Y}(x_{Y},x^{\prime}_{Y})\leq C_{1}.$

Let $\eta:=\operatorname{ctr}({\mathfrak{o}},x,\xi)$ and
$\eta^{\prime}:=\operatorname{ctr}({\mathfrak{o}},x^{\prime},\xi)$. Since
$\mathcal{C}Y$ is hyperbolic, the dependence of $\eta_{Y}$ on $x_{Y}$ is
Lipschitz, that is, there exists a uniform constant $C_{2}>0$ such that

$\forall\,Y\subseteq S,\qquad d_{Y}(\eta_{Y},\eta^{\prime}_{Y})\leq C_{2}.$

Now, Proposition 4.2 implies that

$d_{w}(\Pi_{\xi}(x),\Pi_{\xi}(x^{\prime}))\prec(C_{2})^{p+1}$

which means $\Pi_{\xi}$ is coarsely Lipschitz. Similarly, if
$x\in\mathcal{Z}({\mathfrak{o}},\xi)$ then for
$\eta=\operatorname{ctr}({\mathfrak{o}},x,\xi)$ we have
$d_{Y}(x_{Y},\eta_{Y})\leq C_{2}$ for all elements of $\Lambda$ and hence,
$d_{Y}(x,\Pi_{\xi}(x))\prec(C_{2})^{p+1}$. ∎

######  Lemma 4.5.

There exists $L>0$ such that the following holds. Let $x,y\in G$, and let
$\gamma\in G$ be an infinite geodesic ray based at ${\mathfrak{o}}$, and let
$x_{\gamma}:=c_{\gamma,y}(x)$. For any $Y\in\Lambda$, if
$d_{Y}(x,x_{\gamma})\geq L$, we have

$d_{Y}(x,x_{\gamma})\leq d_{Y}(x,y)+L.$

###### Proof.

Let $y_{\gamma}:=\pi_{\gamma}(y)$. Since $Y$ is hyperbolic, there exists
$L_{1},R_{0}$ such that, if $d_{Y}(\pi_{\gamma}(x),\pi_{\gamma}(y))\geq
L_{1}$, the geodesic $\gamma_{1}:=[x,y]$ in $G$ and the broken geodesic

$\gamma_{2}:=[x,\pi_{\gamma}(x)]\cup[\pi_{\gamma}(x),\pi_{\gamma}(y)]\cup[\pi_{\gamma}(y),y]$

lie in a $R_{0}$-neighborhood of each other for the metric $d_{Y}$. Let
$p_{1},p_{2}$ be nearest point projections (in $G$), respectively, of
$x_{\gamma}$ and $y_{\gamma}$ onto $[x,y]$. This implies

(4) $\displaystyle d_{Y}(x,y)$ $\displaystyle\geq d_{Y}(x,p_{1})\geq
d_{Y}(x,x_{\gamma})-R_{0}$

which proves the claim, as long as $L\geq R_{0}$.

∎

######  Proposition 4.6.

Let $\gamma$ be a hierarchy path that is a geodesic ray with
$\kappa$-excursion. Then, the map $\Pi_{\gamma}$ defined above is a
$\kappa$-projection map. Furthermore, there exist $D_{1}<1,D_{2}>1$ such that
for any two points $x,y\in G$ we have

$d_{G}(x,y)\leq D_{1}\cdot d_{G}(x,\gamma)\qquad\Longrightarrow\qquad
d_{S}(\pi_{\gamma}(x),\pi_{\gamma}(y))\leq D_{2}.$

###### Proof.

We now fix $L$ as given by Lemma 4.5, and we start by contradiction, by
assuming that $d_{S}(\pi_{\gamma}(x),\pi_{\gamma}(y))\geq L$.

By Lemma 4.5, $d_{S}(x,x_{\gamma})\prec d_{S}(x,y)$, and
$d_{Y}(x,x_{\gamma})\prec d_{Y}(x,y)$ whenever $d_{Y}(x,x_{\gamma})$ is large
enough. Now, applying the Distance Formula (Theorem 4.5 [BHS19]) to the pair
of points $(x,y)$ we have:

$\displaystyle d_{G}(x,y)$ $\displaystyle\asymp\sum_{Y\in\Lambda}\lfloor
d_{Y}(x,y)\rfloor_{L}+d_{S}(x,y)$ $\displaystyle\succ\sum_{Y\in\Lambda}\lfloor
d_{Y}(x,x_{\gamma})\rfloor_{L}+d_{S}(x,x_{\gamma})-O(\delta)\qquad\textup{ by
Eq.}~{}\eqref{deltathin}$ $\displaystyle\asymp d_{G}(x,x_{\gamma}).$

That is to say, there exists $D_{1}=D_{1}(L,\delta)$ such that

$d_{G}(x,y)\geq D_{1}\cdot d_{G}(x,x_{\gamma}),$

which is a contradiction since $x_{\gamma}\in\gamma$. Therefore, setting
$D_{2}=L$ yields

$d_{G}(x,y)\leq D_{1}\cdot d_{G}(x,\gamma)\qquad\Longrightarrow\qquad
d_{S}((\pi_{\gamma}(x),\pi_{\gamma}(y))\leq D_{2}.\qed$

### Logarithmic projections

We now consider the set of elements in $G$ whose projection to $\mathcal{C}S$
are quasi-geodesic rays in $\partial\mathcal{C}S$ that have _logarithmically
bounded projection_ to all elements of $\Lambda$. Recall in Definition 2.7,
When $V\sqsubseteq W$ then there is a specific subset
$\rho_{W}^{V}\subset\mathcal{C}W$ such that
$\operatorname{diam}(\rho_{W}^{V})\leq\xi$.

Given a non-maximal element $Y\in\Lambda$, let

$\lVert Y\rVert_{S}:=d_{S}({\mathfrak{o}}_{S},\rho^{Y}_{S}).$

In comparison, for $x\in G$, define

$\lVert x\rVert_{S}:=d_{S}\big{(}{\mathfrak{o}}_{S},\pi_{S}(x))\big{)}.$

######  Definition 4.7.

For a constant $c>0$, let $\mathcal{L}$ be the set of elements in $G$ whose
shadows to an infinite quasi-geodesic ray $\xi\in\partial\mathcal{C}S$ such
that

(5) $d_{Y}({\mathfrak{o}},\xi)\leq c\cdot\log\lVert Y\rVert_{S}$

for every non-maximal element $Y\in\Lambda$.

######  Proposition 4.8.

For any $\xi\in\mathcal{L}$, the set $\mathcal{Z}({\mathfrak{o}},\xi)$ is
$\kappa$-weakly contracting, where $\kappa(r)=\log^{p}(r)$. Furthermore, any
realization $\mathcal{G}({\mathfrak{o}},\xi)$ of the hierarchy
$H({\mathfrak{o}},\xi)$ is also $\kappa$-Morse.

###### Proof.

In this proof, we use the notations $\prec_{c}$ and $O_{c}$ to mean that the
implicit constants additionally depend on $c$. Let
$\mathcal{Z}=\mathcal{Z}({\mathfrak{o}},\xi)$. Given $x,x^{\prime}\in X$ where

$D_{1}\cdot d_{G}(x,x^{\prime})<d_{G}(x,\mathcal{Z}),$

let $y=\Pi_{\xi}(x)$, $y^{\prime}=\Pi_{\xi}(x^{\prime})$. We claim that, for
every proper element of $\Lambda$,

$d_{Y}(y,y^{\prime})\prec_{c}\log\lVert x\rVert_{S}.$

Since $\mathcal{C}S$ is hyperbolic, nearest point projection in $\mathcal{C}S$
is coarsely distance decreasing, hence $\lVert y\rVert_{S}\prec\lVert
x\rVert_{S}$. Also, by Theorem 4.6

(6) $d_{S}(y,y^{\prime})\leq D_{2}$

therefore, $\lVert y^{\prime}\rVert_{S}\prec\lVert x\rVert_{S}$. Which means,
for every curve $\alpha$ in the geodesic segment $[y,y^{\prime}]_{S}$ in
$\mathcal{C}S$ we have $d_{S}({\mathfrak{o}},\alpha)\prec\lVert x\rVert_{S}$.
Axiom (7) (Bounded Geodesic Image) implies that if $d_{Y}(y,y^{\prime})$ is
large then $d_{S}([y,y^{\prime}]_{S},\partial Y)\prec 1$, hence

$\lVert Y\rVert_{S}\prec\lVert x\rVert_{S}.$

By the definition of $\Pi_{\xi}$, $y_{Y}$ and $y^{\prime}_{Y}$ are $D$-close
to the geodesic segment $[{\mathfrak{o}},\xi]_{Y}$ in $\mathcal{C}Y$ and, by
assumption, the length of this segment is at most a uniform multiple of
$\log\lVert Y\rVert_{S}$. Therefore,

$d_{Y}(y,y^{\prime})\prec\big{|}[{\mathfrak{o}},\xi]_{Y}\big{|}\prec_{c}\log\lVert
Y\rVert_{S}\prec\log\lVert x\rVert_{S}.$

In view of Equation (6) and Proposition 4.2, we get

$d_{G}(y,y^{\prime})\prec_{c}\log^{p}\lVert x\rVert_{S}.$

Now, by Theorem A.1 in [QRT20], $\mathcal{Z}({\mathfrak{o}},\xi)$ is
$\kappa$-Morse. Let $m_{\mathcal{Z}}$ be the associated Morse gauge for
$\mathcal{Z}({\mathfrak{o}},\xi)$.

Now we show $\mathcal{G}({\mathfrak{o}},\xi)$ is also $\kappa$-Morse. Assume
$\kappa^{\prime}$ and $r>0$ be given (see Definition 2.3) and, using the fact
that $\mathcal{Z}({\mathfrak{o}},\xi)$ is $\kappa$-Morse, let $R$ be a radius
such that, for any $(q,Q)$-quasi-geodesic ray $\beta$ in $G$ with
$m_{\mathcal{Z}}(q,Q)$ small compared to $r$, we have

$d_{G}(\beta_{R},\mathcal{Z}({\mathfrak{o}},\xi))\leq\kappa^{\prime}(R)\qquad\Longrightarrow\qquad\beta|_{r}\subset\mathcal{N}_{\kappa}(\mathcal{Z}({\mathfrak{o}},\xi),m_{\mathcal{Z}}(q,Q)).$

Also, assume

$d_{G}(\beta_{R},\mathcal{G}({\mathfrak{o}},\xi))\leq\kappa^{\prime}(R).$

We need to show that every $x\in\beta|_{r}$ is close to
$\mathcal{G}({\mathfrak{o}},\xi)$.

Since $\mathcal{G}({\mathfrak{o}},\xi)\subset\mathcal{Z}({\mathfrak{o}},\xi)$
we can still conclude that there is a point
$y\in\mathcal{Z}({\mathfrak{o}},\xi)$ with

$d_{G}(x,y)\leq m_{\mathcal{Z}}(q,Q)\cdot\kappa(x).$

In fact $y$ can be taken to be $\Pi_{\xi}(x)$ and hence $\lVert
y\rVert_{S}\prec\lVert x\rVert_{S}$. Let $z$ be a point in
$\mathcal{G}({\mathfrak{o}},\xi)$ where $d_{S}(z_{S},y_{S})\leq D$ (such a
point exists since the shadow of $\mathcal{G}({\mathfrak{o}},\xi)$ to
$\mathcal{C}S$ is the geodesic ray $[{\mathfrak{o}},\xi)_{S}$). Since
$y,z\in\mathcal{Z}({\mathfrak{o}},\xi)$, we have for every element of
$\Lambda$ that

$d_{Y}(y,z)\prec_{c}\log\max(\lVert y\rVert_{S},\lVert
z\rVert_{S})\prec\log(\lVert x\rVert_{S}+D)\prec\log\lVert x\rVert_{S}.$

Therefore, by Proposition 4.2, we have

$d_{G}(y,z)\prec_{c}\log^{p}\lVert x\rVert_{S}\prec\kappa(x).$

And hence,

$d_{G}(x,z)\leq
d_{G}(x,y)+d_{G}(y,z)\prec_{c}m_{\mathcal{Z}}(q,Q)\cdot\kappa(x).$

We have shown

$\beta|_{r}\subset\mathcal{N}_{\kappa}\Big{(}\mathcal{G}({\mathfrak{o}},\xi),O_{c}\big{(}m_{\mathcal{Z}}(q,Q)\big{)}\Big{)}.$

That is, $\mathcal{G}({\mathfrak{o}},\xi)$ is $\kappa$-Morse with a Morse
gauge $m_{\mathcal{G}}=O_{c}(m_{\mathcal{Z}})$. ∎

###  4.1. Convergence to the $\kappa$-Morse boundary

Let $\mu$ be a probability measure on $G$. We say that $\mu$ is _non-
elementary_ if the semigroup generated by its support contains two loxodromic
elements with disjoint fixed sets in $\partial\mathcal{C}S$.

Let us recall some useful facts on random walks on any HHS group.

######  Theorem 4.9.

Let $\mu$ be a finitely supported, non-elementary probability measure on an
HHS group $G$. Then:

  1. (1)

For almost every sample path $\omega=(w_{n})$, the sequence $(w_{n})_{S}$
converges to a point $\xi_{\omega}$ in the Gromov boundary of $\mathcal{C}S$.

  2. (2)

Moreover, there exists $l>0,c<1$ such that

$\mathbb{P}\left(d_{S}({\mathfrak{o}},w_{n})\geq ln\right)\geq 1-c^{n}$

for any $n$.

  3. (3)

Further, for any $k>0$ there exists $C>0$ such that

$\mathbb{P}\left(d_{S}(w_{n},\gamma_{\omega})\geq C\log n\right)\leq n^{-k}$

for any $n$, where $\gamma_{\omega}=[{\mathfrak{o}},\xi_{\omega})_{S}$.

Claim (2) is proven by Maher ([Mah10], [Mah12]), while (1) and (3) are proven
by Maher-Tiozzo in [MT18]. Also, exactly the same proof as in [QRT19, Theorem
A.17] yields for HHS groups that for finitely supported, non-elementary
probability measure on $G$. Then for any $k>0$ there exists $C>0$ such that
for all $n$ we have

$\mathbb{P}\left(\sup_{Y}d_{Y}({\mathfrak{o}},w_{n})\geq C\log n\right)\leq
Cn^{-k},$

where the supremum is taken over all (proper) elements of $\Lambda$ of $S$. As
a consequence, for almost every sample path there exists $C>0$ such that for
all $n$

$\sup_{Y}d_{Y}({\mathfrak{o}},w_{n})\leq C\log n.$

It follows that almost every sample path converges to a point in the
$\kappa$-Morse boundary of the HHS group, where $\kappa(r)=\log^{p}(r)$. We
now complete the proof of Theorem B by identifying the $\kappa$-Morse boundary
with the Poisson boundary.

######  Theorem 4.10.

Let $\mu$ be a non-elementary, finitely supported measure on an HHS group $G$.
Then for $\kappa(r):=\log^{p}(r)$, where $p$ is the complexity of the
hierarchy, the $\kappa$-Morse boundary is a topological model for the Poisson
boundary of $(G,\mu)$.

###### Proof.

Since almost every sample path sublinearly tracks a $\kappa$-Morse geodesic
ray, with $\kappa(r)=\log^{p}(r)$, so we can apply Theorem 2.6. ∎

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