# The geometry of genericity in mapping class groups and Teichmüller spaces
via CAT(0) cube complexes

Matthew Gentry Durham  Department of Mathematics, University of California,
Riverside, CA<EMAIL_ADDRESS>and  Abdul Zalloum  Department of Mathematics,
Queen’s University, Kingston, ON<EMAIL_ADDRESS>
###### Abstract.

Random walks on spaces with hyperbolic properties tend to sublinearly track
geodesic rays which point in certain hyperbolic-like directions. Qing-Rafi-
Tiozzo recently introduced the sublinearly Morse boundary and proved that this
boundary is a quasi-isometry invariant which captures a notion of generic
direction in a broad context.

In this article, we develop the geometric foundations of sublinear Morseness
in the mapping class group and Teichmüller space. We prove that their
sublinearly Morse boundaries are visibility spaces and admit continuous
equivariant injections into the boundary of the curve graph. Moreover, we
completely characterize sublinear Morseness in terms of the hierarchical
structures of these spaces.

Our techniques include developing tools for modeling the hulls of median rays
in hierarchically hyperbolic spaces via CAT(0) cube complexes. Part of this
analysis involves establishing direct connections between the geometry of the
curve graph and the combinatorics of hyperplanes in the approximating cube
complexes.

##  1\. Introduction

The _sublinearly Morse boundary_ was introduced by Qing-Rafi-Tiozzo [QR19,
QRT20] to capture the asymptotic behavior of random walks on finitely
generated groups with hyperbolic-like properties. Building on work of
Kaimanovich [Kai00], Tiozzo [Tio15], Sisto [Sis17], Mathieu-Sisto [MS20],
Sisto-Taylor [ST19], among others, they prove that the $\log^{p}$-Morse
boundary (for some $p=p(S)>0$) of the mapping class group of a finite-type
surface $S$ is a model for the Poisson boundary for its random walks, and that
random walks $\log^{p}$-track $\log^{p}$-Morse geodesics (see also [NQ22a]).
In this sense, $\log^{p}$-Morse geodesics are _generic directions_ in the
mapping class group. This boundary has subsequently been studied in the
context of CAT(0) cube complexes [MQZ20, IMZ21], where the picture is more
transparent than the general context and its details relevant to us.

In this article, we establish various connections between the sublinearly
Morse boundary of the mapping class group and the hierarchical geometry of the
curve graph. More generally, we develop several foundational properties of the
sublinearly Morse boundaries of _hierarchically hyperbolic spaces_
(abbreviated as HHSes), of which mapping class groups and Teichmüller spaces
are examples.

Our arguments exploit the cubical geometry of HHSes, building on work of
Behrstock-Hagen-Sisto [BHS21]. They showed that the median hull of a finite
set of points in an HHS is approximated by a CAT(0) cube complex via a median
quasi-isometry. We develop a limiting version of their machinery (Theorem B)
which is suitable for studying the median hulls of median quasi-geodesic rays,
allowing us to analyze the asymptotic geometry of the curve graph via the
combinatorics of hyperplanes in cube complexes (Theorem C). These quasicubical
results are separate from the $\kappa$-Morseness discussion and are therefore
of independent interest.

For a sublinear function $\kappa$, a quasi-geodesic ray $q$ in an HHS
$\mathcal{X}$ has a _$\kappa$ -persistent shadow_ in its top-level hyperbolic
space $\mathcal{C}(S)$ if $d_{S}(q(s),q(t))\succ\frac{t-s}{\kappa(t)}$ for all
$s,t\in[0,\infty)$ with $t\geq s$. The following theorem summarizes our main
results about sublinearly Morse rays in HHSes:

######  Theorem A.

Let $\mathcal{X}$ be a proper hierarchically hyperbolic space with unbounded
products, $\mathcal{C}(S)$ be the top-level hyperbolic space, $\kappa$ be a
sublinear function and let $\partial_{\kappa}\mathcal{X}$ denote the
$\kappa$-Morse boundary. We have the following.

  1. (1)

The subsurface projection map $\pi_{S}:\mathcal{X}\rightarrow\mathcal{C}(S)$
induces an $\text{Aut}(\mathcal{X})$-equivariant continuous injection,
$i_{\kappa}:\partial_{\kappa}\mathcal{X}\rightarrow\partial\mathcal{C}(S).$

  2. (2)

There exists $p=p(\mathcal{X})>0$ so that for any quasi-geodesic ray $q$, the
following holds:

    1. (a)

$q$ is $\kappa$-Morse if and only if $q$ is $\kappa$-contracting.

    2. (b)

If $q$ is $\kappa$-Morse, then $q$ has a $\kappa^{p}$-persistent shadow.

    3. (c)

If $q$ is a median ray with a $\kappa$-persistent shadow, then $q$ is
$\kappa$-Morse.

  3. (3)

For any proper geodesic metric space $X$, the $\kappa$-boundary
$\partial_{\kappa}X$ is a visibility space, i.e. points in
$\partial_{\kappa}X$ are connected by bi-infinite $\kappa$-Morse geodesics.

In particular, having a $\kappa$-persistent shadow completely characterizes
sublinear Morseness for median rays. We also prove a related characterization
in terms of bounding the growth of subsurface projections (Theorem J).
Moreover, we establish both of these characterizations for Teichmüller
geodesic rays (see Theorem K), which are not median in general. We discuss
some potential implications to the vertical foliations of $\kappa$-Morse
Teichmüller rays in Subsection 1.7 below.

Theorem A is new for mapping class groups [MM99, MM00, BHS17] and Teichmüller
spaces of finite-type surfaces [Raf07, Dur16], extra-large type Artin groups
[HMS21], the genus two handlebody group [Mil20], surface group extensions of
lattice Veech groups [DDLS21b, DDLS21a] and multicurve stabilizers [Rus21],
and the fundamental groups of 3–manifolds without Nil or Sol components
[BHS19], as well as various combinations and quotients of these objects
[BHMS20, BR20].

We remark that Theorem A is a direct generalization of the Morse case (where
$\kappa=1$) [Cor17, ACGH16, ABD21]. However, our techniques are substantially
different and it is far from clear that the original approaches in HHSes
avoiding cubical techniques would work, at least not without substantial new
technical complications.

Although the statements of items (1), (2b), and (2c) only involve the
hierarchichal structure of an HHS $\mathcal{X},$ the proofs heavily rely on
the cubical approximations of median hulls in $\mathcal{X}$ (Theorem B).
Moreover, the characterization in (2a) of Theorem A proceeds entirely via
quasicubical arguments, and thus holds in the broader class of _locally quasi-
cubical_ coarse median spaces.

The rest of the introduction explains this context, including some of the
tools we develop, and gives various refinements of the above stated results.

###  1.1. Capturing hyperbolicity in HHSes via CAT(0) cube complexes

A geodesic metric space $X$ is _coarse median_ if, roughly, triangles have
coarsely well-defined barycenters. The canonical examples are hyperbolic
spaces, mapping class groups [BM11] and HHSes [BHS19], and CAT(0) cube
complexes; see Definition 2.28 and Bowditch [Bow13, Bow22].

For two coarse median spaces $X,Y$, a map $f:X\rightarrow Y$ is said to be
_$\lambda$ -median_ if $f$ preserves the median up to an additive error of
$\lambda,$ for some $\lambda\geq 0.$

A _$\lambda$ -median path_ in a coarse median space $X$ is a
$(\lambda,\lambda)$-quasi-isometric embedding $\gamma:[a,b]\rightarrow X$
which is $\lambda$-median. More generally, any finite set $F\subset X$ admits
a _median hull_ , denoted $\mathrm{hull}_{X}(F)$, which encodes all the median
paths between the points in $F$ (Definition 2.34) and is median convex
(Definition 2.30).

A coarse median space $X$ is said to be locally quasi-cubical (LQC) if for
each finite set of points $F$, there exist some $\lambda=\lambda(|F|)$, a
CAT(0) cube complex $Q$ of a uniform dimension, and a
$(\lambda,\lambda)$-quasi-isometry $f:Q\rightarrow\mathrm{hull}_{X}(F)$ which
is $\lambda$-median.

Local quasi-cubicality is a higher-rank generalization of Gromov’s
characterization of hyperbolicity [Gro87], i.e., hyperbolic spaces are exactly
the locally quasi-arboreal spaces.

The LQC property was originally established for HHSes in [BHS21] (as “cubical
approximations”), extended to a more general class of coarse median spaces in
[Bow19], and stabilized in [DMS20] (as “cubical models”). It was studied as an
abstract property for hulls of pairs of points in [HHP20] (as “quasi-cubical
intervals”), and we are using the terminology from the forthcoming [PSZ23],
where those authors will systematically study the class of LQC spaces.

One of our main tools, Theorem 4.3, allows us to study the hulls of any finite
number of median rays via cube complexes:

######  Theorem B.

Let $X$ be a proper LQC space, $F\subset X$ nonempty and finite, and $Y$ a
finite set of median rays. Then $\mathrm{hull}_{X}(F\cup Y)$ admits a
$\lambda$-median $(\lambda,\lambda)$-quasi-isometry to a uniformly finite
dimensional CAT(0) cube complex $Q$, with $\lambda=\lambda(X,|F|,|Y|)>0$.

Among our contributions to this quasi-cubical machinery are the techniques we
develop for comparing the approximating CAT(0) cube complexes (cubical models
for short) for finite sets $Y^{\prime}\subsetneq Y$ and $F^{\prime}\subsetneq
F$. This allow us to, for instance, redefine certain median (and hierarchical)
gate maps as cubical gate maps (Lemma 2.57 and Theorem 4.9), which are crucial
for our various characterizations of $\kappa$-Morseness in HHSes.

More importantly, we obtain a cubical interpretation for the well-established
philosophy that hyperbolicity in an HHS is encoded in its top-level hyperbolic
space, which we will now discuss.

In [Gen17, Gen20], Genevois introduced well-separated hyperplanes (Definition
2.23) and showed that, similarly to how the top-level hyperbolic space records
the hyperbolicity of an HHS, these hyperplanes capture the hyperbolic aspects
of a CAT(0) cube complexe. For instance, he showed that Morse geodesic rays
are characterized by crossing such hyperplanes at a uniform rate, and that
well-separated hyperplanes can be used to build hyperbolic models for the
CAT(0) cube complex under consideration. This philosophy was further
reinforced by Murray-Qing-Zalloum [MQZ20] and Incerti-Medici–Zalloum [IMZ21],
in work relevant to our paper.

In this article, we show that hyperbolicity of the top-level curve graph is
witnessed by the crossing of well-separated hyperplanes in the approximating
CAT(0) cube complex. In particular, we establish a direct quantitative
connection between the speed of the shadow of a median ray in an HHS in its
top-level hyperbolic space and the speed at which its image in the
corresponding cubical model of its median hull crosses uniformly well-
separated hyperplanes. More precisely, let $h$ be a median ray (equivalently,
hierarchy ray) in an HHS $\mathcal{X}$ with unbounded products. Let
$H=\mathrm{hull}_{\mathcal{X}}(h)$ be its median hull and $f:Q\to H$ the
cubical model provided by Theorem B above, and $g:H\to Q$ its coarse inverse.

######  Theorem C.

For any proper HHS $\mathcal{X}$ with unbounded products, there exist
$L=L(\mathcal{X})>0$ so that for any median ray $h$, the following holds. For
any $a,b\in h$, let $\mathcal{H}(a,b)$ be a maximal collection of
pairwise-$L$-well-separated hyperplanes in $Q$ separating $g(a),g(b)$. Then

$|\mathcal{H}(a,b)|\succ d_{S}(\pi_{S}(a),\pi_{S}(b)).$

One consequence of this theorem is that if $\pi_{S}(h)$ has infinite diameter
in $\mathcal{C}(S)$, then $g(h)$ crosses an infinite sequence of $L$-well-
separated hyperplanes $\\{H_{i}\\}_{i}$ in $Q$. Moreover, the rate at which
$g(h)$ crosses the $H_{i}$ is controlled by the rate of progress of
$\pi_{S}(h)$ in $\mathcal{C}(S)$. See Corollary 7.18 for a precise statement
and Subsection 1.8 for a detailed sketch of the proof.

In fact, by increasing the constant $L$, this statement holds for the cubical
model of the median hull of any finite number of median rays and interior
points via Theorem B. This flexibility is crucial for this paper and future
applications.

We note that Petyt [Pet21] recently proved the surprising result that mapping
class groups and other _colorable_ HHGs [DMS20, Hag21] are median quasi-
isometric to CAT(0) cube complexes. While this very nice fact eliminates the
need for Theorem B for colorable HHGs in a few instances (e.g., item (2a) of
Theorem A), one would still need something akin to our arguments to derive the
other main results. Moreover, the mapping class group is not _equivariantly_
quasi-cubical [KL96, Bri10], and Petyt’s cube complex need not admit a _factor
system_ [BHS17]. As a consequence, one cannot obtain the equivariance
conclusion in item (1) of Theorem A by avoiding Theorem B, nor can one obtain
the injection by passing through the work of [IMZ21], which requires the
ambient cube complex to admit a factor system.

###  1.2. Sublinear Morseness in LQC spaces

Recall that a function $\kappa:[0,\infty)\to[1,\infty)$ is _sublinear_ if
$\lim_{t\to\infty}\kappa(t)/t=0$. The $\kappa$-_neighborhood_ of a geodesic
ray $\alpha:[0,\infty)\to\mathcal{X}$ is, roughly, a cone whose radius grows
in $\kappa$ with its distance to $\alpha(0)$ (Definition 2.1). We say $\alpha$
is _weakly_ $\kappa$-_Morse_ if quasi-geodesics with endpoints on $\alpha$
stay in its $\kappa$-neighborhood (Definition 2.2). Two rays
$\alpha,\alpha^{\prime}$ are $\kappa$-_asymptotic_ ,
$\alpha\sim_{\kappa}\alpha^{\prime}$, if each is contained in the other’s
$\kappa$-neighborhood; see Figure 2. Note that $\alpha$ is _Morse_ [Cor17,
Definition 1.3] when $\kappa=1$.

In fact, the above (rough) definition is not the main one used in [QRT20,
Definition 3.2], where they were seeking to topologize the set of such rays
along the lines of [CM19]. For that purpose, they used an a priori stronger
notion from [QR19], which they call _$\kappa$ -Morse_. They also defined a
useful notion of a _$\kappa$ -contracting_ set (Definition 2.5), and showed
that it implies $\kappa$-Morseness, though not vice versa.

While these notions are equivalent in CAT(0) spaces [QR19], they are not known
to be equivalent in general. We prove that they are for quasi-geodesic rays in
LQC spaces:

######  Theorem D.

Let $X$ be a proper LQC space and $q$ a quasi-geodesic ray. The following are
equivalent:

  1. (1)

$q$ is $\kappa$-Morse.

  2. (2)

$q$ is $\kappa$-contracting.

  3. (3)

$q$ is weakly $\kappa$-Morse.

Thus, for the purposes of this introduction, the reader can take
$\kappa$-Morse to mean any of these equivalent definitions.

Theorem D is new for Teichmüller space with the Teichmüller metric and more
generally proper HHSes which are not CAT(0). As mentioned above, it follows
for mapping class groups by combining [Pet21] and [QR19].

We note that Qing-Rafi-Tiozzo [QRT20] proved that, in any proper geodesic
metric space, $\kappa$-Morse geodesic rays are $\kappa^{\prime}$-contracting
for some (probably larger) sublinear function $\kappa^{\prime}$. The main
utility of Theorem D is then that one can convert between the three properties
without changing the associated sublinear function.

Moreover, the following corollary, due to [ABD21] in the HHS setting, is an
immediate consequence [ACGH16, RST18]:

######  Corollary E.

Let $X$ be a proper LQC space and let $\alpha$ be a quasi-geodesic ray. The
following are all equivalent:

  1. (1)

$\alpha$ is Morse

  2. (2)

$\alpha$ is contracting.

  3. (3)

$\alpha$ has at least quadratic divergence.

Our proof of the above equivalences is new for HHSes, as it avoids any
hierarchical arguments via Theorem B, and also more general, as it does not
require the ambient HHS to possess the bounded domain dichotomy (Definition
2.54).

Aside from HHSes and CAT(0) spaces, proper LQC spaces are the first known
class where Morse quasi-geodesic rays have at least quadratic divergence.

In CAT(0) spaces, Murray-Qing-Zalloum [MQZ20] proved that $\kappa$-Morse
geodesics are characterized by having divergence bounded below by a quadratic
function. It would be interesting to know whether some version of this works
for LQC spaces:

######  Question 1.

Are $\kappa$-Morse quasi-geodesic rays characterized by having some form of
quadratic divergence in the context of locally quasi-cubical groups?

###  1.3. Morseness in HHSes

Associated to any hierarchically hyperbolic space $\mathcal{X}$ is a robust
collection of machinery which largely controls its coarse geometry. This
machinery is built out of a family of uniformly $\delta$-hyperbolic spaces
$\left\\{\mathcal{C}(U)\right\\}_{U\in\mathfrak{S}}$, along with a family of
uniformly Lipschitz coarse projections $\pi_{U}:\mathcal{X}\to\mathcal{C}(U)$
for each $U\in\mathfrak{S}$. One typically studies an object
$A\subset\mathcal{X}$ by projecting it to each of the $\mathcal{C}(U)$,
implementing hyperbolic geometric arguments therein, and then combining the
results via certain consistency inequations. HHSes are coarse median [BHS19]
and LQC [BHS21]; in fact, they are the canonical non-cubical examples of LQC
spaces.

This framework generalizes work of Masur-Minsky [MM99, MM00] from the context
of the mapping class group $\mathcal{MCG}(S)$ of a finite-type surface $S$. In
that setting, the index set $\mathfrak{S}$ is the collection of isotopy
classes of essential subsurfaces $U\subset S$, with $\mathcal{C}(U)$ the curve
graph of $U$ and $\pi_{U}:\mathcal{X}\to\mathcal{C}(U)$ is the standard
subsurface projection [MM00]. Curve graphs are uniformly $\delta$-hyperbolic
via [Aou13] and $\mathcal{MCG}(S)$ is coarse median via [BM11, Bow13].

Every HHS has a unique hyperbolic space sitting atop the hierarchy, which we
denote by $\mathcal{C}(S)$. As one might imagine, things are considerably
easier when one can restrict to working only in $\mathcal{C}(S)$. Prime
examples of this philosophy are the _convex cocompact_ subgroups of
$\mathcal{MCG}(S)$ [FM02], which are characterized by having quasi-
isometrically embedded orbits in $\mathcal{C}(S)$ [KL08, Ham05], or
equivalently having orbits with uniformly bounded projections to any
$\mathcal{C}(U)$ for proper subsurfaces $U\subsetneq S$, or being _stable_
[DT15].

One can also see this philosophy with the Morse boundary of
$\mathcal{MCG}(S)$, i.e. $\partial_{1}\mathcal{MCG}(S)$, which admits a
continuous injection into the Gromov boundary of the curve graph,
$\partial\mathcal{C}(S)$ [Cor17]. Many of the above results were generalized
to HHSes in [ABD21].

Our next theorems generalize these hierarchical characterizations to the
sublinearly Morse setting.

###  1.4. Encoding sublinear Morseness in the boundary of the curve graph

Part of the HHS philosophy is that the Gromov boundary of the top level
hyperbolic space $\mathcal{C}(S)$ encodes all of the hyperbolic directions in
the ambient space. Our first theorem implies, for instance, that the Gromov
boundary of the curve graph $\mathcal{C}(S)$ sees the entirety of
$\partial_{\kappa}\mathcal{MCG}(S)$ for any sublinear function $\kappa$:

######  Theorem F.

Let $\mathcal{X}$ be a proper HHS with unbounded products endowed with the HHS
structure from [ABD21], $G$ a group of HHS automorphisms of $\mathcal{X}$, and
$\kappa$ a sublinear function. The projection
$\pi_{S}:\mathcal{X}\to\mathcal{C}(S)$ induces a $G$-equivariant continuous
injection $i_{\kappa}:\partial_{\kappa}\mathcal{X}\to\partial\mathcal{C}(S)$.

The assumption that $\mathcal{X}$ has _unbounded products_ is mild, and
excludes none of the main examples. It is necessary, however, so that we can
assume that $\mathcal{X}$ is endowed with the HHS structure constructed in
[ABD21], which is the standard HHS structure for $\mathcal{MCG}(S)$. See
Remark 6.2 for a discussion about HHS automorphisms.

We remark that the map itself $i_{\kappa}$ is defined via a standard Arzelá-
Ascolí argument to produce a median representative, but showing that the
resulting map is well-defined, injective (Proposition 6.6) and continuous is
nontrivial and uses the cubical models (Theorem B). Moreover, continuity is
the only place in this paper where the topology on
$\partial_{\kappa}\mathcal{X}$ makes an appearance, and it does so in a weak
way. In particular, we prove a weak median criterion (Proposition 6.8) which
we expect will guarantee continuity for any reasonable (e.g., second
countable) topology on $\partial_{\kappa}\mathcal{X}$.

In fact, He [He22] recently proved that the topology from [QRT20] on the
$1-$Morse boundary $\partial_{1}\mathcal{X}$ coincides with the Cashen-Mackay
[CM19]. Hence, we immediately obtain the following corollary, which was
previously unknown:

######  Corollary G.

Let $\mathcal{X}$ be a proper HHS with unbounded products endowed with the HHS
structure from [ABD21], and $G$ a group of HHS automorphisms of $\mathcal{X}$.
The projection $\pi_{S}:\mathcal{X}\to\mathcal{C}(S)$ induces a
$G$-equivariant continuous injection
$i:\partial_{1}\mathcal{X}\to\partial\mathcal{C}(S)$, where
$\partial_{1}\mathcal{X}$ is endowed with the Cashen-Mackay topology.

Another application is to HHSes who top-level hyperbolic spaces are quasi-
trees. Examples include right-angled Artin groups, whose top-level spaces are
the contact graphs [Hag13] of the associated Salvetti complexes [BHS17]. More
generally, one can produce examples via various combination theorems [BHS19,
BR20], e.g. $\mathbb{Z}*(\mathcal{MCG}(S)\times\mathcal{MCG}(S))$ for some
hyperbolic surface $S$. The Gromov boundaries of quasitrees are totally
disconnected, and hence the following is an immediate consequence of Theorem
F, generalizing [IMZ21] in the case of RAAGs:

######  Corollary H.

For any sublinear function $\kappa$ and $G$ an HHG for which
$\partial\mathcal{C}(S)$ is totally disconnected, the $\kappa$-Morse boundary
$\partial_{\kappa}G$ is totally disconnected. In particular, the
$\kappa$-Morse boundary of any right-angled Artin group is totally
disconnected.

Qing-Rafi-Tiozzo proved that $\log^{p}$-Morse boundary is a model for the
Poisson boundary for random walks on $\mathcal{MCG}(S)$. Hence, Theorem F
connects this result to earlier work of Maher [Mah10], which says that
$\partial\mathcal{C}(S)$ plays the same role. Moreover, Theorem C provides a
further connection to work of Fernós [Fer17] and Fernós-Lécure-Mathéus
[FLM18], who showed that a certain subspace of the Roller boundary defined via
a collection of well-separated hyperplanes forms a model for the Poisson
boundary of a CAT(0) cube complex.

Finally, Theorem F connects to the main result of [IMZ21], which gives a
similar continuous injection of the $\kappa$-Morse boundary of any cubical HHS
into the boundary of the hyperbolic graph built from well-separated
hyperplanes, as discussed above.

We will discuss the image of the map for $\mathrm{Teich}(S)$ in more detail in
Subsection 1.7.

###  1.5. Characterizations via projections

Our next theorem controls the growth rate of projections of $\kappa$-Morse
quasi-geodesic rays to the hyperbolic spaces in an HHS, and provides the
converse for median rays. We note that median rays are exactly the _hierarchy
rays_ [BHS19, RST18], which are defined by having unparametrized quasi-
geodesic projections to every $\mathcal{C}(U)$.

We give two characterizations. Recall that a ray $q$ is said to have a
$\kappa$-persistent shadow if for any $s,t\in[0,\infty)$ with $s\leq t$, we
have

$d_{S}(q(s),q(t))\succ\frac{t-s}{\kappa(t)}.$

######  Theorem I.

Let $\mathcal{X}$ be a proper HHS with unbounded products with the HHS
structure from [ABD21], and $\kappa$ a sublinear function. There exists
$p=p(\mathcal{X})>0$ so that for any quasi-geodesic ray $q,$ the following
holds:

  1. (1)

If $q$ is a $\kappa$-Morse, then $q$ has a $\kappa^{p}$-persistent shadow.

  2. (2)

If $q$ is a median ray with a $\kappa$-persistent shadow, then $q$ is
$\kappa$-Morse.

In the case that $\kappa=1$, having a $\kappa$-persistent shadow is equivalent
to requiring that $q$ project to a _parameterized_ quasi-geodesic in
$\mathcal{C}(S)$, which is well-known to be equivalent to being $1$-Morse
[Beh06, DR09]. We recover this case because all $1$-Morse rays are uniformly
close to their median representatives in any proper median space.

The proof of Theorem I uses Theorem C and the cubical model to exploit work of
Murray-Qing-Zalloum [MQZ20], who give a complete characterization of
$\kappa$-Morseness in CAT(0) cube complexes via well-separated hyperplanes.
Roughly speaking, they showed that a geodesic ray is $\kappa$-Morse if and
only if it makes $\kappa$-regular progress through an infinite sequence of
$\kappa$-well-separated hyperplanes; see the related Definition 3.8.

Our second characterization is about the growth rate of the projection
distance to the hyperbolic spaces lower in the hierarchy. We say that a quasi-
geodesic ray $q$ has $\kappa$-bounded projections if there exists $C>0$ so
that

$d_{\mathcal{C}(U)}\left(\pi_{U}(q(0)),\pi_{U}(q(t))\right)<C\cdot\kappa(t),$

for all $t\in[0,\infty)$ and all proper $U\sqsubsetneq S$. That is, subsurface
projections grow $\kappa$-sublinearly along $q$. When $\kappa=1$, this is
equivalent to $q$ having uniformly bounded projections.

######  Theorem J.

Let $\mathcal{X}$ be a proper HHS with unbounded products and the HHS
structure from [ABD21], and $\kappa$ a sublinear function. There exists
$p=p(\mathcal{X})>0$ so that for any quasi-geodesic ray $q,$ the following
holds:

  1. (1)

If $q$ is a $\kappa$-Morse, then $q$ has $\kappa$-bounded projections.

  2. (2)

If $\kappa^{p}$ is sublinear and $q$ is a median ray with $\kappa$-bounded
projections, then $q$ is $\kappa^{p}$-Morse.

Setting $\kappa=1$ recovers the Morse case [Beh06, DR09, ABD21], as every
geodesic ray with uniformly bounded projections is median. In Teichmüller
space with the Teichmüller metric, we obtain a characterization for
Teichmüller geodesics, despite the fact that they are not median; see Theorem
K below.

We note that Qing-Rafi-Tiozzo [QRT20] introduced a combinatorial version of
$\kappa$-bounded projections in mapping class groups, showing that it implies
$\kappa^{p}$-Morseness for median rays. They prove that a sample path along a
random walk $\log^{p}$-tracks a median ray satisfying their combinatorial
condition. While characterizing $\kappa$-Morseness was evidently not their
aim, it does provide an intriguing possibility.

However, this is not the case. In Proposition 7.22, we prove that
$\kappa$-Morse rays need not satisfy this property. In particular, we produce
$\kappa$-Morse rays in the mapping class group which do not have combinatorial
$\kappa^{\prime}$-bounded projections for any sublinear $\kappa^{\prime}$.
Thus their stronger notion does not characterize $\kappa$-Morseness, as we do
in Theorems I and J.

###  1.6. Characterizing sublinear Morseness for Teichmüller geodesics

The last application of our techniques is to Teichmüller geodesics. For this
discussion, let $S$ be a finite-type surface admitting a hyperbolic metric,
$\mathcal{MCG}(S)$ its mapping class group and $\mathrm{Teich}(S)$ its
Teichmüller space with the Teichmüller metric.

Work of Masur-Minsky [MM99] and Rafi [Raf05, Raf07, Raf14] shows that
Teichmüller geodesics project to unparametrized quasi-geodesics in every
subsurface curve graph. This is not enough to make them hierarchy paths,
because the hyperbolic spaces associated to annuli in the HHS structure are
horoballs over annular curve graphs [Dur16], and a simple closed curve can
become short along a Teichmüller geodesic even when no twisting is done around
this curve, causing backtracking in the corresponding horoball.

Nonetheless, work of Rafi [Raf05] and Modami-Rafi [MR22] says that the length
of a simple closed curve along a Teichmüller geodesic is controlled by the
size of the projections of the subsurfaces it bounds. In particular, a
sublinear bound on subsurface projections gives a sublinear bound on the
backtracking in the horoballs over annuli. This allows us to conclude that a
Teichmüller ray with $\kappa$-bounded projections (including to annular
horoballs) must stay $\kappa$-close to the hull of any median representative.
The techniques for Theorems I and J then allow us to deduce the following two
characterizations of $\kappa$-Morseness for Teichmüller geodesics:

######  Theorem K.

There exists $p=p(S)>0$ so that for any sublinear function $\kappa$ the
following hold:

  1. (1)

If $\gamma$ is a $\kappa$-Morse Teichmüller geodesic, then $\gamma$ has
$\kappa$-bounded projections.

  2. (2)

If $\kappa^{2p}$ is sublinear, and $\gamma$ is a Teichmüller geodesic with
$\kappa$-bounded projections, then $\gamma$ is $\kappa^{2p}$-Morse.

  3. (3)

If $\kappa^{p}$ is sublinear and $\gamma$ has $\kappa$-persistent shadow, then
$\gamma$ is $\kappa^{p+1}$-Morse.

Once again, the powers $2p$ and $p+1$ are artifacts of our proofs, and we
suspect they are not optimal.

###  1.7. Unique ergodicity and sublinear Morseness

We end with an intriguing connection to Teichmüller dynamics provided by the
injective map $i:\partial_{\kappa}\mathrm{Teich}(S)\to\partial\mathcal{C}(S)$
from Theorem F and the control of subsurface projection growth along a
$\kappa$-Morse Teichmüller ray via Theorem K.

Work of Klarreich [Kla18] identifies the Gromov boundary of the curve graph
$\partial C(S)$ with the space of _ending laminations_ on $S$. There is a
special subset of uniquely ergodic laminations, those filling minimal
laminations which contain no closed leaves and support a unique (up to
rescaling) ergodic measure. Uniquely ergodic laminations are important in
Teichmüller dynamics, and the existence of non-uniquely ergodic filling
minimal laminations has been known for a long time (see, e.g., [Kea77]). There
has been a recent flurry of interesting examples created using curve graph
techniques [LLR18, BLMR19, BLMR20], see also [CMW19]. We point the reader
toward the introduction of [LLR18] for a thorough discussion of the history
and context.

Our present interest in unique ergodicity relates to random walks. Let $\mu$
be a probability measure on the mapping class group whose support generates a
nonelementary semigroup containing two distinct pseudo-Anosov elements.
Kaimanovich-Masur [KM96] proved that any random walk of the mapping class
group on Teichmüller space with the Teichmüller metric with respect to $\mu$
generically converges to a point in Thurston’s compactification of Teichmüller
space, the space of projectivized measured laminations. Moreover, they showed
that the underlying lamination is (generically) uniquely ergodic.

As noted earlier, Qing-Rafi-Tiozzo [QRT20] proved that almost every random
walk of the mapping class group on itself $\log^{p}$-tracks a $\log^{p}$-Morse
geodesic ray. We suspect that there is an analogous statement for random walks
of the mapping class group on Teichmüller space with the Teichmüller metric,
though it appears to not be immediate from their results.

Translating via work of Rafi [Raf05] and Modami-Rafi [MR22], item (1) of
Theorem K says that the extremal lengths of simple closed curves grow slowly
along a $\kappa$-Morse ray; equivalently, these rays diverge slowly in moduli
space. The fact that sufficiently slow divergence in moduli space implies
unique ergodicity is a well-established theme [Che04, CE07, Tre14, Smi17] in
Teichmüller dynamics, although unique ergodicity is better controlled by the
growth rate of flat length [CT17, Main Theorem 2].

Since Theorem F says that each $\kappa$-Morse geodesic ray in
$\mathrm{Teich}(S)$ picks out an ending lamination (equivalently, foliation),
it is reasonable to ask whether it is possible to diverge sufficiently slowly
to be $\kappa$-Morse while insufficiently slowly to have a uniquely ergodic
limit:

######  Question 2.

Does there exist a $\kappa$-Morse Teichmüller geodesic ray whose vertical
foliation is non-uniquely ergodic?

We expect a positive answer to this question (see, e.g., [Che04, Theorem 1]),
though an example does not appear to exist in the literature. For instance,
one can use Theorem K to show that many of the examples of Teichmüller rays
with nonuniquely ergodic vertical foliations constructed in [LLR18, BLMR19,
BLMR20] do not have $\kappa$-bounded projections for explicit slow-growing
functions (e.g., $\kappa(t)=\sqrt{t}$). These constructions involve a flexible
set of parameters, and while we have not confirmed that any choice of
parameters fails to produce a $\kappa$-Morse example, we suspect that this is
the case.

We also note that the analogous question for Weil-Petersson geodesic rays is
already known, since the rays constructed in [BLMR19, BLMR20] have non-annular
bounded projections and hence are 1-Morse with respect to the Weil-Petersson
metric [Bro03].

Presuming a positive answer to Question 2 for Teichmüller rays, it becomes
reasonable to seek out bounds on sublinear functions $\kappa$ for which unique
ergodicity can be guaranteed:

######  Question 3.

For which sublinear functions $\kappa$ does the image of
$i_{\kappa}:\partial_{\kappa}\mathrm{Teich}(S)\to\partial\mathcal{C}(S)$
consist only of uniquely ergodic laminations?

We suspect that our subsurface projection bounds can be used to provide lower
bounds in line with the above question, but we leave that for a future
investigation.

Finally, we end with the following question to which our techniques might
appear to apply but we found surprisingly stubborn:

######  Question 1.1.

Suppose that $\gamma,\gamma^{\prime}$ are Teichmüller geodesic rays whose
projections to $\mathcal{C}(S)$ are infinite with
$[\pi_{S}(\gamma)]=[\pi_{S}(\gamma^{\prime})]\in\partial\mathcal{C}(S)$. If
$\gamma$ is $\kappa$-Morse for some sublinear function $\kappa$, is
$\gamma^{\prime}$ also $\kappa$-Morse?

Note that if $\gamma$ is $\kappa$-Morse and $\gamma^{\prime}$ is
$\kappa^{\prime}$-Morse with $\kappa^{\prime}\geq\kappa$, then injectivity of
$i_{\kappa^{\prime}}:\partial_{\kappa^{\prime}}\mathrm{Teich}(S)\to\partial\mathcal{C}(S)$
says that $\gamma\sim_{\kappa^{\prime}}\gamma^{\prime}$, which then implies
that $\gamma^{\prime}$ is $\kappa$-Morse by Definition 2.3 of
$\kappa$-Morseness. Hence in the above scenario either $\gamma^{\prime}$ is
$\kappa$-Morse or $\gamma^{\prime}$ is not $\kappa^{\prime}$-Morse for any
sublinear function $\kappa^{\prime}$.

###  1.8. Outline of paper and proof sketches

Section 2 provides the background to the paper, dealing with CAT(0) spaces,
(coarse) median spaces, cube complexes, and hierarchically hyperbolic spaces.

In Section 3, we convert results about $\kappa$-Morse geodesic rays in CAT(0)
cube complexes to results about quasi-geodesics. This is necessary because
when we transport a median ray in an HHS to the cubical model of its hull, it
will only be a quasi-median quasi-geodesic ray. Of particular note in this
section are (1) Theorem 3.13, which characterizes $\kappa$-Morseness for
quasi-geodesics in terms of excursion sequences of well-separated hyperplanes
a la [MQZ20], and (2) Proposition 3.17, which shows that the median hull of
any finite number of $\kappa$-Morse rays in a CAT(0) cube complex is
$\kappa$-contracting.

In Section 4, we prove the limiting model Theorem B via a nonrescaled
ultralimit argument, building on ideas from [BHS21]. Our main contribution
here is connecting the coarse median geometry of the hull of a finite set of
rays and points to the median geometry of its cubical model. In particular,
Theorem 4.9 shows that hierarchical/median gate maps to such hulls are readily
interpreted as median gates in appropriate cubical models.

This result is essential for our proof that weakly $\kappa$-Morse implies
$\kappa$-contracting (Theorem 5.1), which we accomplish in Section 5. The
proof involves observing that any weakly $\kappa$-Morse ray $h$ in an LQC
space $X$ is $\kappa$-close to a median ray $h$, whose image in the cubical
model $Q$ of its hull $H=\mathrm{hull}_{X}(h)$ is weakly $\kappa$-Morse, and
hence $\kappa$-contracting. This remains true if one adds two external points
$x,y\in X-H$. This larger hull $H^{\prime}=\mathrm{hull}_{X}(h,x,y)$ is median
quasi-isometric to a cube complex $Q^{\prime}$ in which $h$ is weakly
$\kappa$-Morse. Moreover, we prove that the cubical hull of $h$ in
$Q^{\prime}$ is $\kappa$-contracting in Proposition 3.17. It follows that the
external points $x,y$ in $Q^{\prime}$ have a $\kappa$-contracting projection
to the hull of $h$ in $Q^{\prime}$ via the median gate. By Theorem 4.9, we can
push this forward to a $\kappa$-contracting projection to the hull of $h$ in
the ambient HHS, showing that $h$ and $q$ are $\kappa$-contracting since
$h\sim_{\kappa}q$.

In Section 7, we prove our hierarchical characterizations of
$\kappa$-Morseness, Theorems I and J. The proof that $\kappa$-Morse geodesic
rays have $\kappa$-bounded projections involves familiar notions of active
intervals and a passing-up argument (Lemma 2.53) converts this into the
$\kappa^{p}$-persistent shadow property.

The reverse implications for median rays require the full power of our cubical
techniques. The main supporting result here is Theorem C, which relates
progress of a median ray in the curve graph to the quality of hyperplane
excursion in its cubical model. With this connection in place, one proves that
if a median ray $h$ has a $\kappa$-persistent shadow in $\mathcal{C}(S)$, then
the image of $h$ in its cubical model must cross a $\kappa$-excursion sequence
of hyperplanes, making it $\kappa$-Morse by Theorem 3.13. Again, this holds
even when adding external points, showing that the median hull of $h$ is
$\kappa$-contracting in $\mathcal{X}$ as above, and hence $h$ is
$\kappa$-Morse by Theorem D. Theorem K is proven by a similar argument, where
one now has to take extra care since the median representative of a
Teichmüller geodesic with a $\kappa$-persistent shadow (or $\kappa$-bounded
projections) need not be $\kappa$-close to the geodesic.

Figure 1. A median ray $h$ which makes infinite progress in the top-level
hyperbolic space $\mathcal{C}(S)$ must cross an infinite sequence of uniformly
well-separated hyperplanes.

We end with a sketch of Theorem C (Corollary 7.18). Suppose that $h$ is a
median ray whose projection $\pi_{S}(h)$ to the top-level hyperbolic space
$\mathcal{C}(S)$ is an infinite diameter quasi-geodesic ray. We can take a
sequence of balls $B^{\prime\prime}_{i}$ centered at points along
$\pi_{S}(h)$, each of which separates the hyperbolic hull,
$\mathrm{hull}_{S}(h)$, in $\mathcal{C}(S)$ and which are pairwise as far
apart as necessary. These balls pull back to disjoint subsets $B_{i}$ in
$\mathrm{hull}_{\mathcal{X}}(h)=H$, whose median hulls are pairwise disjoint,
separate $H$, and, importantly, so that the median gate of $B_{i}$ to $B_{j}$
has uniformly bounded diameter for all $i\neq j$ (this uses that the
$B^{\prime\prime}_{i}$ are sufficiently separated in $\mathcal{C}(S)$). The
$B_{i}$ are sent to median quasiconvex subsets $B^{\prime}_{i}$ of the cubical
model $g:H\to Q$, with pairwise bounded gate diameters via Theorem 4.9. A
result of Genevois [Gen16, Proposition 14] says that these convex subsets are
uniformly well-separated, from which we can produce the desired sequence of
uniformly well-separated hyperplanes. Finally, the rates at which $g(h)$
traverses these hyperplanes is comparable to the rate at which $h$ traverses
the $B_{i}$ because $q$ is a quasi-isometry. See Figure 1.

###  1.9. Acknowledgements

Durham was partially supported by NSF grant DMS-1906487. Durham would also
like to thank Cornell University for its hospitality during the 2021-22
academic year.

The authors would like to thank Shaked Bader, Benjamin Dozier, Talia Fernós,
Mark Hagen, Johanna Mangahas, Babak Modami, Harry Petyt, Kasra Rafi, Jacob
Russell, Jenya Sapir, Davide Spriano, and Sam Taylor for useful discussions.
In particular, we thank Babak Modami for sharing a preliminary draft of
[MR22]. We thank Anthony Genevois for useful comments on an earlier draft of
the paper and pointing out Corollary H. We also thank Matt Cordes and Alex
Sisto for pointing out Corollary G. Durham would also like to thank Ruth
Charney and Rémi Coulon for stimulating discussions about generalizing the
Morse property at MSRI during the fall of 2016.

##  2\. Preliminaries

In this section, we will cover the background for the paper, including CAT(0)
cube complexes (Subsection 2.4), coarse median spaces (Subsection 2.5), local
quasi-cubical spaces (Subsection 2.6), and HHSes (Subsection 2.7).

###  2.1. Notations and assumptions

Throughout this paper, we will be making calculations that hold up to certain
errors. This motivates the following notational scheme:

  1. (1)

For a subset $A\subseteq X$ and a number $r\geq 0,$ we use $N(A,r)$ to denote
the set of points in $X$ at distance at most $r$ from $A$. This is called the
_$r$ -neighborhood_ of $A$. We will often also write $x\underset{r}{\in}A$ to
abbreviate $x\in N(A,r).$

  2. (2)

For two subsets $A,B$ of a metric space $X,$ we will use $d_{Haus}(A,B)$ to
denote the Hausdorff distance between $A,B.$

  3. (3)

If $f:Q\rightarrow H$ is a quasi-isometry with a coarse inverse $g,$ we use
the notation

$(g,f,Q\rightarrow H)$

  4. (4)

If $\alpha$ is a quasi-geodesic and $p,q\in\alpha,$ we use $[p,q]_{\alpha}$ to
denote the subsegment of $\alpha$ connecting the points $p,q.$

  5. (5)

For two quantities $A,B$ and a constant $C$ we will use the following
notation.

     * •

$A\underset{C}{\succ}B,$ if there exists a constant $C^{\prime}\geq 1,$
depending only on $C$ and $X$ such that

$A\geq\frac{1}{C^{\prime}}B-C^{\prime}.$

     * •

$A\underset{C}{\asymp}B,$ if $A\underset{C}{\succ}B$ and
$B\underset{C}{\succ}A.$

     * •

$A\succ B$ if there exists a constant $C$, depending only on $X$, such that
$A\underset{C}{\succ}B$.

     * •

$A\asymp B$ if there exists some $C$, depending only on $X$, such that
$A\underset{C}{\asymp}B$.

We will assume throughout the paper that all quasi-geodesics under
consideration are continuous. If $\beta\colon\thinspace[0,\infty)\to X$ is a
continuous $(q,Q)$–quasi-isometric embedding, and $f\colon\thinspace X\to Y$
is a $(K,K)$–quasi-isometry then the composition
$f\circ\beta\colon\thinspace[t_{1},t_{2}]\to Y$ is a quasi-isometric
embedding, but it may not be continuous. However, using Lemma III.1.11 [BH09],
one can adjust the map slightly to make it continuous. Abusing notation, and
in light of the coarse nature of our calculations, we denote this continuous
new map again by $f\circ\beta$.

We remark that the assumption that quasi-geodesics are continuous may seem at
odds with the standard definition of an HHS, which only requires a quasi-
geodesic space. But any quasi-geodesic space is quasi-isometric to a graph,
and quasi-isometries preserve HHSes structures, so this assumption does not
materially impact the discussion to follow.

###  2.2. Sublinear Morseness

For the rest of the paper, we will assume that $X$ is a proper geodesic metric
space and ${\mathfrak{o}}\in X$ is a fixed point. Moreover, unless mentioned
otherwise, all quasi-geodesic rays under consideration will be assumed to
start at the fixed point ${\mathfrak{o}}.$

Let $\kappa\colon\thinspace[0,\infty)\to[1,\infty)$ be a sublinear function
that is monotone increasing and concave. In particular, sublinearity means
that

$\lim_{t\to\infty}\frac{\kappa(t)}{t}=0.$

The assumption that $\kappa$ is increasing and concave makes certain arguments
cleaner, otherwise they are not needed; see [QRT20, Remark 2.3].

For a point $x\in X,$ we define the _norm_ of $x$ by

$\|x\|:=d(x,{\mathfrak{o}}).$

We will simplify our notation and denote $\kappa(\|x\|)$ by $\kappa(x)$.

######  Definition 2.1 ($\kappa$–neighborhood and $\kappa$–fellow traveling).

For a closed set $Z\subseteq X$ and a constant ${\sf n}$ define the
_$(\kappa,{\sf n})$ –neighbourhood _ of $Z$ to be

$\mathcal{N}_{\kappa}(Z,{\sf n})=\Big{\\{}x\in
X\mathbin{\Big{|}}d_{X}(x,Z)\leq{\sf n}\cdot\kappa(x)\Big{\\}}.$

Figure 2. A $\kappa$-neighbourhood of the closed set $Z$ with multiplicative
constant ${\sf n}$.

We will say that $\alpha$ is in a $\kappa$-neighborhood of $\beta$ if there
exists ${\sf n}$ such that $\alpha\subset\mathcal{N}_{\kappa}(\beta,{\sf n}).$
If $\alpha$ is in some $\kappa$-neighbourhood of $\beta$ and $\beta$ is in
some $\kappa$–neighbourhood of $\alpha$, we say that $\alpha$ and $\beta$
_$\kappa$ –fellow travel_ each other, written $\alpha\sim_{\kappa}\beta$. In
fact, Lemma 3.1 in [QRT20] shows that if
$\alpha\subset\mathcal{N}_{\kappa}(\kappa,{\sf n})$, then
$\beta\subset\mathcal{N}_{\kappa}(\kappa,2{\sf n})$. In particular, two quasi-
geodesic rays $\kappa$-fellow travel if and only if one of them is in some
$\kappa$-neighborhood of the other.

We next give three definitions of properties that a geodesic ray or closed set
might satisfy, in increasing order of strength.

The first definition is, for us, the most intuitive, since it is a direct
sublinear generalization of the definition of Morse:

######  Definition 2.2.

(weakly $\kappa$-Morse, [QRT20, Definition A.7]) A closed set $Z$ is said to
be _weakly $\kappa$-Morse_ if there exists a map ${\sf
m}_{Z}:\mathbb{R}^{+}\times\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}$ such that
for any $(q,Q)$-quasi-geodesic $\beta$ with end points on $Z,$ we have
$\beta\subset\mathcal{N}_{\kappa}(Z,{\sf m}_{Z}(q,Q)).$

We also consider the following definition.

######  Definition 2.3.

($\kappa$-Morse, [QRT20, Definition 3.2]) Let $\kappa$ be a concave sublinear
function and let ${\mathfrak{o}}\in X$. A closed set $Z$ is said to be
_$\kappa$ -Morse_ if there exists a proper function ${\sf
m}_{Z}:\mathbb{R}^{+}\times\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}$ such that
for any sublinear function $\kappa^{\prime}$ and for any $r>0$, there exists
$R\geq r$ such that for any $(q,Q)$-quasi-geodesic ray $\beta$ with
$\beta(0)={\mathfrak{o}}$ and ${\sf m}_{Z}(q,Q)$ small compared to $r$, we
have

$d_{X}(\beta_{R},Z)\leq\kappa^{\prime}(R)\implies\beta|_{r}\subseteq\mathcal{N}_{\kappa}(Z,{\sf
m}_{Z}(q,Q)).$

The function ${\sf m}_{Z}$ will be called a _Morse gauge_ of $Z.$

Recall that for a closed set $Z\subset X,$ we use $\mathcal{P}(Z)$ to denote
the collection of all subsets of $Z.$

######  Definition 2.4.

($\kappa$-projection, [QRT20, Definition 5.1]) Let $(X,d_{X})$ be a proper
geodesic metric space and $Z\subset X$ be a closed subset. Let $\kappa$ be a
concave sublinear function. A map $\pi_{Z}:X\rightarrow\mathcal{P}(Z)$ is said
to be a _$\kappa$ -projection_ if there exists constants $D_{1},D_{2}$
depending only on $Z$ and $\kappa$ such that for all $x\in X$ and $z\in Z$ we
have

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

We can now state the sublinear contraction property:

######  Definition 2.5.

($\kappa$-contracting [QRT20, Definition 5.3]) Let $Z$ be a closed set in $X$
and let $\pi_{Z}:X\rightarrow\mathcal{P}(Z)$ be a $\kappa$-projection. The set
$Z$ is said to be _$\kappa$ -contracting with respect to $\pi_{Z}$_ if there
are constants $C_{1},C_{2}$, depending only on $Z$ and $\kappa,$ such that for
any $x,y\in X$ we have

$d(x,y)<C_{1}d(x,Z)\implies\text{diam}(\pi_{Z}(x)\cup\pi_{Z}(y))\leq
C_{2}\kappa(x).$

A set is said to be _$\kappa$ -contracting_ if there exists a
$\kappa$-projection $\pi_{Z}$ such that $Z$ is $\kappa$-contracting with
respect to $\pi_{Z}.$ The constant $C_{2}$ in the definition above will be
referred to as the _$\kappa$ -contraction constant_ or simply the _contraction
constant_. In the special case where $C_{1}=1,$ we say that $Z$ is _$\kappa$
-strongly contracting_.

######  Lemma 2.6.

Let $Z,Z^{\prime}\subset X$ be two closed sets such that $Z^{\prime}$ is in
some $\kappa$-neighborhood of $Z$. We have the following:

  1. (1)

$Z$ is weakly $\kappa$-Morse $\iff$ $Z^{\prime}$ is weakly $\kappa$-Morse.

  2. (2)

$Z$ is $\kappa$-Morse $\iff$ $Z^{\prime}$ is $\kappa$-Morse.

  3. (3)

$Z\text{ is }\kappa\text{-contracting}\iff Z^{\prime}\text{ is
}\kappa\text{-contracting.}$

###### Proof.

The proof of item (2) is in [QRT20]. We provide an outline for the proof of
item (3) and leave item (1) as an exercise for the reader. If $Z$ is
$\kappa$-contracting, then there exists a $\kappa$-projection
$\pi_{Z}:X\rightarrow\mathcal{P}(Z)$ as in Definition 2.4. Composing $\pi_{Z}$
with the nearest point projection map to $Z^{\prime}$ yields a map
$\pi_{Z^{\prime}}:X\rightarrow\mathcal{P}(Z^{\prime})$. Since $Z^{\prime}$ is
in a $\kappa$-neighborhood of $Z,$ it is easy to check that the new map
$\pi_{Z^{\prime}}$ is a $\kappa$-projection. Using the definition of
$\pi_{Z^{\prime}}$, if two points $x,y\in X$ have a large projection to
$Z^{\prime}$ under $\pi_{Z^{\prime}}$, then they must have a large projection
to $Z$ under $\pi_{Z}.$ One can combine such observations to show that
elements of Definition 2.5 are all met. ∎

The following theorem explains how these properties are related in the general
setting:

######  Theorem 2.7 ([QRT20, Theorem 5.5], [QRT20, Lemma 3.10]).

Let $X$ b a proper geodesic metric space and let $\kappa$ be a sublinear
function. We have the following.

  1. (1)

If $Z$ is a closed $\kappa$-contracting set, then it is $\kappa$-Morse.

  2. (2)

If $\alpha$ is a $\kappa$-Morse quasi-geodesic ray, then it is weakly
$\kappa$-Morse.

In Theorem 5.1, we prove that Definitions 2.5, 2.3, and 2.2 are all equivalent
in LQC spaces for quasi-geodesic rays.

In [QRT20], the authors define the $\kappa$-_boundary_ $\partial_{\kappa}X$ of
a geodesic metric space $X$ to be the set of all $\kappa$-Morse rays up to
$\kappa$-fellow-traveling. They use the stronger $\kappa$-Morse property to
allow them to use ideas from [CM19] for defining the topology on
$\partial_{\kappa}X$. This topology only arises in our work in one place,
Subsection 6.2, so we will delay discussing it until then.

We end this subsection with some useful observations:

######  Lemma 2.8 ([QR19, Lemma 3.2]).

Let $X$ be a geodesic metric space and let $\kappa$ be a sublinear function.
For any $D_{0},$ there exists $D_{1},D_{2}$, depending only on $\kappa$ and
$D_{0}$ such that for any $x,y\in X$

$d(x,y)\leq D_{0}\kappa(x)\implies D_{1}\kappa(x)\leq\kappa(y)\leq
D_{2}\kappa(x).$

######  Remark 2.9.

The above Lemma 2.8 will be used frequently in the paper. An immediate
consequence of the statement is that if $x,y$ are within distance
$C\kappa(x)$, then they must also be within distance $C^{\prime}\kappa(y)$,
where $C^{\prime}$ depends only on $\kappa$ and $C$. For instance, any set $Z$
which satisfies Definition 2.2, must also satisfy $d(p,Z)\leq
C\kappa(p^{\prime}),$ where $p^{\prime}\in\pi_{Z}(p),$ and $C$ is a constant
depending only on $\kappa$ and ${\sf m}_{Z}(q,Q).$

The following proposition is a useful consequence of Lemma 2.8.

######  Proposition 2.10.

Let $X$ be a proper geodesic metric space and $h\subset X$ a
$(q^{\prime},Q^{\prime})$-quasi-geodesic ray which is weakly $\kappa$-Morse.
For every constants $q,Q$ there exists a constant $K$ so that if
$\alpha:[0,A]\rightarrow X$ is a $(q,Q)$ quasi-geodesic with end points
$h(t_{1}),h(t_{2})\in h$ then the following hold for all $s\in[0,A]$:

  1. (1)

$d(\alpha(s),h)\leq K\kappa(p_{s}),$ where $p_{s}$ is a closest point
projection of $\alpha(s)$ to $h.$

  2. (2)

$d(\alpha(s),h)\leq K\kappa(t_{2}).$

  3. (3)

$d(\alpha(s),h([t_{1},t_{2}])\leq K\kappa(t_{2}).$

###### Proof.

Item (1) follows immediately from Remark 2.9.

For item (2), observe that since $h$ and $\alpha$ are quasi-geodesics, for any
$s\in[0,A]$, we have $d(\alpha(s),h(t_{2}))\leq B|t_{2}-t_{1}|$ for a constant
$B$ that depends only on $q,Q,q^{\prime},Q^{\prime}.$ Since $p_{s}$ is a
nearest point projection of $p$ to $h$, we have

$d(\alpha(s),p_{s})\leq d(\alpha(s),h(t_{2}))\leq B|t_{2}-t_{1}|\leq Bt_{2}.$

Hence, by the triangle inequality, we have that $d(h(t_{2}),p_{s})\leq
2Bt_{2}.$ It follows that

$\displaystyle\|p_{s}\|$ $\displaystyle=d(h(0),p_{s})$ $\displaystyle\leq
d(h(0),h(t_{2}))+d(h(t_{2}),p_{s})$ $\displaystyle\leq Ct_{2}+2Bt_{2},$

where $C$ is a constant depending only on the quasi-geodesic constants of $h.$
Thus, we have $\|p_{s}\|\leq(C+2B)t_{2}\leq Dt_{2}$, where
$D=\text{max}\\{1,C+2B\\}$, and so

$d(\alpha(s),h)=d(\alpha(s),p_{s})\leq{\sf d}\kappa(p_{s})\leq D{\sf
d}\kappa(t_{2}),$

proving item (2).

For item (3), let $\beta=h|_{[t_{1},t_{2}]}.$ By part (2), we have
$d(\alpha(s),h)\leq E\kappa(t_{2}),$ for any point $s\in[0,A].$ Let
$h_{1}:=h|_{[0,t_{1}]}$, and $h_{2}=h|_{[t_{2},\infty)}$. Notice that
$h=h_{1}\cup\beta\cup h_{2}$. We know that that $\alpha(0)=h(t_{1})$ is in the
$E\kappa(t_{2})$-neighborhood of $h_{1}$. Define

$r=\text{sup}\\{s\in[0,A]\,\,|\,\,\alpha(s)\in N(h_{1},E\kappa(t_{2}))\\}.$

By definition of $r$, for any $0<\epsilon<1$, there exist $0\leq
t^{\prime}\leq t_{1}\leq t^{\prime\prime}\leq A$ so that
$d(h(t^{\prime}),\alpha(r-\epsilon))\leq E\kappa(t_{2})$ and
$d(h(t^{\prime\prime}),\alpha(r+\epsilon))\leq E\kappa(t_{2})$, where
$h(t^{\prime\prime})\in\beta\cup h_{2}.$ By the triangle inequality, we have

$\displaystyle d(h(t^{\prime}),h(t^{\prime\prime}))$ $\displaystyle\leq
d(h(t^{\prime}),\alpha(r-\epsilon))+d(\alpha(r-\epsilon),\alpha(r+\epsilon))+d(\alpha(r+\epsilon),h(t^{\prime\prime}))$
$\displaystyle\leq E\kappa(t_{2})+(2\epsilon q+Q)+E\kappa(t_{2})$
$\displaystyle\leq 2E\kappa(t_{2})+2q+Q.$

For $D^{\prime}=2E\kappa(t_{2})+2q+Q$, the above shows that
$d(h(t^{\prime}),h(t^{\prime\prime}))\leq D^{\prime}$. Since $t^{\prime}\leq
t_{1}\leq t^{\prime\prime}$, we have $d(h(t^{\prime}),h(t_{1}))\leq
PD^{\prime}$ for some constant $P$ depending only on $q^{\prime},Q^{\prime}$.
Hence, we have

$\displaystyle d(\alpha(0),\alpha(r))$ $\displaystyle=d(h(t_{1}),\alpha(r))$
$\displaystyle\leq
d(h(t_{1}),h(t^{\prime}))+d(h(t^{\prime}),\alpha(r-\epsilon))+d(\alpha(r-\epsilon),\alpha(r))$
$\displaystyle\leq PD^{\prime}+E\kappa(t_{2})+\epsilon q+Q$ $\displaystyle\leq
PD^{\prime}+E\kappa(t_{2})+q+Q$ $\displaystyle\leq
P(2E\kappa(t_{2})+2q+Q)+E\kappa(t_{2})+q+Q$
$\displaystyle\leq(P(2E+2q+Q)+E+q+Q)\kappa(t_{2}),$

where the last inequality holds as $\kappa\geq 1$ by definition. Let
$E^{\prime}=(P(2E+2q+Q)+E+q+Q)$. Notice that $E^{\prime}$ depends only on
$q,Q,q^{\prime},Q^{\prime}$ and $\kappa.$ We have just shown that
$d(\alpha(0),\alpha(r))\leq E^{\prime}\kappa(t_{2})$. Further, since $\alpha$
is a $(q,Q)$-quasi-geodesic, we get $r\leq q(E^{\prime}\kappa(t_{2})+Q).$
Hence, for any $l\leq r$, we have

$\displaystyle d(\alpha(0),\alpha(l))$ $\displaystyle\leq ql+Q$
$\displaystyle\leq qr+Q$ $\displaystyle\leq
q^{2}(E^{\prime}\kappa(t_{2})+Q)+Q$
$\displaystyle\leq(q^{2}(E^{\prime}+Q)+Q)\kappa(t_{2}),$

where the last inequality holds as $\kappa\geq 1.$ As $\alpha(0)\in\beta,$ we
have $d(\alpha(l),\beta)\leq(q^{2}(E^{\prime}+Q)+Q)\kappa(t_{2})$ for all
$l\in[0,r]$. An identical argument shows that if
$r^{\prime}=\text{inf}\\{s\in[0,A]\,\,|\,\,\alpha(s)\in
N(h_{2},E\kappa(t_{2}))\\},$ then, for all $l^{\prime}\geq r^{\prime}$, we
have $d(\alpha(l^{\prime}),\beta))\leq M\kappa(t_{2})$ for a constant $M$ that
depends only on $q,Q,\kappa,q^{\prime}$ and $Q^{\prime}$. This finishes the
proof.

∎

###  2.3. Visibility

In this subsection, we prove that for any proper geodesic metric space, the
sublinearly Morse boundary is a visibility space. In order to do so, we will
need the following two statements.

######  Lemma 2.11.

Let $\alpha_{1},\alpha_{2}$ be two quasi-geodesic rays with
$\alpha_{1}(0)=\alpha_{2}(0)={\mathfrak{o}}$ and let
$Z=\alpha_{1}\cup\alpha_{2}$. We have the following:

  1. (1)

If $\alpha_{1},\alpha_{2}$ are $\kappa$-Morse, then $Z$ is $\kappa$-Morse.

  2. (2)

If $\alpha_{1},\alpha_{2}$ are weakly $\kappa$-Morse, then $Z$ is weakly
$\kappa$-Morse.

###### Proof.

To see part (1), we define ${\sf m}_{Z}:=\text{max}\\{{\sf
m}_{\alpha_{1}},{\sf m}_{\alpha_{2}}\\}$ and we let $\kappa^{\prime}$ be any
sublinear function and $r>0.$ Since both $\alpha_{1},\alpha_{2}$ are
$\kappa$-Morse, there exists $R_{\alpha_{1}},R_{\alpha_{2}}\geq r$ such that
the conclusion of Definition 2.3 holds. Define
$R:=\text{min}\\{R_{\alpha_{1}},R_{\alpha_{2}}\\}$. If $\beta^{\prime}$ is a
$(q,Q)$-quasi-geodesic ray starting at ${\mathfrak{o}}$ with ${\sf
m}_{Z}(q,Q)$ small compared to $r$ and
$d_{X}(\beta^{\prime}_{R},Z)\leq\kappa^{\prime}(R),$ then
$d_{X}(\beta^{\prime}_{R},\alpha_{1})\leq\kappa^{\prime}(R)$ or
$d_{X}(\beta^{\prime}_{R},\alpha_{2})\leq\kappa^{\prime}(R)$ as
$Z=\alpha_{1}\cup\alpha_{2}.$ Since $\alpha_{1},\alpha_{2}$ are both
$\kappa$-Morse, we get that
$\beta^{\prime}|_{r}\subseteq\mathcal{N}_{\kappa}(\alpha_{1},{\sf
m}_{\alpha}(q,Q))\subset\mathcal{N}_{\kappa}(Z,{\sf m}_{\alpha}(q,Q))$ or
$\beta^{\prime}|_{r}\subseteq\mathcal{N}_{\kappa}(\alpha_{2},{\sf
m}_{\alpha}(q,Q))\subset\mathcal{N}_{\kappa}(Z,{\sf m}_{\alpha}(q,Q))$ which
proves that $Z$ is $\kappa$-Morse. To see part (2), we let $\beta$ be a finite
$(q,Q)$-quasi-geodesics with end points $p_{1},p_{2}$ on $Z.$ If $p_{1},p_{2}$
are both on $\alpha_{1}$ or $\alpha_{2}$ then the conclusion follows as each
$\alpha_{i}$ is weakly $\kappa$-Morse. It remains to consider the case where
$p_{1}\in\alpha_{1}$ and $p_{2}\in\alpha_{2}.$ In this case, let $p$ be a
nearest point projection of ${\mathfrak{o}}$ to $\beta$ and let
$[{\mathfrak{o}},p]$ be a geodesic connecting ${\mathfrak{o}},p,$ then, using
Lemma 2.5 of [QR19], $\beta_{1}=[{\mathfrak{o}},p]\cup[p,p_{1}]_{\beta}$ is a
$(3q,Q)$-quasi-geodesic with end points on $\alpha_{1}$. Similarly,
$\beta_{2}=[{\mathfrak{o}},p]\cup[p,p_{2}]_{\beta}$ is a $(3q,Q)$-quasi-
geodesic with end points on $\alpha_{2}.$ The conclusion follows as
$\alpha_{1},\alpha_{2}$ are each weakly $\kappa$-Morse and
$\beta\subset\beta_{1}\cup\beta_{2}$.

∎

######  Remark 2.12.

Let $\alpha_{1},\alpha_{2}$ be two quasi-geodesic rays starting at
${\mathfrak{o}}$ such that $\alpha_{2}$ is $\kappa$-Morse, and
$\alpha_{1},\alpha_{2}$ don’t $\kappa$-fellow travel each other. We remark
that for any constant $D$ and any sublinear function $\kappa^{\prime}$ the set
$\alpha\cap\mathcal{N}_{\kappa^{\prime}}(\alpha_{2},D)\\}$ is bounded, as
otherwise, Definition 2.3 would imply that $\alpha_{1}$ and $\alpha_{2}$ do
$\kappa$-fellow travel violating the assumption that they don’t. To summarize,
given $\alpha_{1},\alpha_{2}$ as above, for any constant $D$ and any sublinar
function $\kappa^{\prime},$ there exists a constant $D^{\prime},$ depending
only on $D$, $\kappa^{\prime},$ $\alpha_{1}$ and $\alpha_{2}$ such that
$d_{X}({\mathfrak{o}},\alpha_{1}(t))\leq D^{\prime}$ for all
$\alpha_{1}(t)\in\mathcal{N}_{\kappa^{\prime}}(\alpha_{2},D)).$ This remark
will be used in the proof of Theorem 2.13 below.

We show that the $\kappa$-boundary of any proper geodesic metric space is a
visibility space.

######  Theorem 2.13.

(Visibility of sublinear boundaries) Let $\alpha_{1},\alpha_{2}$ be two
$\kappa$-Morse quasi-geodesic rays starting at ${\mathfrak{o}}$ in $X$ which
do not $\kappa$-fellow travel each other. There exists a geodesic line
$\beta:(-\infty,\infty)\rightarrow X$ such that for any $p\in\beta,$ if
$\beta_{1}:=\beta|_{[p,\infty)}$ and $\beta_{2}:=\beta|_{(-\infty,p]}$, then
$[\beta_{1}]=[\alpha_{1}]$ and $[\beta_{2}]=[\alpha_{2}].$

###### Proof.

Since $\alpha_{i}$ is $\kappa$-Morse for each $i=1,2$, Theorem 2.7 gives that
$\alpha_{i}$ is weakly $\kappa$-Morse for $i=1,2.$ Define
$Z=\alpha_{1}\cup\alpha_{2},$ using Lemma 2.11, the set $Z$ is weakly
$\kappa$-Morse. We let ${\sf m}_{Z}$ denote the function as in Definition 2.2
and we fix $m:={\sf m}_{Z}(1,0).$ We let $\beta_{n}:[a_{n},b_{n}]\rightarrow
X$ be a sequence of geodesic segments starting and ending on
$\alpha_{1}(n),\alpha_{2}(n)$ respectively. Since $Z=\alpha_{1}\cup\alpha_{2}$
is weakly $\kappa$-Morse, we get that
$d_{X}(\beta_{n}(t_{n}),\alpha_{1}\cup\alpha_{2})\leq
m\kappa(\beta_{n}(t_{n}))$ for all $n$ and all $t_{n}\in[a_{n},b_{n}].$ If we
define $c_{n}:=\text{inf}\\{t_{n}|t_{n}\in[a_{n},b_{n}]\text{ and
}\beta_{n}(t_{n})\in\mathcal{N}_{\kappa}(\alpha_{2},m)\\}$, then we have

$d_{X}(\beta_{n}(c_{n}),\alpha_{1}))\leq m\kappa(\beta_{n}(c_{n}))+1$

and

$d_{X}(\beta_{n}(c_{n}),\alpha_{2}))\leq m\kappa(\beta(c_{n}))+1.$

This yields two points $p_{n},p_{n}^{\prime}$ in $\alpha_{1},\alpha_{2}$
respectively such that

$\displaystyle d_{X}(\beta_{n}(c_{n}),p_{n}))$ $\displaystyle\leq
m\kappa(\beta_{n}(c_{n}))+1$
$\displaystyle\leq(m+1)\kappa(\beta_{n}(c_{n})),\text{ and}$ $\displaystyle
d_{X}(\beta_{n}(c_{n}),p_{n}^{\prime}))$ $\displaystyle\leq
m\kappa(\beta_{n}(c_{n}))+1$ $\displaystyle\leq(m+1)\kappa(\beta_{n}(c_{n})).$

Applying Lemma 2.8 to the second equation above gives us that
$\kappa(\beta_{n}(c_{n}))\leq C\kappa(p_{n}^{\prime})$ for a constant $C$
depending only on $\kappa$ and $m$. The triangle inequality gives

$\displaystyle d(p_{n},p_{n}^{\prime})$ $\displaystyle\leq
d_{X}(p_{n},\beta_{n}(c_{n}))+d(\beta_{n}(c_{n}),p_{n}^{\prime})$
$\displaystyle\leq 2m\kappa(\beta_{n}(c_{n}))+2$
$\displaystyle\leq(2m+2)\kappa(\beta_{n}(c_{n}))$
$\displaystyle\leq(2m+2)C\kappa(p_{n}^{\prime}).$

Now, if we let $D:=(2m+2)C$, we get that
$p_{n}\in\mathcal{N}_{\kappa}(\alpha_{2},D)$. Using Remark 2.12 above, there
exists $D^{\prime}$, depending only on $\kappa,D,\alpha_{1}$ and $\alpha_{2}$
such that $d(p_{n},{\mathfrak{o}})\leq D^{\prime}.$ On the other hand, since
$d_{X}(\beta_{n}(c_{n}),p_{n}))\leq(m+1)\kappa(\beta_{n}(c_{n}))$, using Lemma
2.8, we get a constant $C^{\prime}$, depending only on $\kappa$ and $m$ such
that $\kappa(\beta_{n}(c_{n}))\leq C^{\prime}\kappa(p_{n}).$ Hence, we have

$\displaystyle d_{X}(\beta_{n}(c_{n}),p_{n}))$
$\displaystyle\leq(m+1)\kappa(\beta_{n}(c_{n}))$
$\displaystyle\leq(m+1)C^{\prime}\kappa(p_{n})$
$\displaystyle=(m+1)C^{\prime}\kappa(d_{X}(p_{n},{\mathfrak{o}}))$
$\displaystyle\leq(m+1)C^{\prime}\kappa(D^{\prime}).$

Now, the triangle inequality gives us that

$\displaystyle d_{X}(\beta_{n}(c_{n}),{\mathfrak{o}})$ $\displaystyle\leq
d_{X}(\beta_{n}(c_{n}),p_{n}))+d_{X}(p_{n},{\mathfrak{o}}))$
$\displaystyle\leq(m+1)C^{\prime}\kappa(D^{\prime})+D^{\prime}.$

This shows that for each $n$, the point $\beta_{n}(c_{n})\in\beta_{n}$ is at a
bounded distance from ${\mathfrak{o}}$ where the bound is independent of $n.$
Therefore, applying Arzelà–Ascoli to $\\{\beta_{n}\\}$ gives a subsequence
$\\{\beta_{n_{k}}\\}$ and a geodesic line $\beta$ with
$\beta_{n_{k}}\rightarrow\beta$ uniformly on compact sets. The line $\beta$
satisfies the conclusion of the theorem.

∎

###  2.4. CAT(0) cube complexes

Our goal for this section is to recall some definitions and facts regarding
CAT(0) spaces and cube complexes. For a more detailed introduction to
$\mathrm{CAT}(0)$ cube complexes, see [Sag14].

Let $(X,d_{X})$ be a metric space. A metric space is called _proper_ if closed
balls are compact. It is called _geodesic_ if any two points $x,y\in X$ can be
connected by a geodesic segment. A proper, geodesic metric space $(X,d_{X})$
is $\mathrm{CAT}(0)$ if geodesic triangles in $X$ are at least as thin as
triangles in Euclidean space with the same side lengths. To be precise, for
any given geodesic triangle $\triangle pqr$, consider (up to isometry) the
unique triangle $\triangle\overline{p}\overline{q}\overline{r}$ in the
Euclidean plane with the same side lengths. For any pair of points $x,y$ on
the triangle, for instance on edges $[p,q]$ and $[p,r]$ of the triangle
$\triangle pqr$, if we choose points $\overline{x}$ and $\overline{y}$ on
edges $[\overline{p},\overline{q}]$ and $[\overline{p},\overline{r}]$ of the
triangle $\triangle\overline{p}\overline{q}\overline{r}$ so that
$d_{X}(p,x)=d_{\mathbb{E}^{2}}(\overline{p},\overline{x})$ and
$d_{X}(p,y)=d_{\mathbb{E}^{2}}(\overline{p},\overline{y})$, then

$d_{X}(x,y)\leq d_{\mathbb{E}^{2}}(\overline{x},\overline{y}).$

A _cube_ is a Euclidean unit cube $[0,1]^{n}$ for some $n\geq 0$. A _midcube_
of a cube $c$ is a subspace obtained by restricting exactly one coordinate in
$[0,1]^{n}$ to $\frac{1}{2}$. For $n\geq 0$, let $[0,1]^{n}$ be an $n$-cube
equipped with the Euclidean metric. We obtain a _face_ of a $n$–cube by
choosing some indices in $\\{1,\dots,n\\}$ and considering the subset of all
points where we for each chosen index $i$, we fix the $i$-th coordinate either
to be zero or to be one. A _cube complex_ is a topological space obtained by
gluing cubes together along faces, i.e. every gluing map is an isometry
between faces. A CAT(0) cube complex $X$ is said to be _finite dimensional_ if
there is an integer $v$ such that every cube in $X$ is of dimension at most
$v.$

Any cube complex can be equipped with a metric as follows: the $n$–cubes are
equipped with the Euclidean metric, which allows us to define the length of
continuous paths inside the cube complex by partitioning every path into
(finitely-many) segments which lie entirely within one cube and add the
lengths of those segments using the Euclidean metric on each cube. We define

$d^{(2)}(x,y):=\inf\\{length(\gamma)|\gamma\text{ a continous path from $x$ to
$y$}\\}.$

The map $d^{(2)}$ defines a metric on $X$. We sometimes call $d^{(2)}$ the
metric induced by the Euclidean metric on each cube.

######  Definition 2.14 ($\mathrm{CAT}(0)$ cube complexes).

Let $X$ be a cube complex and $d^{(2)}$ the metric induced by the Euclidean
metric on each cube. We say that $X$ is a $\mathrm{CAT}(0)$ cube complex if
$(X,d^{(2)})$ is a $\mathrm{CAT}(0)$ space.

In what follows, we will be interested in the _combinatorial metric_ $d$ on
$X$, which is the metric on the $1$-skeleton of $X$. The combinatorial metric
has an alternative description in terms of _hyperplanes_ , which we now
describe.

######  Definition 2.15.

(Hyperplanes, half spaces and separation) Let $X$ be a $\mathrm{CAT}(0)$ cube
complex. A _hyperplane_ is a connected subspace $h\subset X$ such that for
each cube $c$ of $X$, the intersection $h\cap c$ is either empty or a midcube
of $c$. For each hyperplane $h$, the complement $X\setminus h$ has exactly two
components $h^{+},h^{-}$ called _half-spaces_ associated to $h$ . A hyperplane
$h$ is said to _separate_ the sets $U,V\subseteq X$ if $U\subseteq h^{+}$ and
$V\subseteq h^{-}.$

The following lemma is a standard lemma about how the $l^{1}$-metric relates
to the CAT(0) metric of a CAT(0) cube complex, for example, see [CS11].:

######  Lemma 2.16.

If $X$ is a finite-dimensional $\mathrm{CAT}(0)$ cube complex, then the
$\mathrm{CAT}(0)$ metric $d^{(2)}$ and the combinatorial metric $d$ are bi-
Lipschitz equivalent and complete. In particular, if all cubes in $X$ have
dimension $\leq m$, then $d^{(2)}\leq d\leq\sqrt{m}d^{(2)}$. Furthermore, for
two vertices $x,y\in X^{(0)}$, we have

$d(x,y)=|\\{\text{\emph{hyperplanes} $h\subseteq X$ \emph{which separate the
vertices} }x,y\\}|.$

In light of the above lemma and the coarseness of our calculations, we will
often not distinguish between the two metrics.

######  Definition 2.17.

(Combinatorial geodesics/CAT(0) geodesics) A path in the $1$–skeleton of $X$
is called a _combinatorial geodesic_ if it is a geodesic between vertices of
$X$ with respect to the combinatorial metric. A _CAT(0) geodesic_ is a
geodesic with respect to the CAT(0) metric.

There is an alternative description of a combinatorial geodesic coming from
the median structure on a cube complex, which we now describe.

######  Definition 2.18.

(Medians) We define the _median_ of the vertices $x,y,z\in X$ to be the unique
vertex $m(x,y,z)\in X$ obtained by associating to every hyperplane of $X$ its
half-space that contains the majority of the points $x,y,z$. Equivalently,
$m(x,y,z)$ is the unique point that lives in the intersection of the sets of
all combinatorial geodesics connecting $\\{x,y\\},\\{x,z\\}$ and $\\{y,z\\}.$
For two vertices $x,y\in X,$ we define the _median interval_ $[x,y]$ by

$[x,y]=\\{m(x,y,z)|z\in X^{(0)}\\}.$

Equivalently, $[x,y]$ is the union of all combinatorial geodesics connecting
the vertices $x,y$.

The notion of a median above gives rise to the following.

######  Definition 2.19.

(Convexity) A subset $Y$ of a CAT(0) cube complex $X$ is said to be
_combinatorially convex_ if every combinatorial geodesic connecting vertices
$x,y\in Y$ remains inside $Y$. Equivalently, $Y$ is combinatorially convex if
$m(x,y,z)\in Y$ for any vertices $x,y,z\in X$ with $x,y\in Y.$

In order to define the notion of well-separated hyperplanes, we need to
introduce the following definition.

######  Definition 2.20.

(Facing triples) A _facing triple_ is a collection of three disjoint
hyperplanes such that none of them separates the other two.

######  Remark 2.21.

It is immediate by the definition of a facing triple that a geodesic cannot
cross a facing triple.

In contrast to the notion of a facing triple is that of a chain.

######  Definition 2.22 (Chain).

A sequence of hyperplanes $\\{h_{i}\\}$ is said to form a _chain_ if each
$h_{i}$ separates $h_{i-1}$ from $h_{i+1}.$

The following notion was introduced by Genevois [Gen16] to capture the
hyperbolic-like aspects of a CAT(0) cube complex.

######  Definition 2.23.

(Well-separated sets/hyperplanes) Two disjoint combinatorialy convex sets
$Y_{1},Y_{2}$ are said to be _$L$ -well-separated_ if the number of
hyperplanes meeting them both and containing no facing triple has cardinality
at most $L$.

We will mostly be interested in the special case where $Y_{1},Y_{2}$ in
Definition 2.23 above are both hyperplanes. Given a convex set $Y$ in a CAT(0)
cube complex, one can define a notion of a projection from $X$ to $Y$.

######  Definition 2.24 (Combinatorial gate map).

Let $Y$ be a combinatorially convex set in a CAT(0) cube complex $X,$ and let
$x$ be a vertex in $X$. The _combinatorial projection_ of $x$ to $Y$, denoted
$P_{Y}(x),$ is the vertex minimizing the distance $d(x,Y).$ Such a vertex is
unique (for instance, by Lemma 1.2.3 [Gen15]) and it is characterized by the
property that a hyperplane $h$ separates $x,Y$ if and only if it separates
$x,P_{Y}(x).$ For such a characterization, see Lemma 13.8 in [HW07].

For a CAT(0) cube complex $X$ and for a combinatorially convex set $Y$ in $X$,
the combinatorial nearest point projection can be described purely in terms of
medians. Namely, we have the following, which is immediate from the
description of the combinatorial projection above.

######  Lemma 2.25.

Let $X$ be a CAT(0) cube complex and let $Y$ be a set which is combinatorially
convex. If $P_{Y}:X\rightarrow Y$ is the combinatorial nearest point
projection, then

$P_{Y}(x)\in\underset{y\in Y}{\bigcap}[x,y],$

where $[x,y]$ is the median interval between $x,y.$ Furthermore, $P_{Y}(x)$ is
the only point in $Y$ with such a property.

The following is [Gen16, Proposition 14]. Although it’s stated in the special
case where $Y_{1},Y_{2}$ are hyperplanes, the proof only uses the fact that
hyperplanes are combinatorially convex.

######  Proposition 2.26.

Two combinatorially convex sets $Y_{1},Y_{2}$ in a finite dimensional CAT(0)
cube complex $X$ are $L$-well-separated if and only if $P_{Y_{1}}(Y_{2})$ and
$P_{Y_{2}}(Y_{1})$ have diameters at most $cL$, where $c$ is a constant
depending only on the dimension of $X.$

We recall the following theorem from [MQZ20], which characterizes
$\kappa$-Morseness in terms of well-separated hyperplanes:

######  Theorem 2.27 ([MQZ20, Theorem B]).

Let $X$ be a finite dimensional CAT(0) cube complex and let $\kappa$ be a
sublinear function. A geodesic ray $b$ is $\kappa$-Morse if and only if there
exist a constant $c$, a chain of hyperplanes $\\{h_{i}\\}$ and points
$x_{i}\in b\cap h_{i}$ with:

  1. (1)

$d(x_{i},x_{i+1})\leq c\kappa(x_{i}),$ and

  2. (2)

$h_{i},h_{i+1}$ are $c\kappa(x_{i})$-well-separated.

###  2.5. Coarse median spaces

Coarse median spaces were introduced by Bowditch [Bow13] to simultaneously
generalize properties of hyperbolic spaces and cube complexes. See also
[Bow22] for background on median spaces.

######  Definition 2.28 (Coarse median space).

A metric space $(X,d)$ is said to be a _coarse median space_ if there exist a
map $m_{X}:X^{3}\rightarrow X$ and a function
$h:\mathbb{N}\rightarrow\mathbb{N}$ such that the following holds:

  1. (1)

For any $x,y,z,x^{\prime},y^{\prime},z^{\prime}\in X,$ we have

$d(m_{X}(x,y,z),m(x^{\prime},y^{\prime},z^{\prime}))\leq
h(1)(d(x,x^{\prime})+d(y,y^{\prime})+d(z,z^{\prime})).$

  2. (2)

For any $n\in\mathbb{N},$ if $A\subseteq X$ has cardinality at most $n,$ then
there exist a finite cube complex $Q$ and maps $f:Q\rightarrow A,$
$g:A\rightarrow Q$ such that

    1. (a)

$d(f(m_{Q}(a,b,c),m_{X}(f(a),f(b),f(c)))\leq h(n)$ for all $a,b,c\in Q.$

    2. (b)

$d(a,f\circ g(a))\leq h(n)$ for all $a\in A.$

For two points $x,y$ in a coarse median space, the following definition
describes the collection of median points that lives between $x,y$.

######  Definition 2.29 (Intervals).

Given $x,y$ in a coarse median space $X$, we define the _median interval_ from
$x$ to $y$ by

$[x,y]=\\{m(x,y,z)|z\in X\\}.$

Similarly to the cubical context, we can use the median structure to define a
notion of convexity:

######  Definition 2.30 (Median convexity).

Let $X$ be a coarse median space, a subset $Y\subseteq X$ is said to be _$K$
-median convex_ if there exists a constant $K\geq 0$ such that for any $x,y\in
Y$ and $z\in X,$ we have $d(m(x,y,z),Y)\leq K$. Equivalently, $Y$ is
$K$-median convex if there exist some $K$ such that $[x,y]\in N(Y,K)$ whenever
$x,y\in Y$. We will say that $Y$ is _median convex_ if it is $K$-median convex
for some $K\geq 0.$

For $x,y\in X,$ it is natural to expect that $[x,y]$ is a median convex set.
This is indeed the case.

######  Lemma 2.31 ([NWZ19, Lemma 2.21]).

For any $z\in[x,y],$ we have $d(\mu(x,y,z),z)\leq C$ for a constant $C$
depending only on the parameters of $X$. In particular, median intervals are
themselves median convex.

Starting with a set $A,$ it is natural to ask about the smallest median convex
set containing $A$. This requires introducing the following definition.

######  Definition 2.32.

(Joins) Let $A$ be a subset of a coarse median space $X$. We define the
_median join_ of $A$ by

$J(A):=\underset{x,y\in A}{\bigcup}[x,y].$

We define $J^{n}(A)$ inductively by setting $J^{0}(A)=A$ and
$J^{n}(A)=J(J^{n-1}(A)).$

The process above terminates in finitely many steps. More precisely, we have
the following.

######  Lemma 2.33 ([Bow19, Lemma 6.1]).

If $X$ is a coarse median space of rank $v,$ then there exists a constant $k$
depending only on the parameters of $X$, including $v$, such that
$J^{v+1}(A)\subseteq N(J^{v}(A),k).$ In the special case where $X$ is a CAT(0)
cube complex, the constant $k=0,$ that is, $J^{v+1}(A)=J^{v}(A)$.

In light of Lemma 2.33, one can define the median hull of a finite set $A$ as
follows.

######  Definition 2.34.

(Hulls) Let $X$ be a coarse median space of rank $v$ and let $A$ be a subset
of $X$. The _median hull_ of $A$, denoted $\mathrm{hull}(A)$ is defined by

$\mathrm{hull}(A):=J^{v}(A).$

We remark that in the special case where $X$ is a CAT(0) cube complex,
$\mathrm{hull}(A)$ can be equivalently defined by taking the intersection of
all half spaces which properly contain $A.$ Consequently, a hyperplane $h$ in
$X$ separates points in $A$ if and only if it intersects $\mathrm{hull}(A).$
Further, $\mathrm{hull}(A)$ is convex with respect to both the CAT(0) and
combinatorial metrics on $X.$

It is immediate from Lemma 2.33 that the median hull, $\mathrm{hull}(A)$, is
median convex. In fact, in [Bow19] it is shown that $\mathrm{hull}(A)$ is the
smallest median convex set that contains $A$ in the following sense.

######  Proposition 2.35 ([Bow19, proposition 6.1]).

Let $X$ be a coarse median space with rank $v$. There exists an $r\geq 0,$
depending on the parameters of $X$ such that the following holds:

  1. (1)

(Convexity) The set $\mathrm{hull}(A)$ is $r$-median convex.

  2. (2)

(Comparing hulls) If $H$ is an $s$-median convex set containing $A$, then
$\mathrm{hull}(A)\subseteq N(H,s^{\prime})$ where $s^{\prime}$ depends only on
$s$ and the parameters of $X.$

######  Remark 2.36.

In the special case where $X$ is a CAT(0) cube complex and $A\subseteq X,$ the
set $\mathrm{hull}(A)=J^{v}(A)$ is combinatorially convex (i.e., convex in the
$l^{1}$-metric). In particular, the constant $r$ in Proposition 2.35 is $r=0.$

We will be interested in maps which preserve the median structure, in
particular median convexity.

######  Definition 2.37.

(Median maps) Let $K\geq 0$ and let $X,Y$ be two coarse median spaces. A map
$f:X\rightarrow Y$ is said to be _$K$ -median_ if for any $x,y,z\in X,$ we
have

$d(f(m(x,y,z)),m(f(x),f(y),f(z)))<K.$

When dealing with coarse median spaces, the natural morphism to consider is
the following.

######  Definition 2.38.

Let $X,Y$ be two coarse median spaces. A map $f:X\rightarrow Y$ is said to be
a _$K$ -median quasi-isometric embedding_ if:

  1. (1)

$f$ is a $K$-median map, and

  2. (2)

$f$ is a $(K,K)$-quasi-isometric embedding.

If $f$ also satisfies $d_{Haus}(f(X),Y)\leq K$, then $f$ is said to be a _$K$
-median quasi-isometry_.

The following lemma is left as an exercise for the reader:

######  Lemma 2.39.

Let $X,Y$ be coarse median spaces and let $f:X\rightarrow Y$ be a $K$-median
quasi-isometry. We have the following:

  1. (1)

If $H\subseteq X$ is an $r$-median convex set, then $f(H)$ is an
$r^{\prime}$-median convex set where $r^{\prime}$ depends only on $K,r$ and
the parameters of $X,Y$.

  2. (2)

If $H^{\prime}\subseteq Y$ is an $r$-median convex set, then
$f^{-1}(H^{\prime})$ is an $r^{\prime}$-median convex set where $r^{\prime}$
depends only on $K,r$ and the parameters of $X,Y$.

The following lemma says that taking median hulls coarsely commutes with
applying median quasi-isometries. The proof, which we include for
completeness, is a straight-forward consequence of Proposition 2.35(2) and
Lemma 2.39.

######  Lemma 2.40.

Let $X,Y$ be two coarse median spaces of rank $v$ and let $f:X\rightarrow Y$
be some $K$-median quasi-isometry. There exists a constant $K^{\prime}$,
depending only on $v,K$, such that for any set $A\subset X$, we have

$d_{\text{Haus}}(f(\mathrm{hull}(A)),\mathrm{hull}(f(A))\leq K^{\prime}.$

###### Proof.

Let $f$ be as in the statement. Using Proposition 2.35 and Lemma 2.39, if
$A\subseteq X,$ then $f(\mathrm{hull}(A))$ is $r$-median convex set, where $r$
depends only on $K$ and the parameters of $X,Y.$ Since $f(\mathrm{hull}(A))$
is an $r$-median convex containing $f(A)$ , item (2) of Proposition 2.35
yields

$\mathrm{hull}(f(A))\subseteq N(f(Hull(A)),r^{\prime}),$

for some $r^{\prime}$ depending only on the parameters of $Y$ and $r$.

On the other hand, using Lemma 2.39, the set $f^{-1}(\mathrm{hull}(f(A))$ is
an $s$-median convex set containing $A$, where $s$ depends only on $r,K$ and
the parameters of $X,Y$. Thus, using item (2) of Proposition 2.35, we have
$\mathrm{hull}(A)\subseteq N(f^{-1}(\mathrm{hull}(f(A)),s^{\prime})$ where
$s^{\prime}$ depends only on $K$ and the parameters of $X,Y.$ In other words,
every point of the set $H=\mathrm{hull}(A)$ is within $s^{\prime}$ of
$H^{\prime}=f^{-1}(\mathrm{hull}(f(A))$. Since $f$ is a quasi-isometry, there
exists $s^{\prime\prime}$, depending only on $K$ and $s^{\prime}$ such that
every point in $f(H)$ is within $s^{\prime\prime}$ of $f(H^{\prime})$. But
notice that
$f(H^{\prime})=f(f^{-1}(\mathrm{hull}(f(A)))\subseteq\mathrm{hull}(f(A)).$
Hence, $f(\mathrm{hull}(A))=f(H)\subseteq
N(\mathrm{hull}(f(A),s^{\prime\prime}).$ Hence

$f(\mathrm{hull}(A))\subseteq N(\mathrm{hull}(f(A)),s^{\prime\prime}).$

∎

###  2.6. LQC spaces

The category of objects we will consider in this subsection are coarse median
spaces which are locally approximated by cube complexes: _locally quasi-
cubical_ (LQC) spaces.

Our motivation comes from work of Behrstock-Hagen-Sisto [BHS21, Theorem F],
where they show that hierarchically hyperbolic spaces have this property; see
also [Bow19].

The following two definitions (Definition 2.41 and Definition 2.42) extract
this property to the general setting. It generalizes the perspective of
[HHP20], who studied coarse median spaces with quasicubical median intervals.
This LQC property is strictly stronger than the quasicubical interval
property, and arguably more useful, as exhibited by our techniques that
require the flexibility of adding multiple additional points. The theory of
LQC spaces as interesting spaces in their own right is currently being
developed by Petyt, Spriano and the second author in the forthcoming [PSZ23].

######  Definition 2.41.

(Cubical approximations) Let $X$ be a coarse median space and let
$v\in\mathbb{N}\cup\\{0\\},$ $K\geq 0$. A finite subset $F\subseteq X$ is said
to admit a _ $(v,K)$-cubical approximation_ if there exist a CAT(0) cube
complex $Q$ of dimension $v$ and a $K$-median quasi-isometry
$f:Q\rightarrow\mathrm{hull}(F)$.

We now are ready to introduce the notion of a locally quasi-cubical space.

######  Definition 2.42.

(locally quasi-cubical spaces) A coarse median space $X$ is said to be
_locally quasi-cubical_ if there exists a positive integer $v$ such that every
finite set of points $F\subseteq X$ admits some $(v,K)$-cubical approximation
where $K$ depends only on $|F|.$ The integer $v$ is referred to as the
_dimension_ of $X.$

The natural types of paths to consider in LQC spaces are those which are
coarsely preserve medians.

######  Definition 2.43 (median paths/rays).

Let $\lambda\geq 1.$ A _$\lambda$ -median path_ is a
$(\lambda,\lambda)-$quasi-isometric embedding $\alpha:[a,b]\rightarrow X$
which is $\lambda$-median. Similarly, a _$\lambda$ -median ray_ is a
$(\lambda,\lambda)$-quasi-isometric embedding $h:[0,\infty)\rightarrow X$
which is $\lambda$-median.

We will use the following lemma frequently, and we record its proof for
clarity. Similar arguments have appeared in [BHS21] and [DMS20].

######  Lemma 2.44.

Let $X$ be a locally quasi-cubical space. There exists a $\lambda\geq 1$ such
that every two points of $X$ are connected by some $\lambda$-median path.

###### Proof.

Let $x,y$ be points in $X$, let $f:Q\rightarrow[x,y]$ be a $\lambda$-median
quasi-isometry and let $g$ be a coarse inverse of $f$. For any geodesic
$\alpha$ connecting $g(x),g(y)$ in $Q,$ the path $f\circ\alpha$ is a
$\lambda$-median path. Since $\lambda$ is uniform, we are done. ∎

###  2.7. Hierarchically hyperbolic spaces

We conclude the preliminaries with a brief discussion of _hierarchically
hyperbolic spaces_ (HHSes), which are the main examples of non-cubical LQC
spaces. Our eventual main purpose in this subsection is to describe the coarse
median structure on an HHS, as well as the existence and structure of gate
maps to median convex subsets of HHSes. In light of the complexity of the
definition of an HHS, we will only highlight the features that we require for
our analysis. See [Sis18] for an overview of the theory.

Roughly, an HHS is a pair $(\mathcal{X},\mathfrak{S})$ where $\mathcal{X}$ is
a geodesic metric space and $\mathfrak{S}$ is set indexing a family of
uniformly hyperbolic spaces $\mathcal{C}(U)$ for each $U\in\mathfrak{S}$ such
that the following hold:

  1. (1)

For each _domain_ $W\in\mathfrak{S}$ there exist an $E$-hyperbolic space
$\mathcal{C}W$ and a coarsely surjective $(E,E)$-coarsely Lipschitz map
$\pi_{W}:\mathcal{X}\rightarrow\mathcal{C}W,$ with $E$ independent of $W$.

  2. (2)

The set $\mathfrak{S}$ satisfies the following:

     * •

It is equipped with a partial order called _nesting_ , denoted by $\sqsubset$.
If $\mathfrak{S}\neq\emptyset,$ the set $\mathfrak{S}$ contains a unique
$\sqsubset$-maximal element $S.$

     * •

It has a symmetric and anti-reflexive relation called _orthogonality_ ,
denoted by $\perp$. Furthermore, domains have _orthogonal containers_ : if
$U\in\mathfrak{S}$ and there is a domain orthogonal to $U$ then there is a
domain $W$ such that $V\sqsubset W$ whenever $V\perp U.$

     * •

For any distinct non-orthogonal $U,V\in\mathfrak{S}$, if neither is nested
into the other, then we say $U,V$ are _transverse_ and write $U\pitchfork V.$

     * •

There exists a map $\theta:[0,\infty)\rightarrow[0,\infty)$ such that for any
$D\geq 0,$ if $d_{\mathcal{X}}(x,y)\geq\theta(D),$ then
$d(\pi_{U}(x),\pi_{U}(y))>D$ for some $U\in\mathfrak{S}.$ We will refer to
this property as the _uniqueness_ property.

  3. (3)

There exists an integer called the _complexity_ of $\mathcal{X}$ and denoted
$\xi(\mathfrak{S})$ such that whenever $U_{1},U_{2},\cdots U_{n}$ is a
collection of pairwise non-transverse domains, then $n\leq\xi(\mathfrak{S}).$

  4. (4)

If $U\sqsubset V$ or $U\pitchfork V,$ then there exists a set $\rho^{U}_{V}$
of diameter at most $E$ in $\mathcal{C}V.$

  5. (5)

If $U\sqsubset V$, there is a map
$\rho^{V}_{U}:\mathcal{C}(V)\rightarrow\mathcal{C}(U)$ which satisfies the
following: For $x\in\mathcal{X}$ with $d_{V}(\pi_{V}(x),\rho^{U}_{V})>E,$ we
have $\pi_{U}(x)\underset{E}{\asymp}\rho^{V}_{U}(\pi_{V}(x)).$

The constant $E$ above will be referred to as _the HHS constant._ For two
points $x,y\in\mathcal{X},$ it is standard to use $d_{U}(x,y)$ to denote
$d_{\mathcal{C}U}(\pi_{U}(x),\pi_{U}(y)),$ and similarly for subsets of
$\mathcal{X}.$ For a subset $A\subseteq\mathcal{X},$ we will also use
$\mathrm{diam}_{U}(A)$ to denote the diameter of the set
$\pi_{U}(A)\subset\mathcal{C}(U)$.

We will frequently use the following HHS axiom, so we record it separately:

######  Axiom 2.45.

(Bounded geodesic image) If $U\sqsubset V$ and $\gamma\in\mathcal{C}(V)$ is a
geodesic with $d_{V}(\rho^{U}_{V},\gamma)>E$, then
diam$(\rho^{V}_{U}(\gamma))\leq E.$ Furthermore, for any $x,y\in\mathcal{X}$
and any geodesic $\gamma$ connecting $\pi_{V}(x),\pi_{V}(y)$, if
$d_{V}(\rho^{U}_{V},\gamma)>E$, then $d(\pi_{U}(x),\pi_{U}(y))\leq E.$

The following theorem of Behrstock-Hagen-Sisto is the main inspiration for the
definition of local quasi-cubicality.

######  Theorem 2.46 ([BHS21, Theorem F]).

Every hierarchically hyperbolic space is locally quasi-cubical. Furthermore,
there exists some $\lambda\geq 0$, depending only on the HHS constant $E$ such
that for any domain $U,$ the map
$\pi_{U}:\mathcal{X}\rightarrow\mathcal{C}(U)$ is $\lambda$-median.

See also [DMS20] for a stabilization of the LQC property for most HHSes, and
[Pet21] for a proof that mapping class groups are globally (nonequivariantly)
quasicubical.

Another useful HHS construction are _hierarchy paths_ [MM00, Dur16] and
[BHS19, Theorem 4.4]. These are uniform quasi-geodesics which project to
unparametrized quasi-geodesics in $\mathcal{C}U$ for each $U\in\mathfrak{S}$.
Similarly, _hierarchy rays_ are quasi-geodesic rays which projects to
unparametrized quasi-geodesics in each $\mathcal{C}U$.

######  Lemma 2.47 ([BHS21, Lemma 1.37] and [DHS17, Lemma 3.3]).

Let $\mathcal{X}$ be an HHS, we have the following. Every $\lambda$-median
path in $\mathcal{X}$ is a $\lambda^{\prime}$-hierarchy path, where
$\lambda^{\prime}$ depends only on $\lambda$ and on the constants of
$\mathcal{X}$. Furthermore, for any median ray $h$, there exists
$U\in\mathfrak{S}$ such that $\mathrm{diam}_{U}(h)=\infty.$

In fact, in light of the above lemma and Lemma 2.44, which allows us to build
median paths from the cubical models, we will never mention of hierarchy
paths/rays again in this paper. Our canonical paths connecting pairs of points
are just median paths and those are hierarchy paths by Lemma 2.47.

Another fundamental structure of an HHS is its family of standard product
regions, which correspond to stabilizers of multicurves in the mapping class
group setting [MM00].

######  Definition 2.48.

(Standard product region, dominion). For $U\in\mathfrak{S}$ the _standard
product region_ associated to $U$ is the set
$P_{U}:=\\{x\in\mathcal{X}|\,d_{V}(x,\rho^{U}_{V})\leq E\text{ for }U\sqsubset
V\text{ or }U\pitchfork V\\}.$ Fix a basepoint $p\in\mathcal{X}$. The
_dominion_ of $U$ (relative to $p$), denoted $F_{U}$, is the set of points
$x\in P_{U}$ for which $d_{V}(x,p)\leq E$ for all $V\perp U$ and for the
orthogonal container of $U$.

######  Remark 2.49.

We list some properties of the dominion $F_{U},P_{U}$ for more details, see
[BHS19].

  1. (1)

The sets $F_{U},P_{U}$ are median convex.

  2. (2)

The dominion $F_{U}$ is an HHS with respect to the collection
$\mathfrak{S}_{U}=\\{V\in\mathfrak{S}|V\sqsubseteq U\\}$. In fact, if
$x\in\mathcal{X}$, there exists a coarsely unique $x^{\prime}\in F_{U}$ such
that $d_{V}(x^{\prime},x)\leq E$ for all $V\sqsubseteq U.$

  3. (3)

If $\mathcal{X}$ is an HHS and $U$ is a domain, then there exists a median
convex subset of $\mathcal{X}$, denoted $E_{U}$, such that there is an
$E$-median quasi-isometry $P_{U}\to F_{U}\times E_{U}$, where both
$F_{U},E_{U}$ are given with respect to the restriction of the distance
function on $\mathcal{X}.$

The geometry of the median hull of two points is encoded in its _relevant
domains_ :

######  Definition 2.50.

For two points $x,y\in\mathcal{X}$ and a constant $\theta\geq 0,$ we define

$\mathrm{Rel}_{\theta}(x,y):=\\{U\in\mathfrak{S}|d_{U}(x,y)>\theta\\}.$

The following distance formula, generalizing work of Masur-Minsky [MM00], is
one of the main HHS tools. The second item follows directly from the first,
but we record it for later convenience:

######  Theorem 2.51.

Let $\mathcal{X}$ be an HHS. There exist a constant $\theta_{0}$, and map
$K:[\theta_{0},\infty)\rightarrow[0,\infty)$ such that for each
$\theta\geq\theta_{0}$ the following hold:

  1. (1)

For any $x,y\in\mathcal{X}$, we have

$d_{\mathcal{X}}(x,y)\underset{K(\theta)}{\asymp}\underset{U\in\mathrm{Rel}_{\theta}(x,y)}{\sum}d_{U}(x,y).$

  2. (2)

In particular, for any domain $V$ and $x,y\in F_{V}$, we have

$d_{\mathcal{X}}(x,y)\underset{K(\theta)}{\asymp}\underset{U\in\mathrm{Rel}_{\theta}(x,y)\cap\mathfrak{S}_{V}}{\sum}d_{U}(x,y).$

We remark that Bowditch [Bow19] obtained an analogue of the above distance
formula as a consequence of the LQC property in a slightly more general
axiomatic setting than HHSes.

There is a direct connection between the relevant set
$\mathrm{Rel}_{\theta}(x,y)$ and the product regions that a median path or ray
must visit:

######  Proposition 2.52 (Active sub-paths, Corrected version of [BHS19,
proposition 5.17]).

For all sufficiently large $D$ (in terms of the HHS constants), there exists
$v$ such that the following holds. Let $x,y\in\mathcal{X}$, let $\gamma$ be a
$D$-median path connecting $x,y$ path and let $U$ be a domain with
$d_{U}(x,y)\geq 200DE.$ Then, $\gamma$ has a subpath $\beta$ such that:

  1. (1)

$\beta\subseteq\mathcal{N}_{v}(P_{U}).$

  2. (2)

$\pi_{U}$ is $v$-coarsely constant on any sub-path of $\gamma$ disjoint from
$\beta.$

We will frequently use the following “passing up” lemma:

######  Lemma 2.53 ([BHS19, Lemma 2.5]).

Let $\mathcal{X}$ be an HHS with constant $E$. For every $C>0$ there is an
integer $m=m(C)$ such that if $V\in\mathfrak{S}$ and $x,y\in\mathcal{X}$
satisfy $d_{U_{i}}(x,y)>E$ for a collection of domains $\\{U_{i}\\}_{i=1}^{m}$
with $U_{i}\in\mathfrak{S}_{V}$, then there exists $W\in\mathfrak{S}_{V}$ with
$U_{i}\sqsubsetneq W$ for some $i$ such that $d_{W}(x,y)>C.$

In order to get our characterization of the $\kappa$-Morse property for a
median ray in an HHS via the sublinear growth of subsurface projections
(Theorem 7.1), we will need to know when a ray passes close to some $P_{U}$,
then that $P_{U}$ is actually a quasiflat.

The following definition eliminates this potential pathology of general HHSes:

######  Definition 2.54 (Bounded domain dichotomy).

An HHS $\mathcal{X}$ has the _bounded domain dichotomy_ if there exists $B>0$
so that if $U\in\mathfrak{S}$ has $\mathrm{diam}(\mathcal{C}(U))>B$, then
$\mathrm{diam}(\mathcal{C}(U))=\infty$.

All of the relevant HHSes (and all HHGs) satisfy the bounded domain dichotomy.
It was introduced in [ABD21] to prove the following:

######  Theorem 2.55 (ABD structure).

Any hierarchically hyperbolic space $\mathcal{X}$ satisfying the bounded
domain dichotomy admits an HHS structure $(\mathcal{X},\mathfrak{S})$ so that
for all $U\in\mathfrak{S}$ with $U\neq S$, we have that both $F_{U}$ and
$E_{U}$ are infinite diameter.

Going forward, we make the blanket assumption that $\mathcal{X}$ is a proper
HHS with unbounded products and its ABD structure as in the theorem above.

The next lemma explains the basic properties of medians, as well as median
convex sets and their gate maps:

######  Lemma 2.56 ([BHS19, Lemma 5.5], [RST18, Corollary 5.12]).

Let $\mathcal{X}$ be an HHS. There exists
$E^{\prime}=E^{\prime}(\mathcal{X})>0$ so that for any median convex set $Y$,
there exists a map $\mathfrak{g}_{Y}:\mathcal{X}\rightarrow Y$, called the
_gate_ , satisfying the following properties:

  1. (1)

For any $x,y,z\in\mathcal{X}$, the median $m=m_{\mathcal{X}}(x,y,z)$ satisfies
$d_{U}(m,m_{U}(x,y,z))<E^{\prime}$.

  2. (2)

If $Y$ is median convex, then $\pi_{U}(Y)$ is $E^{\prime}$-quasiconvex for
each $U\in\mathfrak{S}$.

  3. (3)

$\mathfrak{g}_{Y}$ is $E^{\prime}$-coarsely Lipschitz, that is
$d_{\mathcal{X}}(\mathfrak{g}_{Y}(x),\mathfrak{g}_{Y}(x^{\prime}))\leq
E^{\prime}d_{\mathcal{X}}(x,x^{\prime})+E^{\prime}$ for all
$x,x^{\prime}\in\mathcal{X}$.

  4. (4)

For each $x\in\mathcal{X},$ the set $\pi_{U}(\mathfrak{g}_{Y}(x))$ coarsely
agrees with the projection of $\pi_{U}(x)$ to $\pi_{U}(Y).$

  5. (5)

For $x\in\mathcal{X},$ if $x^{\prime}$ is a nearest point projection of $x$ to
$Y,$ then

$d(x,x^{\prime})\underset{E^{\prime}}{\asymp}d(x,\mathfrak{g}_{H}(x)).$

Another useful consequence of this fact and the preceding discussion is the
following lemma, which gives an alternative way to define the gate map to a
median convex subset.

######  Lemma 2.57.

Let $Y\subset\mathcal{X}$ be a $K$-median convex subset where $\mathcal{X}$ is
an HHS. There exists a constant $C$, depending only on $K$ and $\mathcal{X}$
such that the following holds.

  1. (1)

For any $x\in\mathcal{X}$ and any $y\in Y$, we have that
$m(x,\mathfrak{g}_{Y}(x),y)\underset{C}{\asymp}\mathfrak{g}_{Y}(x)$. That is,

$\mathfrak{g}_{Y}(x)\underset{C}{\in}\bigcap_{y\in Y}[x,y].$

  2. (2)

For any $z\in Y$ with $\displaystyle z\underset{C}{\in}\bigcap_{y\in Y}[x,y]$,
we have $z\underset{C}{\asymp}\mathfrak{g}_{Y}(x)$.

###### Proof.

Let $x\in\mathcal{X}$, $y\in Y$ and let $z=\mathfrak{g}_{Y}(x)$. We claim that
$m(x,z,y)$ coarsely agrees with $z$. By construction, $\pi_{U}(z)$ coarsely
coincides with the closest point projection of $\pi_{U}(x)$ to $\pi_{U}(Y)$ ,
for every $U$. Furthermore, by hyperbolicity of $\mathcal{C}(U)$, for any
$x^{\prime}\in\pi_{U}(x),y^{\prime}\in\pi_{U}(Y)$ and any $z^{\prime}$ living
in the closest point projection of $\pi_{U}(x)$ to $\pi_{U}(Y)$, the point
$m_{U}(x^{\prime},z^{\prime},y^{\prime})$ coarsely agrees with $z^{\prime}$.
Hence $z$ coarsely coincides with $m(x,z,y)$ by the uniqueness property (• ‣
2) of HHSes. This finishes the proof of (1).

For (2), if $z\in Y$ satisfies $z\underset{C}{\in}\underset{y\in
Y}{\bigcap}[x,y]$, we have $z\in[x,\mathfrak{g}_{Y}(x)]$. Hence, we have
$\mathfrak{g}_{Y}(x)\underset{C}{\asymp}m(x,\mathfrak{g}_{Y}(x),z)\underset{C}{\asymp}z$
where the last equality holds by Lemma 2.31. This completes the proof.

∎

Combining item (2) of Lemma 2.57 with Lemma 2.25 shows that for a
combinatorialy convex set $Y$ of a CAT(0) cube complex $Q$ which is also an
HHS, the map $\mathfrak{g}_{Y}$ coarsely agrees with the combonatorial nearest
point projection $P_{Y}.$

Item (2) above yields a description of gates via medians, as it says that if
$Y$ is a median convex set, the gate of a point $x\in\mathcal{X}$ to $Y$ is
coarsely the unique point in $Y$ which lives in the median interval $[x,y]$
for all $y\in Y$. Hence median quasi-isometries not only preserve distances,
but also gate maps. In particular, this yield the following corollary.

######  Corollary 2.58.

Let $\mathcal{X},\mathcal{Y}$ be two HHSes, and
$f:\mathcal{Y}\rightarrow\mathcal{X}$ be $K$-median quasi-isometric embedding.
Suppose that $Y_{1},Y_{2}\subset\mathcal{Y}$ are two $K$-median convex sets.
There exists a constant $C$, depending only on $K,$ $\mathcal{X}$, and
$\mathcal{Y}$ such that:

  1. (1)

For each $x\in Y_{1}$, we have
$f(\mathfrak{g}_{Y_{2}}(x))\underset{C}{\asymp}\mathfrak{g}_{f(Y_{2})}(f(x)).$

  2. (2)

$\mathrm{diam}_{\mathcal{Y}}(\mathfrak{g}_{Y_{2}}(Y_{1}))\underset{C}{\asymp}\mathrm{diam}_{\mathcal{X}}(\mathfrak{g}_{f(Y_{2})}(f(Y_{1}))).$

###### Proof.

Since $Y_{1},Y_{2}$ are $K$-median convex, we get a constant $C$ such that
Lemma 2.57 holds. Let $y^{\prime}\in f(Y_{2})$ and let $y\in Y$ with
$f(y)=y^{\prime}.$ Using Lemma 2.57, we have
$\mathfrak{g}_{Y_{2}}(x)\underset{C}{\asymp}m(x,y,\mathfrak{g}_{Y_{2}}(x))$.
Thus,
$f(\mathfrak{g}_{Y_{2}}(x))\underset{C}{\asymp}f(m(x,y,\mathfrak{g}_{Y_{2}}(x)))\underset{C}{\asymp}m(f(x),y^{\prime},f(\mathfrak{g}_{Y_{2}}(x)))$.
This shows that for an arbitrary $y^{\prime}\in f(Y_{2}),$ we have
$f(\mathfrak{g}_{Y_{2}}(x))\underset{C}{\asymp}m(f(x),y^{\prime},f(\mathfrak{g}_{Y_{2}}(x))).$
Hence, by part (2) of Lemma 2.57, we get that

$f(\mathfrak{g}_{Y_{2}}(x))\underset{C}{\asymp}\mathfrak{g}_{f(Y_{2})}(f(x)).$

Part (2) follows immediately from part (1) and the assumption that $f$ is a
quasi-isometry. ∎

##  3\. Preliminaries on sublinear Morseness in CAT(0) spaces and cube
complexes

In this section, we will work in a fixed CAT(0) space $X$. The first
subsection assumes that $X$ is a CAT(0) space, but the last two subsections
are about CAT(0) cube complexes.

Since most of the results in this paper involve exporting problems to
appropriate CAT(0) cube complexes via median quasi-isometries, the work in
this section plays an important supporting role.

###  3.1. Simple description of Morseness in CAT(0) spaces

The following theorem states that in a CAT(0) space $X,$ all notions of
$\kappa$-Morseness are equivalent.

######  Theorem 3.1 ([QR19, Theorem 3.10]).

Let $\alpha$ be a quasi-geodesic ray in a CAT(0) space. The following are all
equivalent:

  1. (1)

$\alpha$ is $\kappa$-Morse.

  2. (2)

$\alpha$ is $\kappa$-contracting.

  3. (3)

$\alpha$ is weakly $\kappa$-Morse.

###  3.2. Median of quasi-geodesic rays in CAT(0) cube complexes

The main goal of this subsection is to characterize $\kappa$-fellow traveling
of two quasi-geodesic rays in a cube complex via the median (Corollary 3.7).
It is essential for establishing injectivity of the map from the
$\kappa$-boundary of an HHS into the boundary of its top level curve graph
(Theorem 6.1).

For the rest of this section, we now assume that $X$ is a finite dimensional
CAT(0) cube complex.

We note that versions of many of the statements in this subsection were proven
in [IMZ21] for geodesics, though we need them for quasi-geodesics, which
requires some extra work.

######  Lemma 3.2.

Let $q,q^{\prime}$ be two quasi-geodesic rays in $X$ starting at the same
vertex ${\mathfrak{o}}$ and let $h_{1},h_{2}$ be two $L$-well-separated
hyperplanes with ${\mathfrak{o}}\in h_{i}^{-}$ for $i=1,2.$ If there exists
$t_{0}$ such that $q(t)\in h_{2}^{+}$ for $t\geq t_{0}$ and $q^{\prime}(t)\in
h_{1}^{-}$, then $d(q(t),q^{\prime}(t))$ is bounded below by a linear function
in $t.$

###### Proof.

Let $t_{0}$ be as in the statement. Notice that among every hyperplane
separating $q(t_{0}),q(t)$ at most $L$ can intersect both $h_{1},h_{2}.$
Therefore, if $t\geq t_{0},$ then, the vertices $q(t),q^{\prime}(t)$ are
separated by at least $(d(q(t),q(t_{0}))-L)\asymp t$ hyperplanes, this gives
the desired statement. ∎

This gives the following corollary.

######  Corollary 3.3.

Let $q$ be a continuous weakly $\kappa$-Morse quasi-geodesic ray which crosses
two $L$-well-separated hyperplanes $h_{1},h_{2}$ in that order. Then $q$ can
cross both $h_{1},h_{2}$ only finitely many times.

###### Proof.

Suppose for the sake of contradiction that $q$ crosses $h_{1},h_{2}$
infinitely many times. Since $h_{1},h_{2}$ are $L$-well-separated, they must
be disjoint. Thus, without loss of generality, we can assume that
$(h_{2})^{+}\subset(h_{1})^{+}$. Since $q$ crosses both $h_{1},h_{2}$
infinitely many times, there must exist two infinite sequences
$\\{s_{i}\\},\\{t_{j}\\}$ such that $q(s_{i})\in(h_{1})^{-}$ and
$q(t_{j})\in(h_{2})^{+}$. We may assume that
${\mathfrak{o}}=q(0)\in(h_{1})^{-}$.

Now, let $g_{i}$ be a sequence of combinatorial geodesic segments connecting
$q(0)$ to $q(s_{i})$. Let $p$ be the combinatorial projection of the vertex
${\mathfrak{o}}$ to $h_{2}.$ Let $w_{j}$ be a sequence of combinatorial
geodesic segments connecting $p$ to $q(t_{j})$ and let $g_{j}$ be the
concatenation of $w_{j}$ with a geodesic connecting ${\mathfrak{o}}$ to $p$.
Since $p$ is the combinatorial projection of ${\mathfrak{o}}$ to $h_{2},$ this
concatenation is a geodesic. That is, the path $g_{j}$ is a geodesic segment
starting at ${\mathfrak{o}}$ and ending at $q(t_{j}).$ Applying Arzelà–Ascoli
to the two sequences of geodesic segments $g_{i},g_{j}$ yields two
$\kappa$-fellow traveling combinatorial geodesic rays $g_{1},g_{2}$ such that
$g_{1}\in(h_{1})^{-}$ and $g_{2}[t_{0},\infty)\in h_{2}^{+}$ for some $t_{0}.$
However, using Lemma 3.2, the geodesics $g_{1},g_{2}$ must diverge linearly
which is a contradiction. ∎

######  Lemma 3.4.

Let $q$ be a continuous $\kappa$-Morse quasi-geodesic ray and let
$Y=\mathrm{hull}(q)$ be its median hull. If $\partial Y$ denotes the visual
boundary of $Y$, then $|\partial Y|=1.$

###### Proof.

Recall that $Y=\mathrm{hull}(q)$ is convex in both the CAT(0) and the
combinatorial metrics. Let $[q(0),q(i)]$ be a sequence of CAT(0) geodesic
segments. Applying Arzelà–Ascoli gives a $\kappa$-Morse CAT(0) geodesic ray
$b$ living in $Y$.

Let $c$ be another geodesic ray in $Y$ starting at $b(0)$ (not necessarily
$\kappa$-Morse). We claim that $c=b.$ Suppose not. Since $\kappa$-Morse
geodesic rays define visibility points in the visual boundary, we get a
combinatorial geodesic line, denoted $l$, connecting the points
$b(\infty),c(\infty)$ in the visual boundary of $Y$. Without loss of
generality, suppose that $l(\infty)=b(\infty)$ and $l(-\infty)=c(\infty).$
Since the geodesic ray $b^{\prime}=[l(0),l(\infty))$ is $\kappa$-Morse, by
Theorem 2.27, it must cross an infinite sequence of well-separated hyperplanes
$\\{h_{1},h_{2},h_{3},\dots\\}$. In particular, the hyperplanes $h_{1},h_{2}$
are $L$-well-separated for some $L$. Using Corollary 3.3, there exists some
$t_{0}$ such that $b([t_{0},\infty))$ does not cross $h_{2}.$ On the other
hand, the geodesic ray $c^{\prime}=[l(0),l(\infty))$ crosses an infinite chain
of hyperplanes $\\{k_{1},k_{2},\dots\\}$ without any facing triples. Since
$h_{1},h_{2}$ are $L$-well-separated and since $\\{k_{1},k_{2},\dots\\}$
contain no facing triples, at most $L$ hyperplanes among
$\\{k_{1},k_{2},\dots\\}$ can cross both $h_{1},h_{2}$. This implies that
there exists an integer $n$ such that $\\{k_{n},k_{n+1},k_{n+2}\dots\\}$ does
not cross $h_{2}$. Therefore, $h_{2}$ separates
$\\{k_{n},k_{n+1},k_{n+2}\dots\\}$ from $b([t_{0},\infty))$. This contradicts
the fact that the hyperplanes $\\{k_{n},k_{n+1},k_{n+2}\dots\\}$ are all met
by $q$, by definition of $\mathrm{hull}(q)$.

∎

######  Lemma 3.5.

If $q,q^{\prime}$ are quasi-geodesic rays starting at ${\mathfrak{o}}$ such
that $m({\mathfrak{o}},q(t_{i}),q^{\prime}(t_{j}))\rightarrow\infty$ as
$t_{i},t_{j}\rightarrow\infty$, then both $q,q^{\prime}$ cross infinitely many
of the same hyperplanes.

###### Proof.

Recall that the median $m(x,y,z)$ is characterized by being the unique vertex
resulting from orienting every hyperplane $h\subset X$ towards the majority of
$x,y,z$ and taking the intersection of the resulting half-spaces (Definition
2.18). Let $a_{i,j}=m({\mathfrak{o}},q(t_{i}),q(t_{j}))$ and let
$\alpha_{i,j}$ be a geodesic connecting ${\mathfrak{o}}$ to $a_{i,j}$. Using
the definition of the median, every hyperplane crossing $\alpha_{i,j}$ must
separate ${\mathfrak{o}}$ from $\\{q(t_{i}),q^{\prime}(t_{j})\\}$. Hence,
every such hyperplane crosses both $q,q^{\prime}.$ ∎

######  Lemma 3.6.

If $q,q^{\prime}$ are $\kappa$-Morse quasi-geodesic rays that cross infinitely
many of the same hyperplanes, then $q\sim_{\kappa}q^{\prime}$.

###### Proof.

Let ${\mathfrak{o}}=q(0)=q^{\prime}(0)$ and let $Y_{1},Y_{2}$ denote
$\mathrm{hull}(q_{1})$, $\mathrm{hull}(q_{2})$ respectively. As the quasi-
geodesics $q_{1},q_{2}$ cross infinitely many of the same hyperplanes, these
same hyperplanes also cross both of $Y_{1},Y_{2}$, by definition of the hull.
Denote these hyperplanes by $\\{h_{1},h_{2},h_{3},\dots\\}$.

Since $Y_{1}\cap Y_{2},Y_{1}\cap h_{i},Y_{2}\cap h_{i}$ are all non-empty
convex subcomplexes, the Helly property implies that there exists a common
intersection point $x_{i}\in Y_{1}\cap Y_{2}\cap h_{i}$. Now, consider the
sequence of CAT(0) geodesic segments $[{\mathfrak{o}},x_{i}]$. Using convexity
of both $Y_{1},Y_{2}$, we have $[{\mathfrak{o}},x_{i}]\in Y_{1}\cap Y_{2}$. Up
to passing to a subsequence, the sequence $[{\mathfrak{o}},x_{i}]$ converges
to a geodesic ray $b\in Y_{1}\cap Y_{2}$. Since $|\partial Y_{1}|=1$ and $q$
is $\kappa$-Morse, the geodesic ray $b$ must be $\kappa$-Morse and it
$\kappa$-fellow travels $q.$ Since $q^{\prime}$ is also $\kappa$-Morse, we
have $|\partial Y_{2}|=1$, therefore, $b$ is the unique geodesic ray living in
$Y_{2}$ with $b(0)={\mathfrak{o}}.$ Hence, $q^{\prime}$ must $\kappa$-fellow
travel $b$ as well. Since both $q$ and $q^{\prime}$ do $\kappa$-fellow travel
$b$, they must $\kappa$-fellow travel, completing the proof. ∎

As a corollary of the previous two lemmas, we obtain the main result of this
subsection.

######  Corollary 3.7.

Let $q,q^{\prime}$ be continuous $\kappa$-Morse quasi-geodesic rays with
$q(0)=q^{\prime}(0)$. We have

$d({\mathfrak{o}},m(q(t_{i}),q^{\prime}(t_{j}),{\mathfrak{o}}))\rightarrow\infty\iff
q\sim_{\kappa}q^{\prime}.$

###### Proof.

The forward direction follows by combining Lemma 3.5 and Lemma 3.6. For the
backwards direction, we argue by contradiction. Assume that there exist points
$t_{i},t_{i}\rightarrow\infty$ and a bounded set $M$ containing
${\mathfrak{o}}$ and such that $m(t_{i},t_{j},{\mathfrak{o}})\in M$ for all
$i,j$. This implies that there exists a combinatorial geodesic $g_{i,j}$
connecting $t_{i},t_{j}$ which goes through $M$, hence, by applying Arzelà-
Ascoli we get a geodesic line $l:(-\infty,\infty)\rightarrow X$. After
possibly enlarging $M$ by a finite amount, we can choose a point $l(s)\in M$.
It is immediate from the definitions that $l_{[s,\infty)}$ and
$l_{(-\infty,s]}$ are $\kappa$-Morse geodesic rays representing
$q,q^{\prime}$, hence, $l_{[s,\infty)},l_{(-\infty,s]}$ $\kappa$-fellow travel
each other which is not possible because $l$ is a geodesic.

∎

###  3.3. Excursions

In this subsection, we give a characterization of $\kappa$-Morse quasi-
geodesics in a CAT(0) cube complex via a sequence of a certain type of
hyperplanes we call _excursion_.

The definition is motivated by the definition of $\kappa$-excursion for
geodesics in [MQZ20] (see the statement of Theorem 2.27). In our setting, we
need to work with quasi-geodesics, and it is more convenient to encode the
excursion property into hyperplanes, as follows.

######  Definition 3.8.

Let $X$ be a CAT(0) cube complex and let $\kappa$ be a sublinear function. A
chain of hyperplanes $\\{h_{i}\\}_{i\in\mathbb{N}}$ is said to be _$\kappa$
-excursion_ if there exists a constant $c$ and points $x_{i}\in h_{i}$ such
that:

  1. (1)

$h_{i},h_{i+1}$ are $c\kappa(x_{i})$-well-separated.

  2. (2)

$d(x_{i},x_{i+1})\leq c\kappa(x_{i}).$

We note that the assumption that a collection of $\kappa$-excursion
hyperplanes is a chain (Definition 2.22) is necessary and important.

######  Remark 3.9.

Given a chain of $\kappa$-excursion hyperplanes
$\\{(h_{i},x_{i})\\}_{i\in\mathbb{N}}$, we note that since the $h_{i},h_{i+1}$
are $c\kappa(x_{i})$-well-separated, the images of the combinatorial gate map
$P_{h_{i+1}}|_{h_{i}}:h_{i}\to h_{i+1}$ has diameter bounded above by
$c^{\prime}c\kappa(x_{i})$ where $c^{\prime}$ depends only on the dimension of
the cube complex (Proposition 2.26). Since the gate map is a closest point
projection and $d(x_{i},x_{i+1})\leq c\kappa(x_{i})$, it follows that
$d(x_{i+1},P_{h_{i+1}}(h_{i}))\leq 2c\kappa(x_{i})$. That is, $x_{i+1}$ is
$\kappa$-close to $P_{h_{i+1}}(h_{i})$.

This aligns philosophically with how we use the cubical models in Section 7 to
characterize $\kappa$-Morse median paths in an HHS.

The points $x_{i}$ above are referred to as the _excursion points_. Similarly,
the constant $c$ is called the _excursion constant_. We will sometime use the
notation $\\{(h_{i},x_{i})\\}_{i\in\mathbb{N}}$ to refer to the hyperplanes
$h_{i}$ and the points $x_{i}$ above.

The first lemma says that any geodesic crossing an infinite chain of excursion
hyperplanes crosses them $\kappa$-close to the excursion points.

######  Lemma 3.10.

Let $\\{h_{i}\\}_{i\in\mathbb{N}}$ be a chain of $\kappa$-excursion
hyperplanes with excursion points $\\{x_{i}\\}_{i\in\mathbb{N}}$ and excursion
constant $c.$ If $b$ is a geodesic connecting two points $x,y$ such that $x$
is between $h_{i-2},h_{i-1}$ and $y$ is between $h_{i+1},h_{i+2}$, then every
geodesic (combinatorial or CAT(0)) connecting $x,y$ must cross $h_{i}$ at a
point $z$ with $d(z,x_{i})\leq 4c\kappa(x_{i}).$

###### Proof.

The argument for this is essentially the same as the proof of [IMZ21, Lemma
3.2]; see also [MQZ20, Lemma 4.15], but we include it for completeness. Let
$[x,y]$ be a geodesic which crosses $h_{i}$ at $z$, as in the statement. We
need to bound the number of hyperplanes separating $x_{i}$ and $z.$ Denote the
collection of such hyperplanes by $\mathcal{H}.$ Every hyperplane in
$\mathcal{H}$ which separates neither $x_{i-1},x_{i}$ nor $x_{i},x_{i+1}$ must
cross either $h_{i-1}$ or $h_{i+1}$. Hence, the cardinality of such
hyperplanes is at most $c\kappa(x_{i-1})+c\kappa(x_{i})\leq 2c\kappa(x_{i})$
This gives us

$\displaystyle|\mathcal{H}|$ $\displaystyle\leq
d(x_{i-1},x_{i})+d(x_{i},x_{i+1})+2c\kappa(x_{i})$ $\displaystyle\leq
c\kappa(x_{i-1})+c\kappa(x_{i})+2c\kappa(x_{i})$ $\displaystyle\leq
4c\kappa(x_{i}).$

Figure 3. A picture for Lemma 3.10: If a hyperplane separates $z$ from $x_{i}$
must either cross $h_{i-1},h_{i}$ or $h_{i},h_{i+1}$ like $j_{2}$, or separate
$x_{i-1},x_{i}$ or $x_{i},x_{i+1}$ like $j_{1}$. In each case, the number of
such hyperplanes is $\kappa$-bounded.

∎

We remark the following.

######  Remark 3.11.

Observe that if $h$ is a $\lambda$-median path in a CAT(0) cube complex $X$
and $h$ intersects a hyperplane $k$ at points $h(t_{1}),h(t_{2})$, then for
any $h(t)$ between $h(t_{1}),h(t_{2}),$ we have

$h(t)\underset{\lambda}{\asymp}m_{X}(h(t_{1}),h(t),h(t_{2}))\underset{\lambda}{\in}k.$

In words, the above equation states that if a $\lambda$-median path $h$
crosses a hyperplane $k$ twice, then the distance it can travel in a given
half space of $k$, say $k^{+},$ is bounded above by a constant depending only
on $\lambda.$ This can be thought of as a coarsening of the property that _no
geodesic can cross a hyperplane twice_. In light of this, the conclusion of
Lemma 3.10 still holds if the geodesic $b$ is replaced by a continuous
$\lambda$-median path $h$. That is, given $\\{(h_{i},x_{i})\\},x,y$ as in
Lemma 3.10, if $h$ is a continuous $\lambda$-median path connecting $x,y$,
then $h$ crosses $h_{i}$ at a point $z$ with $d(z,x_{i})\leq
c^{\prime}\kappa(x_{i})$ where $c^{\prime}$ depends only on the excursion
constant $c$ and on $\lambda.$

The next lemma says that a $\kappa$-excursion chain of hyperplanes can be
represented by a geodesic.

######  Lemma 3.12.

Let $\\{h_{i}\\}_{i\in\mathbb{N}}$ be a chain of $\kappa$-excursion
hyperplanes with excursion points $\\{x_{i}\\}_{i\in\mathbb{N}}$ and excursion
constant $c.$ There exists a $\kappa$-Morse, CAT(0) geodesic ray $b$ such that
$x_{i}\in\mathcal{N}_{\kappa}(b,4c).$

###### Proof.

The proof is a straight-forward application of Arzelà-Ascoli and Lemma 3.10.
In particular, if we let $b_{i}=[x_{1},x_{i}]$ be the unique CAT(0) geodesic
segment connecting $x_{1},x_{i}$, then up to passing to a subsequence $b_{i}$
converges to a geodesic ray $b.$ By Lemma 3.10, the geodesic ray $b$ must
cross each $h_{i}$ at a point $z_{i}$ with $d(z_{i},x_{i})\leq
4c\kappa(x_{i}).$ This implies that $d(z_{i},z_{i+1})\leq
4c\kappa(x_{i})+c\kappa(x_{i})+4c\kappa(x_{i+1})\leq 9c\kappa(x_{i+1}).$ Since
$d(x_{i+1},z_{i+1})\leq 4c\kappa(x_{i+1})$, Lemma 2.8 gives us that
$d(z_{i},z_{i+1})\leq d\kappa(z_{i+1})$, for a constant $d$, depending only on
$c$ and $\kappa.$ Thus, the ray $b$ must be $\kappa$-Morse by 2.27. ∎

We are now ready to characterize $\kappa$-Morseness for quasi-geodesics in
terms of crossing a chain of $\kappa$-excursion hyperplanes.

######  Theorem 3.13.

A quasi-geodesic ray $\alpha$ is $\kappa$-Morse if and only if there exists an
infinite collection of $\kappa$-excursion hyperplanes such that the
$\kappa$-excursion points $x_{i}\in\alpha.$

###### Proof.

Suppose that $\alpha$ and $\\{x_{i}\\}_{i\in\mathbb{N}}$ are as in the
assumption of the backwards direction. Lemma 3.12 provides a $\kappa$-Morse
geodesic ray $b$ such that $x_{i}\in\mathcal{N}_{\kappa}(b).$ As each
$x_{i}\in\alpha,$ and $b$ is $\kappa$-Morse, we get that
$\alpha\subseteq\mathcal{N}_{\kappa}(b).$ Since $b$ is $\kappa$-Morse,
$\alpha$ is also $\kappa$-Morse.

For the forwards direction, assume that $\alpha$ is $\kappa$-Morse with gauge
${\sf m}_{\alpha}$, and $H=\mathrm{hull}(\alpha)$. Then $\alpha$ is
$\kappa$-Morse in the CAT(0) cube complex $H.$ Hence, by [QR19, Proposition
3.10], there exists a $\kappa$-Morse CAT(0) geodesic ray $b$ in $H$
representing $\alpha.$ In particular, we have
$\alpha\subset\mathcal{N}_{\kappa}(b,{\sf n}),$ where ${\sf n}$ depends only
on the gauge ${\sf m}_{\alpha}$. Hence, using Theorem 2.27, $b$ crosses an
infinite sequence of $\kappa$-excursion hyperplanes
$\\{h_{i}\\}_{i\in\mathbb{N}}$ with excursion points $y_{i}\in b$ and
excursion constant $c$ depending only on $X$, $\kappa$ and ${\sf m}_{\alpha}.$

Since $b\in H,$ every hyperplane crossing $b$ must also cross $\alpha$. Let
$x_{i}\in\alpha\cap h_{i}$. Using item (3) of [MQZ20, Corollary 4.16], if
$x_{i}^{\prime}$ is the projection of $x_{i}$ to $b,$ then
$d(y_{i},x_{i}^{\prime})\leq D\kappa(y_{i}),$ where $D$ depends only on $X,$
$\kappa$ and ${\sf m}_{\alpha}.$ Hence, for each $i$ we have

$\displaystyle d(x_{i},y_{i})$ $\displaystyle\leq
d(x_{i},x_{i}^{\prime})+d(x_{i}^{\prime},y_{i})$ $\displaystyle\leq{\sf
n}\kappa(x_{i})+D\kappa(y_{i})$ $\displaystyle\leq({\sf
n}+D)\kappa(\max\\{\|x_{i}\|,\|y_{i}\|\\}).$

Lemma 2.8 gives us a constant $D_{1}\geq 1$, depending only on $\kappa,$ ${\sf
m}_{\alpha}$, and $X$ (and not on $\max\\{\|x_{i}\|,\|y_{i}\|\\}$) such that
$\kappa(y_{i})\leq D_{1}\kappa(x_{i})$ for each $i.$ Hence

$\displaystyle d(x_{i},x_{i+1})$ $\displaystyle\leq
d(x_{i},x_{i}^{\prime})+d(x_{i}^{\prime},y_{i})+d(y_{i},y_{i+1})+d(y_{i+1},x_{i+1}^{\prime})+d(x_{i+1}^{\prime},x_{i+1})$
$\displaystyle\leq{\sf
n}\kappa(x_{i})+D\kappa(y_{i})+c\kappa(y_{i})+D\kappa(y_{i+1})+{\sf
n}\kappa(x_{i+1})$ $\displaystyle\leq(2n+2D+c)D_{1}\kappa(x_{i+1}).$

Applying Lemma 2.8 again gives us a constant $D_{2}$ such that
$d(x_{i},x_{i+1})\leq D_{2}\kappa(x_{i})$ which concludes the proof of the
forward direction.

∎

######  Remark 3.14.

Using the proof of Theorem 3.13, we observe the following:

  1. (1)

In the backwards direction of the proof of Theorem 3.13, the Morse gauge of
$\alpha$, denoted ${\sf m}_{\alpha}$, depends only on $\kappa$, $X$, the
excursion constant $c,$ and $d(\alpha(0),h_{1}).$ Conversely, for a fixed
$\kappa$, the Morse gauge ${\sf m}_{\alpha}$ determines the excursion constant
$c$ and $d(\alpha(0),h_{1}).$

  2. (2)

Suppose that $\alpha$ is a $\kappa$-Morse quasi-geodesic ray, the forward
direction of the previous theorem provides a sequence of $\kappa$-excursion
hyperplanes $\\{h_{i}\\}_{i\in\mathbb{N}}$ and excursion points $x_{i}\in
h_{i}\cap\alpha.$ The proof of the forward direction shows that for any points
$y_{i}\in h_{i}\cap\alpha,$ we must have $d(y_{i},y_{i+1})\leq
D^{\prime}\kappa(y_{i}),$ for a constant $D^{\prime}$ depending only on ${\sf
m}_{\alpha}$, $\kappa$ and $X$. In particular, if $t_{i}$ is the first time
$\alpha$ meets $h_{i}$ and $t_{i}^{\prime}$ is the last time $\alpha$ meets
$h_{i+1},$ then $d(\alpha(t_{i}),\alpha(t_{i}^{\prime}))$ is bounded above by
$D^{\prime}\kappa(y_{i}).$ Consequently, the number of hyperplanes that can
intersect the subpath $[\alpha(t_{i}),\alpha(t_{i}^{\prime})]_{\alpha}$ is
bounded above by $D^{\prime\prime}\kappa(y_{i}),$ where $D^{\prime\prime}$
depends only on $D^{\prime}$ and $({\sf q},{\sf Q}),$ where ${\sf q},{\sf Q}$
are the quasi-geodesic constants of $\alpha.$

###  3.4. Contraction of hulls

The main statement we prove in this subsection is Proposition 3.17. It states
that if $\alpha$, $\beta$ are two $\kappa$-Morse quasi-geodesic rays, then the
set $H=\mathrm{hull}(\alpha\cup\beta)$ is $\kappa$-strongly contracting with
respect to the combinatorial nearest point projection $P_{H}:X\rightarrow H.$

The proof proceeds as follows: Each ray determines a family of $\kappa$-well-
separated hyperplanes, which can be used to partition the hull $H$ into a
family of subsets whose diameters are bounded by $\kappa$. Any point in $X$
lives in the intersection of the half-spaces associated to two adjacent
hyperplanes in the family, and this determines the subset of $H$ to which it
projects; see Figure 5. When considering two points $x,y$ outside of the hull,
one uses $\kappa$-well-separation to bound the number of hyperplanes that can
separate any geodesic between $x,y$ from the hull. This says that such a
geodesic passes $\kappa$-close to the hull, proving that the hull is
$\kappa$-contracting.

The following lemma is the main technical statement supporting the proof:

######  Lemma 3.15.

Let $\alpha,\beta$ be two $\kappa$-Morse quasi-geodesic rays starting at
${\mathfrak{o}}$ and let $\\{(h_{i},x_{i})\\}_{i\in\mathbb{N}}$
$\\{(k_{i},y_{i})\\}_{i\in\mathbb{N}}$ be the two sequences of
$\kappa$-excursion hyperplanes corresponding to $\alpha,\beta.$ If $\alpha$
and $\beta$ do not $\kappa$-fellow travel each other, and
$H=\mathrm{hull}(\alpha\cup\beta),$ then there exist constants $c,L$ and some
$n_{0}$ such that for any fixed $i>n_{0}$, we have:

  1. (1)

The hyperplanes $k_{i},h_{i}$ are $L$-well-separated.

  2. (2)

The hyperplanes $\cdots k_{i-2},k_{i-1},k_{i},h_{i},h_{i+1},h_{i+2}\cdots$
form a chain.

  3. (3)

$\mathrm{diam}(k_{i}^{-}\cap h_{i}^{-}\cap H)<\infty,$ where
$k_{i}^{-},h_{i}^{-}$ are the half spaces containing ${\mathfrak{o}}.$

  4. (4)

For any $x,y\in H,$ if $x,y$ are between $h_{i},h_{i+1},$ then $d(x,y)\leq
c\kappa(x_{i}).$

  5. (5)

For any $x,y\in H,$ if $x,y$ are between $k_{i},k_{i+1},$ then $d(x,y)\leq
c\kappa(y_{i}).$

###### Proof.

Since $\alpha,\beta$ do not $\kappa$-fellow travel, Lemma 3.6 gives us that
only finitely many hyperplanes can intersect both $\alpha$ and $\beta.$ Let
$m$ be large enough to ensure that $h_{i}\cap\beta=\emptyset$ and
$k_{i}\cap\alpha=\emptyset$ for all $i\geq m.$ Observe that as $h_{m},h_{m+1}$
are disjoint and neither of them crosses $\beta$, if $h_{m+1}$ crosses a
hyperplane $k_{i}$, then so does $h_{m}.$ Since $h_{m},h_{m+1}$ are well-
separated, only finitely many hyperplanes, say $K$, from the collection
$\\{k_{i}\\}_{i\in\mathbb{N}}$ can be crossed by $h_{m+1}.$ Therefore, since
the $h_{i},k_{i}$ form chains, the hyperplanes $h_{m+2},k_{m+K}$ are $L$-well-
separated for some $L$. Choosing $n_{0}=\max\\{m+2,m+K\\}$ concludes the proof
of parts (1) and (2).

Figure 4. Lemma 3.15, parts (1) and (2): If $h_{m+1}$ crosses some $k_{i}$,
then $h_{m}$ must as well, since both avoid $\beta$.

For part (3), fix some $i\geq n_{0}$ and let $p$ be the last point on $\alpha$
which intersects $h_{i+1}$. Similarly, we define $p^{\prime}$ to be the last
point on $\beta$ which intersects $k_{i+1}$ and let $A$ be the number of
hyperplanes intersecting the union of $\alpha\cup\beta$ between $p^{\prime}$
and $p$. Since $h_{i},h_{i+1}$ are well-separated, at most $L_{1}$
hyperplanes, with no facing triples, can meet them. Similarly, at most $L_{2}$
hyperplanes, with no facing triples, can meet $k_{i},k_{i+1}$.

We will now bound $\mathrm{diam}(k_{i}^{-}\cap h_{i}^{-}\cap H)$ in terms of
$A,L_{1}$ and $L_{2}$. Let $x,y\in k_{i}^{-}\cap h_{i}^{-}\cap H,$ and let
$\mathcal{H}$ be the collection of separating $x,y.$ Such a collection
contains no facing triple as each hyperplane in $\mathcal{H}$ separates $x,y$.
Since every hyperplane in $\mathcal{H}$ intersects $H$, it must intersect
$\alpha\cup\beta$. There are exactly three possibilities for where a
hyperplane $h\in\mathcal{H}$ crosses $\alpha\cup\beta$:

  1. (1)

In $h_{i+1}^{+}$, and hence crosses both $h_{i},h_{i+1}$,

  2. (2)

In $k_{i+1}^{+}$, and hence crosses both $k_{i},k_{i+1}$, or

  3. (3)

In $h_{i+1}^{-}\cap k_{i+1}^{-}\cap H$, of which there are $A$ such
hyperplanes.

Since $h_{i},h_{i+1}$ and $k_{i},k_{i+1}$ are well-separated with constants
$L_{1},L_{2}$, we have the following:

$\displaystyle|\mathcal{H}|$ $\displaystyle\leq|\mathcal{H}\cap
h_{i+1}^{+}|+|\mathcal{H}\cap k_{i+1}^{+}|+|\mathcal{H}\cap h_{i+1}^{-}\cap
k_{i+1}^{-}|$ $\displaystyle\leq L_{1}+L_{2}+A.$

Hence, $\mathrm{diam}(k_{i}^{-}\cap h_{i}^{-}\cap H)\leq L_{1}+L_{2}+A.$

Figure 5. Lemma 3.15, part (3): Hyperplanes which separate $x$ and $y$ have to
intersect $\alpha\cup\beta$, and this forces them either between
$p,p^{\prime}$ or to cross $k_{i},k_{i+1}$ or $h_{i},h_{i+1}$.

Now we prove part (4). Let $n_{0}$ be as above and consider the chain

$\cdots k_{n_{0}+2},k_{n_{0}+1},h_{n_{0}+1},h_{n_{0}+2},\cdots.$

We denote this chain by $\mathcal{J}.$

Let $i>n_{0}+1$ be fixed and suppose $x,y$ are between $h_{i},h_{i+1}$, as in
the part (4) of the statement. The combinatorial distance between $x,y$ is the
number of hyperplanes separating them. Denote by $\mathcal{H}(x,y)$ the
collection of such hyperplanes, and notice that it contains no facing triple
as each hyperplane in $\mathcal{H}$ separates $x,y$. Using the definition of
$\mathrm{hull}(\alpha\cup\beta)$, every hyperplane in $\mathcal{H}(x,y)$ must
meet either $\alpha$ or $\beta$. Since $\beta\subset h_{j}^{-}$ for each
$j>n_{0}$ and since $\mathcal{J}$ is a chain, every hyperplane
$h\in\mathcal{H}(x,y)$ must satisfy one of the following:

  1. (1)

$h$ meets $\alpha$ between $h_{i-1},h_{i+2}$,

  2. (2)

$h$ meets $(\alpha\cup\beta)\cap h_{i-1}^{-}$ and thus crosses both
$h_{i-1},h_{i}$, or

  3. (3)

$h$ meets $\alpha\cap h_{i+2}^{+}$ and thus crosses both $h_{i+1},h_{i+2}.$

Let $\mathcal{J}_{1},\mathcal{J}_{2},\mathcal{J}_{3}$ denote the hyperplanes
$h\in\mathcal{H}(x,y)$ satisfying conditions (1),(2), and (3), respectively.
In particular, we have
$\mathcal{J}=\mathcal{J}_{1}\cup\mathcal{J}_{2}\cup\mathcal{J}_{3}.$ Using
Theorem 3.13 and part (2) of Remark 3.14, $|\mathcal{J}_{1}|$ is bounded above
by $D_{1}\kappa(x_{i+1}),$ for a constant $D_{1}$ depending only on $\kappa$,
${\sf m}_{\alpha}$, $X$ and the quasi-geodesic constants of $\alpha.$ Further,
using the well-separation of $h_{i},h_{i-1},$ and $h_{i+1},h_{i+2},$ we have
$|\mathcal{J}_{2}|\leq D_{2}\kappa(x_{i-1})$ and $|\mathcal{J}_{3}|\leq
D_{2}\kappa(x_{i+1})$. Hence, we get the following:

$\displaystyle d(x,y)$ $\displaystyle=|\mathcal{H}(x,y)|$
$\displaystyle\leq|\mathcal{J}_{1}|+|\mathcal{J}_{2}|+|\mathcal{J}_{3}|$
$\displaystyle\leq
D_{1}\kappa(x_{i+1})+D_{2}\kappa(x_{i-1})+D_{2}\kappa(x_{i+1})$
$\displaystyle\leq\text{max}(D_{1},D_{2})\kappa(x_{i+1}).$

Using Lemma 2.8, we get a constant $D_{4}$, depending only on $\kappa,{\sf
m}_{\alpha},$ $X$ and the quasi-geodesic constants of $\alpha$ such that
$d(x,y)\leq D_{4}\kappa(x_{i}).$ This concludes the proof of (4). The proof of
(5) is identical.

Figure 6. Lemma 3.15, part (4): Hyperplanes separating $x,y$ either cross the
$\kappa$-bounded piece of $\alpha$ between $h_{i-1}$ and $h_{i+2}$ like
$j_{2}$, cross $h_{i-2},h_{i-1}$ like $j_{1}$, or $h_{i+1},h_{i+2}$ like
$j_{3}$.

∎

We remark that an identical argument to the one given in item (4) of Lemma
3.15 above shows the following.

######  Lemma 3.16.

Let $\alpha$ be a $\kappa$-Morse quasi-geodesic ray, $H=\mathrm{hull}(\alpha)$
and let $\\{(h_{i},x_{i})\\}_{i\in\mathbb{N}}$ be the sequence of excursion
hyperplanes crossed by $\alpha$. There exists a constant $c$ such that for any
$i\in\mathbb{N},$ if $x,y\in H$ are between $h_{i},h_{i+1}$, then

$d(x,y)\leq c\kappa(x_{i}).$

In particular, $H\subset\mathcal{N}_{\kappa}(\alpha,c).$

We now are ready to prove the main proposition of this subsection.

######  Proposition 3.17.

Let $X$ be a CAT(0) cube complex and let $\alpha,\beta$ be two $\kappa$-Morse
quasi-geodesic rays which begin at $\alpha(0)=\beta(0)={\mathfrak{o}}.$ There
exists a constant $C_{1}$ such that for any $x,y\in X$ if $d(x,y)\leq
d(x,P_{H}(x)),$ then we have $d(P_{H}(x),P_{H}(y))\leq C_{1}\kappa(x)$, where
$P_{H}:X\rightarrow H$ is the combinatorial nearest point projection. In
particular, the set $H$ is $\kappa$-strongly-contracting.

###### Proof.

We will only prove the statement in the case where $\alpha$ and $\beta$ do not
$\kappa$-fellow travel each other. The proof for when they do is similar and
we leave it as an exercise for the reader.

Let $\alpha,\beta$ be two $\kappa$-Morse quasi-geodesic rays starting at
${\mathfrak{o}}$, $\\{(h_{i},x_{i})\\}_{i\in\mathbb{N}}$
$\\{(k_{i},y_{i})\\}_{i\in\mathbb{N}}$ be the two sequences of
$\kappa$-excursion hyperplanes corresponding to $\alpha,\beta$ and let $D$ be
the maximum of the two excursion constants. If $n_{0}$ is the constant
provided by Lemma 3.15, then we get a bi-infinite chain $\mathcal{H}$

$\cdots k_{n_{0}+2},k_{n_{0}+1},h_{n_{0}+1},h_{n_{0}+2},\cdots.$

We claim that the combinatorial nearest point projection of any point $x$ that
lives between any two consecutive hyperplanes of the above chain, say
$h_{i+1},h_{i+2}$, must project to a point $P_{H}(x)$ that satisfies
$d(P_{H}(x),x_{i})\leq c\kappa(x_{i}),$ where the constant $c$ is uniform over
$i\geq n_{0}$. To see this, first observe that the claim holds if $P_{H}(x)$
lies in $h_{i}^{+}\cap h_{i+3}^{-}$ by Lemma 2.8. Otherwise, let $b$ be a
combinatorial geodesic connecting $x$ to $P_{H}(x)$, and assume, without loss
of generality, that $P_{H}(x)$ lies in $h^{-}_{j}$ for $j<i-1$. Then $b$
crosses $h_{i-1}$, and so Lemma 3.10 says that the point $z_{i-1}$ at which
$b$ crosses $h_{i-1}$ must satisfy $d(z_{i-1},x_{i})<c\kappa(x_{i})$. Since
$x_{i}\in H$, this says that $P_{H}(x)\in h^{+}_{i-2}$. Lemma 2.8 then says
that $P_{H}(x)$ satisfies $d(P_{H}(x),p)\leq c_{0}\kappa(x_{i})$ for some
point $p\in H$ between $h_{i-1},h_{i}$ and some constant $c_{0}$ that is
uniform over $i$. However, item (4) of Lemma 3.15 along with Lemma 2.8 gives
us a uniform constant $c$ such that $d(P_{H}(x),x_{i})\leq c\kappa(x_{i})$
concluding the proof of the claim.

To see that $H$ is $\kappa$-strongly contracting, we let $x,y\in X$ with
$d(x,y)<d(x,P_{H}(x))$. The point $x$ must lie between two consecutive
hyperplanes in $\mathcal{H}$. Since the rest of the proof only uses the fact
that we have a chain of excursion hyperplanes, we can assume that $x$ lies
between $h_{i},h_{i+1}$. In particular, the argument when $x$ lies between
$k_{i},k_{i+1}$ or $k_{n_{0}+1},h_{n_{0}+1}$ are identical.

If the point $y$ lies between $h_{i-2},h_{i+2}$, then, we are done using the