Each $V\in\mathcal{V}_{i}^{\prime}$ is $\sqsubset_{\mathcal{U}}$-minimal;

  * •

For each $V\in\mathcal{V}^{\prime}$, we have $\delta^{V}_{U}\subset
C_{i}^{\prime}$;

  * •

$\\{\delta^{V}_{U}|V\in\mathcal{V}_{i}\\}$ is a $(2r+4R,r)$-net on
$C_{i}^{\prime}$ (see Definition 8.4 below).

Note that the increase in the net constants in the third item is required to
guarantee the second item.

In particular, we have
$\\#\mathcal{V}_{i}\succ\mathrm{diam}_{T_{U}}(C_{i}^{\prime})\geq\mathrm{diam}_{T_{U}}(C_{i})-2E^{\prime}$.

Let $V\in\mathcal{V}_{i}^{\prime}$. Then, by construction,
$d_{T_{U}}(\delta^{V}_{U},x_{U})>\alpha_{0}+E^{\prime}$ and
$d_{T_{U}}(\delta^{V}_{U},y_{U})>\alpha_{0}+E^{\prime}$. Moreover, up to
ignoring at most $N_{1}=N_{1}(\mathfrak{S},|F\cup\Lambda|)>0$ domains
$V\in\mathcal{V}$, Lemma 8.5 provides $a,b\in F\cup\Lambda$ where $C^{\prime}$
separates $a_{U}$ from $b_{U}$ in $T_{U}$ and $V\in\mathrm{Rel}_{K}(a,b)$.
Hence the BGI property (5) of Lemma 6.13 says that
$d_{T_{V}}(\delta^{U}_{V}(a_{V}),\delta^{U}_{V}(x_{V}))<E^{\prime}$ and
$d_{T_{V}}(\delta^{U}_{V}(b_{V}),\delta^{U}_{V}(y_{V}))<E^{\prime}$.

Lemma 6.16 says that the tuples $(a_{U}),(b_{U}),(x_{U}),(y_{U})$ are all
$\alpha_{0}$-consistent. Thus since
$d_{T_{U}}(a_{U},\delta^{V}_{U})>E^{\prime}+\alpha_{0}$ and similarly for
$b_{U},x_{U},y_{U}$, $\alpha_{0}$-consistency forces that
$d_{T_{V}}(a_{V},\delta^{U}_{V}(a_{U}))<\alpha_{0}$, and similarly for
$b_{V},x_{V},y_{V}$. As such, we get that
$d_{T_{V}}(x_{V},a_{V})<\alpha_{0}+E^{\prime}$ and
$d_{T_{V}}(y_{V},b_{V})<\alpha_{0}+E^{\prime}$, and so that
$d_{T_{V}}(x_{V},y_{V})\geq d_{T_{V}}(a_{V},b_{V})-2\alpha_{0}-2E^{\prime}$,
from which we can conclude that $\hat{a}_{V}=\hat{x}_{V}$ and
$\hat{b}_{V}=\hat{y}_{V}$ by requiring $R>\alpha_{0}+E^{\prime}$.

On the other hand, since each $V\in\mathcal{V}_{i}^{\prime}$ is
$\sqsubset_{\mathcal{U}}$-minimal and $d_{V}(a,b)>K$, item (4) of Lemma 7.14
says that $d_{\widehat{T}_{V}}(\hat{a}_{V},\hat{b}_{V})\succ K$. Hence we
obtain that
$d_{\widehat{T}_{V}}(\hat{x}_{V},\hat{y}_{V})=d_{\widehat{T}_{V}}(\hat{a}_{V},\hat{b}_{V})\succ
K$.

Thus for each $U\in\mathcal{W}$, we have

$\displaystyle d_{T_{U}}(x_{U},y_{U})$ $\displaystyle=$ $\displaystyle
d_{\widehat{T}_{U}}(\hat{x}_{U},\hat{y}_{U})+\sum_{i=1}^{n}\mathrm{diam}_{T_{U}}(C_{i}\cap[x_{U},y_{U}]_{T_{U}})$
$\displaystyle\prec$ $\displaystyle
d_{\widehat{T}_{U}}(\hat{x}_{U},\hat{y}_{U})+\sum_{i=1}^{n}\\#\mathcal{V}_{i}^{\prime}$
$\displaystyle<$ $\displaystyle
d_{\widehat{T}_{U}}(\hat{x}_{U},\hat{y}_{U})+\sum_{i=1}^{n}K\cdot\\#\mathcal{V}_{i}^{\prime}$
$\displaystyle\prec$ $\displaystyle
d_{\widehat{T}_{U}}(\hat{x}_{U},\hat{y}_{U})+\sum_{i=1}^{n}\left(\sum_{V\in\mathcal{V}_{i}^{\prime}}d_{\widehat{T}_{V}}(\hat{x}_{V},\hat{y}_{V})\right).$

Finally, the Covering Lemma 4.8 says that any domain $V\in\mathcal{U}$ can
nest into only boundedly-many domains in $\mathcal{U}$, with bounds determined
by $\mathfrak{S},|F\cup\Lambda|$ since Lemma 4.8 applies to each pair $a,b\in
F\cup\Lambda$. Hence, each $\sqsubset_{\mathcal{U}}$-minimal domain
$V\in\mathcal{U}$ must account for at most boundedly-many $U\in\mathcal{W}$ in
the above way. Thus we can account for the distance lost by collapsing
clusters in each domain in $\mathcal{W}$ with these large
$\sqsubset_{\mathcal{U}}$-minimal domains by increasing the multiplicative
constant for the estimate in (6) by a bounded amount. This completes the
proof.

Figure 9. Proof of Proposition 8.3: The subcluster $C^{\prime}$ isolates the
portion of $C$ between the $R$-neighborhood of $x_{U}$,
$\mathcal{N}_{R}(x_{U})$, and $y_{U}$. The point $z\in H$ coarsely projects
the same as $y$ and the rest of the ray $b$ to all domains with shadows in
$C^{\prime}$.

∎

The following definition was used in the proof of Proposition 8.3:

######  Definition 8.4 (Net).

Given constants $a,A\geq 0$, a $(a,A)$-_net_ on a subspace $Y\subset X$ of a
metric space is a collection $Z\subset Y$ of points so that:

  1. (1)

$Z$ is $a$-coarsely dense in $Y$, and

  2. (2)

For any $z,z^{\prime}\in Z$, we have $d_{X}(z,z^{\prime})\geq A$.

The following lemma was also used in the proof, and we use the same notation:

######  Lemma 8.5.

There exists $N_{1}=N_{1}(\mathfrak{S},|F\cup\Lambda|)>0$ so that for each $i$
and for all but $N_{1}$-many domains $V\in\mathcal{V}_{i}^{\prime}$, there
exists $a,b\in F\cup\Lambda$ so that $d_{T_{V}}(a_{V},b_{V})\succ K$.

###### Proof.

By construction, the set of $\delta^{V}_{U}$ for
$V\in\mathcal{V}_{i}^{\prime}$ is a $(2r+4R,r)$-net in $C^{\prime}$
(Definition 8.4), and hence only boundedly-many $\delta^{V}_{U}$ sets are
within $E^{\prime}$ of a branch point of $T_{U}$ depending on the uniform
bound on branching of $T_{U}$. Each remaining $\delta^{V}_{U}$ lies along an
edge of $T_{U}$, and hence separates $T_{U}=T_{x}\cup T_{z}$ into two
components, the component containing $x_{V}$ and the component containing
$z_{V}$. Let $l_{x}$ be the marked points and rays of $T_{U}$ in $T_{x}$, and
similarly for $l_{z}$. Since each such $\delta^{V}_{U}$ is $E^{\prime}$-away
from any branch point, the BGI property in item (5) of Lemma 6.13 implies that
the geodesics in $T_{U}$ between any pair of marked points or rays in $p,q\in
l_{x}$ must avoid the $E^{\prime}$-neighborhood of $\delta^{V}_{U}$, and thus
$d_{T_{V}}(p_{V},q_{V})<E^{\prime}$, and similarly for marked points and rays
of $l_{z}$.

Yet $V\in\mathcal{U}=\mathrm{Rel}_{K}(F\cup\Lambda)$, and so there exists
$a,b\in F\cup\Lambda$ so that $d_{T_{V}}(a_{V},b_{V})\succ K$. By the above
argument, we must have $a\in l_{x}$ and $b\in l_{z}$, as required. ∎

######  Remark 8.6.

In the proof above, we used the observation that all but boundedly many
shadows for $V\in\mathcal{V}^{\prime}$ avoided the branch points of $T_{U}$.
In fact, this something much stronger is true, namely that all but boundedly-
many domains in $U\in\mathcal{U}$ have the property that $\delta^{U}_{W}$
avoids the branch points of $T_{W}$ for each $W\in\mathcal{U}$ with
$U\sqsubset W$. See Subsection 9.1 below.

##  9\. The map $\Omega:\mathcal{Q}\to\mathcal{Y}$

The purpose of this section is to define a map
$\Omega:\mathcal{Q}\to\mathcal{Y}$ and analyze its features. In particular,
the main goal is to prove the following:

######  Proposition 9.1.

There exist $\alpha_{\omega}=\alpha_{\omega}(\mathfrak{S},|F\cup\Lambda|)>0$
and a map $\Omega:\mathcal{Q}\to\mathcal{Y}$ so that for any
$\hat{x}\in\mathcal{Q}$, the tuple
$\Omega(\hat{x})\in\mathcal{Z}_{\alpha_{\omega}}$. Moreover, if
$\Omega(\hat{x})=(x_{U})$, then $q_{U}(x_{U})=\hat{x}_{U}$ for each
$U\in\mathcal{U}$.

With $\Omega:\mathcal{Q}\to\mathcal{Z}_{\alpha_{\omega}}$ in hand, we will be
able to use our realization result (Proposition 6.17) to define the map
$\widehat{\Omega}:\mathcal{Q}\to H$. In Section 11, we will show that
$\widehat{\Psi}$ is a coarse inverse of $\widehat{\Omega}$, and therefore
$\widehat{\Psi}$ is surjective. Combining this with the distance estimates
from Section 8 will prove that $\widehat{\Psi}:H\to\mathcal{Q}$ is a quasi-
isometric embedding.

The proof of Proposition 9.1 involves analyzing the structure of clusters in
depth. This is arguably the most technical part of the construction of the
cubical model, and undoubtedly where cubical arguments first appear in spirit
though not explicitly.

###  9.1. Bipartite domains

The first step is to make the observation that for all but boundedly-many
domains in $\mathcal{U}$, we have that $\widehat{T}_{U}$ is an interval.

######  Definition 9.2.

A domain $U\in\mathcal{U}$ is called $\chi$-_bipartite_ if there is some
$V\in\mathcal{U}$ with $U\sqsubset V$ so that
$T_{V}-\mathcal{N}_{\chi}(\delta^{U}_{V})$ has two components which contain
all marked points and branch points in $T_{V}$.

  * •

We denote the set of bipartite domains by $\mathcal{B}(\chi)$.

  * •

If $U\in\mathcal{B}(\chi)$ and $V\in\mathcal{U}$ are as above, then we refer
to $V$ as a _witness_ for $U$.

The following lemma explains the bipartite terminology, since it says their
collapsed trees are bipartite graphs, i.e. intervals:

######  Lemma 9.3.

For any $\chi>E^{\prime}+\alpha_{0}$, if $U\in\mathcal{B}(\chi)$ and
$V\in\mathcal{U}$ is a witness for $U$, then the complement
$T_{V}-\mathcal{N}_{E^{\prime}}(\delta^{U}_{V})=C\sqcup C^{\prime}$ has two
components. Moreover, the collapsed tree $\widehat{T}_{U}$ is a finite-length
interval whose endpoints $l_{C},l_{C^{\prime}}$ coincide with the projections
under $\hat{\delta}^{U}_{V}$ of the marked points and rays in $q_{V}(C)$ and
$q_{V}(C^{\prime})$, respectively. In particular, if $U\in\mathcal{B}(\chi)$,
then $U\notin\mathrm{supp}(\lambda)$ for any $\lambda\in\Lambda$.

###### Proof.

By definition of $U\in\mathcal{B}(\chi)$ and $V$ being a witness for $U$, the
complement $T_{V}-\mathcal{N}_{\chi}(\delta^{U}_{V})$ has two components
$C^{\prime}\sqcup C^{\prime\prime}$. By the BGI property in item (5) of Lemma
6.13 and the assumption that $\chi>E^{\prime}$, the marked points and leaves
of $C^{\prime}$ which are marked points or rays of $T_{V}$, denoted
$l_{C^{\prime}}$, project $E^{\prime}$-close in $T_{U}$, and hence coincide in
$\widehat{T}_{U}$, and similarly for $C^{\prime\prime}$. Hence
$\widehat{T}_{U}$ is an interval with endpoints
$q_{U}(\delta^{V}_{U}(l_{C^{\prime}}))$ and
$q_{U}(\delta^{V}_{U}(l_{C^{\prime\prime}}))$, as required.

To see that the diameter of $\widehat{T}_{U}$ is finite, suppose not. Without
loss of generality, we may assume that the endpoint
$q_{U}(\delta^{V}_{U}(l_{C^{\prime}}))$ coincides with $\hat{\lambda}_{U}$ for
some $\lambda\in\Lambda$. Hence the only marked point in $C^{\prime}$ is
$\lambda_{V}$. But by assumption
$d_{T_{V}}(\delta^{U}_{V},\lambda_{V})>E^{\prime}+\alpha_{0}$, and this
violates $\alpha_{0}$-consistency of $(\lambda_{U})$ (Lemma 6.16), which
requires that $d_{T_{V}}(\delta^{U}_{V},\lambda_{V})<\alpha_{0}$ since
$\lambda_{U}\in\partial T_{U}$. This completes the proof. ∎

Figure 10. $V$ is a bipartite domain with witness $U$ because $\delta^{V}_{U}$
is far away from all branch and marked points of $T_{U}$. The two endpoints of
$\widehat{T}_{V}$ correspond to the way that $\delta^{V}_{U}$ partitions the
marked points of $T_{U}$.

We are focusing on bipartite domains is because almost every domain in
$\mathcal{U}$ is bipartite:

######  Proposition 9.4.

For any $\chi>E^{\prime}$, there exists
$B_{\chi}=B_{\chi}(\mathfrak{S},|F\cup\Lambda|,\chi)>0$ so that
$\\#(\mathcal{U}-\mathcal{B}(\chi))<B_{\chi}$.

###### Proof.

Let $\mathcal{V}\subset\mathcal{U}$. We will prove that if $\\#\mathcal{V}$ is
sufficiently large, then $\mathcal{V}$ contains a bipartite domain.

Up to dividing $\\#\mathcal{V}$ by at most $2^{|F\cup\Lambda|}$, we may assume
that $\mathcal{V}\subset\mathrm{Rel}_{K}(a,b)\subset\mathrm{Rel}_{50E}(a,b)$
for some fixed $a,b\in F\cup\Lambda$ while maintaining $\\#\mathcal{V}$ to be
as large as necessary.

Let $\sigma=\sigma(\mathfrak{S},|F\cup\Lambda|)>0$ and
$\nu=\nu(\mathfrak{S},|F\cup\Lambda|)>0$ be as in Lemma 9.5 below. Let
$\mathbf{b}=\mathbf{b}(\mathfrak{S},|F\cup\Lambda|)>0$ be the uniform bound on
the number of branch points of $T_{U}$ for any $U\in\mathcal{U}$. Let
$a=a_{0},a_{1},a_{2},\dots,a_{k}=b$ be a $\sigma$-subdivision of
$\gamma=[a,b]_{W}$.

Now we invoke Proposition 4.3, which provides
$P_{2}(50E,K_{2},\sigma,100\mathbf{b}\nu|F\cup\Lambda|,\chi)>0$ (which depends
only on $\mathfrak{S},|F\cup\Lambda|$, and $\chi$) so that if
$\\#\mathcal{V}>P_{2}$, then there exists $W\in\mathrm{Rel}_{K_{2}}(a,b)$ and
a subcollection $\mathcal{V}^{\prime}\subset\mathcal{V}$ so that

$\mathrm{diam}_{W}(\bigcup_{V\in\mathcal{V}^{\prime}}\rho^{V}_{W})>K_{2}.$

Moreover, if
$\mathcal{W}_{i}^{\prime}=\\{V\in\mathcal{V}^{\prime}|p_{\gamma}(\delta^{V}_{U})\cap[a_{i},a_{i+1}]\neq\emptyset\\}$,
then the proposition also implies that $\\#\\{1\leq i\leq
k-1|\mathcal{W}^{\prime}_{i}\neq\emptyset\\}\geq
100\mathbf{b}\nu|F\cup\Lambda|\chi$.

Since the $\delta^{V}_{W}$ for $V\in\mathcal{V}^{\prime}$ can only intersect
at most two subintervals in the $10E^{\prime}\chi$-subdivision of
$[a_{W},b_{W}]_{T_{W}}$, Lemma 9.5 and the lower bound above implies that the
number of subintervals containing some $\delta^{V}_{U}$ far exceeds
$2\mathbf{b}|F\cup\Lambda|$, which bounds the number of subintervals of
$[a_{W},b_{W}]_{T_{W}}$ containing branch points and marked points. Hence we
were able to force $\mathcal{V}$ to contain at least one $\chi$-bipartite
domain only by requiring it to have at most boundedly-many domains, as
required.

∎

The following lemma was used in the proof above:

######  Lemma 9.5.

For any $\chi>10E^{\prime}$, there exist
$\sigma=\sigma(\mathfrak{S},|F\cup\Lambda|,\chi)>0$ and
$\nu=\nu(\mathfrak{S},|F\cup\Lambda|,\chi)>0$ so that the following holds:

Suppose $a,b\in F\cup\Lambda$ and $W\in\mathrm{Rel}_{50E}(a,b)$. Let
$a=a_{0},a_{1},a_{2},\dots,a_{k}=b$ be a $\sigma$-subdivision of a
$(1,20\delta)$-quasi-geodesic $\gamma$ between $a_{U},b_{U}$ in
$\mathcal{C}(W)$. Set $a_{i}^{\prime}=\phi_{W}^{-1}\circ p_{W}(a_{i})$ for
each $i$, where $p_{W}:\mathcal{C}(W)\to\phi_{W}(T_{W})$ is closest point
projection, and $\phi_{W}:T_{W}\to\mathcal{C}(W)$ is any uniform quality tree
modeling the hull $H_{W}$ of $\pi_{W}(F\cup\Lambda)\subset\mathcal{C}(W)$.

Then for any $10E^{\prime}\chi$-subdivision of $[a_{W},b_{W}]_{T_{W}}\subset
T_{W}$, each $a_{i}^{\prime},a_{j}^{\prime}$ are contained in distinct
subintervals when $i\neq j$. Moreover, we have that
$d_{T_{W}}(a^{\prime}_{i},a^{\prime}_{i+1})<\nu$ for each $0\leq i\leq k-1$.

###### Proof.

The proof is a straight-forward application of the techniques from Section 6.
The one subtle point is choosing $\sigma$ to be as small enough to guarantee
that the subdivision points of $\gamma$ land in separate subintervals along
the tree geodesic, but large enough to guarantee that their images are at a
bounded distance controlled by $\mathfrak{S},|F\cup\Lambda|$. The point is
that the subdivision width of $[a_{W},b_{W}]_{T_{W}}\subset T_{W}$ is fixed
(at $10E^{\prime}\chi$), while we get to control both the subdivision width
$\sigma$ in $\mathcal{C}(W)$ and its separation constant $\nu$ under the
uniform quality quasi-inverse $\phi^{-1}_{W}:\phi_{W}(T_{W})\to T_{W}$. We
leave the details to the interested reader. ∎

The reason for the extra variable $\chi$ will become clear in what follows.
That is, we need to be able to choose $\chi$ sufficiently large, depending on
the relevant context, but always depending only on
$\mathfrak{S},|F\cup\Lambda|$.

Hence for the rest of this section, we fix
$\chi=\chi(\mathfrak{S},|F\cup\Lambda|)>0$, to be determined below. In the
end, taking $\chi>5E^{\prime}+\alpha_{0}$ works.

###  9.2. Honing clusters

We will now explain how to define a map $\Omega:\mathcal{Q}\to\mathcal{Y}$, so
that:

  * •

$\Omega(\mathcal{Q})\subset\mathcal{Z}_{\omega}$ for some
$\omega=\omega(\mathfrak{S},|F\cup\Lambda|)>0$, and

  * •

when $\hat{x}=(\hat{x}_{U})\in\mathcal{Q}$ and $(x_{U})=\Omega(\hat{x})$, then
$q_{U}(x_{U})=\hat{x}_{U}$ for each $U\in\mathcal{U}$.

In this subsection, we set notation and give a description of the map
$\Omega$. We will prove that it has the desired properties in Proposition 9.1
below. Unsurprisingly, the definition is domain-wise in the $U\in\mathcal{U}$.

Let $\hat{x}=(\hat{x}_{U})\in\mathcal{Q}$. When $U\in\mathcal{U}$ has
$\hat{x}_{U}\in\widehat{T}^{e}_{U}$, then the definition is simple:
$x_{U}=q_{U}^{-1}(\hat{x}_{U})$, which is a unique point. For the other
domains, we need some more setup.

Suppose that $U\in\mathcal{U}$ and $\hat{x}_{U}\notin\widehat{T}^{e}_{U}$. The
first step is to associate a collection of marked points $l_{V}\subset
F\cup\Lambda$ to each $\sqsubset_{\mathcal{U}}$-minimal bipartite domain
$V\in\mathcal{B}(\chi)$. We then use these $l_{V}$ to hone the clusters of the
larger $U\in\mathcal{U}$ in which such $V$ are involved.

Suppose then that $U\in\mathcal{U}$ is $\sqsubset_{\mathcal{U}}$-minimal.
Since $\widehat{T}_{U}$ has no internal cluster points, $\hat{x}_{U}$ is
either a marked cluster point or a point in $\widehat{T}^{e}_{U}$. If the
former, let $l_{U}$ be the set of marked points coming from points in
$F\cup\Lambda$ identified in the cluster associated to $\hat{x}_{U}$. If the
latter, set $l_{U}=\emptyset$.

Now suppose that $U\in\mathcal{U}$ is not $\sqsubset_{\mathcal{U}}$-minimal.
We have that either $\hat{x}_{U}\in\widehat{T}^{e}_{U}$, in which case we set
$l_{U}=\emptyset$ and $x_{U}=q_{U}^{-1}(\hat{x}_{U})$. Otherwise,
$\hat{x}_{U}$ is contained in a collapsed cluster $q_{U}(C)$ for a cluster
$C\subset T_{U}$. Let $\mathcal{B}_{\min}(C)$ be the domains
$V\in\mathcal{B}(\chi)$ so that

  * •

$V\sqsubset U$ with $\delta^{V}_{U}\subset C$, and

  * •

$V$ is $\sqsubset_{\mathcal{U}}$-minimal.

  * •

$U$ is a witness for $V$.

Observe that if $\mathcal{B}_{\min}(C)=\emptyset$, then
$\mathrm{diam}_{T_{U}}(C)$ is bounded in terms of $R,r$ and the total
branching of $T_{U}$, and hence in terms of $\mathfrak{S}$ and
$|F\cup\Lambda|$.

We now explain the honing process for domains in $\mathcal{B}_{\min}(C)$,
which will associate to $\hat{x}_{U}$ a bounded diameter subset of the cluster
$C$ in $T_{U}$ (Lemma 9.12).

Let $V\in\mathcal{B}_{\min}(C)$. Then we set

$C^{V}_{U}=\mathrm{hull}_{C}(C\cap(\mathcal{N}_{r}(\delta^{V}_{U})\cup
l_{V})),$

which is a subtree of $C$, see Figure 11. Finally, let
$B_{U}(\hat{x})=\bigcap_{V\in\mathcal{B}_{\min}(C)}C^{V}_{U}$ be the total
intersection of the $C^{V}_{U}$.

Figure 11. The cluster $C$ (green) in $T_{U}$ is identified with $\hat{x}_{U}$
in $\widehat{T}_{U}$. Since $V\in\mathcal{B}_{\min}(C)$, its shadow
$\delta^{V}_{U}$ (orange) avoids all branches of $T_{U}$. Lemma 9.7 implies
all marked points and rays in $T_{U}$ are identified to the two marked
clusters of $\widehat{T}_{V}$. The marked points and leaves $l_{V}$ (pink)
chosen by $\hat{x}_{V}$ determine the half-subtree $C^{V}_{U}$ (blue) of $C$
chosen by $V$.

In Lemma 9.9 we show that if $V,W\in\mathcal{B}_{\min}(C)$, the $C^{V}_{U}\cap
C^{W}_{U}\neq\emptyset$. Since each $C^{V}_{U}$ is a subtree, we can make an
argument in Lemma 9.10 using the Helly property for trees to conclude that
$B_{U}(x)$ is nonempty, though this takes some care because $C$ and the
$C^{V}_{U}$ can be infinite-diameter when $\Lambda\neq\emptyset$. Finally, in
Lemma 9.12 we prove that this intersection has diameter bounded in terms of
$\mathfrak{S}$ and $|F\cup\Lambda|$.

Thus we can define a tuple $\Omega(\hat{x})=(x_{U})_{U\in\mathcal{U}}$ as
follows:

  * •

For $U\in\mathcal{U}$ with $U\in E(\hat{x})$, we set
$x_{U}=q_{U}^{-1}(\hat{x}_{U})$.

  * •

For $U\in\mathcal{U}$ with $U\notin E(\hat{x})$ and
$U\sqsubset_{\mathcal{U}}$-minimal, we let $x_{U}$ be any point in
$q_{U}^{-1}(\hat{x}_{U})$.

  * •

Otherwise, we let $x_{U}\in B_{U}(\hat{x})$.

The following is immediate from the construction:

######  Lemma 9.6.

For every $\hat{x}\in\mathcal{Q}$ and $U\in\mathcal{U}$, we have
$q_{U}(x_{U})=\hat{x}_{U}$.

In Proposition 9.1 below, we prove that $(x_{U})$ is a $\omega$-consistent
tuple for $\omega=\omega(\mathfrak{S},|F\cup\Lambda|)>0$.

###  9.3. Bounding intersections

We carry forward the notation from the Subsection 9.2.

We begin with two simple but important observations. The first is a direct
application of Lemma 9.3 and the definition of $C^{V}_{U}$:

######  Lemma 9.7.

Let $U,V\in\mathcal{U}$ and $C\subset T_{U}$ a cluster with
$V\in\mathcal{B}_{\min}(C)$. Let
$T_{U}-\mathcal{N}_{E^{\prime}}(\delta^{V}_{U})=C^{\prime}\cup
C^{\prime\prime}$, and let $l_{C^{\prime}},l_{C^{\prime\prime}}$ denote the
marked points and rays contained in $C^{\prime},C^{\prime\prime}$
respectively.

Then $l_{V}\in\\{l_{C^{\prime}},l_{C^{\prime\prime}}\\}$, and if
$l_{V}=l_{C^{\prime}}$ then
$C^{V}_{U}=\mathcal{N}_{r}(\delta^{V}_{U})\cup(C^{\prime}\cap C)$, and
similarly when $l_{V}=l_{C^{\prime\prime}}$.

The second follows directly from the construction of the clusters (Subsection
7.2), the definition of $\mathcal{B}_{\min}(C)$, and the uniform bound on
branching of the trees $T_{U}$ and the size of $|F\cup\Lambda|$:

######  Lemma 9.8.

There exists $r^{\prime}=r^{\prime}(\mathfrak{S},|F\cup\Lambda|)>0$ so that
the following holds: Suppose that $C\subset T^{c}_{U}$ is a cluster for some
$U\in\mathcal{U}$, and $C^{\prime}\subset C$ is a (connected) subtree. Then
either:

  * •

$\mathcal{B}_{\min}(C^{\prime})=\emptyset$ and
$\mathrm{diam}_{T_{U}}(C^{\prime})<4r^{\prime}$, or

  * •

$\mathcal{B}_{\min}(C^{\prime})\neq\emptyset$ and then there exists
$\mathcal{V}\subset\mathcal{B}_{\min}(C^{\prime})$ so that
$\\{\delta^{V}_{U}|V\in\mathcal{V}\\}$ forms a $(2r^{\prime},r^{\prime})$-net
on $C^{\prime}$ (Definition 8.4).

We can now prove that the $C^{V}_{U}$ for $V\in\mathcal{B}_{\min}(C)$ have
pairwise nonempty intersection:

######  Lemma 9.9.

If $V,W\in\mathcal{B}_{\min}(C)$, then $C^{V}_{U}\cap C^{W}_{U}\neq\emptyset$.
Moreover, if $d_{T_{U}}(\delta^{V}_{U},\delta^{W}_{U})>r$, then we have the
following:

  1. (1)

At least one of $\delta^{V}_{U}\subset C^{W}_{U}$ or $\delta^{W}_{U}\subset
C^{V}_{U}$ holds.

  2. (2)

If $\delta^{V}_{U}$ is not a subset of $C^{W}_{U}$, then $\delta^{V}_{U}\cap
C^{W}_{U}=\emptyset$, and similarly for $\delta^{W}_{U}$.

###### Proof.

Item (2) is an immediate consequence of the main part of the statement, and we
shall observe item (1) during the course of the proof of the statement.

By definition of
$C^{V}_{U}=\mathrm{hull}_{C}((\mathcal{N}_{r}(\delta^{V}_{U})\cup l_{V})\cap
C)$, we may assume that
$\mathcal{N}_{r}(\delta^{V}_{U})\cap\mathcal{N}_{r}(\delta^{W}_{U})=\emptyset$,
and hence that $V\pitchfork W$ by item (3) of Lemma 6.13.

Let $\gamma_{V}$ be the marked points and rays of $T_{U}$ which are also the
marked points and rays in the component of
$T_{U}-\mathcal{N}_{E^{\prime}}(\delta^{W}_{U})$ containing $\delta^{V}_{U}$.
Define $\gamma_{W}$ similarly.

A standard argument using Lemma 9.7 and the BGI property in item (7) of Lemma
7.14 show that $\widehat{T}_{V}$ and $\widehat{T}_{W}$ are intervals with the
labels of the endpoints of $\widehat{T}_{V}$ coinciding with
$\gamma_{V}-\gamma_{W}$ and $\gamma_{W}$, while the labels of the endpoints of
$\widehat{T}_{W}$ coincide with $\gamma_{W}-\gamma_{V}$ and $\gamma_{V}$.

Now if $a\in l_{V}\cap\gamma_{W}\subset F\cup\Lambda$, then
$\hat{x}_{V}=\hat{a}_{V}$, and thus $l_{V}=\gamma_{W}$. Hence the labels in
$l_{V}$ choose the half-tree of $T_{U}$ containing $\delta^{W}_{U}$ precisely
when $l_{V}\cap\gamma_{W}\neq\emptyset$, and similarly for $l_{W}$. Note that
this proves item (1).

Hence, by the Lemma 9.7, we may assume, for a contradiction, that
$l_{V}\cap\gamma_{W}=l_{W}\cap\gamma_{V}=\emptyset$.

Choose labels $a\in\gamma_{V}-\gamma_{W}$ and $b\in\gamma_{W}-\gamma_{V}$.
Since $a,b$ determine distinct marked clusters or rays in all of
$\widehat{T}_{U},\widehat{T}_{V},\widehat{T}_{W}$, their distances in the
trees $T_{U},T_{V},T_{W}$ are at least $r$, and hence by choosing $r$
sufficiently large depending only on $\mathfrak{S},|F\cup\Lambda|$, we can
arrange that $d_{T_{U}}(a_{U},b_{U})>10E^{\prime}$, and similarly for $V,W$.
Item (6) of Lemma 6.13 then says that
$d_{T_{V}}(\delta^{U}_{V}(\delta^{W}_{U}),\delta^{W}_{V})<E^{\prime}$, which
implies that $\hat{\delta}^{W}_{V}=\hat{b}_{V}=q_{V}(\gamma_{W})$, and
similarly $\hat{\delta}^{V}_{W}=\hat{a}_{W}=q_{W}(\gamma_{V})$.

Since $V\pitchfork W$, item (8) of Lemma 7.14 says that we cannot have both of
$V,W\in E(\hat{x})$. Hence without loss of generality, there are now two cases
(1) $W\in E(\hat{x})$ and $V\notin E(\hat{x})$, and (2) when $V,W\notin
E(\hat{x})$.

In case (1), then $l_{V}\neq\emptyset$, so that $a\in
l_{V}=\gamma_{V}-\gamma_{W}$, and hence
$\hat{x}_{V}=\hat{a}_{V}=q_{V}(l_{V})$. But the above says that
$\hat{x}_{V}\neq\hat{\delta}^{W}_{V}$, while $\hat{x}_{W}$ is not in a marked
cluster, and this contradicts $0$-consistency of $(\hat{x}_{U})$.

In case (2), now $a\in l_{V}$ and $b\in l_{W}$, and the above forces that
$\hat{x}_{V}=\hat{a}_{V}$ is in a different marked cluster than
$\hat{\delta}^{W}_{V}=\hat{b}_{V}$, so that
$\hat{x}_{V}\neq\hat{\delta}^{W}_{V}$. Similarly
$\hat{x}_{W}\neq\hat{\delta}^{V}_{W}$. Together, these also contradict
$0$-consistency.

This completes the proof. ∎

Figure 12. The impossible situation in which $V\pitchfork
W\in\mathcal{B}_{\min}(C)$ choose non-intersecting sides $C^{V}_{U}$ (pink)
and $C^{W}_{U}$ (blue) of $C$ (green). This forces inconsistency of
$\hat{x}_{V}$ and $\hat{x}_{W}$.

The next step is to conclude that the total intersection is nonempty, which
uses but does not follow directly from the Helly property for finite trees:

######  Lemma 9.10.

For each $U\in\mathcal{U}$, we have

$B_{U}=\bigcap_{V\in\mathcal{B}_{\min}(C)}C^{V}_{U}\neq\emptyset.$

###### Proof.

If $C$ has bounded diameter, then $C$ and all of the $C^{V}_{U}$ are finite
diameter trees, and the conclusion is immediate from the Helly property for
finite trees.

Suppose instead that $C$ has infinite diameter. We will produce a finite
diameter subtree $C_{fin}\subset C$ satisfying $C^{V}_{U}\cap C^{W}_{U}\cap
C_{fin}\neq\emptyset$ for all $V,W\in\mathcal{B}_{\min}(C)$, while also
$B_{U}\subset C_{fin}$. It will then follow from the Helly property for trees
that $B_{U}\neq\emptyset$.

Let $\Lambda_{C}=\\{\lambda\in\Lambda|U\in\mathrm{supp}(\lambda)\text{ and
}\lambda_{U}\in C\\}$ denote all of the rays in $\Lambda$ which determine rays
in $T_{U}$ whose ends are contained in $C$. For each $\lambda\in\Lambda_{C}$,
choose $V_{\lambda}\in\mathcal{B}_{\min}(C)$ so that:

  * •

$l_{V_{\lambda}}=(F\cup\Lambda)-\\{\lambda\\}$;

  * •

$\mathcal{N}_{r}(\delta^{V_{\lambda^{\prime}}}_{U})\subset
C^{V_{\lambda}}_{U}$ for all $\lambda\neq\lambda^{\prime}\in\Lambda(C)$

  * •

If $W\in\mathcal{B}_{\min}(C)$ so that some point in $\delta^{W}_{U}$
separates $\delta^{V_{\lambda}}_{U}$ from $\lambda_{U}$ in $T_{U}$, then
$l_{W}=l_{V_{\lambda}}=(F\cup\Lambda)-\\{\lambda\\}$.

Note that such a $V_{\lambda}$ exists for each $\lambda\in\Lambda_{C}$ because
of the canonical property in Definition 7.9 for the tuple $(\hat{x}_{U})$,
since otherwise there would be an infinite sequence
$\\{V_{i}\\}\subset\mathcal{B}_{\min}(C)$ with the sets $\delta^{V_{i}}_{U}$
running out the end of $T_{U}$ corresponding to $\lambda_{U}$ with
$\hat{x}_{V_{i}}=\hat{\lambda}_{V_{i}}$ for all $i$, thereby violating the
finiteness condition. Hence we may choose such a $V_{\lambda}$ by running far
out the end of $C$ corresponding to $\lambda_{U}$ for each
$\lambda\in\Lambda(C)$.

Set $C_{fin}=\bigcap_{\lambda\in\Lambda_{C}}C^{V_{\lambda}}_{U}$, see Figure
13. Observe that $B_{U}\subset C_{fin}$. It suffices to prove the following
claim:

######  Claim 9.11.

For all $V,W\in\mathcal{B}_{\min}(C)$, we have $C^{V}_{U}\cap C^{W}_{U}\cap
C_{fin}\neq\emptyset$.

Figure 13. The proof of Lemma 9.10: For each of the finitely-many
$\lambda\in\Lambda$ determining rays in $T_{U}$, we choose a domain
$V_{\lambda}$ which is located past every branch point along that ray, then
take the intersection of the $C^{V_{\lambda}}_{U}$ across all such $\lambda$.

To see the claim, let $E(\Lambda,C)$ denote the set of
$W\in\mathcal{B}_{\min}(C)$ so that $\delta^{W}_{U}$ contains a point which
separates $\lambda_{U}$ from $\delta^{V_{\lambda}}_{U}$ for some
$\lambda\in\Lambda(C)$.

Observe the following:

  1. (1)

Each $W\in E(\Lambda,C)$ uniquely determines such a $\lambda$, and that
$l_{W}=l_{V_{\lambda}}=(F\cup\Lambda)-\\{\lambda\\}$ by construction. Hence by
convexity of the $\delta$-sets, we have that $C_{fin}\subset
C^{V_{\lambda}}_{U}\subset C^{W}_{U}$.

  2. (2)

If $W\notin E(\Lambda,C)$, then for every $\lambda\in\Lambda(C)$, some point
of $\delta^{V_{\lambda}}_{U}$ separates $\delta^{W}_{U}$ from $\lambda$ in
$T_{U}$. It follows from the definition of $C^{V_{\lambda}}_{U}$ that
$N_{r}(\delta^{W}_{U})\subset C^{V_{\lambda}}_{U}$ for all $\lambda$, and
hence that $N_{r}(\delta^{W}_{U})\subset C_{fin}$.

Following these observations, we can now argue by cases. If $V,W\in
E(\Lambda,C)$, then observation (1) shows that $C_{fin}\subset C^{V}_{U}\cap
C^{W}_{U}$, and we are done. If instead $V\in E(\Lambda,C)$ while $W\notin
E(\Lambda,C)$, then combining both observations provides that
$\delta^{W}_{U}\subset C_{fin}\subset C^{V}_{U}$, and we are done.

Finally, if $V,W\notin E(\Lambda,C)$, then one of two things holds. The first
possibility is that $d_{T_{U}}(\delta^{V}_{U},\delta^{W}_{U})>r$, in which
case Lemma 9.9 says, with loss of generality, that $\delta^{V}_{U}\subset
C^{W}_{U}$ while the second observation says that $\delta^{V}_{U}\subset
C_{fin}$, and hence $\delta^{V}_{U}\subset C^{V}_{U}\cap C^{W}_{U}\cap
C_{fin}$, as required. Otherwise,
$d_{T_{U}}(\delta^{V}_{U},\delta^{W}_{U})<r$, and we may also assume that
neither $\delta^{V}_{U}\subset C^{W}_{U}$ nor $\delta^{W}_{U}\subset
C^{V}_{U}$. But then $C^{W}_{U}\cap
C^{V}_{U}\subset\mathcal{N}_{r}(\delta^{V}_{U})\cap\mathcal{N}_{r}(\delta^{W}_{U})$,
which is contained in $C_{fin}$ by the second observation above. This
completes the proof. ∎

Finally, we prove that $B_{U}=\bigcap_{V\in\mathcal{B}_{\min}(C)}C^{V}_{U}$
has bounded diameter:

######  Lemma 9.12.

There exists $B_{1}=B_{1}(\mathfrak{S},|F\cup\Lambda|)>0$ such that for all
$U\in\mathcal{U}$, we have $\mathrm{diam}_{T_{U}}(B_{U})<B_{1}$.

###### Proof.

By Lemma 9.8, there exists
$r^{\prime}=r^{\prime}(\mathfrak{S},|F\cup\Lambda|)>0$ so that either
$\mathcal{B}_{\min}(C)=\emptyset$ and $\mathrm{diam}_{T_{U}}(C)<2r^{\prime}$,
or for there exists a collection $\mathcal{V}\subset\mathcal{B}_{\min}(C)$ so
that $\\{\delta^{V}_{U}|V\in\mathcal{V}\\}$ forms a
$(2r^{\prime},r^{\prime})$-net on $C$ (Definition 8.4). Since we are done in
the former case, we may assume the latter holds.

We may further suppose that $B_{U}$ contains an edge of $T_{U}$ with diameter
more than $8r^{\prime}$ that does not contain any marked or branch points, for
we are done if it does not. Then the net provides distinct $V,W\in\mathcal{V}$
with $r^{\prime}<d_{T_{U}}(\delta^{V}_{U},\delta^{W}_{U})<2r^{\prime}$ with
$\delta^{V}_{U},\delta^{W}_{U}\subset B_{U}$.

By Lemma 9.9, we have that either $\delta^{V}_{U}\subset C^{W}_{U}$ or
$\delta^{W}_{U}\subset C^{V}_{U}$, and possibly both. Supposing only the first
containment holds, then in fact $\delta^{W}_{U}\cap C^{V}_{U}=\emptyset$ by
item (1) of Lemma 9.9, and hence $\delta^{W}_{U}$ is not a subset of $B_{U}$,
which contradicts our assumption that $\delta^{W}_{U}\subset B_{U}$.

Hence both containments must hold. Then and moreover $C^{V}_{U}\cap
C^{W}_{U}\subset\mathrm{hull}_{C}(\mathcal{N}_{r}(\delta^{V}_{U})\cup\mathcal{N}_{r}(\delta^{W}_{U}))$,
which has diameter bounded by $2r^{\prime}+2r$ since
$d_{T_{U}}(\delta^{V}_{U},\delta^{W}_{U})<2r^{\prime}$.

Either way, the diameter of $B_{U}$ is bounded and we are done. ∎

###  9.4. Lemmas supporting consistency of $\Omega(\hat{x})$

We record the following lemmas which will be useful in the proof of
consistency in Proposition 9.1 below.

In the following, if $A\subset F\cup\Lambda$, then we let $A_{U}$ denote the
corresponding collection of marked points and rays of $T_{U}$. If in addition
$C$ is a cluster, then we let $A_{U}(C)$ be the collection of points and rays
obtained by closest-point projecting $A_{U}$ to $C$ in $T_{U}$. The first
lemma is used in the arguments for transverse consistency, and the second
lemma for nested consistency.

######  Lemma 9.13.

Suppose $\hat{x}_{U}$ is contained in a cluster $C$ in $\widehat{T}_{U}$. If
$A=\bigcap_{V\in\mathcal{B}_{\min}(C)}l_{V}\neq\emptyset$, then
$\mathrm{hull}_{C}(A_{U}(C))\subset B_{U}$. In particular,
$\mathrm{diam}_{T_{U}}(\mathrm{hull}_{C}(A_{U}(C))<B_{1}$.

###### Proof.

By assumption, $A\subset l_{V}$ for all $V\in\mathcal{B}_{\min}(C)$. Moreover,
since $A\neq\emptyset$, we have that $\mathrm{hull}_{C}(A_{U}(C))\subset
C^{V}_{U}$ for all $V\in\mathcal{B}_{\min}(C)$, and hence
$\mathrm{hull}_{C}(A_{U}(C))\subset B_{U}$, which has diameter bounded by
$B_{1}$ by Lemma 9.12. ∎

######  Lemma 9.14.

Let $\hat{x}\in\mathcal{Q}$, and $W,U\sqsubset V\in\mathcal{U}$ with
$d_{T_{V}}(\delta^{W}_{V},\delta^{U}_{V})>4E^{\prime}$. If $C\subset T_{V}$ is
the cluster with $\delta^{U}_{V}\subset C$ and $q_{V}(C)=\hat{x}_{V}$, and
$W\in\mathcal{B}_{\min}(C)$ with $\hat{x}_{W}=\hat{\delta}^{U}_{W}$, then
$\delta^{U}_{V}\subset C^{W}_{V}$.

###### Proof.

Since $W\in\mathcal{B}_{\min}(C)$, Lemma 9.7 gives that $\widehat{T}_{W}$ is
an interval with two endpoints labeled by the marked points and rays $l_{W}$
of $T_{V}$ in the two complementary components of
$T_{V}-\mathcal{N}_{E^{\prime}}(\delta^{V}_{U})$. By assumption,
$\delta^{U}_{V}$ is contained in one of those complementary components, and
item (6) of Lemma 6.13 gives that
$\hat{\delta}^{V}_{W}(\hat{\delta}^{U}_{V})=\hat{\delta}^{U}_{W}$. Since
$\hat{x}_{W}=\hat{\delta}^{U}_{W}$ by assumption, it follows from item (5) of
Lemma 6.13 that $\delta^{U}_{V}$ is contained in the same component
$C^{\prime}$ of $T_{V}-\mathcal{N}_{E^{\prime}}(\delta^{W}_{V})$ chosen by
$\hat{x}_{W}$. It follows that $\delta^{U}_{V}\subset
C^{W}_{V}=\mathrm{hull}_{C}\left(C\cap\mathcal{N}_{r}(\delta^{W}_{U}))\cup
l_{W}\right)$, as required. ∎

###  9.5. Proof of Proposition 9.1

Our goal is to prove that there is a consistency constant
$\alpha_{\omega}=\alpha_{\omega}(\mathfrak{S},|F\cup\Lambda|)>0$ so that
$\Omega(\mathcal{Q})\subset\mathcal{Z}_{\alpha_{\omega}}$. The “moreover” part
of the statement, namely $q_{U}(x_{U})=\hat{x}_{U}$ for each
$U\in\mathcal{U}$, where $\hat{x}\in\mathcal{Q}$ and
$\Omega(\hat{x})=(x_{U})$, is the conclusion of Lemma 9.6.

We check the consistency inequalities (Definition 6.15) by a case-wise
analysis. Let $U,V\in\mathcal{U}$.

(1) $U\pitchfork V$ and $\hat{x}_{U}\neq\hat{\delta}^{V}_{U}$: Then we must
have that $\hat{x}_{V}=\hat{\delta}^{U}_{V}$ by $0$-consistency. Let $C$
denote the marked point cluster of $T_{V}$ containing $\delta^{U}_{V}$, so
that $B_{V}\subset C$ by construction. We will prove that $B_{V}$ and
$\delta^{U}_{V}$ are at a bounded distance.

If $V$ is $\sqsubset$-minimal in $\mathcal{U}$, then $C$ is a marked point
cluster with diameter bounded in terms of $\mathfrak{S},|F\cup\Lambda|$, so we
may assume otherwise. Moreover, we may assume that
$\mathcal{B}_{\min}(C)\neq\emptyset$ by Lemma 9.8, otherwise again $C$ has
bounded diameter.

Let $a\in F\cup\Lambda$ be some marked point or ray with
$d_{T_{V}}(a_{V},\delta^{U}_{V})<E^{\prime}$, which exists by item (1) of
Lemma 6.13. Note that since $\delta^{V}_{U}\subset C$, we have $a_{V}\in C$ by
construction of the clusters.

If $W\in\mathcal{B}_{\min}(C)$, then
$d_{T_{V}}(\delta^{W}_{V},a_{V})>\chi>5E^{\prime}+\alpha_{0}$, by choosing the
bipartite threshold
$\chi=\chi(\mathfrak{S},|F\cup\Lambda|)>5E^{\prime}+\alpha_{0}$. So
$d_{T_{V}}(\delta^{W}_{V},\delta^{U}_{V})>3E^{\prime}$, and thus $W\pitchfork
U$. Since $W\sqsubset V$, we have
$d_{T_{U}}(\delta^{W}_{U},\delta^{V}_{U})<E^{\prime}$, so
$\hat{x}_{U}\neq\hat{\delta}^{W}_{U}=\hat{\delta}^{V}_{U}$, forcing that
$\hat{x}_{W}=\hat{\delta}^{U}_{W}$ by $0$-consistency of $(\hat{x}_{U})$.

We claim that $a\in l_{W}$. The reason for this is that $\widehat{T}_{W}$ is
an interval whose endpoints partition $F\cup\Lambda$ into two sets by Lemma
9.7, where $l_{W}$ labels the points in $F\cup\Lambda$ coinciding with
$\hat{x}_{W}$. As such, if $a\notin l_{W}$ and hence
$\hat{a}_{W}\neq\hat{x}_{W}$, then
$d_{T_{W}}(a_{W},\delta^{V}_{W}(a_{V}))>r>\alpha_{0}$, while already
$d_{T_{V}}(\delta^{W}_{V},a_{V})>5E^{\prime}+\alpha_{0}$, which contradicts
the fact that $(a_{U})$ is an $\alpha_{0}$-consistent tuple by Lemma 6.16.
Thus we must have $a\in l_{W}$, and so
$\hat{a}_{W}=\hat{\delta}^{U}_{W}=\hat{x}_{W}$. Hence $a\in l_{W}$ for all
$W\in\mathcal{B}_{\min}(C)$. We are now done by Lemma 9.13, which implies that
$a_{V}\in B_{V}$, while $d_{T_{V}}(\delta^{U}_{V},a_{V})<E^{\prime}$ and
$\mathrm{diam}_{T_{V}}(B_{V})<B_{1}=B_{1}(\mathfrak{S},|F\cup\Lambda)$ by
Lemma 9.12.

(2) $U\pitchfork V$ with both $\hat{x}_{U}=\hat{\delta}^{V}_{U}$ and
$\hat{x}_{V}=\hat{\delta}^{U}_{V}$: Suppose instead that $\hat{x}_{U}$ is
contained in the cluster $C$ of $T_{U}$ containing $\delta^{V}_{U}$. Moreover,
assume that $B_{U}$ (and hence $x_{U}$) is at least $10r^{\prime}\cdot
B_{1}\alpha_{0}$ away from any projection of a marked point or ray of $T_{U}$
to $C$, where $B_{1}$ is as in Lemma 9.12 and $r^{\prime}$ is from Lemma 9.8.
We will bound the distance between $B_{V}$ (and hence $x_{V}$) and
$\delta^{U}_{V}$ by producing a domain $Z\sqsubset U$ with $Z\pitchfork V$ and
$\hat{x}_{Z}\neq\hat{\delta}^{V}_{Z}$, thereby allowing us to apply the
previous case.

Figure 14. The proof of case (2) of Proposition 9.1: By assuming that $x_{U}$
is far from $\delta^{V}_{U}$, we can produce a bipartite domain
$Z\in\mathcal{B}_{\min}(C)$ which chooses the half-tree not containing
$\delta^{V}_{U}$, thereby forcing $\hat{x}_{Z}\neq\hat{\delta}^{V}_{Z}$,
allowing us to invoke case (1) with $V,Z$ playing the roles of $U,V$,
respectively.

Let $A\subset F\cup\Lambda$ be the set of marked points and rays $a$
satisfying $d_{T_{U}}(\delta^{V}_{U},a_{U})<E^{\prime}$, with $A\neq\emptyset$
by item (1) of Lemma 6.13. Let
$C^{\prime}=\mathrm{hull}_{C}(B_{U},A_{U})\subset C$. By assumption
$\mathrm{diam}_{T_{U}}(C^{\prime})>10r^{\prime}B_{1}$, so Lemma 9.8 provides a
bipartite domain $Z\in\mathcal{B}_{\min}(C^{\prime})$ so that
$d_{T_{U}}(\delta^{Z}_{U},B_{U})>r^{\prime}$ and
$d_{T_{U}}(\delta^{Z}_{U},A_{U})>r^{\prime}$, and so that $\delta^{Z}_{U}$
separates $B_{U}$ from $a_{U}$ for all $a\in A$. Since $B_{U}\subset
C^{V}_{U}$ by definition, it follows that $l_{Z}\cap A=\emptyset$, and hence
$\hat{x}_{Z}\neq\hat{a}_{Z}$ for all $a\in A$.

Moreover, item (6) of Lemma 6.13 implies that
$d_{T_{Z}}(\delta^{V}_{Z},\delta^{U}_{Z}(\delta^{V}_{U}))<E^{\prime}$ while
item (5) of that lemma implies
$d_{T_{Z}}(\delta^{U}_{Z}(\delta^{V}_{U}),\delta^{U}_{Z}(a_{U}))<E^{\prime}$.
On the other hand, $\alpha_{0}$-consistency of $(a_{U})$ and the assumption
that $d_{T_{U}}(\delta^{Z}_{U},a_{U})>10r^{\prime}B_{1}\alpha_{0}>\alpha_{0}$
implies that $d_{T_{Z}}(a_{Z},\delta^{V}_{Z})<\alpha_{0}$. Hence we have
$d_{T_{Z}}(\delta^{V}_{Z},a_{Z})<2E^{\prime}+\alpha_{0}<R$, and it follows
that $\hat{\delta}^{V}_{Z}=\hat{a}_{Z}$. In particular, we have
$\hat{x}_{Z}\neq\hat{\delta}^{V}_{Z}$.

Finally, since $\hat{x}_{V}=\hat{\delta}^{U}_{V}$ by assumption, we let
$C^{\prime\prime}$ be the marked point cluster of $T_{V}$ containing
$\delta^{U}_{V}$, and hence also $\delta^{Z}_{U}\subset C^{\prime\prime}$
since $Z\sqsubset U$. But now we are in the situation from the first case,
where $V,Z$ play the roles of $U,V$ there, since $Z\pitchfork V$ and
$\hat{\delta}_{Z}\neq\hat{x}_{Z}$. The conclusion of that case was that
$d_{T_{V}}(B_{V},\delta^{Z}_{V})$ was bounded distance, while we have
$d_{T_{V}}(\delta^{Z}_{V},\delta^{U}_{V})<E^{\prime}$ since $Z\sqsubset U$,
and we are done.

(3) $U\sqsubset V$ and $\hat{x}_{U}\neq\hat{\delta}^{V}_{U}(\hat{x}_{V})$: By
$0$-consistency, we have $\hat{x}_{V}=\hat{\delta}^{U}_{V}$. We will prove
that $C^{W}_{V}\cap\delta^{U}_{V}\neq\emptyset$ for all
$W\in\mathcal{B}_{\min}(C)$. It will follow then that
$d_{T_{V}}(x_{V},\delta^{U}_{V})<4r$.

Let $C$ denote the cluster of $T_{V}$ containing $\delta^{U}_{V}$ and $x_{V}$.
First observe that if $W\in\mathcal{B}_{\min}(C)$ and
$d_{T_{V}}(\delta^{W}_{V},\delta^{U}_{V})<4E^{\prime}$, then
$\delta^{U}_{V}\cap C^{W}_{V}\neq\emptyset$ by definition of $C^{W}_{V}$.

On the other hand, if $W\in\mathcal{B}_{\min}(C)$ with
$4E^{\prime}<d_{T_{V}}(\delta^{W}_{V},\delta^{U}_{V})$, then $W\pitchfork U$.
Then item (6) of Lemma 6.13 gives that
$d_{T_{W}}(\delta^{U}_{W},\delta^{V}_{W}(\delta^{U}_{V}))<3E^{\prime}$. Hence
$\hat{\delta}^{W}_{U}=\hat{\delta}^{V}_{U}(\hat{\delta}^{W}_{V})=\hat{\delta}^{V}_{U}(\hat{x}_{V})\neq\hat{x}_{U}$,
and so since $\hat{x}_{U}\neq\hat{\delta}^{W}_{U}$, $0$-consistency implies
that $\hat{x}_{W}=\hat{\delta}^{U}_{W}$. Lemma 9.14 gives that
$\delta^{U}_{V}\subset C^{W}_{V}$. Consequently, we have
$C^{W}_{V}\cap\delta^{U}_{V}\neq\emptyset$ for all
$W\in\mathcal{B}_{\min}(C)$, and it follows that
$d_{T_{V}}(x_{V},\delta^{U}_{V})<4r$.

(4) $U\sqsubset V$ and $\hat{x}_{V}\neq\hat{\delta}^{U}_{V}$: In this final
case, $0$-consistency provides
$\hat{\delta}^{V}_{U}(\hat{x}_{V})=\hat{x}_{U}$. We will bound
$\mathrm{diam}_{T_{U}}(x_{U}\cup\delta^{V}_{U}(x_{V}))$.

We may assume that $d_{T_{V}}(x_{V},\delta^{U}_{V})>\alpha_{0}+E^{\prime}$,
otherwise we are done. Let $a\in F\cup\Lambda$ be such that $a_{V}$ is
contained in the same component of
$T_{V}-\mathcal{N}_{\alpha_{0}+E^{\prime}}(\delta^{U}_{V})$ as $x_{V}$. By the
BGI property in item (5) of Lemma 6.13, we get
$d_{T_{U}}(\delta^{V}_{U}(x_{V}),\delta^{V}_{U}(a_{V}))<E^{\prime}$. But since
$d_{T_{V}}(a_{V},\delta^{U}_{V})>\alpha_{0}$ and $(a_{U})$ is
$\alpha_{0}$-consistent by Lemma 6.16, we get
$d_{T_{U}}(a_{U},\delta^{V}_{U}(a_{V}))<\alpha_{0}$. Hence
$d_{T_{U}}(\delta^{V}_{U}(x_{V}),a_{U})<\alpha_{0}+E^{\prime}$.

Since $\hat{x}_{U}=\hat{\delta}^{V}_{U}(\hat{x}_{V})$ by assumption, we get
that $a_{U},x_{U}$ are in the same cluster $C$ in $T_{U}$. We may assume, for
a contradiction, that there is some $W\in\mathcal{B}_{\min}(C)$ such that
$a_{U},x_{U}$ are in separate components of
$T_{U}-\mathcal{N}_{E^{\prime}}(\delta^{W}_{U})$, for otherwise we are done.

Since $W\in\mathcal{B}_{\min}(C)$, Lemma 9.7 says that $\widehat{T}_{W}$ is an
interval with two endpoints labeled by the marked points and rays in $T_{U}$
in the two complementary components of
$T_{U}-\mathcal{N}_{E^{\prime}}(\delta^{W}_{U})$. Since $\delta^{W}_{U}$
separates $a_{U}$ from $x_{U}$ in $T_{U}$, it follows that
$\hat{a}_{W},\hat{x}_{W}$ are not in the same endpoint cluster. Moreover,
since $x_{U}\in C^{W}_{U}$ by definition of $B_{U}$, we have
$\hat{x}_{W}\neq\hat{a}_{W}$.

But this contradicts $0$-consistency of $\hat{x}$, since
$\hat{x}_{V}\neq\hat{\delta}^{U}_{V}=\hat{\delta}^{W}_{V}$ (because
$W\sqsubset U\sqsubset V$) while
$\hat{x}_{W}\neq\hat{a}_{W}=\hat{\delta}^{V}_{W}(\hat{x}_{V})$, with the
latter equality holding by a similar argument for why
$\hat{\delta}^{V}_{U}(\hat{x}_{V})=\hat{a}_{U}$. This completes the proof of
this case, and hence of the proposition.

Figure 15. The proof of case (4) of Proposition 9.1: If $x_{V}$ (blue) is far
enough away from $\delta^{U}_{V}$ (orange), then we can find a marked point or
ray $a\in F\cup\Lambda$ (pink) so that $a_{U}\sim\delta^{V}_{U}(x_{V})$. If
also the cluster $C\subset T_{U}$ containing $x_{U}$ (and $a_{U}$) is wide
enough, we can find a domain $W\in\mathcal{B}_{\min}(C)$ so that
$\delta^{W}_{U}$ separates $x_{U}$ from $a_{U}$, forcing $a_{W}$ and $x_{W}$
to be at opposite ends of $T_{W}$, because $C^{W}_{U}$ (red) has to contain
$x_{U}$ by definition of $B_{U}$.

##  10\. Tree trimming

In this section, we give a general construction that allows us to trim the
trees $\widehat{T}_{U}$ in various ways while preserving the metric on
$\mathcal{Q}$ up to quasi-isometry. Applications of these trimming techniques
include

  * •

surjectivity of $\widehat{\Psi}:H\to\mathcal{Q}$ (Theorem 11.1),

  * •

the ability to convert all of the $\widehat{T}_{U}$ to simplicial trees while
preserving $\mathcal{Q}$ up to quasi-isometry (Corollary 10.9), which is
crucial for proving that $\mathcal{Q}$ is quasi-median quasi-isometric to a
cube complex (Theorem B), and

  * •

modifying the cubical model construction to encode curve graph distance in
their hyperplanes (Theorem F).

As we will see in the Sections 12 and 13, the hierarchical geometry of $H$ is
directly encoded in the cubical geometry of $\mathcal{Q}$, and so tree
trimming will also allow to directly alter the cubical structure of
$\mathcal{Q}$.

Roughly speaking, the proposition says that if we collapse boundedly-many
subtrees of bounded diameter in each component of $\widehat{T}^{e}_{U}$ for
each $U\in\mathcal{U}$, then the resulting collapsing map induces a quasi-
isometry from $\widehat{\mathcal{Y}}$ to the resulting product of trees, and
this map restricts to a map between the $0$-consistent sets which preserves
(non-)canonicality. First we need some notation.

Let $K$ be our largeness threshold for the construction (Subsection 5.2) and
suppose that $B=B(\mathfrak{S},|F\cup\Lambda|)>0$ is some other constant with
respect to which we can make $K$ as large as we like (as is the case in
applications).

Suppose for each $U\in\mathcal{U}$ and each component
$E\subset\widehat{T}^{e}_{U}$ that $A_{1},\dots,A_{n}\subset E$ is a pairwise
disjoint collection of subtrees, with $\mathrm{diam}_{E}(A_{i})<B$ and $n<B$
for each $i$. Let $\widehat{T}^{\prime}_{U}$ be the tree obtained by
collapsing each of the $A_{i}$ to a point across all components
$E\subset\widehat{T}^{e}_{U}$, and let
$\Delta_{U}:\widehat{T}_{U}\to\widehat{T}^{\prime}_{U}$ be the quotient map.
We think of this process as “trimming” the trees $\widehat{T}_{U}$, even
though the collapsed sets can be in the interior of edges of the trees.

As with the original quotients $q_{U}:T_{U}\to\widehat{T}_{U}$, we get induced
relative projections $(\hat{\delta}^{\prime})^{V}_{U}$ as follows:

  * •

If $V\sqsubset U$ or $V\pitchfork U$, then
$(\hat{\delta}^{\prime})^{V}_{U}=\Delta_{U}(\hat{\delta}^{V}_{U})$.

  * •

If $U\sqsubset V$, then
$(\hat{\delta}^{\prime})^{V}_{U}=\Delta_{U}\circ\hat{\delta}^{V}_{U}\circ\Delta_{V}^{-1}:\widehat{T}^{\prime}_{V}\to\widehat{T}^{\prime}_{U}$.

We get a notion of $0$-consistency relative to these projections in the
product
$\overline{\mathcal{Y}}^{\prime}=\prod_{U\in\mathcal{U}}\widehat{T}^{\prime}_{U}\cup\partial\widehat{T}^{\prime}_{U}$,
and can define
$\overline{\mathcal{Q}}^{\prime}\subset\overline{\mathcal{Y}}^{\prime}$ to be
the set of $0$-consistent tuples in the sense of Definition 7.9. Moreover, it
is clear that if
$\Delta:\overline{\mathcal{Y}}\to\overline{\mathcal{Y}}^{\prime}$ is the
corresponding quotient map, then the restriction
$\Delta|_{\overline{\mathcal{Q}}}:\overline{\mathcal{Q}}\to\overline{\mathcal{Y}}^{\prime}$
satisfies
$\Delta(\overline{\mathcal{Q}})\subset\overline{\mathcal{Q}}^{\prime}$, for
trimming a tree will not change the fact that a coordinate coincides with a
relative projection. With these definitions in place, we observe that the
conclusions of the Collapsed Tree Control Lemma 7.14 hold for this trimmed
setup.

For each $a\in F$ and $U\in\mathcal{U}$, we set
$\hat{a}^{\prime}_{U}=\Delta_{U}(\hat{a}_{U})$. For $\lambda\in\Lambda$, we
can similarly define
$\hat{\lambda}^{\prime}_{U}=\Delta_{U}(\hat{\lambda}_{U})$ for all
$U\in\mathcal{U}$, but the following remark is a word of caution:

######  Remark 10.1 (Collapsing rays).

Notice that when $\Lambda\neq\emptyset$, then the component quotient maps need
not extend to bijections
$\partial\Delta_{U}:\partial\widehat{T}_{U}\to\partial\widehat{T}^{\prime}_{U}$.
This can occur when, for some $\lambda\in\Lambda$ with
$U\in\mathrm{supp}(\lambda)$, any ray in $\widehat{T}_{U}$ representing
$\hat{\lambda}_{U}\in\partial\widehat{T}_{U}$ is contains an infinite sequence
of cluster points with consecutive pairs separated by an edge (in
$\widehat{T}^{e}_{U}$) of diameter less than $B^{2}$. As a result, any such
ray can be collapsed to a bounded diameter set. In particular, it follows that
there is some infinite collection of domains $\mathcal{V}\subset\mathcal{U}$
with $V\sqsubset U$ for all $V\in\mathcal{V}$, and so that
$(\hat{\delta}^{V}_{U})^{\prime}=(\hat{\delta}^{W}_{U})^{\prime}$ for all
$V,W\in\mathcal{V}$. In particular,
$\hat{\lambda}^{\prime}_{U}=(\hat{\delta}^{V}_{U})^{\prime}$ for all
$V\in\mathcal{V}$. As in the example in Subsection 1.7 from the introduction,
the set $\\{\hat{\delta}^{V}_{U}|V\in\mathcal{V}\\}$ represents
$\hat{\lambda}_{U}\in\partial\widehat{T}_{U}$.

We can now state our tree trimming theorem:

######  Theorem 10.2 (Tree Trimming).

Using the above notation, there exists
$B_{tt}=B_{tt}(\mathfrak{S},|F\cup\Lambda|,K)>0$, so that for any $B<B_{tt}$
there exists $L=L(\mathfrak{S},|F\cup\Lambda|,B)>0$ so that the quotient map
$\Delta:\overline{\mathcal{Y}}\to\overline{\mathcal{Y}}^{\prime}$ satisfies
the following:

  1. (1)

$\Delta(\overline{\mathcal{Q}})=\overline{\mathcal{Q}}^{\prime}$,
$\Delta(\mathcal{Q})=\mathcal{Q}^{\prime}$, and
$\Delta(\mathcal{Q}^{\infty})=(\mathcal{Q}^{\infty})^{\prime}$.

  2. (2)

The restriction $\Delta|_{\mathcal{Q}}:\mathcal{Q}\to\mathcal{Q}^{\prime}$ is
an $(L,L)$-quasi-isometry.

  3. (3)

If the number of domains $U\in\mathcal{U}$ with collapsed subtrees is bounded
by $B^{\prime}$, then
$\Delta|_{\mathcal{Q}}:\mathcal{Q}\to\mathcal{Q}^{\prime}$ is a
$(1,BB^{\prime})$-quasi-isometry.

  * •

Moreover, the map $\Delta|_{\mathcal{Q}}:\mathcal{Q}\to\mathcal{Q}^{\prime}$
is $0$-median.

###### Proof.

The median “moreover” is proven in Corollary 15.16 below, see that section for
the relevant definitions.

We begin by proving item (1). As noted before the statement, the fact that
$\Delta$ preserves consistency follows from the fact that we are only
collapsing subtrees, so coordinates which were relative projections remain
relative projections. Hence
$\Delta(\overline{\mathcal{Q}})\subset\overline{\mathcal{Q}}^{\prime}$.

(Non-)canonicality is preserved: To see that
$\Delta(\mathcal{Q})\subset\Delta(\mathcal{Q}^{\prime})$, we confirm
canonicality (as in Definition 7.9) is also preserved. Suppose that
$\hat{x}\in\mathcal{Q}$. Then $\hat{x}_{U}\notin\partial\widehat{T}_{U}$ for
all $U$, and it follows that
$\Delta_{U}(\hat{x}_{U})\notin\partial\widehat{T}^{\prime}_{U}$, since
interior points of $\widehat{T}_{U}$ cannot become boundary points of
$\widehat{T}^{\prime}_{U}$ by collapsing subtrees. On the other hand, since
$\\#\\{U|\hat{x}_{U}\neq\hat{a}_{U}\\}<\infty$ for every $a\in F$ and we are
only collapsing subtrees, it follows that the same property holds for
$\hat{x}^{\prime}$.

To see that
$\Delta(\mathcal{Q}^{\infty})\subset(\mathcal{Q}^{\infty})^{\prime}$, let
$\hat{x}\in\mathcal{Q}^{\infty}$ be non-canonical. Then either (a) there
exists $U\in\mathcal{U}$ so that $\hat{x}_{U}\in\partial\widehat{T}_{U}$, or
(b) there exists an infinite collection of domains $\mathcal{V}_{U}$ and $a\in
F$ so that $\hat{x}_{V}\neq\hat{a}_{V}$ for all $V\in\mathcal{V}_{U}$.

In case (a), set $\hat{x}^{\prime}=\Delta(\hat{x})$ and suppose that
$\hat{x}^{\prime}_{U}\notin\partial\widehat{T}^{\prime}_{U}$, for we are done
otherwise. Hence we are in the situation discussed before the statement of the
proposition. Let $\lambda\in\Lambda$ be so that
$\hat{x}_{U}=\hat{\lambda}_{U}\in\partial\widehat{T}_{U}$. Since the map
$\Delta_{U}:\widehat{T}_{U}\to\widehat{T}^{\prime}_{U}$ collapses at most
$B$-many subtrees of diameter at most $B$ in each component of
$\widehat{T}^{e}_{U}$, it follows that there exists an infinite sequence of
$\sqsubset_{\mathcal{U}}$-minimal bipartite domains $V_{i}\sqsubset U$ so that

  * •

the sequence $\hat{\delta}^{V_{n}}_{U}$ monotonically escapes out the end of
$\widehat{T}_{U}$ corresponding to $\hat{\lambda}_{U}$,

  * •

each consecutive pair $\hat{\delta}^{V_{i}}_{U},\hat{\delta}^{V_{i+1}}_{U}$
satisfies
$d_{\widehat{T}_{U}}(\hat{\delta}^{V_{i}}_{U},\hat{\delta}^{V_{i+1}}_{U})<B^{2}$,

  * •

$(\hat{\delta}^{V_{i}}_{U})^{\prime}=\hat{x}^{\prime}_{U}$ for all $i$, and

  * •

$\hat{\lambda}_{V_{i}}=\hat{x}_{V_{i}}$ for all $i$, with $\lambda$ the unique
element of $F\cup\Lambda$ labeling $\hat{x}_{V_{i}}$ for all $i$.

As a consequence of the last item, we see that
$\hat{x}^{\prime}_{V_{i}}\neq\hat{a}^{\prime}_{V_{i}}$ for any $a\in F$, and
hence $\hat{x}^{\prime}$ is not canonical, as required.

For the rest of item (1), it remains to prove that
$\Delta|_{\mathcal{Q}}:\mathcal{Q}\to\mathcal{Q}^{\prime}$ and
$\Delta|_{\mathcal{Q}^{\infty}}:\mathcal{Q}^{\infty}\to(\mathcal{Q}^{\infty})^{\prime}$
are surjective.

Surjectivity of $\Delta$: Let $\hat{y}\in\overline{\mathcal{Q}}^{\prime}$. We
will build $\hat{x}\in\overline{\mathcal{Q}}$ with $\Delta(\hat{x})=\hat{y}$
by a domain-wise argument, by a construction closely related to the
construction of $\widehat{\Omega}:\mathcal{Q}\to\mathcal{Y}$ in Section 9. As
in that construction, the idea is to pick out a point of
$C_{U}\subset\widehat{T}_{U}$ by using $\sqsubset_{\mathcal{U}}$-minimal
domains, and then show that the corresponding tuple is $0$-consistent. Set
$C_{U}=\Delta^{-1}_{U}(\hat{y}_{U})$.

We begin by dealing with two easy cases:

  * •

If $U\in\mathcal{U}$ be such that
$\hat{y}_{U}=\hat{\lambda}^{\prime}_{U}\in\partial\widehat{T}^{\prime}_{U}$,
then we set $\hat{x}_{U}=\hat{\lambda}_{U}\in\partial\widehat{T}_{U}$.

  * •

If $\hat{y}_{U}\in\widehat{T}^{\prime}_{U}$ is not a marked or cluster point,
then $C_{U}$ contains no marked or cluster points, and we let $\hat{x}_{U}$ be
any point therein.

The general case is when $\hat{y}_{U}\in\widehat{T}^{\prime}_{U}$ is a marked
or cluster point, and we construct $\hat{x}_{U}$ by induction on
$\sqsubset$-level in $\mathcal{U}$.

The base case is when $U$ is $\sqsubset_{\mathcal{U}}$-minimal, and hence
$\hat{y}_{U}$ is a marked point of $\widehat{T}^{\prime}_{U}$ (since
$\widehat{T}^{\prime}_{U}$ has no cluster points). Note that
$\mathrm{diam}_{\widehat{T}_{U}}(C_{U})<B^{2}$, but it is possible that
$\Delta_{U}:\widehat{T}_{U}\to\widehat{T}^{\prime}_{U}$ identifies distinct
marked points of $\widehat{T}_{U}$. We need to choose one of them.

######  Claim 10.3.

If $W,Z\in\\{V\in\mathcal{U}|V\pitchfork U,\mathrm{and}\hskip
7.22743pt\hat{y}_{V}\neq(\hat{\delta}^{U}_{V})^{\prime}\\}$, then
$\hat{\delta}^{W}_{U}=\hat{\delta}^{Z}_{U}$.

###### Proof of Claim 10.3.

Suppose instead that $\hat{\delta}^{W}_{U}\neq\hat{\delta}^{Z}_{U}$. Then
$W\pitchfork Z$, and Lemma 10.8 (which appears after the proof of the theorem)
implies that $\hat{\delta}^{U}_{W}=\hat{\delta}^{Z}_{W}$ and
$\hat{\delta}^{U}_{Z}=\hat{\delta}^{W}_{Z}$. But since
$\hat{y}_{W}\neq\hat{\delta}^{U}_{W}=\hat{\delta}^{Z}_{W}$ and
$\hat{y}_{Z}\neq\hat{\delta}^{U}_{Z}=\hat{\delta}^{W}_{Z}$, we get a
contradiction of $0$-consistency of $\hat{y}$. ∎

Observe that we did not require that $U$ was $\sqsubset_{\mathcal{U}}$-minimal
for the above claim, which will be useful later.

What Claim 10.3 allows us to do is define $\hat{x}_{U}$ for
$\sqsubset_{\mathcal{U}}$-minimal $U$ as follows:

  * •

If $\mathcal{W}_{U}^{\pitchfork}=\\{V\in\mathcal{U}|V\pitchfork U\hskip
7.22743pt\mathrm{and}\hskip
7.22743pt\hat{y}_{W}\neq(\hat{\delta}^{V}_{U})^{\prime}\\}\neq\emptyset$, then
we set $\hat{x}_{U}=\hat{\delta}^{W}_{U}$ for any (and hence all, by the
claim) $W\in\mathcal{W}_{U}^{\pitchfork}$.

  * •

If $\mathcal{W}_{U}^{\pitchfork}=\emptyset$, then we let $\hat{x}_{U}$ be any
marked point in $C_{U}$.

At this point, we also want to define $l_{U}\in F\cup\Lambda$ to be the
collection of elements $a\in F\cup\Lambda$ so that $\hat{x}_{U}=\hat{a}_{U}$.
We set $l_{U}=\emptyset$ when $\hat{y}_{U}$ (and hence $\hat{x}_{U}$) is not a
marked point. These labels will play a similar role as they did in the
construction of $\widehat{\Omega}:\mathcal{Q}\to H$, see especially Subsection
9.2.

For the inductive step, suppose that $U$ is not
$\sqsubset_{\mathcal{U}}$-minimal and that we have defined $\hat{x}_{V}$ for
all $V\sqsubset U$ and, in particular, for all
$\sqsubset_{\mathcal{U}}$-minimal $V\in\mathcal{U}$.

Now if $V\sqsubset U$ and
$\hat{x}_{V}=\hat{\lambda}_{V}\in\partial\widehat{T}_{V}$ for some
$\lambda\in\Lambda$, then we set $\hat{x}_{U}=\hat{\delta}^{V}_{U}$.
Similarly, if there exists $V\sqsubset U$ so that
$\hat{x}_{V}\in\widehat{T}_{V}$ is not a marked, ray, or cluster point, then
we set $\hat{x}_{U}=\hat{\delta}^{V}_{U}$. Note that in these cases, we either
have that $\hat{y}_{V}\in\partial\widehat{T}_{V}^{\prime}$ or $\hat{y}_{V}$ is
not a marked point, ray, or cluster point, and so
$\hat{y}_{U}=(\hat{\delta}^{V}_{U})^{\prime}$ by $0$-consistency of $\hat{y}$,
meaning that $\hat{\delta}^{V}_{U}\in C_{U}$, so that we have $\hat{x}_{U}\in
C_{U}$, as required.

In the general case, $\hat{x}_{V}\in\widehat{T}_{V}$ is either a marked or
cluster point (and not a ray) for every $V\sqsubset U$. Set $C=C_{U}$ and let
$\mathcal{V}_{U}$ denote the set of $\sqsubset_{\mathcal{U}}$-minimal domains
$V\sqsubset U$ so that $\hat{\delta}^{V}_{U}\in C$ (or equivalently,
$(\hat{\delta}^{V}_{U})^{\prime}=\hat{y}_{U}$). For each
$V\in\mathcal{V}_{U}$, set
$C^{V}_{U}=\mathrm{hull}_{C}(\hat{\delta}^{V}_{U}\cup\pi_{C}(l_{V}))$, namely
the convex hull in $C$ of $\hat{\delta}^{V}_{U}$ and the projections of the
marked points or rays labeled by the elements of $l_{V}$ down to the convex
subset $C$. The following claim is an “exact” analogous version of Lemma 9.9:

######  Claim 10.4.

For every $V,W\in\mathcal{V}_{U}$, we have $C^{V}_{U}\cap
C^{W}_{U}\neq\emptyset$. Moreover:

  1. (1)

At least one of $\hat{\delta}^{V}_{U}\in C^{W}_{U}$ or
$\hat{\delta}^{W}_{U}\in C^{V}_{U}$ holds.

  2. (2)

If $\hat{\delta}^{V}_{U}\in C^{W}_{U}$ but $\hat{\delta}^{W}_{U}\notin
C^{V}_{U}$, then $l_{V}\subset l_{W}$ and $C^{V}_{U}\subset C^{W}_{U}$ (and
hence $C^{V}_{U}=C^{V}_{U}\cap C^{W}_{U}$).

###### Proof of Claim 10.4.

We may assume that $\hat{\delta}^{V}_{U}\neq\hat{\delta}^{W}_{U}$, and hence
that $V\pitchfork W$. If $C^{V}_{U}\cap C^{W}_{U}=\emptyset$, then
$\hat{\delta}^{V}_{U}\notin C^{W}_{U}$ and $\hat{\delta}^{W}_{U}\notin
C^{V}_{U}$, and it follows that $l_{V}\cap l_{W}=\emptyset$. The BGI property
of Lemma 7.14 and Item (6) of Lemma 6.13 imply that
$\hat{\delta}^{W}_{V}\notin\Delta^{-1}_{V}(\hat{y}_{V})$ and
$\hat{\delta}^{V}_{W}\notin\Delta^{-1}_{W}(\hat{y}_{W})$, and hence
$\hat{y}_{V}\neq(\hat{\delta}^{W}_{V})^{\prime}$ and
$\hat{y}_{W}\neq(\hat{\delta}^{V}_{W})^{\prime}$, which is a contradiction.
This proves the main part of the claim and also item (1).

For item (2), suppose that $\hat{\delta}^{V}_{U}\in C^{W}_{U}$ but
$\hat{\delta}^{W}_{U}\notin C^{V}_{U}$. It follows that $\hat{\delta}^{V}_{U}$
is contained in one of the components of
$\widehat{T}_{U}-\\{\hat{\delta}^{W}_{U}\\}$ which is chosen by $l_{W}$, while
the components of $\widehat{T}_{U}-\\{\hat{\delta}^{V}_{U}\\}$ chosen by
$l_{V}$ do not contain $\hat{\delta}^{W}_{U}$. Suppose that $a\in
l_{V}-l_{W}$. Then $\hat{a}_{U}$ is a marked point in a component of
$\widehat{T}_{U}-\\{\hat{\delta}^{V}_{U}\\}$ not containing
$\hat{\delta}^{W}_{U}$, and so $\hat{\delta}^{V}_{U}$ separates $\hat{a}_{U}$
from $\hat{\delta}^{W}_{U}$. Since every leaf of $\widehat{T}_{U}$ is a marked
point, it follows that there exists some $b\in l_{W}$ so that $\hat{b}_{U}$ is
contained in the component of $\widehat{T}_{U}-\\{\hat{\delta}^{W}_{U}\\}$
containing $\hat{\delta}^{V}_{U}$, with $\hat{\delta}^{V}_{U}$ separating
$\hat{b}_{U}$ from $\hat{\delta}^{W}_{U}$. It follows then that the BGI
property of Lemma 7.14 implies that $\hat{a}_{W}=\hat{b}_{W}=\hat{x}_{W}$,
which is a contradiction. Hence $l_{V}\subset l_{W}$. The fact that
$C^{V}_{U}\subset C^{W}_{U}$ now follows from the facts that $l_{V}\subset
l_{W}$ and $\hat{\delta}^{V}_{U}\in C^{W}_{U}$, completing the proof of (2)
and the lemma.

∎

With this claim in hand, we have two main subcases, whose specifics we state
as the following claim, part of which is analogous to Lemma 9.10:

######  Claim 10.5.

The following hold:

  1. (1)

If $\bigcap_{V\in\mathcal{V}_{U}}l_{V}\neq\emptyset$, then either

    1. (a)

The diameter of $C_{U}$ is infinite and there exists some $\lambda\in\Lambda$
so $\hat{\lambda}_{U}\in\partial C_{U}$ and
$\bigcap_{V\in\mathcal{V}_{U}}l_{V}=\\{\lambda\\}$, or

    2. (b)

Otherwise, every $a\in\bigcap_{V\in\mathcal{V}_{U}}l_{V}$ labels a marked
point of $C_{U}$ (i.e., does not label an infinite end of $C$).

  2. (2)

If $\bigcap_{V\in\mathcal{V}_{U}}l_{V}=\emptyset$ and
$\mathrm{diam}(C_{U})=\infty$, then there exist
$\lambda_{1},\dots,\lambda_{n}\in\Lambda$ labeling the infinite ends of
$C_{U}$, and $V_{1},\dots,V_{n}\in\mathcal{V}_{U}$, so that

    1. (a)

$l_{V_{i}}=F\cup\Lambda-\\{\lambda_{i}\\}$ for each $i$,

    2. (b)

If $C_{fin}=\bigcap_{i}C^{V_{i}}_{U}$, then $C_{fin}\neq\emptyset$ and
$\mathrm{diam}(C_{fin})<\infty$, and

    3. (c)

$C^{V}_{U}\cap C^{W}_{U}\cap C_{fin}\neq\emptyset$ for all
$V,W\in\mathcal{V}_{U}$.

  * •

Consequently, combining the above with the Helly property for trees provides
that $\bigcap_{V_{U}}C^{V}_{U}\neq\emptyset$ and has finite diameter.

###### Proof of Claim 10.5.

Suppose first that $\bigcap_{V\in\mathcal{V}_{U}}l_{V}\neq\emptyset$. Suppose
that the diameter of $C_{U}$ is infinite and that $\lambda\in\Lambda$ with
$\hat{\lambda}_{U}\in\partial C_{U}$ (by which we mean that
$\hat{\lambda}_{U}$ represents a ray in the Gromov boundary of the tree
$C_{U}$), with $\lambda\in\bigcap_{V\in\mathcal{V}_{U}}l_{V}$. We will prove
that $\bigcap_{V\in\mathcal{V}_{U}}l_{V}=\\{\lambda\\}$.

Since we are assuming that
$\hat{y}_{U}\notin\partial\widehat{T}^{\prime}_{U}$, there exists a sequence
$V_{n}\in\mathcal{V}_{U}$ with $\hat{\delta}^{V_{n}}_{U}$ monotonically
escaping out the end of $C_{U}$ corresponding to $\hat{\lambda}_{U}$, as
earlier in the proof of this theorem. Taking an element $V_{n}$ of this
sequence sufficiently far out the end corresponding to $\hat{\lambda}_{U}$
gives that $\hat{\delta}^{V_{n}}_{U}$ partitions $\hat{\lambda}_{U}$ away from
every other marked point and branched point of $\widehat{T}_{U}$. As a
consequence $C^{V_{n}}_{U}$ contains only $\hat{\lambda}_{U}$, and hence
$l_{V_{n}}=\\{\lambda\\}$, which forces
$\bigcap_{V\in\mathcal{V}_{U}}l_{V}=\\{\lambda\\}$, as required.

As a consequence, if (1)(a) does not hold, then the only other possibility is
that every $a\in\bigcap_{V\in\mathcal{V}_{U}}l_{V}$ labels a marked point of
$C_{U}$ (i.e., $a\in F\cup\Lambda$ but
$\hat{a}_{U}\notin\partial\widehat{T}_{U}$), proving the dichotomy of items
(1)(a) and (1)(b).

To deduce (2), suppose that $\bigcap_{V\in\mathcal{V}_{U}}l_{V}=\emptyset$ and
$\mathrm{diam}(C_{U})=\infty$. Let $\lambda_{1},\dots,\lambda_{n}\in\Lambda$
label the infinite ends of $C_{U}$. For each $i$, there exists a sequence of
elements of $\mathcal{V}_{U}$ whose $\hat{\delta}$-sets monotonically escape
out the end corresponding to $(\hat{\lambda}_{i})_{U}$. Since
$\lambda_{i}\notin\bigcap_{V\in\mathcal{V}_{U}}l_{V}=\emptyset$, it follows
that infinitely-many of the elements $V$ in this sequence must satisfy
$l_{V_{i}}=F\cup\Lambda-\\{\lambda_{i}\\}$. Choose such a
$V_{i}\in\mathcal{V}_{U}$ for each $i$. Observe that
$C_{fin}=\bigcap_{i}C^{V_{i}}_{U}\neq\emptyset$, as $\lambda_{j}\in l_{V_{i}}$
for all $i\neq j$, and also $\mathrm{diam}(C_{fin})<\infty$, as every infinite
end of $C_{U}$ is cut off in this intersection.

As in the proof of Claim 9.11, set $E(\Lambda,C_{U})$ to be the set of
$W\in\mathcal{V}_{U}$ so that $\hat{\delta}^{W}_{U}$ separates
$(\hat{\lambda}_{i})_{U}$ from $\hat{\delta}^{V_{i}}_{U}$ for some $i$. As in
that proof, we have the following observations:

  1. (1)

Each $W\in E(\Lambda,C_{U})$ uniquely determines an $i$, and we necessarily
have $l_{W}=F\cup\Lambda-\\{\lambda_{i}\\}$ and so $C_{fin}\subset
C^{V_{i}}_{U}\subset C^{W}_{U}$.

  2. (2)

On the other hand, if $W\notin E(\Lambda,C_{U})$, then the definition of each
$V_{i}$ implies that $\hat{\delta}^{W}_{U}\in C^{V_{i}}_{U}$ for each $i$, and
hence $\hat{\delta}^{W}_{U}\in C_{fin}$.

With these observations in hand, let $V,W\in\mathcal{V}_{U}$ and we prove that
$C^{V}_{U}\cap C^{W}_{U}\cap C_{fin}\neq\emptyset$ for all
$V,W\in\mathcal{V}_{U}$, which will complete the proof of the claim. If
$V,W\in E(\Lambda,C_{U})$, then observation (1) implies that $C_{fin}\subset
C^{V}_{U}\cap C^{W}_{U}$. If $V\in E(\Lambda,C_{U})$ while $W$ is not, then
combining both observations gives $\hat{\delta}^{W}_{U}\in C_{fin}\subset
C^{V}_{U}$, which is what we wanted. Finally, if $V,W\notin E(\Lambda,C_{U})$,
then Claim 10.4 implies that, without loss of generality,
$\hat{\delta}^{V}_{U}\in C^{W}_{U}$, while observation (2) says that
$\hat{\delta}^{V}_{U}\in C_{fin}$, implying $\hat{\delta}^{V}_{U}\in
C^{V}_{U}\cap C^{W}_{U}\cap C_{fin}$. This completes the proof of part (2) and
hence of the claim. ∎

Observe that $\bigcap_{V\in\mathcal{V}_{U}}C^{V}_{U}$ may contain many marked
points and cluster points, and we want to be able to choose a unique
coordinate $\hat{x}_{U}$ inside of this intersection. To do so, we need to use
the other information encoded in these marked and cluster points. Claim 10.3
allows us to decide between the marked points, and the following claim allows
us to choose between the cluster points:

First, set $\mathcal{W}_{U}^{\sqsubset}=\\{V\sqsubset
U|(\hat{\delta}^{U}_{V})^{\prime}(\hat{y}_{U})\neq\hat{y}_{V}\\}$. We need a
claim which is the nesting version of Claim 10.3:

######  Claim 10.6.

Suppose $V,W\in\mathcal{W}_{U}^{\sqsubset}$, then
$\hat{\delta}^{V}_{U}=\hat{\delta}^{W}_{U}$.

###### Proof of Claim 10.6.

Suppose for a contradiction that
$\hat{\delta}^{W}_{U}\neq\hat{\delta}^{V}_{U}$, so that $V\pitchfork W$. Then
item (10) of Lemma 7.14 implies that
$\hat{\delta}^{V}_{W}=\hat{\delta}^{U}_{W}(\hat{\delta}^{V}_{U})$ and
$\hat{\delta}^{W}_{V}=\hat{\delta}^{U}_{V}(\hat{\delta}^{W}_{U})$. But then
$\hat{y}_{W}\neq(\hat{\delta}^{V}_{W})^{\prime}=(\hat{\delta}^{U}_{W})^{\prime}((\hat{\delta}^{V}_{U})^{\prime})$
and
$\hat{y}_{V}\neq(\hat{\delta}^{W}_{V})^{\prime}=(\hat{\delta}^{U}_{V})^{\prime}((\hat{\delta}^{W}_{U})^{\prime})$,
which contradicts $0$-consistency of $\hat{y}$, completing the proof. ∎

What Claim 10.6 allows us to do is choose between cluster points in
$\bigcap_{V\in\mathcal{V}_{U}}C^{V}_{U}$, in the same way Claim 10.3 will
allow us to choose between marked points.

We need one final claim, which will allow us to deal with the situation when
$\bigcap_{V\in\mathcal{V}_{U}}C^{V}_{U}$ contains both marked points and
cluster points, since it says that we are forced into a consistent choice:

######  Claim 10.7.

The following holds:

  1. (1)

If there exists $V\pitchfork U$ so that
$\hat{y}_{V}\neq(\hat{\delta}^{U}_{V})^{\prime}$, then for all
$W\in\mathcal{V}_{U}$ with $\hat{\delta}^{W}_{U}\neq\hat{\delta}^{V}_{U}$, we
have $\hat{x}_{W}=\hat{\delta}^{V}_{W}$.

  2. (2)

If there exists $W\in\mathcal{V}_{U}$ with
$\hat{y}_{W}\neq(\hat{\delta}^{U}_{W})^{\prime}(\hat{y}_{U})$, then for all
$V\pitchfork U$ with $\hat{\delta}^{W}_{U}\neq\hat{\delta}^{V}_{U}$, we have
$\hat{y}_{V}=(\hat{\delta}^{U}_{V})^{\prime}$.

  * •

In particular, if $V\in\mathcal{W}_{U}^{\pitchfork}$ and
$W\in\mathcal{W}_{U}^{\sqsubset}$, then
$\hat{\delta}^{V}_{U}=\hat{\delta}^{W}_{U}$.

###### Proof.

Suppose $V\pitchfork U$ as in item (1), and let $W\in\mathcal{V}_{U}$ with
$\hat{\delta}^{W}_{U}\neq\hat{\delta}^{V}_{U}$. Then $W\pitchfork V$, and
since $W\sqsubset U$, we have
$(\hat{\delta}^{W}_{V})^{\prime}=(\hat{\delta}^{U}_{V})^{\prime}\neq\hat{y}_{V}$,
and hence $\hat{x}_{W}=\hat{\delta}^{V}_{W}$, by definition of $\hat{x}_{W}$.

Now suppose $W\in\mathcal{V}_{U}$, as in item (2), and that $V\pitchfork U$
with $\hat{\delta}^{W}_{U}\neq\hat{\delta}^{V}_{U}$. Then $W\pitchfork V$. By
$0$-consistency of $\hat{y}$, we have that
$\hat{y}_{U}=(\hat{\delta}^{W}_{U})^{\prime}$. But since
$\hat{\delta}^{V}_{U}\neq\hat{\delta}^{W}_{U}$, we have
$(\hat{\delta}^{V}_{U})^{\prime}\neq(\hat{\delta}^{W}_{U})^{\prime}$, and so
$\hat{y}_{V}=(\hat{\delta}^{U}_{V})^{\prime}=(\hat{\delta}^{W}_{V})^{\prime}$
since $W\sqsubset U$. This completes the proof. ∎

We are finally ready to define our tuple $\hat{x}$, which we do domain-wise
for each $U\in\mathcal{U}$:

  1. (1)

If
$\hat{y}_{U}=\hat{\lambda}^{\prime}_{U}\in\partial\widehat{T}^{\prime}_{U}$,
then we set $\hat{x}_{U}=\hat{\lambda}_{U}\in\widehat{T}_{U}$.

  2. (2)

If $\hat{y}_{U}$ is not a marked or cluster point of
$\widehat{T}^{\prime}_{U}$, then we let $\hat{x}_{U}\in C_{U}$ be any point in
$\Delta^{-1}_{U}(\hat{y}_{U})$.

  3. (3)

For any $U$ of types (1) or (2): If $V\pitchfork U$ or $U\sqsubset V$, we set
$\hat{x}_{V}=\hat{\delta}^{U}_{V}$, and if $U\sqsubset V$, we set
$\hat{x}_{V}=\hat{\delta}^{U}_{V}(\hat{x}_{U})$.

  4. (4)

If $U$ is $\sqsubset_{\mathcal{U}}$-minimal and $\hat{y}_{U}$ is a marked
point of $\widehat{T}^{\prime}_{U}$, then either

    1. (a)

The set $\mathcal{W}_{U}^{\pitchfork}=\\{W\in\mathcal{U}|W\pitchfork U\hskip
7.22743pt\textrm{and }\hskip
7.22743pt(\hat{\delta}^{U}_{W})^{\prime}\neq\hat{y}_{W}\\}$ is nonempty, and
we set $\hat{x}_{U}=\hat{\delta}^{W}_{U}$ for any (and hence all by Claim
10.3) $W\in\mathcal{W}_{U}^{\pitchfork}$;

    2. (b)

$\mathcal{W}_{U}^{\sqsubset}=\\{V\sqsubset
U|(\hat{\delta}^{U}_{V})^{\prime}(\hat{y}_{U})\neq\hat{y}_{V}\hskip
7.22743pt\textrm{and}\hskip
7.22743pt(\hat{\delta}^{U}_{W})^{\prime}(\hat{y}_{U})\neq\hat{y}_{W}\\}$ is
nonempty, and we set $\hat{x}_{U}=\hat{\delta}^{V}_{U}$ for any (and hence all
by Claim 10.6) $V\in\mathcal{W}_{U}^{\sqsubset}$; or

    3. (c)

$\mathcal{W}_{U}^{\pitchfork}$ and $\mathcal{W}_{U}^{\sqsubset}$ are both
empty, and we set $\hat{x}_{U}$ to be any point of $\widehat{T}^{e}_{U}$
contained in $C_{U}$, in particular a point which is not a cluster or marked
point.

  5. (5)

Proceeding inductively, if $U$ is not in cases (1)–(4) above, then Claim 10.5
gives the following possibilities:

    1. (a)

The diameter of $C_{U}$ is infinite and there is some $\lambda\in\Lambda$ so
that $\hat{\lambda}_{U}\in\partial C_{U}$ and
$\bigcap_{V\in\mathcal{V}_{U}}l_{V}=\\{\lambda\\}$. In this case, we set
$\hat{x}_{U}=\hat{\lambda}_{U}\in\partial\widehat{T}_{U}$.

    2. (b)

We have $\bigcap_{V\in\mathcal{V}_{U}}C^{V}_{U}\neq\emptyset$ and has finite
diameter. Then there are three cases:

      1. (i)

$\hat{x}_{U}=\hat{\delta}^{W}_{U}$ for any $W\in\mathcal{W}_{U}^{\pitchfork}$
if $\mathcal{W}_{U}^{\pitchfork}\neq\emptyset$,

      2. (ii)

$\hat{x}_{U}=\hat{\delta}^{V}_{U}$ for any $V\in\mathcal{W}_{U}^{\sqsubset}$
when $\mathcal{W}_{U}^{\sqsubset}\neq\emptyset$, or

      3. (iii)

Otherwise, both $\mathcal{W}_{U}^{\pitchfork}$ and
$\mathcal{W}_{U}^{\sqsubset}$ are empty, and we set $\hat{x}_{U}$ to be any
point in $\widehat{T}^{e}_{U}\cap C_{U}$.

Note that the choice in item (5) above is well-defined by Claim 10.7.

Finally, we prove that $\hat{x}=(\hat{x}_{U})$ is $0$-consistent by a case-
wise analysis, similar to the proof of Proposition 9.1. Showing this will
complete the proof of surjectivity of
$\Delta:\overline{\mathcal{Q}}\to\overline{\mathcal{Q}}^{\prime}$.

$U\pitchfork V$: We may assume that $\hat{x}_{V}\neq\hat{\delta}^{U}_{V}$, for
we are done otherwise. Then $\hat{y}_{V}\neq(\hat{\delta}^{U}_{V})^{\prime}$,
and so $\hat{y}_{U}=(\hat{\delta}^{V}_{U})^{\prime}$ by $0$-consistency of
$\hat{y}$. It follows that $\hat{\delta}^{V}_{U}\in
C_{U}=\Delta^{-1}_{U}(\hat{y}_{U})$, and note that
$\hat{\delta}^{V}_{U}=\hat{a}_{U}$ for some $a\in F\cup\Lambda$ by Lemma 7.14,
with $\hat{a}_{U}$ not representing a ray in $\widehat{T}_{U}$. If $U$ is
$\sqsubset_{\mathcal{U}}$-minimal, then our choice of $\hat{x}_{U}$ in (4)
says that $\hat{x}_{U}=\hat{\delta}^{V}_{U}$, as required.

Now suppose that $U$ is not $\sqsubset_{\mathcal{U}}$-minimal. Set
$\mathcal{V}_{U}$ to be the set of $\sqsubset_{\mathcal{U}}$-minimal
$W\sqsubset U$ with $\hat{\delta}^{W}_{U}\in C_{U}$. Let $W\in\mathcal{V}_{U}$
be so that $\hat{\delta}^{W}_{U}\neq\hat{\delta}^{V}_{U}$. Then $W\pitchfork
V$ and
$(\hat{\delta}^{W}_{V})^{\prime}=(\hat{\delta}^{U}_{V})^{\prime}\neq\hat{y}_{V}$,
and so $\hat{x}_{W}=\hat{\delta}^{V}_{W}=\hat{a}_{W}$, with the first equality
following from our choice in (4) and the last equality by $0$-consistency of
$a$. But this says that $a\in l_{W}$, and hence $\hat{a}_{U}\in C^{W}_{U}$ for
all $W\in\mathcal{V}_{U}$ (including those for which
$\hat{\delta}^{W}_{U}=\hat{\delta}^{V}_{U}=\hat{a}_{U}$).

It follows then that $\hat{a}_{U}\in\bigcap_{W\in\mathcal{V}_{U}}C^{W}_{U}$
and so $\hat{x}_{U}=\hat{a}_{U}$ by definition of $\hat{x}$ in item (5(b)i)
above.

$V\sqsubset U$: We may assume that
$\hat{x}_{V}\neq\hat{\delta}^{U}_{V}(\hat{x}_{U})$, for we are done otherwise.
Then $\hat{y}_{V}\neq(\hat{\delta}^{U}_{V})^{\prime}(\hat{y}_{U})$, and
$0$-consistency of $\hat{y}$ implies that
$\hat{y}_{U}=(\hat{\delta}^{V}_{U})^{\prime}$, and thus
$\hat{\delta}^{V}_{U}\in C_{U}=\Delta^{-1}_{U}(\hat{y}_{U})$. We claim that
for all $W\in\mathcal{V}_{U}$, we have $\hat{\delta}^{V}_{U}\in C^{W}_{U}$.

To see this, is suffices to consider $W\in\mathcal{V}_{U}$ with
$\hat{\delta}^{W}_{U}\neq\hat{\delta}^{V}_{U}$. Then $W\pitchfork V$. But
$\hat{x}_{V}\neq\hat{\delta}^{U}_{V}(\hat{x}_{U})$, so if also
$\hat{\delta}^{V}_{U}\notin C^{W}_{U}$, then
$\hat{x}_{V}\neq\hat{\delta}^{U}_{V}(\hat{\delta}^{W}_{U})=\hat{\delta}^{W}_{V}$,
while also
$\hat{x}_{W}\neq\hat{\delta}^{U}_{W}(\hat{\delta}^{V}_{U})=\hat{\delta}^{V}_{W}$,
which both use item (10) of Lemma 7.14. But then
$\hat{y}_{V}\neq(\hat{\delta}^{W}_{V})^{\prime}$ and
$\hat{y}_{W}\neq(\hat{\delta}^{V}_{W})^{\prime}$, which contradicts
$0$-consistency of $\hat{y}$.

Hence $\hat{\delta}^{V}_{U}\in C^{W}_{U}$ for all $W\in\mathcal{V}_{U}$. It
follows that $\hat{\delta}^{V}_{U}\in\bigcap_{W\in\mathcal{V}_{U}}C^{W}_{U}$,
and then (5(b)ii) says that $\hat{x}_{U}=\hat{\delta}^{V}_{U}$.

Hence $(\hat{x})$ is $0$-consistent. Since $\Delta(\hat{x})=\hat{y}$ because
$\hat{x}_{U}\in\Delta^{-1}_{U}(\hat{y}_{U})$ for all $U\in\mathcal{U}$, we are
done with the proof of surjectivity of
$\Delta:\overline{\mathcal{Q}}\to\overline{\mathcal{Q}}$.

$\Delta|_{\mathcal{Q}}:\mathcal{Q}\to\mathcal{Q}^{\prime}$ is a quasi-
isometry: Since $\Delta|_{\mathcal{Q}}$ is surjective by the above, it
suffices to relate distances. For that, note that
$d_{\widehat{\mathcal{Y}}}\geq d_{\widehat{\mathcal{Y}}^{\prime}}$, so we only
need to prove that $d_{\mathcal{Q}}\prec d_{\mathcal{Q}^{\prime}}$, and for
this it suffices to bound the distances in each $U\in\mathcal{U}$.

Let $\hat{x},\hat{y}\in\mathcal{Q}$ with $\hat{x}^{\prime}=\Delta(\hat{x})$
and $\hat{y}^{\prime}=\Delta(\hat{y})$. We may assume that
$\hat{x}_{U}\neq\hat{y}_{U}$ for any $U\in\mathcal{U}$ that we will consider.
We begin by making a useful reduction.

Observe that item (8) of Lemma 7.14 implies that there are at most boundedly-
many domains $U\in\mathcal{U}$ for which $\hat{x}_{U},\hat{y}_{U}$ are in the
same component of $\widehat{T}^{e}_{U}$, where the bound is controlled by
$\mathfrak{S}$ and $|F\cup\Lambda|$. For such $U$, we have
$d_{\widehat{T}_{U}}(\hat{x}_{U},\hat{y}_{U})\leq
d_{\widehat{T}^{\prime}_{U}}(\hat{x}^{\prime}_{U},\hat{y}^{\prime}_{U})+B^{2}$
because at most $B$-many subtrees of diameter $B$ have been deleted from that
edge. Hence we may deal with these boundedly-many domains with an additive
constant in the distance estimate comparing $d_{\mathcal{Q}}(\hat{x},\hat{y})$
and $d_{\mathcal{Q}^{\prime}}(\hat{x}^{\prime},\hat{y}^{\prime})$. Hereafter,
we assume that at least one of $\hat{x}_{U},\hat{y}_{U}$ coincides with a
cluster, or they are separated by at least one cluster.

Figure 16. The proof of Theorem 10.2: In case (1), we consider a
$\sqsubset_{\mathcal{U}}$-minimal domain like $V$, which is guaranteed to have
a large diameter while also only containing a bounded number of trimming
subtrees, and hence trimming only affects the distance between
$\hat{x}_{V},\hat{y}_{V}$ by a bounded amount. In case (2), the geodesic in
$\widehat{T}_{U}$ between $\hat{x}_{U},\hat{y}_{U}$ (pink) must cross a number
of marked cluster points and hence potentially arbitrarily many trimmed
subtrees (blue). Some of these may be sporadic (green), but there are only
boundedly-many of these across all $\widehat{T}_{U}$. Hence the cluster points
labeled by $\sqsubset_{\mathcal{U}}$-minimal bipartite domains decompose
$[\hat{x}_{U},\hat{y}_{U}]_{\widehat{T}_{U}}$ into components which contain
only boundedly-many trimmed subtrees, allowing for these domains to account
for the lost distance by a bounded multiplicative constant.

There are two cases, (1) $U$ is $\sqsubset_{\mathcal{U}}$-minimal and (2)
otherwise.

Case (1): $\widehat{T}^{e}_{U}$ has boundedly-many components, where the bound
depends on $|F\cup\Lambda|$, as each collapsed point is a marked point. So
there are at most $B|F\cup\Lambda|$-many subtrees of diameter at most $B$ in
$\widehat{T}^{e}_{U}$ being collapsed. By item (8) of Lemma 7.14 and
Proposition 9.4, there are at most boundedly-many (in
$\mathfrak{S},|F\cup\Lambda|$) such $\sqsubset_{\mathcal{U}}$-minimal $U$ for
which either one of $\hat{x}_{U},\hat{y}_{U}$ is not a cluster point or $U$ is
not bipartite. We may deal with such $U$ by increasing the additive constant
in the distance estimate.

The general case is where $U$ is bipartite and $\hat{x}_{U},\hat{y}_{U}$ are
the distinct endpoint clusters of the interval $\widehat{T}_{U}$ (Lemma 9.3).
In this case,

$d_{\widehat{T}^{\prime}_{U}}(\hat{x}^{\prime}_{U},\hat{y}^{\prime}_{U})\geq
d_{\widehat{T}_{U}}(\hat{x}_{U},\hat{y}_{U})-B^{2}\geq K-B^{2}-N,$

where $N=N(\mathfrak{S},|F\cup\Lambda|)>0$ by item (6) of Lemma 7.14. Hence by
choosing $K=K(\mathfrak{S},|F\cup\Lambda|)>0$ sufficiently large, we can
guarantee that

$d_{\widehat{T}_{U}}(\hat{x}_{U},\hat{y}_{U})\leq
d_{\widehat{T}^{\prime}_{U}}(\hat{x}^{\prime}_{U},\hat{y}^{\prime}_{U})+B^{2}\leq(1+B^{2})d_{\widehat{T}^{\prime}_{U}}(\hat{x}^{\prime}_{U},\hat{y}^{\prime}_{U})$

and thus account for these domains via a bounded multiplicative error.
Moreover, by choosing $K$ sufficiently large, we can guarantee that
$\mathrm{diam}(\widehat{T}^{\prime}_{U})$ is as large as necessary, which will
allow us to use a multiplicative error and these
$\sqsubset_{\mathcal{U}}$-minimal bipartite domains to account for domains up
the $\sqsubset$-lattice in the other case.

Case (2): Suppose that $U\in\mathcal{U}$ is not
$\sqsubset_{\mathcal{U}}$-minimal. Since $\hat{x}_{U}\neq\hat{y}_{U}$, there
are some $k\leq 2$, intervals $E_{1},\dots,E_{k}\subset\widehat{T}^{e}_{U}$,
and cluster points $C_{1},\dots,C_{k-1}$ with $E_{i},E_{i+1}$ connected at
$C_{i}$ for $1\leq i\leq k-1$, and each $C_{i}$ separating $\hat{x}_{U}$ from
$\hat{y}_{U}$.

Note that it is possible that each $C_{i}$ is _sporadic_ , namely $C_{i}$
either is

  * •

a marked point cluster, or

  * •

an internal cluster containing no $10E^{\prime}$-bipartite domains for which
$U$ is a witness (Definition 9.2).

However, the set of such sporadic clusters across $\widehat{T}_{U}$ for all
$U\in\mathcal{U}$ is bounded in terms of $|F\cup\Lambda|$ and $10E^{\prime}$,
which is bounded in terms of $\mathfrak{S},|F\cup\Lambda|$. In particular,
$k<N_{s}=N_{s}(\mathfrak{S},|F\cup\Lambda|)$. In particular, the number of
edge components adjacent to such clusters is bounded, and so the total
contributed distance loss between $\hat{x},\hat{y}$ and
$\hat{x}^{\prime},\hat{y}^{\prime}$ from collapsing at most $B$-many subtrees
in any of these components is bounded in terms of
$\mathcal{X},|F\cup\Lambda|$, and $B$. Thus we can deal with this lost
distance with an additive constant.

Hence, for every $U\in\mathcal{U}$ which is not
$\sqsubset_{\mathcal{U}}$-minimal, the geodesic
$[\hat{x}_{U},\hat{y}_{U}]\subset\widehat{T}_{U}$ can be decomposed along non-
sporadic cluster points $C^{\prime}_{1},\dots,C^{\prime}_{n}$ so that each
edge in the complement of the $C^{\prime}_{i}$ has only boundedly-many bounded
diameter subintervals collapsed. In other words, there exists
$B^{\prime\prime}=B^{\prime\prime}(\mathcal{X},|F\cup\Lambda|,B)>0$ so that

$d_{\widehat{T}_{U}}(C^{\prime}_{i},C^{\prime}_{i+1})\leq
d_{\widehat{T}^{\prime}_{U}}(\Delta_{U}(C^{\prime}_{i}),\Delta_{U}(C^{\prime}_{i+1}))+B^{\prime\prime}.$

As the $C^{\prime}_{i}$ are non-sporadic clusters, each coincides with
$\hat{\delta}^{V_{i}}_{U}$ for some $\sqsubset_{\mathcal{U}}$-minimal
$10E^{\prime}$-bipartite domain $V_{i}\in\mathcal{U}$ for which $U$ is a
witness. This means (Definition 9.2) that the $10E^{\prime}$-neighborhood of
$\delta^{V_{i}}_{U}$ avoids all marked points and branched points of $T_{U}$.
Since $\hat{x}_{U},\hat{y}_{U}\neq\hat{\delta}^{V_{i}}_{U}$ for each $i$, we
have that $\hat{x}_{V_{i}},\hat{y}_{V_{i}}$ must coincide with the marked
point clusters contained in the two components of $T_{U}-\delta^{V_{i}}_{U}$.
In particular, $\hat{x}_{V_{i}}\neq\hat{y}_{V_{i}}$ are at distinct marked
point clusters, and thus
$d_{\widehat{T}_{V_{i}}}(\hat{x}_{V_{i}},\hat{y}_{V_{i}})\succ K$ because
$V_{i}$ is bipartite and $\sqsubset_{\mathcal{U}}$-minimal (this uses item (6)
of Lemma 7.14). Moreover, by choosing $K$ sufficiently large, we can guarantee
that
$d_{\widehat{T}^{\prime}_{V_{i}}}(\hat{x}^{\prime}_{V_{i}},\hat{y}^{\prime}_{V_{i}})>1$.
As a consequence, we have

$\displaystyle d_{\widehat{T}_{U}}(C^{\prime}_{i},C^{\prime}_{i+1})$
$\displaystyle\leq$ $\displaystyle
d_{\widehat{T}^{\prime}_{U}}(\Delta_{U}(C^{\prime}_{i}),\Delta_{U}(C^{\prime}_{i+1}))+B^{\prime\prime}$
$\displaystyle\leq$ $\displaystyle
d_{\widehat{T}^{\prime}_{U}}(\Delta_{U}(C^{\prime}_{i}),\Delta_{U}(C^{\prime}_{i+1}))+B^{\prime\prime}d_{\widehat{T}^{\prime}_{V_{i}}}(\hat{x}^{\prime}_{V_{i}},\hat{y}^{\prime}_{V_{i}}).$

Hence we can account for the lost distance along $[\hat{x}_{U},\hat{y}_{U}]$
between each pair of successive non-sporadic clusters
$C^{\prime}_{i},C^{\prime}_{i+1}$ by distances in the
$\sqsubset_{\mathcal{U}}$-minimal bipartite $V_{i}$. Similarly, we can use
$V_{1},V_{k}$ to account for any lost distance between
$\hat{x}_{U},C^{\prime}_{1}$ and $C^{\prime}_{n},\hat{y}_{U}$, respectively.
Since each domain in $\mathcal{U}$ nests into at most boundedly-many domains
in $\mathcal{U}$ by the Covering Lemma 4.8, the proof of case (2) is complete.

Finally, the “moreover” part is clear: if only boundedly-many trees are
trimmed, then the global distance is only decreased by the sum of all of the
diameters of the collapsed subtrees. Hence only an additive error is
necessary. This completes the proof. ∎

The following lemma was used in the proof:

######  Lemma 10.8.

Suppose that $U,V,W\in\mathcal{U}$ are pairwise transverse and
$\hat{\delta}^{V}_{U}\neq\hat{\delta}^{W}_{U}$. Then
$\hat{\delta}^{U}_{V}=\hat{\delta}^{W}_{V}$ and
$\hat{\delta}^{U}_{W}=\hat{\delta}^{V}_{W}$.

###### Proof.

By choosing the cluster separation constant $r$ sufficiently large, we can
guarantee that $d_{U}(\rho^{V}_{U},\rho^{W}_{U})>100E$ since
$\hat{\delta}^{V}_{U}\neq\hat{\delta}^{W}_{U}$. Thus if also
$\hat{\delta}^{U}_{V}\neq\hat{\delta}^{W}_{V}$, we also have
$d_{V}(\rho^{U}_{V},\rho^{W}_{V})>100E$. Now choose $x\in\mathbf{P}_{W}$ the
product region in $\mathcal{X}$ for $W$ (Subsection 2.6), making
$d_{U}(x,\rho^{W}_{U})<E$ and $d_{V}(x,\rho^{W}_{V})<E$. It follows that
$d_{U}(x,\rho^{V}_{U})>E$ and $d_{V}(x,\rho^{U}_{V})>E$, which contradicts the
$E$-consistency of $x$. This completes the proof. ∎

###  10.1. Replacing $\widehat{T}_{U}$ with simplicial trees

In this subsection, we quickly observe how to use Tree Trimming 10.2 to
replace all $\widehat{T}_{U}$ with simplicial trees which encode the relative
projection data into the vertices while only changing $\mathcal{Q}$ up to
quasi-isometry with controlled constants. This refinement will be necessary
produce the cubical structure of $\mathcal{Q}$ via a direct argument later in
Section 13.

The goal is to replace each $\widehat{T}_{U}$ with a simplicial tree (i.e., a
1-dimensional cube complex), where all cluster points and marked points are
vertices. The issue is that the intervals between cluster and marked points
can have non-integer lengths, and the idea is to trim any such interval by
collapsing a subinterval of length less than $1$. Tree Trimming 10.2 will tell
us that the resulting quotient map is a uniform quasi-isometry.

To setup notation, fix $U\in\mathcal{U}$. Observe that each component
$E\subset\widehat{T}^{e}_{U}$ is a tree with valence bounded by
$|F\cup\Lambda|$ and $\mathcal{X}$, with leaves all cluster points or marked
points of $\widehat{T}_{U}$, and with no internal marked or cluster points. By
choosing the cluster separation constant $r$ sufficiently large, we we may
choose intervals $J_{1},\dots,J_{n}$, with

  * •

$n$ controlled by $\mathcal{X}$ and $|F\cup\Lambda|$;

  * •

each $J_{i}$ adjacent to a distinct leaf of $E$;

  * •

$|J_{i}|<1$ for each $i$;

  * •

the distances between each pair of leaves in $E$ is larger than $2$;

  * •

and if we collapse each $J_{i}$ to a point, the resulting tree $E\to
E^{\prime}$ has the property that all distances between leaves in $E^{\prime}$
are integers.

Let $\Delta_{U}:\widehat{T}_{U}\to\widehat{T}^{\prime}_{U}$ denote the map
that quotients each $J_{i}$ to a point for all such intervals across all edge
components $E\subset\widehat{T}^{e}_{U}$ for every $U\in\mathcal{U}$. Observe
that each collapsed interval $J$ is identified with the cluster point to which
it was adjacent. We again refer to the images of marked and cluster points in
the $\widehat{T}^{\prime}_{U}$ using the same terminology. Note that by
construction, every pair of cluster points or marked points in each
$\widehat{T}^{\prime}_{U}$ are at integer distance, and thus we can subdivide
the edges of each $\widehat{T}^{\prime}_{U}$ to make it a simplicial tree with
marked and cluster points at vertices.

Finally, set
$\mathcal{Y}^{\prime}=\prod_{U\in\mathcal{U}}\widehat{T}^{\prime}_{U}$, let
$\overline{\mathcal{Q}}^{\prime}$ be the set of $0$-consistent tuples, and let
$\Delta:\mathcal{Y}\to\mathcal{Y}^{\prime}$ all be as defined in the previous
subsection. It follows from Tree Trimming 10.2 that the map
$\Delta:\mathcal{Q}\to\mathcal{Q}^{\prime}$ is a uniform quasi-isometry, since
only boundedly-many (in $\mathcal{X}$ and $|F\cup\Lambda|$) subtrees of
diameter at most $1$ are collapsed in each edge component.

The following summarizes this discussion:

######  Corollary 10.9.

For any finite collection $F\cup\Lambda$ of interior points, boundary points,
and hierarchy rays in an HHS $\mathcal{X}$, let
$\\{\widehat{T}_{U}\\}_{U\in\mathcal{U}}$ and $\mathcal{Q}$ be the family of
collapsed trees and $0$-consistent set as produced in Section 7. Then up to a
uniform quasi-isometry of $\mathcal{Q}$ (in $\mathcal{X}$ and
$|F\cup\Lambda|)$, we may assume that every $\widehat{T}_{U}$ is a simplicial
tree with all collapsed points and marked points being vertices.

##  11\. Coarse surjectivity of $\widehat{\Psi}:H\to\mathcal{Q}$

In this section, we prove that $\widehat{\Psi}:H\to\mathcal{Q}$ is coarsely
surjective, which we do by proving that the composition
$\widehat{\Psi}\circ\widehat{\Omega}:\mathcal{Q}\to\mathcal{Q}$ is coarsely
constant. Namely:

######  Theorem 11.1.

There exists $B_{0}=B_{0}(\mathfrak{S},|F\cup\Lambda|,K)>0$ and a map
$\widehat{\Omega}:\mathcal{Q}\to H$ so that for any $\hat{x}\in\mathcal{Q}$,
we have
$d_{\mathcal{Q}}(\widehat{\Psi}\circ\widehat{\Omega}(\hat{x}),\hat{x})<B_{0}$.

As a corollary of this and Theorem 8.1, we get the following:

######  Corollary 11.2.

There exists $L^{\prime}=L^{\prime}(\mathfrak{S},|F\cup\Lambda|,K)>0$ so that
$\Phi:H\to\mathcal{Q}$ is an $(L^{\prime},L^{\prime})$-quasi-isometry.
Moreover, $\widehat{\Psi}$ and $\widehat{\Omega}$ are quasi-inverses.

###### Proof.

The fact that $\Phi$ is a quasi-isometric embedding follows from Theorem 8.1
plus the Distance Formula Theorem 2.9, while Theorem 11.1 says that $\Phi$ is
coarsely-dense, making it a quasi-isometry. ∎

######  Remark 11.3.

Note that our proof of the Distance Formula 2.9 (in Corollary 14.3 below) does
not rely on either of the above results.

We will break the proof of Theorem 11.1 into two steps, first analyzing the
composition at the level of the trees $T_{U}$, and then the collapsed trees
$\widehat{T}_{U}$.

###  11.1. Coarse surjectivity of $\Psi$

We first prove that $\Psi:H\to\mathcal{Z}_{\alpha_{0}}$ is coarsely
surjective, where $\Psi(H)\subset\mathcal{Z}_{\alpha_{0}}$ for
$\alpha_{0}=\alpha_{0}(\mathfrak{S},|F\cup\Lambda|)>0$ by Lemma 6.16. To do
this, we need to define a map from $\Phi_{\alpha}:\mathcal{Z}_{\alpha}\to H$
for any $\alpha\geq\alpha_{0}$, which we do as follows:

Recall that Proposition 6.17 provides a
$\beta=\beta(\mathcal{X},|F\cup\Lambda|,K,\alpha)>0$ so that
$\Phi(\mathcal{Z}_{\alpha})\subset\prod_{U\in\mathfrak{S}}\mathcal{C}(U)$
consists of $\beta$-consistent tuples. Thus we can define a map from
$\Phi_{\alpha}:\mathcal{Z}_{\alpha}\to H$ by

$\Phi_{\alpha}=\mathrm{ret}_{H}\circ\mathrm{Real}_{\beta}\circ\Phi:\mathcal{Z}_{\alpha}\to
H,$

where $\mathrm{Real}_{\beta}$ assigns a point of $\mathcal{X}$ to any
$\beta$-consistent tuple, and $\mathrm{ret}_{H}:\mathcal{X}\to H$ is the
coarse Lipschitz retraction provided by Lemma 6.6, whose quality is controlled
by $\mathcal{X}$ and $|F\cup\Lambda|$.

We now state and prove our first main step of the coarse inverse theorem. One
interpretation is that, with a distance formula-like “metric” on
$\mathcal{Z}_{\alpha}$, the composition $\Psi\circ\Phi_{\alpha}$ is coarsely
the identity. But in fact, the proposition gives much more information, since
we get explicit bounds on component-wise distances, including a global bound
on all component-wise distances (depending on $K$), and a bound on the number
of domains satisfying a medium-size distance bound (with this size independent
of $K$).

We note that, in the statement below, we need the extra flexibility of dealing
with $\mathcal{Z}_{\alpha}$ for $\alpha\geq\alpha_{0}$, because Proposition
9.1 provides an
$\alpha_{\omega}=\alpha_{\omega}(\mathfrak{S},|F\cup\Lambda|)>0$ so that
$\Omega:\mathcal{Q}\to\mathcal{Z}_{\alpha_{\omega}}$, and likely
$\alpha_{\omega}>\alpha_{0}$.

######  Proposition 11.4.

For any $\alpha\geq\alpha_{0}$, there exist
$D_{1}=D_{1}(K,\alpha,\mathfrak{S},|F\cup\Lambda|)>0$ so that for all
$x=(x_{V})\in\mathcal{Z}_{\alpha}$, we have

  1. (1)

For all $U\in\mathcal{U}$, we have
$d_{T_{U}}(x_{U},\psi_{U}\circ\Phi_{\alpha}(x))<D_{1}$

  2. (2)

$\\#\\{U\in\mathcal{U}|d_{T_{U}}(x_{U},\psi_{U}\circ\Phi_{\alpha}(x))>2\alpha+3E^{\prime}\\}<D_{1}$.

  * •

In particular, by choosing $L>2\alpha+E^{\prime}$, we get

$\sum_{U\in\mathcal{U}}[d_{T_{U}}(x_{U},\psi_{U}\circ\Phi_{\alpha}(x)]_{L}<D_{1}^{2}.$

###### Proof.

The proof of (1) is essentially a definition chase: Let
$x=(x_{U})\in\mathcal{Z}_{\alpha}$, let $(x^{\prime}_{V})=\Phi_{\alpha}(x)$ be
the $\beta=\beta(\mathcal{X},|F\cup\Lambda|,K,\alpha)>0$ consistent tuple
provided by Proposition 6.17, and let
$y^{\prime}\in\mathrm{Real}_{\beta}(\Phi_{\alpha}(x))$ be a realization.

Let $\theta_{\beta}=\theta_{\beta}(\alpha_{0},\mathfrak{S})>0$ be the constant
produced by the Realization Theorem 2.4 for the $\beta$-consistent tuple
$(x^{\prime}_{V})$, so that $d_{V}(y^{\prime},x^{\prime}_{V})<\theta_{\beta}$
for each $V\in\mathfrak{S}$, where we recall that $\mathfrak{S}$ is the full
domain index set for the ambient HHS $(\mathcal{X},\mathfrak{S})$.

Observe that since the image of $\phi_{V}:T_{V}\to\mathcal{C}(U)$ is within
Hausdorff distance $C=C(\mathcal{X},|F\cup\Lambda|)>0$ of
$H_{V}=\mathrm{hull}_{V}(F\cup\Lambda)$, we get that
$x^{\prime}_{V}\in\mathcal{N}_{C}(H_{V})$ for each $V\in\mathfrak{S}$. It
follows that
$y^{\prime}\in\mathrm{hull}^{\mathcal{X}}_{\theta_{\beta}+C}(F\cup\Lambda)$
(see Definition 6.4). In particular, if $y\in\mathrm{ret}_{H}(y^{\prime})$,
then $d_{V}(y,y^{\prime})<\theta_{0}=\theta_{0}(\beta,\mathfrak{S})$ for all
$V\in\mathfrak{S}$. It follows that
$d_{V}(y,x^{\prime}_{V})<\theta_{0}+\theta_{\beta}+C$ for all
$V\in\mathfrak{S}$.

On the other hand, $y=\mathrm{ret}_{H}(y^{\prime})\in H$, and so
$d_{U}(y,H_{U})<\theta$, where $\theta=\theta(\mathcal{X},|F\cup\Lambda|)$ is
the hull constant defining $H$. Then since $d_{Haus}(\phi_{U}(T_{U}),H_{U})<C$
and $\phi^{-1}_{U}$ is a $(C,C)$-quasi-isometry, we see that
$d_{T_{U}}(x_{U},\psi_{U}\circ\Phi_{\alpha}(x))<d_{T_{U}}(x_{U},\phi^{-1}_{U}\circ
p_{U}(y)))<C(\theta_{0}+\theta_{\beta}+\theta+2C)+C$, giving us the desired
bound for (1).

For the proof of (2), let $y=(y_{U})\in\Psi\circ\Phi_{\alpha}(x)$. Let
$\mathcal{V}=\\{U\in\mathcal{U}|d_{T}(x_{U},y_{U})>2\alpha+3E^{\prime}\\}$.
Set $K_{1}=\alpha+E^{\prime}$ and choose
$K_{2}=K_{2}(\mathfrak{S},|F|,D_{1},\alpha)>0$ to be determined shortly. By
assuming, for a contradiction, that $\\#\mathcal{V}$ is sufficiently large, we
will be able to employ Strong Passing-up 4.7 to produce a domain
$W\in\mathcal{U}$ for which $d_{T_{W}}(x_{U},y_{U})>D_{1}$, violating the
first conclusion of the current proposition.

By passing to a subset if necessary, we may assume that
$\mathcal{V}\subset\mathrm{Rel}_{K}(a,b)$ for some fixed $a,b\in F\cup\Lambda$
and that $\\#V>P_{1}(K_{1},K_{2})$, where $P_{1}$ is the function provided by
Proposition 4.3, since otherwise $\\#\mathcal{V}$ is bounded as desired.

Hence by Proposition 4.7, there exists a subset
$\mathcal{V}^{\prime}\subset\mathcal{V}$ and $W\in\mathrm{Rel}_{K_{2}}(a,b)$
so that $V\sqsubset W$ for all $V\in\mathcal{V}^{\prime}$ and

$\mathrm{diam}_{W}\left(\bigcup_{V\in\mathcal{V}^{\prime}}\rho^{V}_{W}\right)>K_{2}.$

Figure 17. The proof of case (2) of Proposition 11.4: Not both of $x_{U}$ and
$y_{U}$ (both blue) can be close to $\delta^{V}_{U}$ in $T_{U}$, and similarly
for $V$. The BGI property then forces $x_{W}$ and $y_{W}$ to be relatively
close to $\delta^{U}_{W}$ and $\delta^{V}_{W}$, respectively, which then
forces $d_{T_{W}}(x_{W},y_{W})$ to be larger then $D_{1}$, violating the bound
from item (1).

Observe that we may choose $K_{2}=K_{2}(\mathfrak{S},|F|,D_{1},\alpha)>0$
sufficiently large so that
$\mathrm{diam}_{T_{W}}\left(\bigcup_{V\in\mathcal{V}^{\prime}}\delta^{V}_{W}\right)>D_{1}+2E^{\prime}$.
Let $U,V\in\mathcal{V}^{\prime}$ realize this diameter, hence forcing
$U\pitchfork V$.

Now if $d_{T_{U}}(x_{U},\delta^{V}_{U})>\alpha$ and
$d_{T_{U}}(y_{U},\delta^{V}_{U})>\alpha$, then
$d_{T_{V}}(x_{V},y_{V})<2\alpha$ since both are $\alpha$-consistent,
contradicting the assumption that
$d_{T_{V}}(x_{V},y_{V})>2\alpha+3E^{\prime}$. Consequently, exactly one of
$x_{U}$ and $y_{U}$ is $\alpha$-close to $\delta^{V}_{U}$ in $T_{U}$, and
similarly for $V$. Without loss of generality, assume that
$d_{T_{V}}(x_{V},\delta^{U}_{V})<\alpha$ and
$d_{T_{V}}(y_{V},\delta^{U}_{V})<\alpha$.

Also observe that the geodesic $[a_{W},b_{W}]$ in $T_{W}$ (which is possibly a
bi-infinite geodesic) between $a_{W},b_{W}$ must pass within $E^{\prime}$ of
both $\delta^{U}_{W},\delta^{V}_{W}$ by the BGI property (item (5) of Lemma
6.13). Since $d_{T_{W}}(\delta^{U}_{W},\delta^{V}_{W})>K_{2}$, we may assume,
without loss of generality, that $[a_{W},b_{W}]$ first passes close to
$\delta^{U}_{W}$ then $\delta^{V}_{W}$. It follows then from item (6) of Lemma
6.13 that $d_{T_{U}}(b_{U},\delta^{V}_{U})<2E^{\prime}$ and
$d_{T_{V}}(a_{V},\delta^{U}_{V})<2E^{\prime}$.

It follows then that
$d_{T_{U}}(x_{U},b_{U})>2\alpha+3E^{\prime}-2E^{\prime}>E^{\prime}$ and
$d_{T_{V}}(y_{V},a_{V})>2\alpha+3E^{\prime}-2E^{\prime}>E^{\prime}$. Hence the
BGI property (item (5) of Lemma 6.13) says that the geodesics in $T_{W}$
between $x_{W},b_{W}$ and $y_{W},a_{W}$ must pass within $E^{\prime}$ of
$\delta^{U}_{W}$ and $\delta^{V}_{W}$, respectively. It follows then that
$d_{T_{W}}(x_{W},y_{W})>d_{T_{W}}(\delta^{U}_{W},\delta^{V}_{W})-2E^{\prime}\geq
D_{1}+2E^{\prime}-2E^{\prime}=D_{1}$, which is a contradiction.

∎

###  11.2. The map $\widehat{\Omega}:\mathcal{Q}\to H$ and coarse surjectivity

We are now ready to define the map $\widehat{\Omega}:\mathcal{Q}\to H$ using
our work from Section 9 as follows. Given a tuple
$(\hat{x}_{U})\in\mathcal{Q}$, let
$(x_{U})_{U\in\mathcal{U}}\in\mathcal{Z}_{\alpha_{\omega}}$ be a
$\alpha_{\omega}$-consistent partial tuple, where $x_{U}\in B_{U}$ as defined
in Subsection 9.2 for each $U\in\mathcal{U}$ and
$\alpha_{\omega}=\alpha_{\omega}(\mathcal{X},|F\cup\Lambda|)>0$ is provided by
Proposition 9.1. Since $(x_{U})\in\mathcal{Z}_{\alpha_{\omega}}$, we can apply
the map $\Phi_{\alpha_{\omega}}:\mathcal{Z}_{\alpha_{\omega}}\to H$ as defined
above in Subsection 11.1. Hence by composing these we obtain a map:

$\widehat{\Omega}=\Phi_{\alpha_{\omega}}\circ\Omega:\mathcal{Q}\to H.$

Since $\widehat{\Psi}(H)\subset\mathcal{Q}$ by Lemma 7.15, we get that
$\widehat{\Psi}\circ\widehat{\Omega}:\mathcal{Q}\to\mathcal{Q}$. We are now in
a position to prove that this composition is coarsely the identity:

###### Proof of Theorem 11.1.

We need to prove that there exists
$B_{0}=B_{0}(\mathfrak{S},|F\cup\Lambda|,K)>0$ so that for any
$\hat{x}\in\mathcal{Q}$, if
$\hat{y}\in\widehat{\Psi}\circ\widehat{\Omega}(\hat{x})$, then we have

$d_{\mathcal{Q}}(\hat{x},\hat{y})<B_{0}.$

For this, we will use Proposition 11.4, which provides
$D_{1}=D_{1}(K,\alpha_{\omega},\mathfrak{S},|F\cup\Lambda)>0$ so if
$(x_{U})=\Omega(\hat{x})$ and $(y_{U})\in\Psi\circ\widehat{\Omega}(\hat{x})$
then

  1. (1)

$d_{T_{U}}(x_{U},y_{U})<D_{1}$ for all $U\in\mathcal{U}$, and

  2. (2)

$\\#\\{U\in\mathcal{U}|d_{T_{U}}(x_{U},y_{U})>2\alpha_{\omega}+3E^{\prime}\\}<D_{1}$.

Let $\mathcal{V}=\\{U\in\mathcal{U}|\hat{x}_{U}\neq\hat{y}_{U}\\}$. Item (9)
of Lemma 7.14 says that the number of domains $U\in\mathcal{U}$ so that
$\hat{x}_{U}$ and $\hat{y}_{U}$ are both not contained in marked point
clusters is bounded solely in terms of $\mathfrak{S},|F\cup\Lambda|$. Hence by
the global bound $D_{1}$ in property (1) above, we may assume that for every
$V\in\mathcal{V}$, both $\hat{x}_{V},\hat{y}_{V}$ are at marked point
clusters.

Moreover, by Proposition 9.4, all but boundedly-many domains in $\mathcal{V}$
are bipartite, so we may again use property (1) to further reduce to the case
where every $V\in\mathcal{V}$ is bipartite.

Hence if we consider $\mathcal{V}_{min}$ the set of
$\sqsubset_{\mathcal{U}}$-minimal bipartite domains $V\in\mathcal{V}$, then
each $\widehat{T}_{V}$ is a non-degenerate interval by Lemma 9.7. Thus if
$V\in\mathcal{V}_{min}$, then the assumption that $\hat{x}_{V}\neq\hat{y}_{V}$
are at distinct marked cluster points implies that
$d_{\widehat{T}_{V}}(\hat{x}_{V},\hat{y}_{V})\succ K$ by item (6) of Lemma
7.14. Since $2\alpha_{\omega}+E^{\prime}$ is independent of our choice of $K$,
we can choose $K$ large enough to guarantee that $\\#\mathcal{V}_{min}<D_{1}$.

The next step is to bound the number of domains $V$ higher in the
$\sqsubset$-lattice where $\hat{x}_{V}\neq\hat{y}_{V}$, though we will not do
so directly. Instead, we observe that in all but boundedly-many domains in
$\mathcal{V}$, we have that
$d_{T_{V}}(x_{V},y_{V})<2\alpha_{\omega}+3E^{\prime}$, which is a constant
independent of $K$. The idea is to now use the Tree Trimming Theorem 10.2 as
follows.

Set $B=2\alpha_{\omega}+3E^{\prime}$. Let
$\mathcal{V}^{\prime}\subset\mathcal{V}-\mathcal{V}_{min}$ be the set of
domains $V\in\mathcal{U}$ where $\hat{x}_{V}\neq\hat{y}_{V}$ are at distinct
marked cluster points and $d_{T_{V}}(x_{V},y_{V})<B$, and so that at least one
of the cluster points contained in $[\hat{x}_{V},\hat{y}_{V}]$ is labeled by a
$\sqsubset_{\mathcal{U}}$-minimal bipartite domain not in $\mathcal{V}_{min}$.
Observe that all but boundedly-many cluster points satisfy this last property,
by the bound on $\\#\mathcal{V}_{min}$ and the Covering Lemma 4.8, which
bounds the number of domains in $\mathcal{U}$ into which the domains in
$\mathcal{V}_{min}$ can nest. In particular,
$\\#(\mathcal{V}-\mathcal{V}^{\prime})$ is bounded in terms of
$D_{1},\mathfrak{S},|F\cup\Lambda|$ by the above reductions.

For each $V\in\mathcal{V}^{\prime}$, let $A_{V}$ denote the set of these
segments and let $\Delta_{V}:\widehat{T}_{V}\to\widehat{T}^{\prime}_{V}$
denote the map which collapses each segment to a point. Theorem 10.2 says that
the combined map $\Delta:\mathcal{Q}\to\mathcal{Q}^{\prime}$ is a quasi-
isometry, with constants depending only on $\mathfrak{S},|F\cup\Lambda|$ and
$B$. Moreover, by its construction we have
$d_{\widehat{T}^{\prime}_{U}}(\Delta_{U}(\hat{x}_{U}),\Delta_{U}(\hat{y}_{U}))\leq
d_{\widehat{T}_{U}}(\hat{x}_{U},\hat{y}_{U})$ for each $U\in\mathcal{U}$.

Observe that $\Delta_{V}(\hat{x}_{V})=\Delta_{V}(\hat{y}_{V})$ by
construction. Hence we have

$\displaystyle d_{\mathcal{Q}}(\hat{x},\hat{y})$ $\displaystyle=$
$\displaystyle\sum_{U\in\mathcal{U}}d_{\widehat{T}_{U}}(\hat{x}_{U},\hat{y}_{U})$
$\displaystyle\asymp$
$\displaystyle\sum_{U\in\mathcal{U}}d_{\widehat{T}^{\prime}_{U}}(\Delta_{U}(\hat{x}_{U}),\Delta_{U}(\hat{y}_{U}))$
$\displaystyle=$
$\displaystyle\sum_{U\in\mathcal{U}-\mathcal{V}^{\prime}}d_{\widehat{T}^{\prime}_{U}}(\Delta_{U}(\hat{x}_{U}),\Delta_{U}(\hat{y}_{U}))+\sum_{V\in\mathcal{V}^{\prime}}d_{\widehat{T}^{\prime}_{V}}(\Delta_{V}(\hat{x}_{V}),\Delta_{V}(\hat{y}_{V}))$
$\displaystyle=$
$\displaystyle\sum_{U\in\mathcal{U}-\mathcal{V}^{\prime}}d_{\widehat{T}^{\prime}_{U}}(\Delta_{U}(\hat{x}_{U}),\Delta_{U}(\hat{y}_{U}))$

and this final sum is bounded since each term is at most $D_{1}$ by property
(1) above, and $\\#(\mathcal{U}-\mathcal{V}^{\prime})$ is bounded by
$D_{1},\mathfrak{S},|F\cup\Lambda|$, as observed above.

Hence we get that $d_{\mathcal{Q}}(\hat{x},\hat{y})$ is bounded in terms of
$\mathfrak{S},|F\cup\Lambda|$, and $K$, as required, completing the proof.

∎

##  12\. The induced wall-structure on $\overline{\mathcal{Q}}$

In this section, we analyze the combinatorial structure of
$\overline{\mathcal{Q}}$, the $0$-consistent set in the product of simplicial
trees
$\overline{\mathcal{Y}}=\prod_{U\in\mathcal{U}}\widehat{T}_{U}\cup\partial\widehat{T}_{U}$,
as produced in Corollary 10.9.

Since each of the $\widehat{T}_{U}$ is a tree and thus a cube complex,
$\overline{\mathcal{Y}}$ naturally admits the structure of a cube complex, and
we can use its hyperplanes to induce walls on $\overline{\mathcal{Q}}$. The
main goal of this section is to study the resulting wall-space, in particular
to prove that it forms a pocset (Proposition 12.18). Much of this discussion
is straight-forward in the interior case (i.e., when $\Lambda=\emptyset$), so
the bulk of the work happens in the infinite case.

###  12.1. Hierarchical families of trees

It is useful to axiomatize our setup in the finite case, and indeed this
proves useful later for the applications in Section 17. The following
definition axiomatizes the properties we need for Proposition 12.18 and its
supporting lemmas when $\Lambda=\emptyset$.

######  Definition 12.1 (Hierarchical family of trees).

A _hierarchical family of trees_ is the following collection of objects and
properties:

  1. (1)

A finite index set $\mathcal{U}$ of _domains_ with _relations_
$\sqsubset,\pitchfork,\perp$ so that

     * •

$\sqsubset$ is anti-reflexive and anti-symmetric, and gives a partial order
with a bound on the length of chains;

     * •

$\pitchfork$ is symmetric and anti-reflexive;

     * •

There is a unique $\sqsubset$-maximal element $S$;

     * •

Any pair of domains $U,V\in\mathcal{U}$ satisfies exactly one of the
relations.

  2. (2)

There is a bound on the size of any pairwise $\perp$-comparable subset of
$\mathcal{U}$.

  3. (3)

To each $U\in\mathcal{U}$ there is an associated finite simplicial tree
$\widehat{T}_{U}$.

  4. (4)

A finite collection of labels $F$ and, on each tree, a finite set of _marked
points_ , which are labeled by $F$, with each element of $F$ labeling exactly
one marked point of $\widehat{T}_{U}$. Moreover, for each $U\in\mathcal{U}$,
each leaf of $\widehat{T}_{U}$ is a marked point.

  5. (5)

A family of _relative projections_ determining:

     * •

For each $U,V\in\mathcal{U}$ with $V\sqsubset U$ or $V\pitchfork U$, there is
a vertex $\hat{\delta}^{V}_{U}\in\widehat{T}^{(0)}_{U}$. Moreover, each
component of $\widehat{T}_{U}-\hat{\delta}^{V}_{U}$ contains a marked point.

     * •

If $U,V,W\in\mathcal{U}$ with $U\perp V$ and $V\sqsubset W$ and either
$U\sqsubset W$ or $U\pitchfork W$, then
$\hat{\delta}^{U}_{W}=\hat{\delta}^{V}_{W}$.

  6. (6)

(BGI) There exist projection maps
$\hat{\delta}^{U}_{V}:\widehat{T}_{U}\to\widehat{T}_{V}$ when $V\sqsubset U$,
satisfying the following bounded geodesic image property:

     * •

If $C\subset\widehat{T}_{U}-\hat{\delta}^{V}_{U}$ is a component, then
$\hat{\delta}^{U}_{V}(C)$ coincides with $\hat{f}_{V}$ for any $f\in F$ with
$\hat{f}_{U}\in C$.

From the above set of data, one can define consistency equations and the _$0$
-consistent subset_ $\mathcal{Q}$ of the _total space_
$\mathcal{Y}\subset\prod_{U\in\mathcal{U}}\widehat{T}^{(0)}_{U}$ as in
Definition 7.9.

We observe that our hierarchical setup satisfies this set of properties:

######  Lemma 12.2.

The family of trees $\\{\widehat{T}_{U}\\}_{U\in\mathcal{U}}$ with associated
collapsed relative projections $\hat{\delta}^{V}_{U}$, as produced in Section
7 starting from a reduced tree system (Definition 7.1 associated to the
hierarchical hull of a finite set of internal points in an HHS $\mathcal{X}$
satisfies Definition 12.1.

###### Proof.

Items (1) and (2) follow from the Definition 2.1 of an HHS. Items (3), (4),
and (5) are contained in Definition 7.5, while item (6) is in Definition 7.7.

∎

######  Remark 12.3 (Infinite case, $\Lambda\neq\emptyset$).

While it is possible to axiomatize the infinite setting (when
$\Lambda\neq\emptyset$), we were unable to agreeably and succinctly package
the required extra axioms, which all essentially encode various structural
applications of Strong Passing-Up 4.7. As with much of this article, the
infinite setting asserts itself several times throughout the discussion, with
the finite setting being relatively straight-forward.

######  Remark 12.4 (Other generalizations).

It seems possible that the construction in this section works in greater
generality. For instance, one could define a _hierarchical family of cube
complexes_ , where the $\widehat{T}_{U}$ are higher dimensional cube
complexes, and perhaps the relative projections are convex subcomplexes, not
necessarily points. As we do not have a direct application in mind, we have
not pursued this more general version of this machinery.

######  Remark 12.5 (Proofs in this section).

We will proceed with the proofs in this section by using the various parts of
Definition 12.1 to prove the (usually much simpler) finite case when
$\Lambda\neq\emptyset$, and then supplement this setup with hierarchical facts
(such as Strong Passing-Up 4.7) when dealing with the infinite case. The more
general framework in the interior case is necessary for the proof of Theorem F
in Section 17.

###  12.2. A quick primer on CAT(0) cube complexes, wallspaces, and Sageev’s
construction

As we have arrived at the point in the paper where we begin constructing cube
complexes, we review some basics. For a detailed introduction to cube
complexes, see Sageev’s notes [Sag14] as well as Hruska-Wise [HW14].
Wallspaces were introduced by Haglund-Paulin [HP98], though we most closely
follow Roller’s approach [Rol16] to Sageev’s cubulation machine [Sag95].

A _cube complex_ is a CW complex obtained by gluing a collection of unit
Euclidean cubes of various dimensions along their faces by a collection of
(Euclidean) isometries. A cube complex $\Sigma$ is _non-positively curved_
when it satisfies Gromov’s link condition, and every non-positively curved
simply connected cube complex admits a unique CAT(0) metric.

A _hyperplane_ in a CAT(0) cube complex $\Sigma$ is a connected subspace whose
intersection with each cube $\sigma=[0,1]^{n}$ is either $\emptyset$ or the
subspace obtained by restricting exactly one coordinate to $\frac{1}{2}$. In
particular, every hyperplane partitions $\Sigma$. This motivates the
definition of a half-spaces and pocsets, following Roller [Rol16].

######  Definition 12.6 (Pocset).

Let $(\Sigma,<)$ be a partially-ordered set. A _pocset_ structure on $\Sigma$
has the additional structure of an order-reversing involution $A\mapsto A^{*}$
satisfying

  1. (1)

$A\neq A^{*}$ and $A,A^{*}$ are $<$-incomparable;

  2. (2)

$A<B\implies B^{*}<A^{*}$.

  * •

Given $A<B$, the _interval_ between $A$ and $B$ is $[A,B]=\\{C|A<C<B\\}$.

  * •

The pocset $(\Sigma,<)$ is _locally finite_ if every interval is finite.

  * •

Given $A,B\in\Sigma$, we say $A$ is _transverse_ to $B$ if none of
$A<B,A<B^{*},A^{*}<B,A^{*}<B^{*}$ holds. The pocset $(\Sigma,<)$ has _finite
width_ if the maximal size of a pairwise transverse subset is finite.

In a CAT(0) cube complex, every vertex is contained in exactly one half-space
associated to each hyperplane. The following notion abstracts the
combinatorial structure of such a family of half-spaces:

######  Definition 12.7 (Ultrafilters and the DCC).

Given a locally finite pocset $(\Sigma,<)$, an _ultrafilter_ $\alpha$ on
$\Sigma$ is a subset of $\Sigma$ satisfying

  1. (1)

(Choice) Exactly one element of each pair $\\{A,A^{*}\\}$ is an element of
$\alpha$;

  2. (2)

(Consistency) $A\in\alpha$ and $A<B\implies B\in\alpha$.

  * •

An ultrafilter $\alpha$ satisfies the _descending chain condition_ (DCC) if
every descending chain of elements terminates.

We can now define the dual cube complex to a locally finite, discrete, finite-
width pocset:

######  Definition 12.8 (Dual cube complex).

Given a locally finite, discrete, finite-width pocset $(\Sigma,<)$, we define
a CW complex $X(\Sigma)$ as follows:

  1. (1)

The $0$-cells $X^{(0)}$ of $X$ are the DCC ultrafilters on $(\Sigma,<)$.

  2. (2)

Two $0$-cells $\alpha,\beta$ are connected by an edge in $X^{(1)}$ if and only
if $|\alpha\triangle\beta|=2$, i.e. $\alpha=(\beta-\\{A\\})\cup\\{A^{*}\\}$
for some $A\in\Sigma$.

  3. (3)

Define $X^{(n)}$ inductively by adding an $n$-cube whenever the boundary of
one appears in the $X^{(n-1)}$-skeleton.

See Sageev’s notes [Sag14] for a thorough discussion of the above.

The following is a theorem of Roller [Rol16], building on Sageev [Sag95]:

######  Theorem 12.9 (Sageev’s machine).

Given a locally finite, finite-width pocset $(\Sigma,<)$, the CW complex
$X(\Sigma)$ is a finite dimensional CAT(0) cube complex, where the dimension
is equal to the width of $(\Sigma,<)$.

  * •

We call $X(\Sigma)$ the _dual cube complex_ to $(\Sigma,<)$.

###  12.3. Inducing walls on $\overline{\mathcal{Q}}$

For the rest of this section, fix a finite set $F\cup\Lambda$ of internal
points and hierarchy rays in an HHS $\mathcal{X}$, along with their associated
structures, such as the $K$-relevant set $\mathcal{U}$ and its associated
simplicial trees $\widehat{T}_{U}$, as we have been working with in most of
this paper. Our goal is to induce a wallspace structure on the $0$-consistent
set $\overline{\mathcal{Q}}$, and then prove that it gives a pocset. In
Section 13, we prove that the dual cube complex to this wallspace on
$\mathcal{Q}$ is isometric to $\mathcal{Q}$, as described in Theorem 13.6.

To begin, observe that $\mathcal{Y}=\prod_{U\in\mathcal{U}}\widehat{T}_{U}$ is
a product of cube complexes. It is a general fact that a product of cube
complexes admits a natural cubical structure, where the global hyperplanes
come from the hyperplanes in each component in a natural way. Using this
observation, we can induce a wallspace structure on $\overline{\mathcal{Q}}$
as follows.

For each $U\in\mathcal{U}$, a _component hyperplane_ $h_{U}$ in
$\widehat{T}_{U}$ is simply the midpoint of an edge in $\widehat{T}_{U}$. Each
such hyperplane $h_{U}$ then determines a _global hyperplane_
$\bar{h}_{U}\subset\overline{\mathcal{Y}}$ of the following form:

$\bar{h}_{U}=h_{U}\times\prod_{V\in\mathcal{U}-\\{U\\}}\widehat{T}_{V}\cup\partial\widehat{T}_{V}.$

Set $\mathbf{h}_{U}=\bar{h}_{U}\cap\overline{\mathcal{Q}}$, which we refer to
as a $\overline{\mathcal{Q}}$-_hyperplane_. We make some basic observations
about these $\overline{\mathcal{Q}}$-hyperplanes:

######  Lemma 12.10.

Let $U\in\mathcal{U}$, and $h_{U}\in\widehat{T}_{U}$ be a hyperplane. The
following hold:

  1. (1)

If $\hat{z}\in\mathbf{h}_{U}$, then the coordinates $\hat{z}_{V}$ of $\hat{z}$
satisfy:

    1. (a)

If $U\sqsubset V$ or $U\pitchfork V$, then $\hat{z}_{V}=\hat{\delta}^{U}_{V}$.

    2. (b)

If $V\sqsubset U$, then $\hat{z}_{V}=\hat{f}_{V}$ for some $f\in
F\cup\Lambda$.

  2. (2)

If $h_{V}\subset\widehat{T}_{V}$ is another component hyperplane, then
$\mathbf{h}_{U}\cap\mathbf{h}_{V}\neq\emptyset$ if and only if $U\perp V$.

###### Proof.

For (1), first assume that $U\pitchfork V$. Then $h_{U}=\hat{z}_{U}$ is not a
marked or cluster point, and hence $0$-consistency implies that
$\hat{z}_{U}\neq\hat{\delta}^{V}_{U}$. Similarly, if $U\sqsubset V$ but
$\hat{z}_{V}\neq\hat{\delta}^{U}_{V}$, then the BGI property (6) of Definition
12.1 implies that $\hat{\delta}^{V}_{U}(\hat{z}_{V})=\hat{f}_{V}$ for some
$f\in F\cup\Lambda$. But $\hat{z}_{U}=h_{U}$ is not a marked point, which is a
contradiction, forcing $\hat{z}_{V}=\hat{\delta}^{U}_{V}$.

Now if $V\sqsubset U$, then $\hat{z}_{U}=h_{U}$ is not a marked or cluster
point, and so $\hat{z}_{U}\neq\hat{\delta}^{V}_{U}$, and so
$\hat{z}_{U}=\hat{\delta}^{U}_{V}(\hat{z}_{U})$ by $0$-consistency, while the
latter equals $\hat{f}_{U}$ for some $f\in F\cup\Lambda$ by the BGI property
(6) of Definition 12.1.

Finally, Now (2) follows from (1), for (1) says that if $U\notperp V$ and
$\hat{z}\in\mathbf{h}_{U}\cap\mathbf{h}_{V}$, then one of
$\hat{z}_{U},\hat{z}_{V}$ must be at a marked or cluster point, which is a
contradiction. This completes the proof. ∎

We observe the following immediate corollary of item (2) of Lemma 12.10 and
(2) of Definition 12.1, namely finite dimensionality of
$\overline{\mathcal{Q}}$:

######  Corollary 12.11.

The number of pairwise intersecting $\overline{\mathcal{Q}}$-hyperplanes is
bounded by the number of pairwise orthogonal domains in $\mathcal{U}$, and is
hence finite.

###  12.4. Half-spaces in $\overline{\mathcal{Q}}$

We next observe that the $\overline{\mathcal{Q}}$-hyperplanes form a wallspace
structure on $\overline{\mathcal{Q}}$.

If $h_{U}\in\widehat{T}_{U}$ is a component hyperplane and
$\widehat{\pi}_{U}:\mathcal{Y}\to\widehat{T}_{U}$ is the coordinate
projection, then $\mathbf{h}_{U}=\widehat{\pi}_{U}^{-1}(h_{U})$. Since
$\widehat{T}_{U}-\\{h_{U}\\}$ has two components
$\widehat{T}_{U}^{+},\widehat{T}_{U}^{-}$, we get a partition

$\overline{\mathcal{Q}}^{+}_{\mathbf{h}_{U}}\sqcup\overline{\mathcal{Q}}^{-}_{\mathbf{h}_{U}}\sqcup\mathbf{h}_{U}=\overline{\mathcal{Q}}$

where
$\overline{\mathcal{Q}}^{\pm}_{\mathbf{h}_{U}}=\widehat{\pi}_{U}^{-1}(\widehat{T}_{U}^{\pm})\cap\overline{\mathcal{Q}}$,
which we call the $\overline{\mathcal{Q}}$-_half-spaces_ associated to
$\mathbf{h}_{U}$.

######  Notation 12.12.

In what follows, we will usually not need to refer to both a half-space and
its dual. Thus we will usually refer to the $\overline{\mathcal{Q}}$-half-
space associated to some tree hyperplane $h_{U}$ as
$\overline{\mathcal{Q}}_{U}$, and the corresponding half-tree of
$\widehat{T}_{U}-\\{h_{U}\\}$ as $\widehat{T}^{\prime}_{U}$.

The following lemma gives a description of these half-spaces, and in
particular shows both are nonempty:

######  Lemma 12.13.

For each $\overline{\mathcal{Q}}$-hyperplane $\mathbf{h}_{U}$, the following
description holds:

  1. (1)

If $U\perp V$, then
$\widehat{\pi}_{V}(\mathbf{h}_{U})=\widehat{\pi}_{V}(\overline{\mathcal{Q}}_{U})=\widehat{T}_{V}$.

  2. (2)

If $U\pitchfork V$, then

     * •

$\widehat{\pi}_{V}(\mathbf{h}_{U})=\hat{\delta}^{U}_{V}$.

     * •

If $\hat{\delta}^{V}_{U}\in\widehat{T}^{\prime}_{U}$, then
$\widehat{\pi}_{V}(\overline{\mathcal{Q}}_{U})=\widehat{T}_{V}$. Otherwise,
$\widehat{\pi}_{V}(\overline{\mathcal{Q}}_{U})=\hat{\delta}^{U}_{V}$.

  3. (3)

If $V\sqsubset U$, then

     * •

$\widehat{\pi}_{V}(\mathbf{h}_{U})=\hat{\delta}^{U}_{V}(h_{U})$.

     * •

If $\hat{\delta}^{V}_{U}\in\widehat{T}^{\prime}_{U}$, then
$\widehat{\pi}_{V}(\overline{\mathcal{Q}}_{U})=\widehat{T}_{V}$. Otherwise,
$\widehat{\pi}_{V}(\overline{\mathcal{Q}}_{U})=\hat{\delta}^{U}_{V}(\widehat{T}^{\prime}_{U})$.

  4. (4)

If $U\sqsubset V$, then

     * •

$\widehat{\pi}_{V}(\mathbf{h}_{U})=\hat{\delta}^{U}_{V}$.

     * •

If
$\widehat{\pi}_{V}(\overline{\mathcal{Q}}_{U})=\mathrm{hull}_{\widehat{T}_{V}}(\hat{\delta}^{U}_{V}\cup
l_{U})$, where $l_{U}$ are the labels of the marked points in
$\widehat{T}_{U}$ contained in $\widehat{T}^{\prime}_{U}$.

###### Proof.

The proof of the description of the component projections of
$\overline{\mathcal{Q}}_{U}$ is a straightforward exercise in the definitions
utilizing Lemma 12.10 and we leave most of it to the reader. The one exception
is item (4), which deserves a comment.

When $U\sqsubset V$, then $\widehat{T}_{V}-\\{\hat{\delta}^{U}_{V}\\}$
consists of some number of components, each of which contains some marked
point or ray. By the BGI property in item (5) of Definition 12.1, each of
these components projects to a marked cluster point in $\widehat{T}_{U}$. By
construction, $h_{U}$ partitions all marked points and rays in
$\widehat{T}_{U}$, and so each half-space $\widehat{T}^{\prime}_{U}$ contains
some (nonempty) set of marked points or rays, whose labels we denote by
$l_{U}$. Hence each component of the subtree
$\mathrm{hull}_{\widehat{T}_{V}}(\hat{\delta}^{U}_{V}\cup l_{U})$, namely the
convex hull in $\widehat{T}_{V}$ of $\hat{\delta}^{U}_{V}$ and $l_{U}$,
projects to a marked point contained in $\widehat{T}_{U}$ by the BGI property
(6) of Definition 12.1. This completes the proof. ∎

###  12.5. Intersection and nesting of half-spaces

In this subsection, we prove some useful descriptions of half-space
intersections and nesting. We note that the first lemma is an exact version of
[DMS20, Lemma 4.6].

######  Lemma 12.14.

Let $h_{U}\in\widehat{T}_{U}$ and $h_{V}\in\widehat{T}_{V}$ be component
hyperplanes with corresponding $\overline{\mathcal{Q}}$-hyperplanes
$\mathbf{h}_{U},\mathbf{h}_{V}$, and
$\overline{\mathcal{Q}}_{U},\overline{\mathcal{Q}}_{V}$ be choices of half-
spaces corresponding to half-trees
$\widehat{T}^{\prime}_{U}\subset\widehat{T}_{U}-\\{h_{U}\\}$ and
$\widehat{T}^{\prime}_{V}\subset\widehat{T}_{V}-\\{h_{V}\\}$, respectively.
Then $\overline{\mathcal{Q}}_{U}\cap\overline{\mathcal{Q}}_{V}\neq\emptyset$
if and only if one of the following holds:

  1. (1)

$U\perp V$;

  2. (2)

$U=V$ and $\widehat{T}^{\prime}_{U}\cap\widehat{T}^{\prime}_{V}\neq\emptyset$;

  3. (3)

$U\pitchfork V$ and either $\hat{\delta}^{V}_{U}\in\widehat{T}^{\prime}_{U}$
or $\hat{\delta}^{U}_{V}\in\widehat{T}^{\prime}_{V}$;

  4. (4)

$U\sqsubset V$ and either $\hat{\delta}^{U}_{V}\in\widehat{T}^{\prime}_{V}$ or
$\hat{\delta}^{V}_{U}(\widehat{T}^{\prime}_{V})\in\widehat{T}^{\prime}_{U}$.

###### Proof.

Item (1) follows from item (1) of Lemma 12.13. Item (2) follows from the
definition of the half-spaces as inverse images under component-wise
projection. Item (3) is a consequence of item (2) of Lemma 12.13, for if
$\hat{\delta}^{V}_{U}\notin\widehat{T}^{\prime}_{U}$, then
$\widehat{\pi}_{V}(\overline{\mathcal{Q}}_{U})=\hat{\delta}^{U}_{V}$, so if
$\hat{\delta}^{V}_{U}\notin\widehat{T}^{\prime}_{V}$, we cannot have
$\overline{\mathcal{Q}}_{U}\cap\overline{\mathcal{Q}}_{V}\neq\emptyset$.
Finally, item (4) is a consequence of items (3) and (4) of Lemma 12.13, for if
$\hat{\delta}^{U}_{V}\notin\widehat{T}^{\prime}_{V}$, then
$\widehat{\pi}_{U}(\overline{\mathcal{Q}}_{V})=\hat{\delta}^{V}_{U}(\widehat{T}^{\prime}_{V})$
by item (3) of that lemma. Hence if
$\overline{\mathcal{Q}}_{U}\cap\overline{\mathcal{Q}}_{V}\neq\emptyset$, then
we must have that
$\hat{\delta}^{V}_{U}(\widehat{T}^{\prime}_{V})\in\widehat{T}^{\prime}_{U}$,
as required. This completes the proof. ∎

A similar analysis gives a description of when half-spaces are nested:

######  Lemma 12.15.

Using the same notation as above, we have
$\overline{\mathcal{Q}}_{U}\subset\overline{\mathcal{Q}}_{V}$ if and only if
one of the following holds:

  1. (1)

$U=V$ and $\widehat{T}^{\prime}_{U}\subset\widehat{T}^{\prime}_{V}$;

  2. (2)

$U\pitchfork V$ and $\hat{\delta}^{U}_{V}\in\widehat{T}^{\prime}_{V}$ and
$\hat{\delta}^{V}_{U}\notin\widehat{T}^{\prime}_{U}$;

  3. (3)

$U\sqsubset V$ and $\hat{\delta}^{U}_{V}\in\widehat{T}^{\prime}_{V}$ and
$\widehat{T}^{\prime}_{U}\subset\hat{\delta}^{V}_{U}(\widehat{T}^{\prime}_{V})$;

  4. (4)

$V\sqsubset U$ and both $\hat{\delta}^{V}_{U}\notin\widehat{T}^{\prime}_{U}$
and
$\hat{\delta}^{U}_{V}(\widehat{T}^{\prime}_{U})\in\widehat{T}^{\prime}_{V}$.

  * •

In particular, if $\overline{\mathcal{Q}}_{U}\neq\overline{\mathcal{Q}}_{V}$
and $U,V$ are not orthogonal or equal, then one of
$\overline{\mathcal{Q}}_{U}\subset\overline{\mathcal{Q}}_{V}$,
$\overline{\mathcal{Q}}_{U}\subset\overline{\mathcal{Q}}^{*}_{V}$,
$\overline{\mathcal{Q}}^{*}_{U}\subset\overline{\mathcal{Q}}_{V}$, or
$\overline{\mathcal{Q}}^{*}_{U}\subset\overline{\mathcal{Q}}^{*}_{V}$ holds.
On the other hand, if $\overline{\mathcal{Q}}_{U}=\overline{\mathcal{Q}}_{V}$
or $U,V$ are orthogonal, then none of the above relations holds.

###### Proof.

Item (1) follows from the definition of
$\overline{\mathcal{Q}}_{U},\overline{\mathcal{Q}}_{V}$ as preimages under
component-wise projection to $\widehat{T}_{U}=\widehat{T}_{V}$. Items (2)–(4)
are direct consequences of items (2)–(4) of Lemma 12.13, respectively.

We now prove the last statement. The statement follows immediately from (1)
when the domain labels for
$\overline{\mathcal{Q}}_{U},\overline{\mathcal{Q}}_{V}$ are the same. Suppose
then that $\overline{\mathcal{Q}}_{U},\overline{\mathcal{Q}}_{V}$ correspond
to half-trees $\widehat{T}^{\prime}_{U},\widehat{T}^{\prime}_{V}$ for tree-
hyperplanes $h_{U},h_{V}$ in $\widehat{T}_{U},\widehat{T}_{V}$, respectively.
When $U\pitchfork V$, if
$\overline{\mathcal{Q}}_{U}\nsubset\overline{\mathcal{Q}}_{V}$, then
$\hat{\delta}^{V}_{U}$ is contained in the complement of
$\widehat{T}^{\prime}_{U}$, and hence item (2) above implies that
$\overline{\mathcal{Q}}_{U}\subset\overline{\mathcal{Q}}^{*}_{V}$. For
$U\sqsubset V$, if
$\overline{\mathcal{Q}}_{U}\nsubset\overline{\mathcal{Q}}_{V}$, then
$\hat{\delta}^{U}_{V}\notin\widehat{T}^{\prime}_{V}$. It follows that
$\hat{\delta}^{U}_{V}\in\widehat{T}_{V}-\widehat{T}^{\prime}_{V}$, and hence
item (3) implies that
$\overline{\mathcal{Q}}_{U}\subset\overline{\mathcal{Q}}^{*}_{V}$. See Figure
18. Finally, the claim when
$\overline{\mathcal{Q}}_{U}=\overline{\mathcal{Q}}_{V}$ follows immediately
from the definitions, while when $U\perp V$, the claim also requires item (1)
of Lemma 12.13. This completes the proof.

Figure 18. The four nesting possibilities in Lemma 12.15: In (a), the domain
labels for $\overline{\mathcal{Q}}_{U},\overline{\mathcal{Q}}_{V}$ are the
same and we have $\overline{\mathcal{Q}}_{U}\subset\overline{\mathcal{Q}}_{V}$
because $\widehat{T}^{\prime}_{U}\subset\widehat{T}^{\prime}_{V}$. In (b),
$U\pitchfork V$ and
$\overline{\mathcal{Q}}_{U}\subset\overline{\mathcal{Q}}_{V}$ because
$\hat{\delta}^{U}_{V}\in\widehat{T}^{\prime}_{V}$ while
$\hat{\delta}^{V}_{U}\notin\widehat{T}^{\prime}_{U}$. In (c), we have
$U\sqsubset V$ and
$\overline{\mathcal{Q}}_{U}\subset\overline{\mathcal{Q}}_{V}$ because
$\widehat{T}^{\prime}_{U}\subset\hat{\delta}^{V}_{U}(\widehat{T}^{\prime}_{V})$.
In (d), we have $U\sqsubset V$ and
$\overline{\mathcal{Q}}_{V}\subset\overline{\mathcal{Q}}_{U}$ because
$\hat{\delta}^{V}_{U}(\widehat{T}^{\prime}_{V})\subset\widehat{T}^{\prime}_{U}$.

∎

###  12.6. Half-spaces are nonempty

In this subsection, we prove that every $\overline{\mathcal{Q}}$-half-space
contains a point in $\mathcal{Q}$. This fact is almost immediate when
$\Lambda=\emptyset$, but requires some work in the infinite case. It is
crucial for proving local finite-ness in the proof that the induced wall-space
on $\overline{\mathcal{Q}}$ forms a pocset (Proposition 12.18), as well as
proving that only points in $\mathcal{Q}^{\infty}$ are contained in infinite
descending chains of half-spaces (Proposition 12.17).

######  Lemma 12.16.

For any $\overline{\mathcal{Q}}$-half-space $\overline{\mathcal{Q}}_{U}$, we
have that $\overline{\mathcal{Q}}_{U}\cap\mathcal{Q}$ and
$\overline{\mathcal{Q}}^{*}_{U}\cap\mathcal{Q}$ are nonempty. Moreover, if
$\widehat{T}^{\prime}_{U}$ is the half-tree of $\widehat{T}_{U}$ corresponding
to $\overline{\mathcal{Q}}_{U}$, then the following hold:

  1. (1)

If $a\in F$ with $\hat{a}_{U}\in\widehat{T}^{\prime}_{U}$, then
$\hat{a}\in\overline{\mathcal{Q}}_{U}\cap\mathcal{Q}$.

  2. (2)

If $\lambda\in\Lambda$ with
$\hat{\lambda}_{U}\in\widehat{T}^{\prime}_{U}\cup\partial\widehat{T}^{\prime}_{U}$,
then $\hat{\lambda}\in\overline{\mathcal{Q}}_{U}\cap\mathcal{Q}^{\infty}$.

###### Proof.

Let $\widehat{T}^{\prime}_{U}$ be the half-tree of $\widehat{T}_{U}$
corresponding to $\overline{\mathcal{Q}}_{U}$. Note that there is some $z\in
F\cup\Lambda$ so that
$\hat{z}_{U}\in\widehat{T}^{\prime}_{U}\cup\partial\widehat{T}^{\prime}_{U}$.
First suppose that we can take $z=a\in F$ so that
$\hat{a}_{U}\in\widehat{T}^{\prime}_{U}$, then
$\hat{a}\in\overline{\mathcal{Q}}_{U}\cap\mathcal{Q}$. This is because
$\left(\overline{\mathcal{Q}}_{U}\sqcup\overline{\mathcal{Q}}^{*}_{U}\sqcup\mathbf{h}_{U}\right)\cap\mathcal{Q}=\mathcal{Q}$
is indeed a partition of $\mathcal{Q}$, and by Lemma 12.13 and our assumption
that $\hat{a}_{U}\in\widehat{T}^{\prime}_{U}$, we cannot have
$\hat{a}\in\overline{\mathcal{Q}}^{*}_{U}\cup\mathbf{h}_{U}$. This proves part
(1) of the statement, and the whole statement when $\Lambda=\emptyset$. Also,
observe that part (2) is immediate from this observation as well.

The bulk of the work is in when we must take $z=\lambda\in\Lambda$ with
$\hat{\lambda}_{U}\in\widehat{T}^{\prime}_{U}\cup\partial\widehat{T}^{\prime}_{U}$.
Observe that when this happens, the opposite half-tree of
$\widehat{T}^{\prime}_{U}$ contains some marked point for an element of
$F\neq\emptyset$, and hence
$\overline{\mathcal{Q}}^{*}_{U}\cap\mathcal{Q}\neq\emptyset$. For showing
$\overline{\mathcal{Q}}_{U}\cap\mathcal{Q}\neq\emptyset$, there are a three
main cases:

  1. (1)

$U\in\mathrm{supp}(\lambda)$ and $\hat{\lambda}_{U}\in\partial\widehat{T}_{U}$
represents a ray;

  2. (2)

$U\in\mathrm{supp}(\lambda)$ and $\hat{\lambda}_{U}\in\widehat{T}_{U}$ is a
cluster point;

  3. (3)

$U\notin\mathrm{supp}(\lambda)$.

In each case, the idea is to construct a tuple
$\hat{w}\in\overline{\mathcal{Q}}_{U}$ and prove that it is canonical
(Definition 7.9). The arguments are related, and each invokes Strong Passing-
Up 4.7 at a key moment.

Case (1): In this case, there are infinitely-many vertices in
$\widehat{T}^{\prime}_{U}$ which are not marked, branched, or cluster points.
Choose a vertex $w\in\widehat{T}^{\prime}_{U}$ sufficiently out the ray for
$\hat{\lambda}_{U}$ so that no marked point or branch points separates $w$
from $\hat{\lambda}_{U}$.

Define a tuple $\hat{w}$ domain-wise as follows:

  * •

$\hat{w}_{U}=w$;

  * •

If $V\pitchfork U$ or $U\sqsubset V$, then
$\hat{w}_{V}=\hat{\delta}^{U}_{V}=\hat{\lambda}_{V}$;

  * •

If $V\sqsubset U$, then we set
$\hat{w}_{V}=\hat{\delta}^{U}_{V}(h_{U})=\hat{\lambda}_{V}$;

  * •

Otherwise, we fix $a\in F$ and for any $V\perp U$, we set
$\hat{w}_{V}=\hat{a}_{V}$.

Observe that $\hat{w}\in\overline{\mathcal{Q}}_{U}$ by construction, in
particular it is $0$-consistent. To see canonicality, let $b\in F$ and set
$\mathcal{V}_{b}=\\{V\in\mathcal{U}|\hat{b}_{V}\neq\hat{w}_{V}\\}$. If