The set of cells in $\uchc$ can be partially ordered by inclusion. The result is a graded poset, whose rank function is given by the dimension of the corresponding cell. The height we just defined is induced by this rank function. One could construct the order complex of such a poset, by taking a simplex for every chain. This would result in a simplicial complex, and is not what we are considering here.

Since mirrors are separating subspaces (see  Proposition <ref>), the collection of mirrors can be used to define a wallspace structure $(\uchc, \mirrors)$ on $\uchc$, and one could consider the dual $\cat 0$ cubical complex $\dccm$ associated to this wallspace by Sageev's construction.
We refer the reader to <cit.> for details about this construction, and we only review the main ingredients here.
Given a mirror $M$, any partition of the complementary components into two classes is called a wall associated to $M$. An orientation of a wall is a choice of one of the two classes.
A vertex of $\dccm$ can then be described as a consistent choice of orientation for each mirror.

When $X$ and all mirrors are homeomorphic to manifolds, each mirror of $\uchc$ has exactly two complementary components.
In this quite restrictive case, an orientation of a wall is just a choice of one of the two complementary components.
Therefore vertices of $\dccm$ correspond to tiles (i.e. $n$–cells) in the stratification of $\uchc$, and two vertices are connected by an edge when the corresponding tiles are adjacent along a mirror.
In particular, $\dccm$ is an $n$–dimensional cubical complex that can be subdivided to recover  $\dccx$.
However, if there are mirrors which have more than two complementary components (such as in Figures <ref> and <ref>), then we find vertices in $\dccm$ which do not correspond to tiles  from the stratification of $\uchc$ (they are not canonical vertices, in the terminology of <cit.>).
As a result, the dimension of $\dccm$ is usually higher than that of $X$, and it is more challenging to relate the actions of $\hg$ on $\uchc$ and on $\dccm$.

§.§ Cubes and links

In this section we explore basic facts about the cubical geometry of $\dccx$. While this complex is not locally compact (see Remark <ref>), its dimension is the same as that of $X$ (see Lemma <ref>), and the links of vertices are flag complexes (see Proposition <ref>).

The first two lemmas show that squares and cubes in $\dccx$ admit unique vertices of minimum and maximum height. Recall that the height of a vertex is the dimension of its dual cell, and notice that, by definition of $\dccx$, if $u,v$ are adjacent vertices, then $|\height u - \height v |=1$.

A square in $\dccx$.

Let $S$ be a square of $\dccx$.
Let $v_1,v_2,v_3,v_4$ be its vertices, with $v_2$ and $v_4$ non-adjacent in $S$.
If $\height{v_2} = \height{v_4}$, then $|\height{v_1}-\height{v_3}|=2$.
In particular there is a unique vertex of maximal (respectively minimal) height, and the cell dual to it contains (respectively is contained in) each of the cells dual to the other vertices.

Let $h=\height{v_2}=\height{v_4}$ be the common value of the height of $v_2$ and $v_4$. Since $v_1$ is adjacent to $v_2$ and $v_4$, we have $\height{v_1}=h\pm 1$, and similarly for $v_3$ (see Figure <ref>).
In particular $|\height{v_1}-\height{v_3}|$ is either $0$ or $2$.
By contradiction let us assume that $|\height{v_1}-\height{v_3}|=0$, i.e. $\height{v_1}=\height{v_3}=h\pm 1$.
Without loss of generality we can assume that $\height{v_1}=\height{v_3}=h+1$.  (The case $\height{v_1}=\height{v_3}=h-1$ is completely analogous, via a dual argument).
For $j=1,2,3,4$, let $\sigma_j$ be the cell of $\uchc$ dual to the vertex $v_j$.
Since $v_1$ is adjacent to $v_2$ and $v_4$, and has higher height, $\sigma_1$ contains $\sigma_2$ and $\sigma_4$; the same holds for $\sigma_3$. So $\sigma_1 \cap \sigma_3$ contains $\sigma_2\cup \sigma_4$, contradicting Lemma <ref>.

To prove the final statement, let us assume without loss of generality that $v_1$ is the vertex of maximal height and $v_3$ is the one of minimal height, i.e. $\height{v_1}-1=h=\height{v_3}+1$. Then we have that $\sigma_3\subseteq \sigma_2,\sigma_4 \subseteq \sigma_1$.

In the next lemma we extend this result to higher dimensional cubes of $\dccx$. By  an edge–path in $\dccx$ we will mean a continuous path which is entirely contained in the 1–skeleton (i.e. is a sequence of edges). If an edge–path $p$ goes through vertices $v_0,\dots,v_s$ of $\dccx$, we will write $p=(v_0,\dots,v_s)$; note that the sequence of vertices completely determines the sequence of edges, hence the path. We call $p$ an edge–loop if it is a closed loop, i.e. $v_0=v_s$.
For an edge–path $p=(v_0,\dots,v_s)$ we define $\length p =s$  to be the length of $p$, i.e. the number of edges in it.
We also define the height of $p$ to be $\height p=\max\{\height {v_0},\dots,\height {v_s}\}$. Notice that along each edge of $p$ the height must increase or decrease exactly by 1.

Let $Q$ be a cube of $\dccx$. Then the following hold.

* There is a unique vertex $v\in Q$ of minimal height. The cell dual to it is contained in each of the cells dual to the vertices of $Q$.

* There is a unique vertex $w\in Q$ of maximal height. The cell dual to it  contains each of the cells dual to the vertices of $Q$.

We prove the first statement; the second is obtained by an analogous argument.
Let $k$ be the minimal height of vertices of $Q$, and assume by contradiction that there is at least a pair of vertices of $Q$ of height $k$.
Consider an edge–path $p=(v_0,\dots,v_s)$ in $Q$ such that $\height {v_0}=\height {v_s}=k$, $v_0\neq v_s$, and such that $p$ is an edge–path of minimal height among all edge–paths in $Q$ joining a pair of vertices of height $k$. This is well–defined since the height of such a path can only be an integer between $0$ and $n$. Let $\height p=h$ be the height of $p$.

Lowering a vertex of maximal height on a path in $\dccx$.

Let $v_j$ be a vertex of $p$ of maximal height $\height {v_j}=h=\height p$. Then $\height {v_{j\pm 1}}=h-1$ (notice that $k\geq 0$ and $h\geq k+1\geq 1$). Since $(v_{j-1},v_j,v_{j+1})$ is part of the cube $Q$, it must be contained in a square, i.e. there exists a vertex $u_j$ of $Q$ (not necessarily on $p$), such that $\{v_{j-1},v_j,v_{j+1},u_j\}$ span a square in $Q$.
Lemma <ref> implies that $\height {u_j}=h-2$.
We can construct a new path in $Q$ starting from the path $p$ by lowering the vertex of maximal height, i.e. by replacing $v_j$ with $u_j$ (see Figure <ref>). We repeat the same operation on all vertices of height $h$ along the path, and let $p'$ be the resulting path in $Q$.
We have that $\height {p'}=h-1< h=\height {p}$, contradicting the minimality of the height of $p$.
This concludes the proof by contradiction, and proves the uniqueness of a vertex $v$ of minimal height $k$ in $Q$.

We are left to show that the cell $\sigma$ dual to $v$ is contained in all the cells dual to the other vertices of $Q$.
By contradiction suppose there are vertices in $Q$ whose dual cells do not contain $\sigma$; call such vertices exceptional.
Let $p=(v_0,\dots,v_s)$ be an edge–path in $Q$ with $v_0=v$, $v_s$ an exceptional vertex, and having minimal length among all edge–paths of $Q$ between $v$ and an exceptional vertex.
We have $\height {v_{s-1}}=\height {v_s}\pm 1$.
If $\height {v_{s-1}}=\height {v_s}- 1$, then the cell dual to $v_{s-1}$ is contained in the cell dual to $v_s$. By minimality of $p$, we have that $v_{s-1}$ is not exceptional, so the cell dual to $v_{s-1}$ contains $\sigma$, and hence $v_s$ cannot be exceptional. Therefore $\height {v_{s-1}}=\height {v_s}+ 1$ (as in Figure <ref>).

Walking backwards along a path in $\dccx$ to find a vertex $v_j$ that can be lowered, and then walking forward to lower vertices until the exceptional endpoint $v_s$.

We keep walking backwards along $p$ until we find a triple of vertices $\{v_{j-1},v_j,v_{j+1}\}$ such that $\height {v_j}=\height {v_{j\pm 1}}+1$ (notice $j$ is well defined and positive, since $v$ is the unique vertex of minimal height in the whole $Q$).
Arguing as before we complete to a square in $Q$ with vertices $\{v_{j-1},v_j,v_{j+1},u_j\}$; again by Lemma <ref> we have $\height {u_j}=\height {v_j}-2$ (see Figure <ref>).
By minimality of $p$, $u_j$ must be non–exceptional, and so we can change $p$ by replacing $v_j$ with $u_j$, without changing its length.
Walking forward along $p$, we can keep changing the path without changing its length, until we are able to complete $\{v_{s-1},v_s\}$ to a square $\{u_{s-2},v_{s-1},v_s,u_{s-1}\}$ in $Q$ with $u_{s-1}$ non-exceptional and with height $\height{u_{s-1}}=\height {v_{s-1}}-2=\height {v_s}-1$ (once again, see Figure <ref>).
In particular, the cell dual to $u_{s-1}$ contains $\sigma$ and is contained in the cell dual to $v_s$, which contradicts the fact that the last vertex  $v_s$ was chosen to be exceptional.

As a consequence, we obtain the following statement.

The complex $\dccx$ has dimension $\dim {\dccx}=\dim X = n$.

If $\tau $ is a tile of $\uchc$ and $x$ one of its vertices, then the collection of cells containing $x$ and contained in $\tau$ provides a cube of dimension exactly $n$, so $\dim {\dccx} \geq n$, so we focus on the other inequality.

Let $Q$ be a cube of $\dccx$, and let $v_{\operatorname{min}}$  be the vertex of minimal height in $Q$ (see Lemma <ref>).
We claim that for each vertex $v\in Q$ we have

\height v = \height{v_{\operatorname{min}}}+d_Q(v_{\operatorname{min}},v)

where $d_Q(v_{\operatorname{min}},v)$ is the distance in $Q$ of $v$ from $v_{\operatorname{min}}$.
Since the height can take values only  between $0$ and $n=\dim X$, this directly implies that

\dim Q = \max \{ d_Q(v_{\operatorname{min}},v)\} = \max \{ \height v - \height{v_{\operatorname{min}}}\} \leq n.

In order to prove the claim, pick a vertex $v\in Q$, and let $p=(v_0,\dots,v_s)$ be an edge–path of minimal length $s=d_Q(v_{\operatorname{min}},v)$ in $Q$ from $v_0=v_{\operatorname{min}}$ to $v_s=v$.
Since the height can at most increase by $1$ along each edge of $p$, we have the inequality $\height v \leq \height{v_{\operatorname{min}}}+d_Q(v_{\operatorname{min}},v)$.
Assume by contradiction that the inequality is strict. Then the height is not monotonically increasing along $p$. Let $v_k$ be the first vertex of $p$ which is a local maximum for the height function.
Arguing as above via Lemma <ref>, we look at the triple $v_{k-1},v_k,v_{k+1}$, and complete it to a square with a fourth vertex $u_k$ such that $\height{u_k}=\height{v_k}-2$.
We can even assume without loss of generality that $k=2$ (otherwise we proceed as in the proof of Lemma <ref> and change $p$ along squares walking backwards towards $v_{\operatorname{min}}$).
But then $\height{u_2}=\height{v_2}-2=\height{v_{\operatorname{min}}}$. Minimality of $v_{\operatorname{min}}$ implies $u_2=v_{\operatorname{min}}$, and therefore we get that $v_3$ was already adjacent to $v_{\operatorname{min}}$. This provides a path from $v_{\operatorname{min}}$ to $v$ of length at most  $d_Q(v_{\operatorname{min}},v)-2$, which is a contradiction.

We now turn to the study of links of vertices of $\dccx$.
Recall that $\dccx$ is a cubical complex, hence its links are simplicial complexes (see <ref>).
In particular, if $v\in \dccx$ is a vertex, then vertices in $\lk{v,\dccx}$ correspond to vertices in $\dccx$ which are adjacent to $v$.
We begin with the following combinatorial characterization of simplices in the link of a vertex.

Let $\sigma$ be a $k$-cell of $\uchc$, and let $v$ be the dual vertex in $\dccx$.
Let $v_0,\dots,v_m$ be a collection of vertices of $\dccx$ adjacent to $v$, and let $\tau_0,\dots,\tau_m$ be the dual cells in $\uchc$. Then $v_0,\dots,v_m$ induce a simplex in $\lk {v,\dccx}$ if and only if the following two conditions are satisfied

$( \downarrow )$ there exists a cell $\lambda$ of $\uchc$ such that $\lambda\subseteq \tau_j,j=0,\dots,m$,

$( \uparrow )$ there exists a cell $\mu$ of $\uchc$ such that $\tau_j \subseteq \mu,j=0,\dots,m$.

First of all, note that since $\dccx$ is a cubical complex, the vertices $v_0,\dots,v_m$ induce a simplex in $\lk {v,\dccx}$ if and only if there exists a cube $Q$ of $\dccx$ containing $v,v_0,\dots,v_m$.

Assume that they induce a simplex, and let $Q$ be the corresponding cube.
From Lemma <ref> we know that $Q$ has a unique vertex of minimal height, and a unique vertex of maximal height. Let $\lambda,\mu$ be the dual cells. Lemma <ref> then implies that $\lambda,\mu$ satisfy the conditions $(\downarrow)$ and $(\uparrow)$ in the statement.

Vice versa suppose that the conditions $(\downarrow)$ and $(\uparrow)$ are satisfied.
Notice that we have $\lambda \subseteq \tau_j\subseteq \mu$ for all $j=0,\dots,m$.

Let $C_\lambda=\cdXu(\lambda)$ and $C_\mu=\cdXu(\mu)$ be the corresponding cubes of $X$, under the map $\cdXu=\cdX \circ \piuchc :\uchc \to \hc \to X$.
Notice that $\lk{\lambda,\mu}\cong \lk{C_\lambda, C_\mu}$ by  Lemma <ref>.
In particular, we see that in $\uchc$ there is a collection of cells containing $\lambda$ and contained in $\mu$ (among which we find the cells $\tau_j$) that gives rise to a cube $Q$ in $\dccx$ containing the vertices  $v,v_0,\dots,v_m$.  Therefore $v_0,\dots,v_m$ induce a simplex in $\lk{v,\dccx}$, as desired.

When the conditions ($\downarrow$) and ($\uparrow$) from Lemma <ref> are satisfied, the cells $\lambda,\mu$  can be chosen to be the lower and upper cell provided by Lemma <ref>.

A cell of dimension at least $2$ in $\uchc$ always admits infinitely many codimension-1 cells (see Figure <ref>). Lemma <ref> implies that the link of the dual vertex is neither compact nor connected. In particular the cubical complex $\dccx$ is not locally compact.
As a result, even though $\dccx $ is constructed as a sort of dual cubical barycentric subdivision with respect to the combinatorial decomposition of $\uchc$ into cells, $\dccx$ is not homeomorphic to $\uchc$. Namely, $\uchc$ is locally compact, while $\dccx$ is not locally compact.

As recalled above, if $v\in \dccx$ is a vertex, then the vertices appearing in $\lk{v,\dccx}$ correspond to vertices of $\dccx$ that are adjacent to $v$, and these vertices have height equal to $\height v\pm 1$.
We find it useful to decompose $\lk {v,\dccx}$ into two subcomplexes: we denote by $\lkd {v,\dccx}$ the full subcomplex of $\lk {v,\dccx}$ generated by vertices of height $\height v-1$, and by $\lku {v,\dccx}$  the full subcomplex of $\lk {v,\dccx}$ generated by vertices of height $\height v+1$.
As we will see, their geometry is controlled respectively by a certain Helly property for orthogonal hyperplanes in $\hh^n$, and by the non–positive curvature of $X$.
The following statement provides the Helly property. Notice that orthogonality is a key feature here: without the orthogonality requirement, the statement already fails for three geodesics in $\hh^2$.
On the other hand, the interested reader will notice that the argument generalizes to a collection of pairwise orthogonal and totally geodesic hypersurfaces in a simply connected complete manifold of non–positive curvature. We will not need this generality in this paper.

Let $\mathcal V$ be a collection of pairwise orthogonal hyperplanes in $\hh^n$. Then $|\mathcal V|\leq n$, and for all $k\in \{2, \dots , n\}$ all the $k$–fold intersections are non-empty.

We begin with some preliminary observation about orthogonal subspaces.

Let $V_1,\dots, V_k \in \mathcal V$ be a collection of hyperplanes from $\mathcal V$, and let $N=\cap_{j=1}^k V_j$ be their intersection.
For $x\in N$, let $\tangent{x}{\hh^n}$  denote the tangent space of $\hh^n$ at $x$, and let $v_j \in \tangent{x}{\hh^n}$ be a unit vector orthogonal to $V_j$ (i.e. to all vectors in the tangent space $\tangent{x}{V_j}$).
The fact that $V_i$ and $V_j$ are orthogonal hyperplanes means that $v_i$ and $v_j$ are orthogonal vectors for all $i\neq j$.
Then a direct computation shows that if $\{n_1,\dots,n_m\}$ is an orthonormal basis for the tangent space of $\tangent{x}{N}$, then $\{n_1,\dots,n_m,v_1,\dots,v_k\}$ is an orthonormal basis for $\tangent{x}{\hh^n}$.
This shows in particular that $k\leq n$.

To prove the statement about non–emptiness of intersections, we notice that the case $k=2$ is exactly the hypothesis that any pair of hyperplanes from $\mathcal V$ intersect.
For $k\geq 3$, we argue that if all the $h$–fold intersections of elements from $\{V_1,\dots, V_k\}$  are non–empty for all $h<k$, then the $k$–fold intersection is non–empty too.

Let $N_j=\cap_{i\neq j} V_i$. By assumption we have  $N_j\neq \varnothing$.

Assume by contradiction that $V_1\cap \dots\cap V_k=\varnothing$.
Then for any choice of indices $j_1\neq j_2$ we have that $N_{j_1}\cap N_{j_2}= \varnothing$.
In particular, $N_2$ and $N_3$ are non–empty disjoint subspaces of $V_1$ (see Figure <ref>).
Let $\gamma_1$ be the common perpendicular between them in $V_1$, and let $x_k\in N_k$ be its endpoint for $k=2,3$.
Now in the tangent space $\tangent{x_2}{\hh^n}$ we consider an orthonormal basis $\{n_1,\dots,n_m,v_1,v_3,\dots,v_k\}$ constructed as above by adding to an orthonormal basis $\{n_1,\dots,n_m\}$ for $\tangent{x_2}{N_2}$ unit vectors $v_1,v_3,\dots,v_k$ orthogonal to
If $w$ denotes a tangent vector at $x_2$ along $\gamma_1$, then a direct computation shows that $w$ is orthogonal to
$\{n_1,\dots,n_m\}$, because $\gamma_1$ is orthogonal to $N_2$, and it is also orthogonal to $v_1$, because $\gamma_1 \subseteq V_1$.
Therefore $w$ is in the subspace of $\tangent{x_2}{\hh^n}$ generated by $v_3,\dots,v_k$.
If we define $W_2=\cap_{j=3}^k V_j$, then this means that $\gamma_1$ is orthogonal to $W_2$ at $x_2$.
Arguing in the same way at the point $x_3$, we find that $\gamma_1$ is orthogonal at $x_3$ to the subspace $W_3=\cap_{j=2,j\neq 3}^k V_j$.
Note that $W_2\cap W_3=\cap_{j=2}^k V_j=N_1$ is non–empty.
Moreover, as observed above, it is disjoint from $N_2$ and from $N_3$.
Therefore we can connect $x_2$ (respectively $x_3$) to a point $x_1$ in $N_1$ with a geodesic arc $\gamma_2$ contained in $W_2$ (respectively $\gamma_3$ contained in $W_3$).
Since all the spaces involved are totally geodesic, the arcs $\gamma_1,\gamma_2,\gamma_3$ are geodesic arcs in $\hh^n$, so we have obtained a geodesic triangle with two right angles, which leads to the desired contradiction.

The Helly property in Lemma <ref>.

The next statement completes our investigation of the combinatorial geometry of $\dccx$.
Thanks to Gromov's link condition (see Lemma <ref>), it already implies that $\dccx$ is locally $\cat 0$.
We will show in Theorem <ref> that it is actually $\cat 0$.

Let $\sigma$ be a $k$-cell of $\uchc$, and let $v$ be the dual vertex in $\dccx$. Then the following hold:

*  $\lkd {v,\dccx}$ is a flag simplicial complex.

*  $\lku {v,\dccx}$   is a flag simplicial complex.

*  $\lk {v,\dccx}$  is a flag simplicial complex.

Throughout this proof, $w_j$ will denote a vertex in $\lk {v,\dccx}$, $v_j$ the corresponding vertex of $\dccx$ adjacent to $v$, and $\tau_j$ the cell of $\uchc$ dual to $v_j$.
Notice that two vertices $w_i,w_j$ are adjacent in $\lk {v,\dccx}$ precisely when $v,v_i,v_j$ are contained in a square of $\dccx$.

We first prove (<ref>).
Let $w_0,\dots,w_p$ be pairwise adjacent vertices in $\lkd {v,\dccx}$.
Notice that $\tau_0,\dots,\tau_p$ are all cells of codimension 1 in the boundary of $\sigma$.
For each $i\neq j$, $v,v_i,v_j$ are contained in a square of $\dccx$.
By Lemma <ref>, the fourth vertex of the square is dual to a cell contained in $\tau_i\cap\tau_j$.
This shows that the cells $\tau_j$ intersect pairwise.
We claim that actually $\tau_0\cap \dots \cap\tau_p\neq \varnothing$.
To see this, embed $\sigma$ into a hyperbolic space of   dimension $\dim \sigma$ (as in <ref>).
The family of hyperplanes $V_0,\dots,V_p$ supporting the cells $\tau_0,\dots, \tau_p$ is a collection of pairwise orthogonal hyperplanes, and the boundary of the cell $\tau_j$ in $V_j$ is given by subspaces that are orthogonal to the other $V_i$'s.
Note that by Lemma <ref> we know that $V_0 \cap \dots \cap V_p\neq \varnothing$.
Arguing by induction on $p$, we can assume that $\tau_0\cap \dots \cap \tau_{p-1}\neq \varnothing$, and then we can leverage the orthogonality structure as in the proof of Lemma <ref> to obtain that $\varnothing \neq \tau_0\cap \dots \cap \tau_{p-1} \cap V_p \subseteq \tau_0 \cap \dots \cap \tau_p$, which proves the claim.
Since the latter intersection is non-empty, by Lemma <ref>,  it consists of a single cell $\lambda \subseteq \tau_j$.
We use Lemma <ref> with this cell $\lambda$ and   $\mu=\sigma$ to conclude that $w_0,\dots,w_k$ span a  simplex.

We argue via a dual argument to prove (<ref>). Let $w_0,\dots,w_p$ be pairwise adjacent vertices in $\lku {v,\dccx}$.
Notice that  $\tau_0,\dots,\tau_p$ are cells containing $\sigma$ as a cell of codimension 1 in their boundary.

For each $i\neq j$, $v,v_i,v_j$ are contained in a square of $\dccx$.
By Lemma <ref>, the fourth vertex of the square is dual to a cell  containing   $\tau_i\cup\tau_j$.
So  $\tau_i,\tau_j$ are adjacent in $\lk{\sigma,\uchc}$.
By (<ref>) in Lemma <ref> this link is isomorphic to the link of the corresponding cube in $X$.
Since $X$ is non-positively curved, this link is a flag simplicial complex.
Therefore there is a cell $\mu$ containing all the cells $\tau_j$; this can actually be taken to be the upper cell provided by Lemma <ref>.
We use Lemma <ref> with this cell $\mu$ and   $\lambda=\sigma$ to conclude that $w_0,\dots,w_k$ span a  simplex.

Finally, in order to prove (<ref>), let $w_0,\dots,w_p$ be pairwise adjacent vertices in $\lk {v,\dccx}$, ordered so that for some $m$ we have $w_0\dots,w_m \in \lkd {v,\dccx}$ and $w_{m+1}\dots,w_p \in \lku {v,\dccx}$.

By (<ref>) we know that $w_0\dots,w_m$ span a simplex, hence by Lemma <ref> there exists a cell $\lambda$ in $\cap_{j=0}^m \tau_j$.
Similarly, by (<ref>) we know that $w_{m+1}\dots,w_p$ span a simplex, hence by Lemma <ref> there exists a cell $\mu$ containing $\tau_{m+1},\dots,\tau_p$.
Notice that
$\lambda \subseteq \tau_i\subseteq \sigma \subseteq \tau_j \subseteq \mu$
for all $i=0,\dots,m$ and  $j=m+1,\dots,p$. In particular  we have
$\lambda \subseteq \tau_j \subseteq \mu$ for all $j=0,\dots,p$.
Using Lemma <ref> again we obtain that $w_0,\dots,w_p$ spans a simplex.

§.§ Efficiency

In this section we study a notion of complexity for edge–paths in $\dccx$, which is based on the height function, and use it to find suitable representatives of homotopy classes of edge–paths and edge–loops.
Recall that if $p$ is an edge–path in the $1$–skeleton of a cubical complex, an elementary homotopy of $p$ is a homotopy which is contained in the $2$–skeleton and is obtained by a finite sequence of the following two moves:

* remove a backtracking subpath, i.e. replace $(v_1,v_2,v_1)$ with $v_1$;

* slide across a square, i.e. replace $(v_1,v_2,v_3)$ with $(v_1,v_4,v_3)$ if $v_1,v_2,v_3,v_4$ appear in this order on the boundary of a square (as in Figure <ref>).

An edge–path $p=(v_0,\dots,v_s)$ is said to be efficient if $\exists \ k\in \{0,\dots,s\}$ such that the height strictly increases from $v_0$ to $v_k$ and strictly decreases from $v_k$ to $v_s$, i.e. $v_k$ is the unique point of maximum for the height along $p$.

We allow $k=0$ or $k=s$, i.e. that the height is strictly monotone  along $p$.
In any case, $\height{p}=\height{v_k}$, and the cell dual to  $v_k$ contains the cells dual to all the other vertices of $p$.
This implies that an efficient edge–path is contained in the union of at most two cubes which share at least a vertex.
In particular, an efficient edge–loop  is entirely contained in a single cube. These observations motivate the following definitions and constructions.

If $\tau$ is a tile of $\uchc$, we define the dual tile $\dcc \tau $ to be the full subcomplex of $\dccx$ whose vertices are dual to the cells of $\tau$.
If $v$ is the vertex of $\dccx$ which is dual to $\tau$, then $\dcc \tau$ consists of all the cubes of $\dccx$ that contain $v$, i.e. $\dcc \tau$ is the combinatorial $1$–neighborhood of $v$.
Notice that $v$ is the only vertex of height $n$ in $\dcc \tau$ (see Figure <ref>).

We say that an edge–path $p$ in $\dccx$ stays in a tile if there exists a tile $\tau$ of $\uchc$ such that $p\subseteq \dcc \tau$.

Let $p$ be an edge–path in $\dccx$.
If $p$ stays in a tile, then there is an elementary homotopy relative to endpoints between $p$ and an efficient path.

Let $p=(v_0,\dots,v_s)$.
First of all, notice that if $s=0,1$ then $p$ is already efficient. Moreover, by an elementary homotopy relative to endpoints, we can assume that $p$ has no backtracking subpath.
Since $p$ stays in a tile, $p$ goes through at most one vertex of height $n$ (possibly several times, possibly at the endpoints $v_0,v_s$).

For $0<j<s$, we say $v_j$ is a local minimum (with respect to the height function along $p$) if $\height {v_{j\pm 1}}=\height {v_j}+1$, and we consider the following quantity

$$\mathfrak{h}(p)=\min\{\height {v_j} \ | \ v_j \text{ is a local minimum} \}.$$

If there is no local minimum, set $\mathfrak{h}(p)=\infty$; in this case $p$ is already efficient.
So let us assume that there are some local minima, i.e. $\mathfrak{h} (p)<\infty$.
Notice that $\mathfrak{h} (p)$ is in general larger than the minimum of the height along $p$.
If $\mathfrak{h}(p)=n$ then $p$ is constant, hence efficient.
If $\mathfrak{h}(p)=n-1$, then $p$ has a backtracking subpath, because $p$ goes through at most one vertex of height $n$. By an elementary homotopy relative to endpoints we can remove this local minimum. Repeating this process, we obtain a path $p'$ with $\mathfrak{h}(p')=n$, and we reduce to the previous case. So let us assume in the following that $\mathfrak{h}(p)\leq n-2$.

We now claim that, by deforming $p$ locally at local minima, we can produce an elementary homotopy relative to endpoints to a path $p'$ such that $\mathfrak{h}(p')\geq \mathfrak{h}(p)+1$.
To prove the claim, let $v_j$ be a local minimum, and let its height be $\height{p_j}=h_j$ for some $0<j<s$.
Consider the subpath $(v_{j-1},v_j,v_{j+1})$, and note that the cells dual to $v_{j-1},v_{j+1}$ meet along the cell dual to $v_j$. Since $p$ stays in a tile, there is a cell containing all these cells, namely the tile itself.
By Lemma <ref> we get that $(v_{j-1},v_j,v_{j+1})$ is part of a square in $\dccx$, whose fourth vertex is some $v_j'$, of height $\height{v_j'}=h_j+2$.
Then we can homotope $(v_{j-1},v_j,v_{j+1})$ to the other side $(v_{j-1},v_j',v_{j+1})$ of the square via an elementary homotopy relative to endpoints (see Lemma <ref>).
This process can be applied to all local minima at the same time, since no two local minima can be adjacent along $p$. Then we remove all backtracking subpaths, if needed, keeping endpoints fixed.
The result is an elementary homotopy relative to endpoints between $p$ and an edge–path $p'$ with $\mathfrak{h}(p')\geq \mathfrak{h}(p)+1$. It is even possible that $\mathfrak{h}(p')=\infty$ but in any case this proves the claim.

We repeat this process of elevating local minima, and after a finite number of steps we obtain a path $p''$ with $\mathfrak{h}(p'')\geq n-1$ (again, possibly $\mathfrak{h}(p'')=\infty$). Hence, we reduce to the previously discussed cases to conclude that  $p''$ (hence $p$) admits an elementary homotopy relative to endpoints to an efficient path.

In the previous lemma we allow $p$ to be an edge–loop, i.e. $v_0=v_s$. All the homotopies in it are relative to the base point $v_0=v_s$. In the following statement we consider free homotopies, i.e. homotopies that are not required to fix any point.

Let $p$ be an edge–loop in $\dccx$. If $p$ stays in a tile, then there is an elementary homotopy  between $p$ and a constant path.

Pick a basepoint $v_0$ on $p$ to be a vertex of maximal height on $p$, and write $p=(v_0,\dots,v_s)$, for $v_0=v_s$.
Apply the previous argument (from Lemma <ref>) to $p$.
At every iteration we allow ourselves to change the basepoint on $p$ to always be a vertex of maximal height.
At the end there can be no local minimum, hence the path is constant.

A simple way for an edge-loop to satisfy the condition of Corollary <ref> is to be short.
Recall from  <ref> that the length $\length p$ of an edge–path  $p$ is defined to be the number of edges of $p$.

Let $p$ be an edge–loop in $\dccx$. Then $\length p$ is even. Moreover, if $\length p \leq 4$ then  $p$ stays in a tile, and there is an elementary homotopy between $p$ and a constant path.

The first statement follows from the fact that if an edge $e$ has endpoints $v,w$ then $|h(v)-h(w)|=1$, so if an edge–path has odd length then the endpoints have different height.

Suppose now $\length p\leq 4$.
If $\ell(p)=2$ then $p=(v,w,v)$ for two adjacent vertices $v,w$. In particular the cell dual to $v$ contains the one dual to $w$, or vice versa.
If $\ell(p)=4$ then $p$ is the boundary path of a square. It follows from Lemma <ref> that $p$ contains a unique point of maximal height, and that the cell dual to it contains every other cell.
In either case, there is a cell containing all the cells dual to the vertices of $p$. If $\tau$ is a tile of $\uchc$ containing that cell, then $p$ is entirely contained in $\dcc \tau$ by construction. In particular, $p$ stays in a tile, so the statement follows from Corollary <ref>.

A dual tile in $\dccx$, and a long edge–path that stays in a tile.

From now on, our main goal in this section will be to show that every edge–loop in $\dccx$ can be written as a product of nullhomotopic edge–loops, i.e. $\dccx$ is simply connected.
A naive approach would consist in splitting an edge–loop along mirrors into shorter edge–paths, until they are short enough to be contracted (in the sense of Corollary <ref>).
However, there are arbitrarily long edge–paths that stay in a tile (see Figure <ref>).
Therefore, an inductive argument based on length alone would not suffice, and this idea requires some additional tools which we develop in <ref>, before returning to the problem of simple connectedness of $\dccx$ in <ref>.

Given two cells $\sigma,\sigma'$ contained in the same tile $\tau$, let $\mu=\mu(\sigma,\sigma')$ be their upper cell (i.e. the smallest cell that contains both of them, as defined in Lemma <ref>).
If $v,v'$ and $w$ are the vertices dual to $\sigma,\sigma'$ and $\mu$ respectively, then an edge–path  of minimal length in $\dccx$ from $v$ to $v'$ can be obtained as an efficient path $p$ in $\dcc \tau$ going through $w$.
Such an efficient edge–path is not unique, but the length of any such path is given by

$$\length p = 2\height w - \height v - \height{v'} = 2 \dim \mu - \dim \sigma - \dim {\sigma '}. $$

It should be noted that if $\mu \subsetneq \tau$ then there are edge–paths from $v$ to $v'$ which are strictly longer than $p$ but still efficient.

§.§ Mirror complexity

Here we define an additional notion of complexity for an edge–path, based on the relative position in $\uchc$ between mirrors and the cells dual to the vertices of the edge–path.
We start with the following definition, in analogy to that of a dual tile. If $M$ is a mirror of $\uchc$, we define the dual mirror $\dcc M$ to be the full subcomplex of $\dccx$ whose vertices are dual to the cells of $M$.
Since we have not proved yet that $\dccx$ is simply connected, a priori it is not clear that a dual mirror enjoys properties reminiscent of those of a mirror of $\uchc$; for instance, it is not clear yet whether it is convex.
Nevertheless, we can obtain the following statement about separation (analogous to Proposition <ref>).

Let $M$ be a mirror of $\uchc$ and let $\dcc M$ be the dual mirror in $\dccx$.
Let $z_1,z_2$ be two points in $\uchc \setminus M$, let $\sigma_1,\sigma_2$ be cells in $\uchc$ such that $z_k\in \sigma_k$, and let $v_k$ be the vertex of $\dccx$ dual to $\sigma_k$.
Then $M$ separates $z_1$ and $z_2$ if and only if $\dcc M$ separates $v_1$ and $v_2$.
In particular, $\dcc M$ separates $\dccx$.

Suppose $M$ separates $z_1$ and $z_2$, and assume by contradiction that there is an edge–path $p$ in $\dccx$ from $v_1$ to $v_2$ avoiding $\dcc M$.
Then the union of the cells dual to the vertices of $p$ contains a path–connected subspace of $\uchc \setminus M$ that contains both $z_1$ and $z_2$. This is in contradiction with the fact that $M$ separates $z_1$ from $z_2$.

Vice versa, suppose $\dcc M$ separates $v_1$ and $v_2$, and assume by contradiction that there is a path $\gamma$ in $\uchc$ from $z_1$ to $z_2$ avoiding $M$.
By a small perturbation, we can assume that $\gamma$ intersects the strata of $\uchc$ in such a way that the sequence of the minimal cells that it visits gives rise to an edge–path in $\dccx$ (i.e. their dimension jumps by 1 at a time along $\gamma$). By construction, such an edge–path connects $v_1$ to $v_2$ in the complement of $\dcc M$, which is not possible.

In particular, it follows that $\dcc M$ separates $\dccx$, because $M$ separates $\uchc$ by Proposition <ref>.

This provides a correspondence between complementary components of a mirror $M$ in $\uchc$ and complementary components of the dual mirror $\dcc M$ in $\dccx$.

§.§.§ Crossings

Let $p=(v_0,\dots,v_s)$ be an edge–path (possibly an edge–loop) in $\dccx$, and let $\sigma_0,\dots, \sigma_s$ be the  cells of $\dccx$ dual to its vertices.
Let $M$ be a mirror in $\uchc$, and let $\dcc M$ be the dual mirror in $\dccx$.
Recall from Lemma <ref> that $\dccx\setminus \dcc M$ is disconnected.
We say that $p$ crosses  $M$ if $p \cap \dcc M \neq \varnothing$ and there are at least two connected components $C_1,C_2$ of $\dccx \setminus \dcc M$ such that $p \cap C_k\neq \varnothing$.
This means that among the cells $\sigma_0,\dots, \sigma_s$, some are contained in $M$, but at least two of them are such that their interiors are contained in different complementary components of $M$. (Recall that in our setting cells are closed and complementary components of mirrors are open.)
Let $q=(v_j,\dots,v_{j+m})$ be a subpath of $p$.
We say that $q$ is a ($p,M$)–crossing if $v_j,\dots,v_{j+m} \in \dcc M$, but $v_{j-1}$ and $v_{j+m+1}$  lie in different connected components of $\dccx \setminus \dcc M$.

(See Figure <ref> for some examples.)
We denote by $\mirrorcomplexitymirror{p}{M}$  the number of ($p,M$)–crossings.
The mirror complexity of $p$ is defined by taking into account the family  $\mirrors$ of all mirrors of $\uchc$, i.e. by the following formula:

\mirrorcomplexity p= \sum_{M\in \mirrors}\mirrorcomplexitymirror{p}{M}.

An edge–path $p$ in $\dccx$ crossing some mirrors. Mirror crossings are highlighted. We have $\mirrorcomplexitymirror{p}{M}=2$, $\mirrorcomplexitymirror{p}{N_1}=1$, $\mirrorcomplexitymirror{p}{N_2}=3$, and $\mirrorcomplexitymirror{p}{N_3}=3$. In particular, notice that even if $p$ intersects $\dcc{N_1}$ twice, there is only one $(p,N_1)$-crossing.

The relevance of this notion of complexity with respect to Remark <ref> is showcased by the following two lemmas.

Let $p$ be an edge–path in $\dccx$. Then $p$ stays in a tile if and only if $p$ does not cross any mirror.

Suppose that $p$ stays in a tile, i.e. there exists a tile $\tau$ such that $p \subseteq \dcc \tau$.
Assume by contradiction that $p$ crosses a mirror $M$.
So there are two vertices $v_1, v_2$ of $p$ which are separated by $\dcc M$.
Let $\sigma_k$ be the cell dual to $v_k$.
By Lemma <ref> $M$ separates the interior of $\sigma_1$ from the interior of $\sigma_2$.
In particular, there is no tile of $\uchc$ that contains both of them,  which contradicts the hypothesis that $p$ stays in a tile.

Vice versa, suppose $p$ does not cross any mirror, and assume by contradiction that there are two vertices $v_1,v_2$ on $p$ such that the dual cells are not contained in the same tile.
Let  $\tau_1,\tau_2$ be different tiles containing them.
Up to choosing $v_1,v_2$ closer to each other along $p$, we can assume that the tiles are adjacent, i.e. $\tau_1\cap\tau_2\neq \varnothing$.
In particular, $\sigma=\tau_1\cap\tau_2$ is a cell and it is contained in some mirror $M$.
Then $p$ intersects $M$ between $v_1$ and $v_2$.

Moreover the tiles $\tau_1,\tau_2$ provide a framing in the sense of <ref>.  Proposition <ref> implies that the interiors of $\tau_1,\tau_2$ are separated by $M$. The same holds for the interiors of the cells dual to $v_1,v_2$.
So by Lemma <ref> we have that $v_1,v_2$ are separated by $\dcc M$, i.e. $p$ crosses $M$, a contradiction.

Let $p$ be an edge–path in $\dccx$, and let $M$ be a mirror in $\uchc$. Then the following hold.

*  $\mirrorcomplexitymirror{p}{M}=0$ if and only if $p$ does not cross $M$.

*  $\mirrorcomplexity p=0$ if and only if $p$ stays in a tile.

*  If $p$ is a loop and  $\mirrorcomplexitymirror{p}{M}\geq 1$, then  $\mirrorcomplexitymirror{p}{M}\geq 2$.

*  If $p$ has finite length, then $\mirrorcomplexitymirror{p}{M}$ and $\mirrorcomplexity p$ are finite.

For (<ref>), note that $\mirrorcomplexitymirror{p}{M}$ is by definition the number of ($p,M$)-crossings. For (<ref>), note that $\mirrorcomplexity p$ is a sum of non-negative numbers, so it is zero if and only if  $\mirrorcomplexitymirror{p}{M}=0$ for every mirror $M$. By (<ref>) this is equivalent to saying that $p$ does not cross any mirror. Then the statement follows from Lemma <ref>.
To prove (<ref>), note that if $p$ is a loop that crosses $M$ at least once, then it must cross it at least twice, because $\dcc M$ separates $\dccx$ by Lemma <ref>.

Finally, to prove (<ref>) notice that each $(p,M)$–crossing contributes to at least one vertex of $p$, dual to a cell of $M$. Since $p$ has finite length, there can be only finitely many $(p,M)$–crossings. Then the finiteness of $\mirrorcomplexity p$ follows from the fact that $X$ (hence the collection of mirrors $\mirrors$) is locally finite.

In this framework, Corollary <ref> can be restated by saying that $\length p \leq 4$ implies $\mirrorcomplexity p=0$.

§.§.§ Bridges

Let $p=(v_0,\dots,v_s)$ be an edge–path in $\dccx$, and let $\sigma_0,\dots, \sigma_s$ be the  cells of $\uchc$ dual to its vertices.
Let $M$ be a mirror in $\uchc$, and let $\dcc M$ be the dual mirror in $\dccx$.
We say that $p$ is a bridge if there exists a mirror $M$ of $\uchc$ such that $v_0,v_s\in \dcc M$, but $p\not \subseteq \dcc M$.
In other words, $\sigma_0,\sigma_s\subseteq M$ but some of the other cells $\sigma_1,\dots,\sigma_{s-1}$ are not contained in $M$.
In this case, we say that $p$ is supported by $M$.
We say $p$ is a minimal bridge if none of its subpaths is a bridge (see Figure <ref>).

Let $p$ be an edge–path in $\dccx$. If $p$ is a bridge, then there exists a subpath $q \subseteq p$ that is a minimal bridge.

Let us consider the collection of subpaths of $p$ which  are bridges.
Notice that this collection contains $p$ itself, it is partially ordered by inclusion, and it is finite.
Therefore it contains a minimal element.

A bridge $p$ supported by a mirror $M$, with subpaths $q$ and $q'$ which are bridges supported by the mirrors $N$ and $N'$ respectively. Notice that only $q$ is a minimal bridge.

Let $p$ be a minimal bridge supported by a mirror $M$, and let $N$ be a mirror such that $\mirrorcomplexitymirror{p}{N}>0$. Then the following hold.

*  $\mirrorcomplexitymirror{p}{N} =1$.

*  $\dcc M \cap \dcc N \neq \varnothing$ and $M \cap N \neq \varnothing$.

The assumption that $\mirrorcomplexitymirror{p}{N}>0$ means that $p$ crosses $N$ at least once.
If $p$ crossed $N$ twice, then any subpath between two consecutive $(p,N)$-crossings would be a bridge supported by $N$. But this would contradict minimality, hence $\mirrorcomplexitymirror{p}{N} =1$, which proves (<ref>).
In particular the endpoints of $p$ lie in different connected components of $\dccx \setminus \dcc N$. Since they also live on the same dual mirror $\dcc M$, which is connected, and dual mirrors separate by Lemma <ref>, we can conclude that $\dcc M\cap \dcc N\neq \varnothing$. Finally, the cell dual to a vertex in their intersection is contained in $M\cap N$, hence we obtain (<ref>).

Recall that for a mirror $M$ in $\uchc$ we have a nearest point projection $\pi_M:\uchc \to M$, as discussed in <ref>.
If $p$ is a minimal bridge supported on $M$, then we can use $\pi_M$ to induce a projection of $p$ to $\dcc M$, as established by the next results.

Let $M,N$ be mirrors in $\uchc$ and let $\tau$ be a tile in $\uchc$.

* If $M\cap N\neq \varnothing$, then $\pi_M(N)=M\cap N$.

* If $M\cap \tau \neq \varnothing$, then $\pi_M(\tau)=M\cap \tau$.

We start by proving $\pi_M(N) \subseteq M\cap N$.
By contradiction, let $x\in N$ be such that $\pi_M(x)\not \in N$.
Let $y=\pi_{M\cap N}(x) \in M\cap N$, where $\pi_{M\cap N}$ denotes the nearest point projection to the closed convex subspace $M\cap N$.
Since $\pi_M(x)\not \in N$, we have $y\neq \pi_M(x)$, so we can consider the geodesic triangle with vertices $x,\pi_M(x)$ and $y$.
By convexity of $M$, the geodesic $[\pi_M(x),y]$ is contained in $M$. By convexity of $N$, the geodesic $[x,y]$ is contained in $N$. Moreover, since $\pi_M$ is the nearest point projection to $M$, the angle between $[x,\pi_M(x)]$ and $[\pi_M(x),y]$ at $\pi_M(x)$ is at least $\frac{\pi}{2}$.
Analogously, the angle between $[x,y]$ and any geodesic in $M\cap N$ at $y$ is at least $\frac{\pi}{2}$ too, and since $N$ meets $M$ orthogonally, this is enough to ensure that the angle between $[x,y]$ and $[\pi_M(x),y]$ at $y$ is at least $\frac{\pi}{2}$ too.
We obtained a geodesic triangle with two angles at least $\frac \pi 2$, which is impossible in the $\cat 0$ space $\uchc$.
Vice versa, if $x\in M\cap N$, then $x=\pi_M(x)$, so $x\in \pi_M(N)$ already.

The second statement can be obtained via an analogous argument. Indeed, recall that $\tau$ is isometric to a convex subset of $\hh^n$ bounded by orthogonal hyperplanes (see Lemma <ref>). In particular, the nearest point projection to a boundary face of $\tau$ is entirely contained in $\tau$.

The next lemma is a combinatorial statement about the stratification of $\uchc$ introduced in <ref>, and will be needed in the following lemma.

Let $\tau,\tau'$ be non-disjoint tiles of $\uchc$. Let $W_1,\dots, W_q$ be the collection of mirrors of $\uchc$ that separate $\tau$ and $\tau'$.
Then we have that

*  $W_1,\dots, W_q$ coincides with the collection of mirrors of $\uchc$ that contain $\tau \cap \tau'$.

*  $\tau \cap W_1\cap \dots \cap W_q = \tau\cap \tau' = \tau' \cap W_1\cap \dots \cap W_q.$

First of all, notice that the collection of mirrors is not empty since $\tau$ and $\tau'$ are different tiles.
We start by proving (<ref>).
Let $W$ be a mirror containing $\tau\cap \tau'$. Then the two tiles provide a framing for the cell $\tau\cap \tau'$. In particular we get from Proposition <ref> that $W$ separates the two tiles, hence $W$ is in the collection $\{W_1,\dots,W_q\}$.
Conversely, if $\tau\cap \tau'$ was not inside one $W_i$, then we could connect the two tiles with a path that goes through the intersection but avoids $W_i$, contradicting the fact that $W_i$ separates them.

To prove (<ref>) we argue as follows. By (<ref>) we know that  $\tau\cap \tau'\subseteq W_1\cap \dots\cap W_q$, so we have that $\tau\cap \tau'\subseteq \tau \cap W_1\cap \dots\cap W_q$.
Now note that, by definition of the stratification, if $\tau\cap \tau'$ is a $k$-cell, then it must be contained in $n-k$ mirrors, so $q=n-k$. But then the two sides of the inclusion are cells of the same dimension $k$, so they must be equal. Switching the roles of $\tau$ and $\tau'$ proves the second equality in (<ref>).

Let $\tau$ be a tile and $M$ be a mirror in $\uchc$, such that $ M \cap \tau\neq \varnothing$.
Let $\sigma$ be a cell of $\tau$, and let $N_1,\dots,N_r\neq M$ be all the mirrors  containing $\sigma$ and such that $M \cap N_j \neq \varnothing$ for $j=1,\dots,r$. (Possibly $r=0$ if there are no such mirrors.)
Then the following hold.

*  $\tau\cap M \cap N_1\cap \dots \cap N_r$ is an $(n-1-r)$–cell  that contains $\pi_M(\sigma)$.

*  The cell $\tau\cap M \cap N_1\cap \dots \cap N_r$ only depends on $\sigma$ and $M$.

We start by proving (<ref>).
It follows from Lemma <ref> that $\pi_M(\sigma)\subseteq \tau \cap M\cap N_j$ for each $j=1,\dots,r$.
So,  we obtain that $\pi_M(\sigma)\subseteq \tau \cap M \cap N_1\cap \dots \cap N_r$.
To show that this intersection is a cell, note that $\tau$ is an $n$–cell.
So, by Lemma <ref> we have that $M\cap \tau$ is an $(n-1)$–cell and then for each $j=1,\dots,r$ we have that $\tau\cap M \cap N_1\cap \dots \cap N_j$ is an $(n-1-j)$–cell.

To prove (<ref>) we argue as follows.
Suppose $\tau'$ is another tile as in the statement, i.e. $\sigma \subseteq \tau'$ and $M\cap \tau'\neq \varnothing$.
Let $W_1,\dots, W_q$ be the collection of mirrors of $\uchc$ that separate $\tau$ and $\tau'$. (Note that this collection depends on $\tau$ and $\tau'$, while the collection $N_1,\dots, N_r$ only depends on $\sigma$ and $M$.)

Since $\sigma\subseteq \tau \cap \tau'$, we  also have that $\sigma$ is contained in each $W_i$ thanks to (<ref>) in Lemma <ref>.
We now claim that each $W_i$ meets $M$. This is clear if $\sigma\subseteq M$. On the other hand, if $\sigma$ is not inside $M$, then we can take an efficient edge–path $p$ in $\dccx$ from the vertex dual to $\sigma$ to the vertex dual to $M\cap \tau$ which is contained in $\dcc \tau$ and meets $\dcc M$ only at the endpoint $M\cap \tau$. Take an analogous path $p'$ in $\dcc{\tau '}$, and concatenate $p$ and $p'$ to obtain a minimal bridge $\widehat p$ supported on $M$. Since $W_i$ separates $\tau$ and $\tau'$, we see that $\widehat p$ crosses $W_i$. So by (<ref>) in Lemma <ref> we conclude that $M\cap W_i\neq \varnothing$, which proves the claim.

As a result, we have that the collection $\{W_1, \dots , W_q\}$ is a subcollection of $\{M,N_1, \dots , N_r\}$. (Note that $M$ could be one of the mirrors separating $\tau$ and $\tau'$, but  $N_i\neq M$ by definition.) In particular, using (<ref>) from <ref>, we obtain that

\begin{equation*}
\tau\cap M \cap N_1\cap \dots \cap N_r \subseteq \tau  \cap W_1\cap \dots \cap W_r
\overset{\ref{lem:tiles intersection}}{=} \tau\cap \tau' \subseteq \tau'
\end{equation*}

Therefore it follows that $ \tau\cap M \cap N_1\cap \dots \cap N_r \subseteq \tau'\cap M \cap N_1\cap \dots \cap N_r $. Reversing the roles of $\tau$ and $\tau'$ provides the other inclusion, and shows that the cell defined in (<ref>) does not depend on the choice of the tile.

In the notation and setting of Lemma <ref>, if $v\in \dccx$ is the vertex dual to $\sigma$, then we denote by $\pi_M(v)$ the vertex dual to the cell constructed in (<ref>) of the lemma, and call it the projection of $v$ to $\dcc M$. This is well defined by (<ref>) in the same lemma. Note that in general $0\leq r\leq n-\dim \sigma $, as $\sigma$ could be contained in some mirrors that are disjoint from $M$.
Nevertheless, this provides the desired projection to $\dcc M$ for vertices of $\dccx$ which are contained in the cubical $2$–neighborhood of $\dcc M$, i.e. the union of all the dual tiles corresponding to all the tiles that intersect $M$ in $\uchc$.

The content of the next two lemmas is that a minimal bridge supported by a mirror $M$ is completely contained in such a neighborhood of $\dcc M$ (see Lemma <ref>), so we can define a projection of a minimal bridge to $\dcc M$ (see Lemma <ref>). We note that the minimality assumption is necessary, see the difference between $q$ and $q'$ in Figure <ref>.

Let $p$ be a minimal bridge supported on a mirror $M$.
Then for each vertex $v$ of $p$ there exists a tile $\tau$ such that $v \in \dcc \tau$ and $\tau \cap M\neq \varnothing$.

Let $p=(v_0,\dots,v_s)$, let $\sigma_k$ be the cell of $\uchc$ dual to $v_k$, and assume by contradiction that some vertices do not satisfy the statement.
Let $v_k$ be the first one.
Since $p$ is a bridge, its endpoints are on $\dcc M$, so $k\neq 0,s$.
Let $\tau_\pm$ be tiles such that $v_{k- 1}\in \dcc {\tau_-}$ and $v_{k}\in \dcc {\tau_+}$. In particular this means that $\sigma_{k-1}\subseteq \tau_-$ and $\sigma_k\subseteq \tau_+$.
By construction, we can choose $\tau_-$ so that $\tau_-\cap M\neq \varnothing$, while necessarily $\tau_+$ is disjoint from $M$.
Moreover, if we had $\sigma_k \subseteq \sigma_{k-1}$ then we would have $v_k \in \dcc{\tau_-}$, against the choice of $v_k$.
But since  $v_{k-1}$ and $v_k$ are adjacent in $\dccx$, this forces $\sigma_{k-1}\subseteq \sigma_k$.
In particular, the intersection $\tau_-\cap \tau_+$ is non empty: it contains at least the cell $\sigma_{k-1}$.

Consider the cell $\sigma=\tau_-\cap \tau_+$.
For any mirror $N$ containing $\sigma$, we claim that $N$ must  intersect $M$.
Indeed, the tiles $\tau_\pm$ form a framing for $\sigma$ in the sense of <ref>.
By  Proposition <ref> we know that $\tau_\pm$ belong to the closure of distinct complementary components of $N$.
In particular, a maximal subpath of $p \cap \dcc N$ whose vertices are dual to cells contained in $\sigma$ gives rise to a $(p,N)$-crossing, hence $\mirrorcomplexitymirror{p}{N}>0$.
By (<ref>) in Lemma <ref> we know $M\cap N \neq \varnothing$, as claimed.

Let $N_1,\dots, N_r$ be the collection of all mirrors containing $\sigma$. Since $\sigma$ is a cell of $\tau_-$, we can write $\sigma=\tau_-\cap N_1\cap \dots \cap N_r$.
Using (<ref>) in Lemma <ref>, we have that

$$\pi_M(\sigma)\subseteq \tau_- \cap M  \cap N_1  \cap \dots \cap  N_r =$$

$$=(M \cap \tau_ -) \cap ( \tau_- \cap N_1  \cap \dots \cap  N_r) \subseteq M \cap \sigma \subseteq M \cap \tau_+$$

This contradicts the fact that $\tau_+$ is disjoint from $M$.

In the next lemma we finally obtain a projection of a minimal bridge to a supporting mirror. As it might be expected, such a projection is length–decreasing (see Figure <ref>).

Let $p$ be a minimal bridge supported on a mirror $M$.
Then there exists an edge–path $p^M\subseteq \dcc M$, such that $p^M$ has the same endpoints as $p$ and $\length{p^M} \leq \length p -2$.

Let $p=(v_0,\dots,v_s)$, and let $\sigma_0,\dots,\sigma_s$ be the cells dual to its vertices.
Since $p$ is a minimal bridge supported on $M$, by Lemma <ref> we know that for each vertex $v_k$ there exists a tile $\tau_k$ of $\uchc$ such that $v_k\in \dcc{\tau_k}$ and $\tau_k\cap M\neq \varnothing$.
Let $w_k=\pi_M(v_k)$ be the projection of $v_k$ to $\dcc M$, constructed in  Lemma <ref>.
We claim that  for each $k$ the vertices $w_{k-1}$ and $w_k$ are either the same vertex or adjacent vertices.

To see this, consider two vertices $v_{k-1}$ and $v_k$ adjacent along $p$. Without loss of generality (i.e., possibly reversing the orientation of $p$) we can assume that $\sigma_{k-1}$ is a cell of codimension $1$ in $\sigma_{k}$.
In particular, we can take $\tau_{k-1}=\tau_k$, and there is exactly one mirror $\widehat N_k$ that contains $\sigma_{k-1}$ but not $\sigma_{k}$.
Let $\{N_1,\dots,N_r\}$ be the collection of all the mirrors that contain $\sigma_k$ and intersect $M$, but are different from $M$.
Then the analogous collection for $\sigma_{k-1}$ consists of the same mirrors, possibly with the addition of $\widehat N_k$.
(Note that since $p$ is a minimal bridge supported on $M$, any mirror containing $\sigma_1,\dots,\sigma_{s-1}$ is guaranteed to be different from $M$, while $\widehat N_k=M$ for $k=1$.)
By (<ref>) in Lemma <ref> we have that
$\pi_M(\sigma_k)\subseteq \tau_k\cap M \cap N_1\cap \dots \cap N_r$
and that either
$\pi_M(\sigma_{k-1})\subseteq \tau_k\cap M \cap N_1\cap \dots \cap N_r$
$\pi_M(\sigma_{k-1})\subseteq \tau_k\cap M \cap N_1\cap \dots \cap N_r \cap \widehat N_k$.
In the first case we have that $\pi_M(\sigma_{k-1})$ and $\pi_M(\sigma_{k})$ are contained in the intersection of the same mirrors, hence $w_{k-1}=w_k$; in the second case $\tau_k\cap M \cap N_1\cap \dots \cap N_r \cap \widehat N_k$ is a codimension-$1$ cell of $\tau_k\cap M \cap N_1\cap \dots \cap N_r$, hence $w_{k-1}$ is adjacent to $w_k$.
This proves the claim.

Notice in particular that in the case $k=1$ we have $\widehat N_k=M$, so we have proved that $w_0=w_1$. Analogously, we also have $w_s=w_{s-1}$.
As a result, $(w_0,\dots,w_s)$ is an edge–path in $\dcc M$.
Let $p^M$ be the edge–path obtained from $(w_0,\dots,w_s)$ by removing all backtracking subpaths and repeated vertices.
In particular, since $w_0=w_1$ and $w_s=w_{s-1}$, we have that $\length{p^M}\leq s-2 = \length p -2$.
Moreover,  since $p$ is a bridge supported on $M$, we have that $\sigma_0, \sigma_s\subseteq M$, so that $v_0=w_0,v_s=w_s$, i.e. $p$ and $p^M$ have the same endpoints.

Projection of the minimal bridge $p$ to $\dcc M$, for a supporting mirror $M$.

§.§ Surgeries on edge–loops

We are now ready to apply the above technology to the study of edge–loops in $\dccx$.
The goal is to show that $\dccx$ is simply connected.
The strategy will be to reduce the length and mirror complexity of an edge–loop enough to ensure that it stays in a tile, so that Corollary <ref> can be applied.
The following statement is the key surgery step.
Roughly speaking, we chop an edge–loop $p$ along a mirror $M$ that it crosses, use the projection $p^M$ of $p$ to $M$ to introduce a shortcut along $M$ and obtain two edge–loops $p_1,p_2$ such that $p$ and $p_1p_2$ are elementary homotopic, and finally then check that the lengths have dropped.

Let $p$ be an edge–loop in $\dccx$.
If  $\mirrorcomplexity p >0$, then there exist two edge–loops $p_1,p_2$ in $\dccx$ such that $\length{p_1},\length{p_2} < \length p$, and there is an elementary homotopy between $p$ and $p_1p_2$

By assumption, there is a mirror $M_0$ that is crossed by $p$, so  $\mirrorcomplexitymirror{p}{M_0}\geq 1$, hence by (<ref>) in Lemma <ref> we have that $\mirrorcomplexitymirror{p}{M_0}\geq 2$, i.e. $p$ crosses $M_0$ at least twice.
It follows from the definitions that any subarc of $p$ between any two $(p,M_0)$-crossings is a bridge supported by $M_0$.

Choose a $(p,M_0)$–crossing and a bridge $q$ supported by $M_0$ (in general this cannot be chosen to be minimal, see Figure <ref>). Let $q'$ be the complement of $q$ in $p$, i.e. the edge–path such that $p=qq'$. Of course  we have

\begin{equation}\label{eq:L+L' extra 1}
\length q+\length{q'}=\length p
\end{equation}

Moreover, without loss of generality we can assume that

\begin{equation}\label{eq:L<L'}
\length q \leq \length{q'}.
\end{equation}

By Lemma <ref> we can find a  minimal bridge $q_1\subseteq q\subseteq p$. In particular we have

\begin{equation} \label{eq:bridge<L}
\length{q_1}\leq \length q.
\end{equation}

Let $q_2$ be the complement of $q_1$ in $p$, i.e. the edge–path such that $p=q_1q_2$.
We can compute that

\begin{equation} \label{eq:retrace}
\length{q_2} =
\length p - \length{q_1} \overset{\eqref{eq:bridge<L}}{\geq}
\length p - \length q \overset{\eqref{eq:L+L' extra 1}}{=}
\length{q'}
\overset{\eqref{eq:L<L'}}{\geq}
\length q
\overset{\eqref{eq:bridge<L}}{\geq}
\length{q_1}.
\end{equation}

Let $M$ be a mirror supporting the minimal bridge $q_1$, and let $q_1^M$ be the projection of $q_1$ to $\dcc M$, i.e. the edge–path obtained in Lemma <ref>.
In particular we have

\begin{equation}\label{eq:shortcut}
\length{q_1^M} \leq \length{q_1} -2 < \length{q_1}.
\end{equation}

Define the edge–loops $p_1=q_1\overline{q_1^M}$ and $p_2=q_1^Mq_2$, where $\overline{q_1^M}$ denotes the edge–path $q_1^M$ with the opposite orientation.
There is an elementary homotopy between the edge–loops $p=q_1q_2$ and $p_1p_2=q_1\overline{q_1^M}q_1^M q_2$, obtained by removing the backtracking subpath $\overline{q_1^M}q_1^M$.
We can compute the desired inequality on the length of $p_1$ and $p_2$ as follows:

\begin{equation*}
\length{p_1}  =
\length{q_1} + \length{q_1^M} \overset{\eqref{eq:shortcut}}{<}
\length{q_1} + \length{q_1}
% \overset{\eqref{eq:bridge<L}+\eqref{eq:L<L'}}{\leq}
% \length{q_1} + \length{q'}
\overset{\eqref{eq:retrace}}{\leq}
\length{q_1} + \length{q_2} = \length{p}.
\end{equation*}

\begin{equation*}
\length{p_2} = \length{q_1^M} + \length{q_2} \overset{\eqref{eq:shortcut}}{<}
\length{q_1} +\length {q_2} = \length p .
\end{equation*}

The surgery in Proposition <ref>. Note that all the minimal bridges in $p$ are supported on $M$ but $p$ does not cross $M$, hence the need to first split $p$ into $q$ and $q'$, and then into $q_1$ and $q_2$.

Note that it is possible to have an edge–loop $p$ for which the surgery  from Proposition <ref> strictly reduces the mirror complexity for only one subloop. For an example see Figure <ref>, where the mirror complexity of the loop $p_2$ is the same as that of the original loop $p$.

We are now ready to prove the main result of this section.

The complex $\dccx$ is a connected $\cat 0$ cubical complex.

By construction, the complex $\dccx$ is a cubical complex. Moreover, Gromov's link condition from Lemma <ref> implies that $\dccx$ is non–positively curved, since the link of any vertex is a flag simplicial complex by  Proposition <ref>.

Next, $\dccx$ is path–connected because $\uchc$ is path–connected. Indeed, let $v,w$ be vertices in $\dccx$, and let $\sigma_v,\sigma_w$ be the dual cells. Pick any continuous path $\eta$ connecting the two cells in $\uchc$, and keep track of the list of cells that are intersected by $\eta$. By isotoping $\eta$ into lower-dimensional cells, we can ensure that the difference between the dimension of two consecutive cells in this list is exactly $1$. The dual vertices in $\dccx$ give rise to an edge–path from $v$ to $w$.

To conclude, we need to show that $\dccx$ is simply connected.
To do this, we argue that edge–loops are nullhomotopic by induction on their length.
Let $p$ be an edge–loop in $\dccx$, homotopically non-trivial and of minimal length.
If $p$ does not cross any mirror, then by Lemma <ref> $p$ stays in a tile.
Hence by Corollary <ref> there is an elementary homotopy between $p$ and a constant path.
So let us assume that $\mirrorcomplexity p > 0$.
Then by Proposition <ref> there exist two edge–loops $p_1,p_2$ in $\dccx$ such that $p$ is homotopic to $p_1p_2$ and  $\length{p_1},\length{p_2} < \length p$.
By minimality, both $p_1$ and $p_2$ are homotopically trivial, and so is $p$.

We conclude this section by noting that the action of the hyperbolized group $\hg=\pi_1(\hc)$ on $\uchc$ induces an action on the dual cubical complex $\dccx$.

The group $\hg=\pi_1(\hc)$ acts on $\dccx$ by cubical isometries. Moreover, if $X$ is compact, then the action is cocompact.

The group $\hg$ acts on $\uchc$ preserving the family of mirrors, hence the stratification defined in <ref>.
The action permutes the cells, so $\hg$ acts on vertices of the dual cubical complex $\dccx$ described in <ref>.
Moreover, the action of $\hg$ on $\uchc$ preserves the incidence relations between cells, hence we can extend the action of $\hg$ on vertices to a combinatorial action of $\hg$ on the entire  $\dccx$.
Since $\hg$ acts on $\dccx$ combinatorially, it preserves the standard cubical metric.

When $X$ is compact, $\hc$ is compact as well,  by (<ref>) in Lemma <ref>.
The action of $\hg$ on $\uchc$ has finitely many orbits of cells, so its action on $\dccx$ has finitely many orbits of vertices. Since  $\dccx$ is finite-dimensional (see Lemma <ref>), the quotient has finitely many cubes, so it is compact.

§ SPECIAL CUBULATION

The purpose of this section is to study the action of the hyperbolized group $\hg=\pi_1(\hc)$ on the dual cubical complex $\dccx$ (see Lemma <ref>), and prove that the group $\hg$ is virtually compact special in the sense of <cit.>.
When $X$ is compact and admissible,  $\hg$ is a Gromov hyperbolic group and $\dccx$ is a $\cat 0$ cubical complex (see (<ref>) in Proposition <ref>, and Theorem <ref>).
Therefore, one could hope to obtain virtual specialness directly from Agol's result from <cit.>.
However, the action of $\hg$ on $\dccx$ is not proper (see Remark <ref>).
To remedy this, we will use a result of Groves and Manning from <cit.> that deals with improper actions.
This requires a study of stabilizers of cubes.

In <ref> we show that cube stabilizers for the action of $\hg$ on $\dccx$ coincide with cell stabilizers for the action of $\hg$ on $\uchc$. Then in <ref> we show that such stabilizers are quasiconvex and virtually compact special.
The complex $X$ is always assumed to be admissible in the sense of <ref>. In some statements (such as Theorem <ref>) we also assume that it is compact.

It is worth noting that when $X$ is admissible, $\widetilde X$ is already a $\cat 0$ cube complex.
Moreover the map $\cdX:\hc\to X$ from Proposition <ref> induces a surjection $\hg \to \pi_1(X)$ that can be used to obtain an action of $\hg$ on $\widetilde X$.
However, this action has a very large kernel. For example, in the case in which $X$ is already simply connected the map
$\hg \to \pi_1(X)$ is trivial, but $\hg$ is an infinite group; indeed, it retracts to $\hgq$, as discussed in Remark <ref>.

§.§ Cube stabilizers for the action of  the hyperbolized group on the dual cubical complex

In this section we relate the cube stabilizers for the action of $\hg$ on $\dccx$ to the cell stabilizers for the action of $\hg$ on $\uchc$.

The stabilizer of a vertex in $\dccx$ coincides with the stabilizer of its dual  cell in $\uchc$.

It follows from Lemma <ref> that the action of $\hg$ on $\dccx$ is in general not proper.
Namely, the stabilizer of a vertex dual to a cell of dimension at least $2$ is infinite (compare Remark <ref> and Figure <ref>).

We now proceed to the study of stabilizers of higher–dimensional cubes for the action of $\hg$ on $\dccx$.
Recall that the dual cubical complex $\dccx$ is equipped with a $\hg$-invariant height function: the vertex dual to a $k$-cell has height $k$. We proved in Lemma <ref> that every cube in  $\dccx$ has a unique vertex of minimal height.

The stabilizer of a cube in $\dccx$ coincides with the stabilizer of its vertex of minimal height in $\uchc$.

Let $C$ be a cube in $\dccx$, let $v$ be its vertex of minimal height, and let $\sigma$ be the dual cell in $\uchc$.
Let $g\in \hg$ be an element that stabilizes $C$.
Since the height function is invariant, $g$ must fix $v$, by uniqueness of the vertex of minimal height.

Conversely let $g$ fix $v$. By Lemma <ref> we get that $g$ stabilizes $\sigma$, i.e. $g.\sigma=\sigma$.
Let $w$ be another vertex of $C$ and let $\tau$ be the dual cell.
By Lemma <ref>, we have that $\sigma \subseteq \tau$.
It follows that $\sigma =g.\sigma \subseteq g.\tau$, so that both $\tau$ and $g.\tau$ appear in the link of $\sigma$ in the combinatorial structure of $\uchc$ (see (<ref>) in Lemma <ref>).
Since the covering projection $\piuchc:\uchc \to \hc$ induces  isomorphisms on links, if $\tau$ and $g.\tau$ were distinct, then in $\hc$ we would see a tile $\pi(\tau)=\pi(g.\tau)$ isometric to a copy of $\hq$ with some identification along the boundary (namely along the subspace corresponding to $\pi(\sigma)$).
However, by (<ref>) in Lemma <ref> we know that tiles of $\hc$ are embedded copies of $\hq$, so we must have $g.\tau = \tau$.
By Lemma <ref>, this means $g.w=w$. Therefore $g$ fixes $C$ pointwise.

In the proof of Lemma <ref> we  established that the stabilizer of a cell in $\dccx$ actually fixes the cell pointwise.

§.§ Cell stabilizers are quasiconvex and virtually compact special

The goal of this section is to study the stabilizers of cells for the action of $\hg$ on $\uchc$ by covering transformations.
In particular, note that by Lemma <ref> stabilizers of tiles (i.e. top–dimensional cells) are isomorphic to the fundamental group of the hyperbolizing cube $\hgq=\pi_1(\hq)$.
More precisely, our goal is to show that cell stabilizers for the action of $\hg$ on $\uchc$ are quasiconvex in $\hg$, and virtually compact special.

§.§.§ Quasiconvexity

In the following we say that an action of a group on a metric space is geometric if it is proper, cocompact and isometric.
We will make use of the following standard fact.

Let $Z$ be a proper Gromov hyperbolic geodesic space, and let $Y\subseteq Z$ be a quasiconvex subset. Let $G$ be a finitely generated group acting geometrically on $Z$, and let $H$ be the stabilizer of $Y$ in $G$.
If $H$ acts cocompactly on $Y$, then $H$ is quasiconvex in $G$.

We apply this lemma to the cases $G=\hg, H=\hgq$ and $G=\Gamma, H=\hgq$. As noted, $\hg$ is a Gromov hyperbolic group when $X$ is compact.
In both cases, before using the lemma we need to ensure that $H$ is a subgroup of $G$. This is not obvious, because a priori $H$ is just defined as the fundamental group of the hyperbolizing cube $\hq$.

Let $X$ be compact. Then $\hgq$ is a quasiconvex subgroup of $\hg$.

By Lemma <ref>, we know that a hyperbolized complex   retracts to each of its tiles, each of which is homeomorphic to the hyperbolizing cell. In our setting this means that $\hc$ retracts to $\hq$, so in particular the inclusion $\hq\hookrightarrow \hc$ as a tile induces an injection $\hgq \hookrightarrow \hg$.
Since $X$ is compact, the group $\hg$ acts geometrically on $\uchc$. Moreover, the subgroup $\hgq$ acts geometrically on a tile, which is a closed convex subspace by Lemma <ref>. Therefore $\hgq$ is quasiconvex by Lemma <ref>.

The group $\hgq$ is a quasiconvex subgroup of $\Gamma$.

First of all we will prove that $\hgq$ naturally injects in $\Gamma$, by showing that there exists a (normal) cover $\hcalt Y$ of $\hm=\hh^n / \Gamma$ which retracts to $\hq$ (see Figure <ref>).
This would provide the desired injection

$$\hgq=\pi_1(\hq) \hookrightarrow \pi_1(\hcalt Y) \hookrightarrow \pi_1(\hm)=\Gamma .$$

To construct this cover, consider the  cubical complex $Y$ given by the standard cubulation of $\rr^n$ with vertices on $\zz^n$.
Notice that $Y$ admits a standard folding $f:Y\to \square^n$, and that $Y$ is an admissible cubical complex.
Therefore we can consider the hyperbolized complex $\hcalt Y$.
As in the proof of Lemma <ref>,   Lemma <ref> implies that $\hcalt Y$ retracts onto any of its tiles, hence $\hgq$ injects in $\pi_1(\hcalt Y)$.

We now claim that $\hcalt Y$ is a (normal) covering space of $\hm$.
For each $i=1,\dots,n$ consider the mirror of $Y$ given by $M_i=\{y_i=0\}$, and the hyperplane of $Y$ given by $H_i=\{y_i=\frac{1}{2}\}$. Let $m_{i}$ and $h_{i}$ be the reflections in $M_i$ and $H_i$ respectively, i.e.

\[ m_i:Y\to Y, m_i(y_1,\dots,y_i,\dots,y_n)= (y_1,\dots,-y_i,\dots,y_n)\]

\[ h_i:Y\to Y, h_i(y_1,\dots,y_i,\dots,y_n)= (y_1,\dots,1-y_i,\dots,y_n)\]

For each $i=1,\dots,n$, the group $D_i=\langle m_i,h_i \rangle$ is an infinite dihedral group of cubical isometries of $Y$. The group $D=\langle m_1,h_1,\dots m_n,h_n\rangle$ is isomorphic to the direct product $D_1\times \dots \times D_n$, and admits a representation into the group  $\igq$ of Euclidean isometries of the standard cube $\square^n$, in which $m_i$ acts trivially and $h_i$ acts as the standard reflection of $\square^n$ in the $i$–th coordinate.

By Lemma <ref> we have an action of $\igq$ on $\hq$ by isometries, hence we  can induce an action of $D$ on $\hq$ by isometries such that the Charney–Davis map $\cd:\hq\to \square^n$ is equivariant.
Moreover, the standard folding $f:Y\to \square^n$  is clearly $D$–equivariant too, because it can be obtained by  reflecting in the mirrors of $Y$.

Since the two maps in the pullback square defining $\hcalt Y$ are $D$–equivariant (see Figure <ref>), we obtain an action of $D$ on $\hcalt Y$ by isometries.

Note that $t_i=h_im_i$ is the unit integer translation of $Y$ in the $i^{th}$ direction. As a result, $D$ contains a (normal) subgroup $T$ isomorphic to the group of integer translations $\zz^n$.
The action of $D$ on $\hcalt Y$ restricts to a free and properly discontinuous action of $T$ on $\hcalt Y$. A fundamental domain for this action is given by a single tile. Each tile is isometric to a hyperbolizing cube $\hq$, and the induced action identifies corresponding cells on opposite mirrors, recovering $\hm$ (see  <ref> for more details about the construction of $\hq$.)
In particular $\mu_\Gamma:\hcalt Y \to \hcalt Y/T  \cong  \hm$ realizes the desired covering map, which covers the standard universal covering map $\mu:Y=\rr^n\to (S^1)^n$ (see Figure <ref>).

Finally, let us prove that $\hgq$ is quasiconvex in $\Gamma$. We know $\Gamma$  acts geometrically on $\hh^n$, permuting the stratification induced by the coordinate mirrors and their translates. The subgroup $\hgq$ stabilizes a $\Gamma$–cell, i.e. the closure of a connected component of the complement of such collection. This is a closed convex subspace, on which $\hgq$ acts geometrically (see <ref> for details). In particular $\hgq$ is quasiconvex in $\Gamma$ by Lemma <ref>.

$\hcalt Y$ as a covering space of $\hm$ that retracts to $\hq$.

In Lemma <ref> we have constructed a normal covering space  of $\hm$ by producing an action of $T=\zz^n$ by deck transformations on the hyperbolization $\hcalt Y$ of the standard integral cubulation of $\rr^n$.
This covering space can also be defined as the covering space of $\hm$ corresponding to the kernel of the homomorphism $\Gamma=\pi_1(\hm)\to \zz^n$ induced by the  collapse map $\cd_0:\hm\to (S^1)^n$ obtained by applying the Pontryagin–Thom construction to $\hm$ with respect to suitable codimension–$1$ submanifolds (see <ref> for details, and Figure <ref>).
Compact quotients of $\hcalt Y$, provide examples of closed hyperbolized manifolds which are finite covers of $\hm$. These are all genuine arithmetic hyperbolic manifolds.

§.§.§ Virtual specialness

A cubical complex is special if it admits a local isometry into the Salvetti complex of a right-angled Artin group (see <cit.>).
A group $G$ is virtually compact special if there exist a finite index subgroup $G'\subseteq G$ and a compact special cubical complex $B$ such that $G'=\pi_1(B)$.
This property passes from a Gromov hyperbolic group to its quasiconvex subgroups, as established in the following statement. This kind of arguments has appeared in the literature (see for instance <cit.>). We include a proof for the reader's convenience.

Let $G$ be a Gromov hyperbolic group, and let $H$ be a quasiconvex subgroup. If $G$ is virtually compact special, then so is $H$.

Let $G'$ be a finite index subgroup of $G$ and $B$ a compact special cubical complex such that $G'=\pi_1(B)$.
By <cit.> we can assume without loss of generality that $B$ is also non–positively curved.

The universal cover $\widetilde B$  is a  $\cat 0$ cubical complex. It is finite dimensional, uniformly locally finite, and Gromov hyperbolic, because $G'$ acts geometrically on it by covering transformations.

Let $H'=H\cap G'$. This is a finite–index subgroup of $H$, and a quasiconvex subgroup of $G'$.
Since $G'$ is Gromov hyperbolic and acts geometrically on  $\widetilde B$, it follows that $H'$–orbits are quasiconvex.

By <cit.> or <cit.>, there exists a cocompact convex core for $H'$, i.e. a convex subcomplex $\widetilde Y\subseteq \widetilde B$ on which $H'$ acts cocompactly.
Moreover, $H'$ acts by deck transformations, and the quotient $Y=\widetilde Y / H'$ is a compact non–positively curved cubical complex with $\pi_1(Y)=H'$.
The convex embedding $\widetilde Y \hookrightarrow \widetilde B$ descends to a local isometry $Y\to  B$. Since $ B$ is special, by <cit.> we obtain that $Y$ is special too.
Therefore $H$ is   virtually compact special, as desired.

In the previous proof we have the Gromov hyperbolic group $H'$ acting geometrically on the $\cat 0$ cubical complex $\widetilde Y$, so the fact that $H'$ is virtually compact special also follows from the celebrated theorem of Agol in <cit.>. However here everything happens inside the special group $G'$, so one does not need Agol's result.

We now apply the previous lemma to the cell stabilizers for the action  of $\hq$ on $\uchc$, starting with the stabilizer of a tile.

The group $\hgq$ is virtually compact special.

$\hgq$ is a quasiconvex subgroup of $\Gamma$ by Lemma <ref> and $\Gamma$ is virtually compact special by <cit.>. Indeed, it is an arithmetic lattice in $\so$ by construction (see <ref> or  <cit.> for details).
The statement then follows from  Lemma <ref>.

Finally we prove the same result for all cell stabilizers.

Let $X$ be compact. Then the cell stabilizers for the action of $\hg$ on $\uchc$ are quasiconvex and virtually compact special.

Let $\sigma$ be a cell in $\uchc$ and let $H$ be the stabilizer of $\sigma$ for the action of $\hg$ on $\uchc$.
Since $\sigma$ is a convex subset of $\uchc$ and $H$ acts geometrically on it, we conclude by Lemma <ref> that $H$ is quasiconvex in $\hg$.

Arguing as in the proof of Lemma <ref>, if $\tau$ is a tile containing $\sigma$, and $K$ is its stabilizer, then $H \subseteq K$.
Note that the folding map $\hc\to \hq$ provides an isomorphism of $K \cong \hgq$, under which $H$ is isomorphic to a quasiconvex subgroup of $\hgq$ (again by Lemma <ref>).
We know that $\hgq$ is Gromov hyperbolic (by Lemma <ref> or Lemma <ref>) and virtually compact special (by Lemma <ref>).
So it follows from Lemma <ref> that $H$ is virtually compact special too.

§.§ Specialization

We are now ready to prove that the fundamental group $\hg$ of the hyperbolized complex $\hc$ is virtually compact special, when the original cubical complex $X$ is admissible and compact.
If the action of $\hg$ on $\dccx$ was proper, this would follow from Theorem <ref>, Lemma <ref> and Agol's main result from <cit.>.
However, as observed in Remark <ref>, the action on $\dccx$ is not proper.
We will use a result by Groves and Manning (see Theorem D in <cit.>), which is  designed to deal with this situation. We report here their statement for the reader's convenience.

Suppose that $G$ is a Gromov hyperbolic group acting cocompactly on a $\cat 0$ cubical complex so that cell stabilizers are quasiconvex and virtually compact special. Then $G$ is virtually compact special.

Note that when authors of <cit.> say “virtually special” they imply that the quotient is compact (see page 3 in <cit.>). Also notice that they explicitly do not assume their complexes to be locally compact (see page 2).

If $X$ is a compact admissible cubical complex and $\Gamma$ is a hyperbolizing lattice, then $\hg$ is virtually compact special Gromov hyperbolic group.

First of all, since $X$ is admissible, by Theorem <ref> the dual cubical complex $\dccx$ is a $\cat 0$ cubical complex.
Moreover, since $X$ is compact, $\hg$ is a Gromov hyperbolic group by (<ref>) in Proposition <ref>.
By Lemma <ref> $\hg$ acts on $\dccx$ cocompactly by isometries.

Let $C$ be a cube in $\dccx$ and let $H$ be its stabilizer.
By Lemma <ref> $H$ coincides with the stabilizer of the vertex of minimal height in $C$.
By Lemma <ref> this in turn coincides with the stabilizer of the corresponding dual cell in $\uchc$.
Therefore by Lemma <ref> $H$ is a quasiconvex subgroup of $\hg$ and it is also virtually compact special.
Finally, by <cit.> the group $\hg$ is virtually compact special.

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titleAspherical manifolds that cannot be triangulated
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titleProper affine actions for right-angled Coxeter groups
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titleRight-angled Artin groups are commensurable with
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titleHyperbolization of polyhedra
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titleRelative hyperbolization and aspherical bordisms: an addendum
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titleThe lower central series of the free partially commutative
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Mathematics - Group Theory, Mathematics - Geometric Topology
titleCubulated hyperbolic groups admit Anosov representations
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titleOn the differential form spectrum of negatively curved
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Walter de Gruyter & Co., Berlin

noteWith an editorial by Heinz Bauer
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titleElementary geometry in hyperbolic space

titleFiniteness of maximal geodesic submanifolds in hyperbolic
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Mathematics - Group Theory, 20F65
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titleFinite automorphisms of negatively curved Poincaré
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titleThe topology of four-dimensional manifolds
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titleCubulation of Gromov-Thurston manifolds
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titleHyperbolic groups acting improperly
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titleSpecial IMM groups
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family=Fox Manning,

Mathematics - Group Theory, Mathematics - Differential Geometry,
Mathematics - Geometric Topology, 20F67, 53C23, 20E26, 57Q15
titleRelative cubulation of relative strict hyperbolization
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Springer, New York

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titleHyperbolic groups

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titleFinite index subgroups of graph products
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titleFiniteness conditions and ${\rm PD}_r$-group covers of
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titleSimplicité de groupes d'automorphismes d'espaces à
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titleSpecial cube complexes
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titleA combination theorem for special cube complexes
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titleVirtual specialness of certain graphs of special cube
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titleHyperbolic 5-manifolds that fiber over $S^1$
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titleVanishing theorems for $L^2$-cohomology groups
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titleResidually finite rationally solvable groups and virtual
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titleAnti-trees and right-angled Artin subgroups of braid
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titleRight-angled Artin groups and a generalized isomorphism
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Geom. Topol. Publ., Coventry

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titleNoncoherence of some lattices in ${\rm Isom}(\mathbb H^n)$

Mathematics - Geometric Topology, Mathematics - Group Theory
titleVirtual algebraic fibrations of surface-by-surface groups
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titleThe relation between the Baum-Connes conjecture and the
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title$K$-théorie bivariante pour les algèbres de Banach
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titlePin(2)-equivariant Seiberg-Witten Floer homology and
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titleExamples of Poincaré duality groups
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Princeton University Press, Princeton, N. J.

seriesAnnals of Mathematics Studies, No. 76
titleCharacteristic classes

titleA compact Kähler surface of negative curvature not
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titleRiemannian hyperbolization
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titleClosures of totally geodesic immersions in manifolds of
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Princeton University Press, Princeton, NJ

isbn[9780691170442]; [9780691170459]; [9780691213507]
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titleThe structure of groups with a quasiconvex hierarchy

titleFoldable cubical complexes of nonpositive curvature
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