# Asphericity of cubical presentations: the 2-dimensional case

Macarena Arenas  DPMMS, Centre for Mathematical Sciences, Wilberforce Road,
Cambridge, CB3 0WB, UK<EMAIL_ADDRESS>
###### Abstract.

We show that under suitable hypotheses, the second homotopy group of the
coned-off space associated to a $C(9)$ cubical presentation is trivial, and
use this to provide classifying spaces for proper actions for the fundamental
groups of many quotients of square complexes admitting such cubical
presentations. When the cubical presentations satisfy a condition analogous to
requiring that the relators in a group presentation are not proper powers, we
conclude that the corresponding coned-off space is aspherical.

###### Key words and phrases:

Small Cancellation, Cube Complexes, Asphericity

###### 2010 Mathematics Subject Classification:

20F06, 20F67

The author was supported by a Cambridge Trust & Newnham College Scholarship,
and by the Denman Baynes Junior Research Fellowship at Clare College,
Cambridge.

##  1\. Introduction

The aim of this work is to explore the asphericity of certain cubical
presentations $X^{*}=\langle X|\\{Y_{i}\rightarrow X\\}\rangle$ where $X$ is a
compact, non-positively curved cube complex and $Y_{i}\rightarrow X$ are local
isometries of compact non-positively curved cube complexes. A finite cubical
presentation is a natural generalisation of a finite group presentation
$\mathcal{P}=\langle s_{1},\ldots\,s_{k}|r_{i},\ldots,r_{\ell}\rangle$.

It is a well-known result of Lyndon [Lyn66] that classical $C(6)$
presentations are aspherical if no relators are proper powers, and thus that
the groups admitting such presentations have cohomological dimension at most
equal to $2$. This was generalised to the setting of graphical
$C^{\prime}(\frac{1}{6})$ small-cancellation presentations by Gromov [Gro03]
and Ollivier [Oll06], and graphical $C(6)$ presentations by Gruber [Gru15],
and to the setting of small-cancellation for rotation families of groups by
Coulon [Cou11]. In what follows, we take the first steps towards a
generalisation of these results to the setting of cubical small-cancellation
theory, and show:

######  Theorem 1.1.

Let $X^{*}=\langle X|\\{Y_{i}\\}_{i\in I}\rangle$ be a minimal cubical
presentation that satisfies the $C(9)$ condition. Let
$\pi_{1}X^{*}=\pi_{1}X/\langle\langle\\{\pi_{1}Y_{i}\\}_{i\in
I}\rangle\rangle=:G$. If $dim(X)\leq 2$ and each $Y_{i}$ is homotopy
equivalent to a graph, then $X^{*}$ is a $K(G,1)$, so $G$ is torsion-free and
$gd(G)\leq 2$.

The above hypothesised minimality is the cubical analogue of requiring, in the
classical setting, that none of the relators are proper powers, and is
necessary to avoid torsion. See Definition 3.12 below.

Replacing minimality with the weaker hypothesis that $X^{*}=\langle
X|\\{Y_{i}\\}_{i\in I}\rangle$ be symmetric in the sense of Definition 3.9, we
obtain instead classifying spaces for proper actions for $\pi_{1}X^{*}$. These
spaces arise in connection to the Baum-Connes conjecture, and thus this is
part of the motivation for finding good models for $\underline{E}G$.

######  Theorem 1.2.

Let $X^{*}=\langle X|\\{Y_{i}\\}^{k}_{i=1}\rangle$ be a symmetric cubical
presentation that satisfies the $C(9)$ condition. Let
$\pi_{1}X^{*}=\pi_{1}X/\langle\langle\\{\pi_{1}Y_{i}\\}^{k}_{i=1}\rangle\rangle=:G$.
If $dim(X)\leq 2$ and each $Y_{i}$ is homotopy equivalent to a graph, then
there is a quotient $\bar{X}^{*}$ of $\widetilde{X^{*}}$ that is an
$\underline{E}G$, so $cd_{\mathbb{Q}}(G)\leq dim(\bar{X}^{*})\leq 2$. If, in
addition, $X$ has a finite regular cover where each $Y_{i}\rightarrow X$ lifts
to an embedding, then $vcd(G)\leq 2$.

Since some of the cubically presented groups satisfying the hypotheses above
have torsion, Theorem 1.2 is the best statement possible in this generality.

Theorems 1.1 and 1.2 will follow from Theorem 4.5, which applies to cubical
presentations of arbitrary dimension and is of independent interest.

Assuming minimality or symmetry, and under the hypotheses on the dimension of
$X$ and the $Y_{i}$’s stated above, Theorems 1.1 and 1.2 answer a question
posed by Wise in [Wis21, 4.5]. The general case of the theorem, where the
dimensions of $X$ and $Y_{i}$ are arbitrary, and which we use to derive near-
sharp bounds on the (virtual) cohomological dimension of $X^{*}$, will be
treated in forthcoming work [Are].

Classical and graphical small-cancellation groups are hyperbolic as soon as
they satisfy either the $C^{\prime}(\frac{1}{6})$ or the $C(7)$ conditions,
and the groups produced in [Cou11] are also hyperbolic. Thus, our main theorem
is particularly interesting in that hyperbolicity is neither assumed nor
deduced: while one can conclude hyperbolicity for a sufficiently good cubical
small cancellation presentation if one assumes hyperbolicity of $\pi_{1}X$, it
is nevertheless the case that non-hyperbolic, and even non-relatively-
hyperbolic groups can satisfy strong cubical small-cancellation conditions.
Thus, Theorems 1.1 and 1.2 apply to a large and varied family of quotients of
cubulated groups.

Despite their indisputable value as a source of examples of interesting group-
theoretic behaviour, the classical and graphical small-cancellation theories
are limited in their applicability in that they can only serve to understand
“nice” quotients of free groups; cubical small-cancellation theory transfers
some of the complexity arising from the relators in a group presentation to
the “cubical generator” in the cubical presentation – that is, the cube
complex $X$ that is being quotiented – making it possible to exploit the tools
of cubical geometry to prove more general theorems.

This approach has already proven fruitful in many instances: particularly, it
plays a key role in Agol’s celebrated proofs of the Virtual Haken and Virtual
Fibred Conjectures [Ago13, Ago08], which build on work of Wise [Wis21] and his
collaborators [BW12, HW12, HW15]; it is used by Arzhantseva and Hagen in
[AH22] to show that many groups that arise as quotients of cubulated groups
are acylindrically hyperbolic; it is used by Jankiewicz and Wise in [JW22] to
construct fundamental groups of compact nonpositively curved cube complexes
that do not virtually split, and it is used by the author in [Are23a] to
produce a version of the Rips construction that provides cocompactly cubulated
hyperbolic groups of arbitrarily large finite cohomological dimension, many of
which algebraically fibre.

###  1.1. Structure of the paper

In Section 2, as a sort of base-case for our main theorem, we reprove without
resorting to CAT(0) geometry, the well-known result that CAT(0) cube complexes
are contractible. In Section 3 we present the necessary background on cubical
small cancellation theory, diagrams in cell complexes, and asphericity. In
Section 4 we give the proof of our main technical result, Theorem 4.5, and use
it to deduce Theorems 1.1 and 1.2. Finally, in Section 5 we outline some
examples to which our theorem applies.

### Acknowledgements

I am grateful to my PhD supervisor, Henry Wilton, and to Jack Button, Mark
Hagen, and the anonymous referee for all their comments and suggestions, which
greatly improved the quality of this work.

##  2\. On the asphericity of non-positively curved cube complexes

We will assume that the reader is familiar with the basic background on cube
complexes. Particularly with the definitions of _non-positively curved cube
complex_ , _local isometry_ , _midcube_ , _hyperplane_ , and _hyperplane
carrier_. These can be consulted in [Sag95, BSV14, Are23b], for instance.

A _CAT(0) cube complex_ is a simply connected, non-positively curved cube
complex. CAT(0) cube complexes are homotopy equivalent to CAT(0) spaces in the
metric sense of Cartan–Alexandrov–Toponogov [BH99, Lea13], and are therefore
contractible, which in turn implies that non-positively curved cube complexes
are aspherical spaces. While the CAT(0) metric is sometimes useful, in
practice one can prove most results about non-positively curved cube complexes
without resorting to the CAT(0) metric. Because of this, and since non-
positively curved cube complexes correspond to cubical presentations having no
relators, and as such, satisfy the $C(9)$ condition in the sense of Definition
3.13, it is therefore worthwhile to investigate whether the asphericity of
non-positively curved cube complexes (or equivalently, the contractibility of
their universal covers) can be proved via purely cubical/combinatorial means.
This is the purpose of the present section. We stress that alternative proofs
of the contractibility of CAT(0) cube complexes, while not present in the
literature as far as the author is aware, are well-known to experts.

######  Theorem 2.1.

Non-positively curved cube complexes are aspherical.

######  Definition 2.2.

Let $X$ be a CW complex. Suppose that $c,c^{\prime}$ are two cubes of $X$ such
that $c^{\prime}\subset c$, $c$ is a maximal face of $X$, and no other maximal
face of $X$ contains $c^{\prime}$. Then $c^{\prime}$ is a _free face_ of $X$.
A _collapse_ is the removal of all open cells $c^{\prime\prime}$ such that
$c^{\prime}\subseteq c\subseteq c^{\prime\prime}$ , where $c^{\prime}$ is a
free face. A cell complex is _collapsible_ if there is a sequence of collapses
leading to a point.

All collapsible cell-complexes are contractible [Whi39], but the converse is
not necessarily true, even in the setting of cube complexes: Bing’s _house
with two rooms_ [Hat02, pg. 4] is an example of a contractible cube complex
that isn’t collapsible (and not non-positively curved).

In what follows we will show that finite CAT(0) cube complexes are
collapsible. From this, we will be able to deduce that all CAT(0) cube
complexes are contractible via a nested subcomplex argument. Note that
collapsibility of finite CAT(0) cube complexes follows from more general work
of Adiprasito and Benedetti [AB20]. However, their proof heavily relies on
CAT(0) geometry.

Lurking behind our proof is a form of convexity associated to the $\ell^{1}$
metric on the $1$-skeleton of a CAT(0) cube complex $X$. We define it below in
a manner that elucidates its combinatorial nature.

######  Definition 2.3 (Combinatorial convexity).

A connected subcomplex $Y$ of a CAT(0) cube complex $X$ is _convex_ if, for
each $n$-cube $c$ with $n\geq 2$ in $X$, whenever a corner of $c$ lies in $Y$,
then the cube $c$ lies in $Y$. The _convex hull_ $Hull(Z)$ of a subcomplex
$Z\subset X$ is the smallest convex subcomplex of $X$ containing $Z$.

A consequence of Definition 2.3 and Greendlinger’s Lemma [Gre60] – a general
form of which is stated as Theorem 3.26 below – is the following observation.

######  Remark 2.4.

A convex subcomplex $Y$ of a CAT(0) cube complex $X$ is itself a CAT(0) cube
complex.

The result below can be readily derived from [Hag08, 2.28].

######  Theorem 2.5.

The convex hull of a finite subcomplex of a CAT(0) cube complex is finite.

We also make use of the following proposition, which can be found in [Sag95].

######  Proposition 2.6.

A hyperplane in a CAT(0) cube complex is a CAT(0) cube complex, and separates
it into exactly two connected components.

######  Remark 2.7.

As we are aiming to reprove a basic fact about CAT(0) cube complexes, some
care must be taken to ensure that our arguments are not circular. That is,
that none of the result we utilise hinge on the contractibility of CAT(0) cube
complexes to begin with. In the case of Theorem 2.5 and Proposition 2.6 above,
the proofs only utilise combinatorial convexity, disc diagrams, and the fact
that CAT(0) cube complexes are simply connected.

###### Proof of Theorem 2.1 via collapsibility of finite CAT(0) cube
complexes.

Finite CAT(0) cube complexes: We first show, by induction on dimension, that a
finite CAT(0) cube complex is either a 0-dimensional cube or has a free face.
A connected, 0-dimensional cube complex is necessarily a single vertex; a
finite, 1-dimensional CAT(0) cube complex is a finite tree, and therefore has
a free face (a leaf). Assume that the result holds for all CAT(0) cube
complexes of dimension $<k$. Let $Y$ be a $k$-dimensional CAT(0) cube complex,
and let $H$ be a hyperplane in $Y$. Then $H$ has dimension less than $k$ and
is also a CAT(0) cube complex, so $H$ has a free face $f_{H}$. Note that
$f_{H}$ lies in a face $f_{Y}$ of $Y$, and $f_{Y}$ cannot be glued to any
other cubes of $Y$, as this would imply that $f_{H}$ extends to a midcube in
any such cube. Hence, $f_{Y}$ is a free face of $Y$.

Now we show, by induction on dimension, that for any CAT(0) cube complex $Y$
and for any free face $f$ of $Y$, there is a sequence of collapses starting
with $f$ that results in a subcomplex $Y^{\prime}$ that is again a CAT(0) cube
complex. Note that this claim implies that $Y$ is collapsible, since
$Y^{\prime}$ must have a free face and then the process can be repeated, and
terminates in finitely many steps because $Y$ was finite to begin with. The
base case is $dim(Y)=1$, which is immediate as $Y$ is then a tree, so every
connected subcomplex of $Y$ is a tree and thus CAT(0). Now assume that the
claim holds for all CAT(0) cube complexes of dimension $<k$ and let
$dim(Y)=k$.

Let $f$ be a free face of $Y$ and $F$ be the maximal face containing $f$, let
$H$ be a hyperplane of $Y$ having a free face that is a midcube $m$ of $f$ and
note that $m$ is a free face of $H$. Since $dim(H)<dim(Y)$ and $H$ is a CAT(0)
cube complex, the induction hypothesis implies that there is a sequence of
collapses starting with $m$ that results in a CAT(0) cube complex
$H^{\prime}$, moreover, after repeating this process finitely many times, we
may assume that the sequence of collapses terminates at a point $p$ that is a
vertex of $H$. We claim that this sequence of collapses extends to a sequence
of collapses for the carrier $N(H)$ of $H$ in $Y$ which terminates with an
edge $e$ of $Y$.

Indeed, each collapse of $H$ results in a subcomplex that has a free face
$m^{\prime}$, so the corresponding collapse in $Y$ results in a subcomplex
that has a free face (the face having $m^{\prime}$ as a midcube). Thus, the
sequence of collapses terminating on $p$ extends to a sequence of collapses
terminating in a subcomplex of $Y$ in which $N(H)$ collapses to $N(p)$, which
is an edge $e$. Now, since $Y$ is CAT(0), then the interior $N(H)^{o}$ of
$N(H)$ separates $Y$ by Proposition 2.6, and each connected component
$Z_{1},Z_{2}$ of $Y-N(H)^{o}$ is CAT(0): each $Z_{i}$ is non-positively curved
since for each vertex $v$ of $Y-N(H)^{o}$, the link of $v$ in $Y-N(H)^{o}$ is
the restriction $link(v)|_{Y-N(H)^{o}}$, and is either identical to $link(v)$,
or is obtained from $link(v)$ by removing the flag subcomplex corresponding to
$link(v)|_{N(H)^{o}}$ from it; each $Z_{i}$ is simply connected because
$Y=Z_{1}\cup N(H)\cup Z_{2}$ is simply connected, and is homotopy equivalent
to the wedge $Z_{1}\vee Z_{2}$.

Finally, note that $Y^{\prime}$ is obtained from $Y-N(H)^{o}$ by reattaching
$e$ to it along $(Y-N(H)^{o})\cap e$, so it now suffices to check that
$link(v)$ is flag for each vertex $v$ of $(Y-N(H)^{o})\cap e$ in $Y^{\prime}$,
that is, for the endpoints of $e$. But the links of these vertices are flag in
$Y-N(H)^{o}$, and attaching $e$ only adds a disconnected vertex to each of
these links, so the flag condition continues to hold, and $Y^{\prime}$ is
simply connected because both connected components of $Y-N(H)^{o}$ are, thus
$Y^{\prime}$ is homotopy equivalent to the wedge of $2$ simply connected cube
complexes.

Infinite CAT(0) cube complexes: As $S^{n}$ is compact, any map
$S^{n}\longrightarrow Y$ for $n\geq 1$ has compact image $Im_{Y}(S^{n})$, so
$Im_{Y}(S^{n})$ is contained in a finite subcomplex $Z\subset Y$. Thus,
$Im_{Y}(S^{n})\subset Hull(Z)$, which is finite by Theorem 2.5 and Remark 2.4,
and contractible by the argument above. Hence, $S^{n}\longrightarrow Y$ is
nullhomotopic, and $\pi_{n}(Y)=0$ for all $n\geq 1$. ∎

##  3\. Background

###  3.1. Cubical small-cancellation theory

Unless otherwise noted, all definitions and results concerning cubical small-
cancellation theory recounted in Subsections 3.1 and 3.2 originate in [Wis21].

We begin this section by introducing our main objects of study:

######  Definition 3.1 (Cubical presentation).

A cubical presentation $\langle X|\\{Y_{i}\\}\rangle$ consists of a connected
non-positively curved cube complex $X$ together with a collection of local
isometries of connected non-positively curved cube complexes
$Y_{i}\overset{\varphi_{i}}{\longrightarrow}X$. Local isometries of non-
positively curved cube complexes are $\pi_{1}$-injective, so it makes sense to
define the _fundamental group of a cubical presentation_ as
$\pi_{1}X/\langle\\!\langle\\{\pi_{1}Y_{i}\\}\rangle\\!\rangle$. By the
Seifert-Van Kampen Theorem, this group is isomorphic to the fundamental group
of the space $X^{*}$ obtained by coning off each $Y_{i}$ in $X$. Hereinafter,
all auxiliary definitions and results about cubical presentations are in
practice statements about their associated coned-off spaces.

######  Remark 3.2.

A group presentation $\langle a_{1},\ldots,a_{s}|r_{1},\ldots,r_{m}\rangle$
can be interpreted cubically by letting $X$ be a bouquet of s circles and
letting each $Y_{i}$ map to the path determined by $r_{i}$. On the other
extreme, for every non-positively curved cube complex $X$ there is a “free”
cubical presentation $X^{*}=\langle X|\ \rangle$ with fundamental group
$\pi_{1}X=\pi_{1}X^{*}$.

######  Definition 3.3 (Elevations).

Let $Y\rightarrow X$ be a map where $Y$ is connected, and let
$\hat{X}\rightarrow X$ be a covering map. An _elevation_
$\hat{Y}\rightarrow\hat{X}$ is a map satisfying

  1. (1)

$\hat{Y}$ is connected,

  2. (2)

$\hat{Y}\rightarrow Y$ is a covering map, the composition $\hat{Y}\rightarrow
Y\rightarrow X$ equals $\hat{Y}\rightarrow\hat{X}\rightarrow X$, and

  3. (3)

assuming all maps involved are basepoint preserving, $\pi_{1}\hat{Y}$ equals
the preimage of $\pi_{1}\hat{X}$ in $\pi_{1}Y$.

######  Convention 3.4.

Later in this text we will be dealing exclusively with two kinds of
elevations: elevations of a map $Y\rightarrow X$ to the universal cover
$\widetilde{X}\rightarrow X$ and elevations of $Y\rightarrow X$ to a covering
space $\hat{X}\rightarrow X$ that arises as a subspace of $\widetilde{X^{*}}$.

  1. (1)

Elevations of the first type will be denoted $\widetilde{Y}\rightarrow X$,
since $\widetilde{Y}$ is indeed a copy of the universal cover of $Y$ in
$\widetilde{X}$. At times, we will distinguish various elevations using the
action of $\pi_{1}X$ on $\widetilde{X}$. That is, we choose a base elevation
$\widetilde{Y}$ and tag a translate $g\widetilde{Y}$ by the corresponding
element $g\in\pi_{1}X$.

  2. (2)

Elevations of the second type will be denoted $Y\rightarrow X$. This choice of
notation might seem confusing to the reader now, but it will follow from
Theorem 3.27 that under appropriate small-cancellation conditions –also
introduced below, in Definition 3.13– elevations $Y\rightarrow X$ to
$\hat{X}\rightarrow X$ are embeddings. As in the case of elevations to
$\widetilde{X}$, we might distinguish various elevations to $\hat{X}$ using
the action of $\pi_{1}X^{*}$ on $\widetilde{X^{*}}$.

######  Definition 3.5 (Pieces).

Let $\langle X|\\{Y_{i}\\}\rangle$ be a cubical presentation. An _abstract
contiguous cone-piece_ $P$ of $Y_{j}$ in $Y_{i}$ is an intersection
$\widetilde{Y}_{j}\cap\widetilde{Y}_{i}$ where
$\widetilde{Y}_{j},\widetilde{Y}_{i}$ are fixed elevations to $\widetilde{X}$
and either $i\neq j$ or where $i=j$ but
$\widetilde{Y}_{j}\neq\widetilde{Y}_{i}$. Each abstract contiguous cone-piece
$P$ induces a map $P\rightarrow Y_{i}$ which is the composition
$P\hookrightarrow\widetilde{Y}_{i}\rightarrow Y_{i}$, and a _cone-piece_ of
$Y_{j}$ in $Y_{i}$ is a combinatorial path $p\rightarrow P$ in an abstract
contiguous cone-piece of $Y_{j}$ in $Y_{i}$.

An _abstract contiguous wall-piece_ $P$ of $Y_{i}$ is an intersection
$N(H)\cap\widetilde{Y}_{i}$ where $\widetilde{Y}_{i}$ is a fixed elevation and
$N(H)$ is the carrier of a hyperplane $H$ that is disjoint from
$\widetilde{Y}_{i}$. To avoid having to deal with empty pieces, we shall
assume that $H$ is dual to an edge with an endpoint on $\widetilde{Y}_{i}$. As
in the case of cone-pieces, each abstract contiguous wall-piece $P$ induces a
map $P\rightarrow Y_{i}$, and a _wall-piece_ of $Y_{i}$ is a combinatorial
path $p\rightarrow P$ in an abstract contiguous wall-piece of $Y_{i}$.

A _piece_ is either a cone-piece or a wall-piece.

######  Remark 3.6.

In Definition 3.5, two elevations of a cone $Y$ are considered identical if
they differ by an element of $Stab_{\pi_{1}X}(\widetilde{Y})$. This is in
keeping with the conventions of classical small cancellation theory, where
overlaps between a relator and any of its cyclic permutations are not regarded
as pieces.

######  Definition 3.7.

Let $Y\rightarrow X$ be a local isometry. $Aut_{X}(Y)$ is the group of
combinatorial automorphisms $\psi:Y\rightarrow Y$ such that the diagram below
is commutative:

${Y}$${Y}$${X}$$\scriptstyle{\psi}$

If $Y$ is simply connected, then $Aut_{X}(Y)$ is equal to
$Stab_{\pi_{1}X}(Y)$. In general,
$Aut_{X}(Y)\cong(N_{Aut_{X}(\widetilde{Y})}\pi_{1}Y)/\pi_{1}Y$, where
$N_{G}(H)$ is the normaliser of $H$ in $G$.

See [AH22] for a detailed discussion.

######  Convention 3.8.

Let $X^{*}=\langle X|\\{Y_{i}\\}_{I}\rangle$. For convenience we assume, as is
standard to assume in this framework [Wis21, 3.3], that $\pi_{1}Y_{i}$ is
normal in $Stab_{\pi_{1}X}(\widetilde{Y}_{i})$ for each $i\in I$. Throughout,
we will assume also that $X$ is finite dimensional and locally finite, and
that each $Y_{i}$ is connected.

For the purposes of this paper, we will need to impose further restrictions on
a cubical presentation; these are described below.

######  Definition 3.9.

A local-isometry $Y\rightarrow X$ with $Y$ connected and superconvex is
symmetric if for each component $K$ of the fibre product $Y\otimes_{{}_{X}}Y$,
either $K$ maps isomorphically to each copy of $Y$, or
$[\pi_{1}Y:\pi_{1}K]=\infty$. A cubical presentation $\langle
X|\\{Y_{i}\\}\rangle$ is _symmetric_ if each $Y_{i}\rightarrow X$ is
symmetric.

Superconvexity is defined in [Wis21, 2.35], and also in [Are23a, 2.4]. We
refrain from stating the definition here, since we will only use it implicitly
when applying Lemma 3.11 in Section 4.

######  Definition 3.10.

(Commensurator) The _commensurator_ $C_{G}(H)$ of a subgroup $H$ of $G$ is the
subgroup $C_{G}(H)=\\{g\in G:[H:H^{g}\cap H]<\infty\text{ and
}[H^{g}:H^{g}\cap H]<\infty\\}$.

In [Wis21, 8.12], the following is observed:

######  Lemma 3.11.

Let $X^{*}=\langle X|\\{Y_{i}\\}\rangle$ be a cubical presentation. Then
$X^{*}$ is symmetric if and only if
$C_{\pi_{1}X}(\pi_{1}Y_{i})=Stab_{\pi_{1}X}(\widetilde{Y}_{i})$,
$[Stab_{\pi_{1}X}(\widetilde{Y}_{i}):\pi_{1}Y_{i}]<\infty$, and
$\pi_{1}Y_{i}\vartriangleleft Stab_{\pi_{1}X}(\widetilde{Y}_{i})$ for each
$Y_{i}\rightarrow X$.

To the best of our knowledge, the following notion was introduced in [AH22],
where it is used in the course of proving acylindrical hyperbolicity for
certain cubical presentations satisfying strong cubical small-cancellation
conditions.

######  Definition 3.12.

A cubical presentation $\langle X|\\{Y_{i}\\}_{I}\rangle$ is _minimal_ if the
following holds for each $i\in I$: let $\widetilde{Y}_{i}\rightarrow Y_{i}$ be
the universal cover, and let $\widetilde{Y_{i}}\rightarrow Y_{i}\rightarrow X$
be an elevation of $Y_{i}\rightarrow X$. Fix a basepoint $x_{0}$ in $X$. Then
$Stab_{\pi_{1}(X,x_{0})}(\widetilde{Y_{i}})=\pi_{1}(Y_{i},y_{i_{0}})$.

Minimality generalises prohibiting relators that are proper powers in the
classical small-cancellation case. In our setting, minimality is used to avoid
the “obvious” torsion that could be created in a quotient
$\pi_{1}X/\langle\langle\\{\pi_{1}Y^{\prime}_{i}\\}\rangle\rangle$ if the
$Y^{\prime}_{i}\rightarrow X$ are themselves non-trivial finite covers
$Y^{\prime}_{i}\rightarrow Y_{i}\rightarrow X$. For instance, if $Y$ is a
finite degree covering of $X$, the cubical presentation $X^{*}=\langle
X|Y\rangle$ will have finite fundamental group, even if it satisfies the
$C(9)$ condition defined below (or any other small cancellation condition).

######  Definition 3.13.

A cubical presentation $X^{*}$ satisfies the $C(n)$ small-cancellation
condition if no essential closed path $\sigma\rightarrow Y_{i}$ is the
concatenation of fewer than $n$ pieces.

An analogue of the $C^{\prime}(\frac{1}{n})$ condition can also be defined in
this setting, but we won’t need it in the present work. We remark only that,
as in the classical case, the cubical $C^{\prime}(\frac{1}{n})$ condition
implies the cubical $C(n+1)$ condition, but the implication is very much not
reversible: regardless of the choices of $n$ and $n^{\prime}$, the cubical
$C(n)$ condition does not necessarily imply the cubical
$C^{\prime}(\frac{1}{n^{\prime}})$ condition.

###  3.2. Diagrams in cube complexes and cubical presentations

In this section we introduce disc and spherical diagrams, and the general
related jargon – the analysis of the possible diagrammatic behaviours in
$C(9)$ cubical presentations, and the use of the tools already available in
this context, will be the main ingredients utilised in Section 4.

######  Definition 3.14.

A map $f:X\longrightarrow Y$ between 2-complexes is _combinatorial_ if it maps
cells to cells of the same dimension. A complex is _combinatorial_ if all
attaching maps are combinatorial, possibly after subdividing the cells.

######  Definition 3.15 (Disc diagram).

A _disc diagram_ $D$ is a compact contractible combinatorial 2-complex,
together with an embedding $D\hookrightarrow S^{2}$. The _boundary path_
$\partial D$ is the attaching map of the 2-cell at infinity. A _disc diagram
in a complex X_ is a combinatorial map $D\rightarrow X$. A _square disc
diagram_ is a disc diagram that is also a cube complex (though not necessarily
non-positively curved!). Note that any disc diagram in a cube complex is a
square disc diagram.

We introduce some of the phenomena that may arise in square-disc diagrams;
unlike in the general setting of $2$-complexes, problematic behaviour of
diagrams can be described and classified precisely in the cubical setting.

######  Definition 3.16 (Square-disc behaviours).

A _dual curve_ in a square disc diagram is a path that is a concatenation of
midcubes. The 1-cells crossed by a dual curve are _dual_ to it. A _bigon_ is a
pair of dual curves that cross at their first and last midcubes. A _monogon_
is a single dual curve that crosses itself at its first and last midcubes. A
_nonogon_ is a single dual curve of length $\geq 1$ that starts and ends on
the same dual 1-cell, thus it corresponds to an immersed cycle of midcubes. A
_spur_ is a vertex of degree $1$ on $\partial D$.

A _corner_ in a diagram $D$ is a vertex $v$ that is an endpoint of consecutive
edges $a,b$ on $\partial D$ lying in a square $s$. A _cornsquare_ – short for
“generalised corner of a square”– consists of a square $c$ and dual curves
$p,q$ emanating from consecutive edges $a,b$ of $c$ that terminate on
consecutive edges $a^{\prime},b^{\prime}$ of $\partial D$. The _outerpath_ of
the cornsquare is the path $a^{\prime}b^{\prime}$ on $\partial D$. A
_cancellable pair_ in $D$ is a pair of $2$-cells $R_{1},R_{2}$ meeting along a
path $e$ such that the following diagram commutes:

${e}$${\partial R_{1}}$${\partial R_{2}}$${X}$

A cancellable pair leads to a smaller area disc diagram via the following
procedure: cut out $e\cup Int(R_{1})\cup Int(R_{2})$ and then glue together
the paths $\partial R_{1}-e$ and $\partial R_{2}-e$ to obtain a diagram
$D^{\prime}$ with $\text{\sf Area}(D^{\prime})=\text{\sf Area}(D)-2$ and
$\partial D^{\prime}=\partial D$.

By performing the procedure just described to cancellable pairs, diagrams in
non-positively curved cube complexes can often be simplified to avoid certain
pathologies.

######  Lemma 3.17.

[Wis21, 2.3+2.4] Let $D\rightarrow X$ be a disc diagram in a non-positively
curved cube complex. If $D$ contains a bigon or a nonogon, then there is a new
diagram $D^{\prime}$ having the same boundary path as $D$, so $\partial
D^{\prime}\rightarrow X$ equals $\partial D\rightarrow X$, and such that
$Area(D^{\prime})\leq Area(D)-2$. Moreover, no disc diagram in $X$ contains a
monogon, and if $D$ has minimal area among all diagrams with boundary path
$\partial D$, then $D$ cannot contain a bigon nor a nonogon.

Some additional terminology is necessary when describing diagrams in cubical
presentations:

######  Definition 3.18.

Recall that the coned-off space $X^{*}$ introduced in Definition 3.1 consists
of $X$ with a cone on $Y_{i}$ attached to $X$ for each $i$. The vertices of
the cones on $Y_{i}$’s are the _cone-vertices_ of $X^{*}$. The cellular
structure of $X^{*}$ consists of all the original cubes of $X$, and the
pyramids over cubes in $Y_{i}$ with a cone-vertex for the apex. Let
$D\rightarrow X^{*}$ be a disc diagram in a cubical presentation. The vertices
in $D$ which are mapped to the cone-vertices of $X^{*}$ are the _cone-
vertices_ of $D$. Triangles in $D$ are naturally grouped into cyclic families
meeting around a cone-vertex. Each such family forms a subspace of $D$ that is
a cone on its bounding cycle. A _cone-cell_ of $D$ is a cone that arises in
this way. To simplify the theory, when analysing diagrams in a cubical
presentation we “forget” the subdivided cell-structure of a cone-cell $C$ and
regard it simply as a single $2$-cell.

A situation that may occur with diagrams in coned-off spaces is that two cone-
cells might come from the same coned-off relation $Y$. When this happens, it
is often possible to fuse these two cone-cells together into a single cone-
cell, as explained below. This simplification does not arise in the “purely
cubical” setting, but will be useful for analysing diagrams in cubical
presentations.

######  Definition 3.19.

A pair of cone cells $C,C^{\prime}$ in $D$ is _combinable_ if they map to the
same cone $Y$ of $X^{*}$ and $\partial C$ and $\partial C^{\prime}$ both pass
through a vertex $v$ of $D$, and map to closed paths at the same point of $Y$
when regarding $v$ as their basepoint.

As the name suggests, such a pair can be combined to simplify the diagram by
replacing the pair with a single cone-cell mapping to $Y$ and whose boundary
is the concatenation $\partial C\partial C^{\prime}$.

######  Definition 3.20.

A _disc diagram_ $D\rightarrow X^{*}$ is _reduced_ if the following conditions
hold:

  1. (1)

There is no bigon in a square subdiagram of $D$.

  2. (2)

There is no cornsquare whose outerpath lies on a cone-cell of $D$.

  3. (3)

There does not exist a cancellable pair of squares.

  4. (4)

There is no square $s$ in $D$ with an edge on a cone-cell $C$ mapping to the
cone $Y$, such that $(C\cup s)\rightarrow X$ factors as $(C\cup s)\rightarrow
Y\rightarrow X$.

  5. (5)

For each internal cone-cell $C$ of $D$ mapping to a cone $Y$, the path
$\partial C$ is essential in $Y$.

  6. (6)

There does not exist a pair of combinable cone-cells in $D$.

A closely-related notion to that of disc diagram reducibility is that of
complexity:

######  Definition 3.21.

The _complexity_ $Comp(D)$ of a disc diagram $D\rightarrow X^{*}$ is the
ordered pair $(\\#\text{Cone-cells},\\#\text{Squares})$. We order the pairs
lexicographically: namely $(\\#C,\\#S)<(\\#C^{\prime},\\#S^{\prime})$ whenever
$\\#C<\\#C^{\prime}$ or $\\#C=\\#C^{\prime}$ and $\\#S<\\#S^{\prime}$. A disc
diagram $D\rightarrow X^{*}$ has _minimal complexity_ if no disc diagram
$D^{\prime}\rightarrow X^{*}$ having $\partial D=\partial D^{\prime}$ has
$Comp(D^{\prime})<Comp(D)$.

Since the objective of this work is to understand the second homotopy group of
a cubical presentation, it will come as no surprise that in addition to disc
diagrams, their spherical analogues will also play a key role in this
endeavour.

######  Definition 3.22 (Spherical diagram).

A _spherical diagram_ $\Sigma$ is a compact simply-connected combinatorial
2-complex, together with a homeomorphism $\Sigma\hookrightarrow S^{2}$. A
_spherical diagram in a complex X_ is a combinatorial map $\Sigma\rightarrow
X$. We define _reduced_ spherical diagrams and _complexity_ in the same way as
in Definitions 3.20 and 3.21, replacing all instances of “disc diagram” by
“spherical diagram”. A spherical diagram has _minimal complexity_ if no
spherical diagram $\Sigma^{\prime}\rightarrow X^{*}$ homotopic to $\Sigma$ has
$Comp(\Sigma^{\prime})<Comp(\Sigma)$.

All diagrams in this paper are either square disc or square spherical
diagrams, or are disc or spherical diagrams in the coned-off space associated
to a cubical presentation.

######  Remark 3.23.

If a disc diagram or spherical diagram has minimal complexity, then it is also
reduced, as any of the pathologies in Definition 3.20 would indicate a
possible complexity reduction. For instance, a cornsquare whose outerpath lies
on a cone-cell can be absorbed, after a sequence of hexagon moves – each of
which corresponds to pushing a hexagon on one side of a $3$-cube to obtain the
hexagon on the other side – into said cone-cell; a cancellable pair of squares
can be cancelled as described in Definition 3.16; a square bigon leads to a
cancellable pair of squares after a sequence of hexagon moves; and a pair of
combinable cone-cells in $D$ can be combined to reduce the number of cone-
cells by $1$. A detailed case-by-case analysis of all possible reductions can
be found in [Wis21, 3.e].

######  Definition 3.24 (Shell).

A _shell_ of $D$ is a 2-cell $C\rightarrow D$ whose boundary path $\partial
C\rightarrow D$ is a concatenation $QP_{1}\cdots P_{k}$ for some $k\leq 4$
where $Q$ is a boundary arc in $D$ and $P_{1},\ldots,P_{k}$ are non-trivial
pieces in the interior of $D$. The arc $Q$ is the _outerpath_ of $C$ and the
concatenation $S:=P_{1}\cdots P_{k}$ is the _innerpath_ of $C$.

######  Remark 3.25.

Note that if a cubical presentation $X^{*}$ satisfies the $C(n)$ condition and
$D$ is a minimal complexity diagram, then the outerpath of a shell in $D$ is
the concatenation of $\geq n-4$ pieces.

Figure 1. A potential disc diagram with a range of features: a spur, shells,
corners, cornsquares and a cut-vertex.

In almost all forms of small-cancellation theory, the main technical result
that facilitates proving theorems is a form of diagram classification. In the
classical setting, this is known as Greendlinger’s Lemma [Gre60], in the
cubical setting this is known as diagram trichotomy, or as the “Fundamental
Lemma”. We state it in a simplified form that is sufficient for our
applications, and which we shall call _diagram dichotomy_ :

######  Theorem 3.26 (Diagram Dichotomy).

[Jan16] Let $X^{*}=\langle X|\\{Y_{i}\\}\rangle$ be a cubical presentation
satisfying the $C(9)$ condition, and let $D\to X^{*}$ be a disc diagram. Then
one of the following holds:

  * •

$D$ consists of a single cell,

  * •

$D$ has at least two shells and/or corners and/or spurs.

######  Theorem 3.27.

[Jan16] Let $X^{*}=\langle X|\\{Y_{i}\\}\rangle$ be a cubical presentation
satisfying the $C(9)$ condition. Then each $Y_{i}$ embeds in the universal
cover $\widetilde{X^{*}}$ of the coned-off space $X^{*}$.

###  3.3. Asphericity

We will require the following fundamental fact about combinatorial paths in
$2$-complexes.

######  Theorem 3.28 (The Van-Kampen Lemma).

Let $X$ be a combinatorial $2$-complex. Let $P\rightarrow X^{1}$ be a closed
combinatorial path. Then $P$ is nullhomotopic if and only if there exists a
disc diagram $D$ in $X$ with $\partial D\cong P$ so that there is a
commutative diagram:

${\partial D}$${D}$${P}$${X}$

For our purposes, the spherical version of Van-Kampen’s Lemma will also be
extremely useful. That is, we need to be able to assert that all elements of
$\pi_{2}$ have diagram representatives:

######  Theorem 3.29 (The Spherical Van-Kampen Lemma).

Let $X$ be a combinatorial $2$-complex. Then every homotopy class of maps
$S^{2}\rightarrow X$ is represented by a spherical diagram $\Sigma\rightarrow
X$.

Theorem 3.29 is presented (with or without a proof) in slightly different ways
in various sources. The version we use can be extracted from [Fen83, Section
2] via the viewpoint of “pictures”, which are planar graph representations of
homotopy elements and are dual to spherical diagrams.

##  4\. Main theorem

We commence this section with a definition.

######  Definition 4.1 (Classifying space for proper actions).

Let $G$ be a group. A _classifying space for proper actions_ for $G$ is a
$G$-CW-complex $\underline{E}G$ satisfying:

  1. (1)

every cell stabiliser is finite,

  2. (2)

for each finite subgroup $H<G$, the fixed point space $\underline{E}G^{H}$ is
contractible.

Every group $G$ admits an $\underline{E}G$, and all models for
$\underline{E}G$ are $G$-homotopy equivalent; we also have the inequality
$cd_{\mathbb{Q}}(G)\leq dim(\underline{E}G)$, for any group $G$ admitting a
finite dimensional $\underline{E}G$. More details, and a general construction
of classifying spaces for proper actions, can be found in [L0̈5].

To prove our asphericity result, we must slightly simplify the coned-off space
$X^{*}$ associated to a cubical presentation. Intuitively, we do this in
$\widetilde{X^{*}}$ by “collapsing” or “squashing together” various cones into
a single one whenever their base spaces correspond to elevations of a
$Y_{i}\rightarrow X$ having the same preimage in $\widetilde{X}$; at this
point, we remind the reader of Convention 3.4.

The precise construction is as follows:

######  Construction 4.2.

Let $X^{*}=\langle X|\\{Y_{i}\\}_{i\in I}\rangle$ be a cubical presentation
and consider the universal cover $\widetilde{X^{*}}$ of the coned-off space.
Note that $X$ is a subspace of $X^{*}$, so the preimage $\hat{X}$ of $X$ in
$\widetilde{X^{*}}$ is a covering space of $X$, namely the regular cover
corresponding to $ker(\pi_{1}X\rightarrow\pi_{1}X^{*})$. Consider the
universal cover $\widetilde{X}$ of $X$. For each $i\in I$, and for any fixed
elevation $\widetilde{Y}_{i}\rightarrow\widetilde{X}$ of $Y_{i}$, we have that
$\pi_{1}(Y_{i},y_{i_{0}})<Stab_{\pi_{1}(X,x_{0})}(\widetilde{Y}_{i})$. If
$X^{*}=\langle X|\\{Y_{i}\\}_{i\in I}\rangle$ is minimal, then by definition
$\pi_{1}(Y_{i},y_{i_{0}})=Stab_{\pi_{1}(X,x_{0})}(\widetilde{Y}_{i})$, but in
general this is not the case.

Let $\\{g_{\ell}\pi_{1}Y_{i}\\}$ be coset representatives of
$\pi_{1}(Y_{i},y_{i_{0}})$ in $Stab_{\pi_{1}(X,x_{0})}(\widetilde{Y}_{i})$.
The elevations $\\{g_{\ell}Y_{i}\rightarrow\hat{X}\\}$ have the same image in
$\hat{X}\subset\widetilde{X^{*}}$, so their cones are all isomorphic in
$\widetilde{X^{*}}$. Thus, there is a quotient
$\cup_{\ell}g_{\ell}C(Im_{\hat{X}}(Y_{i}))\rightarrow C(Im_{\hat{X}}(Y_{i}))$
where all cones over $Im_{\hat{X}}(Y_{i})$ are identified to a single cone.
This extends to a quotient $\widetilde{X^{*}}\rightarrow\bar{X}^{*}$, which we
call the _reduced space_ of $\widetilde{X^{*}}$.

######  Remark 4.3 (Cubically presenting the trivial group).

Another way to think about $\bar{X}^{*}$ is as the cubical presentation
$\langle\hat{X}|\\{Im_{\hat{X}}(Y_{i})\\}\rangle$, where the $Y_{i}$ range
over all $i\in I$ and over all elevations of $Y_{i}$ with distict image. Of
course, $\hat{X}$ is not compact, and there are infinitely many $Y_{i}$’s, so
this is an “infinitely generated” cubical presentation with infinitely many
“relators”. It presents, in fact, the trivial group, as we show in Lemma 4.4.
It is immediate from Construction 4.2 that if $\widetilde{X^{*}}$ satisfies
the $C(n)$ condition for $n>0$, then so does $\bar{X}^{*}$.

Thus, there are no hidden technicalities in applying the theory from Section 3
to $\bar{X}^{*}$ rather than to $\widetilde{X^{*}}$. In particular, it makes
sense to talk about minimal complexity diagrams and pieces, and Diagram
Dichotomy and Theorem 3.27 all hold for $\bar{X}^{*}$.

As mentioned above, the following is a quick but important observation:

######  Lemma 4.4.

$\bar{X}^{*}$ is simply-connected.

###### Proof.

Choosing a representative $g_{0}C(Im_{\hat{X}}(Y_{i}))$ for each collection
$\\{g_{\ell}C(Im_{\hat{X}}(Y_{i}))\\}_{g_{\ell}}$ described above, we may view
the reduced space $\bar{X}^{*}$ as a subspace of $\widetilde{X^{*}}$. Thus,
$\bar{X}^{*}$ is a retract of $\widetilde{X^{*}}$, and $\pi_{1}\bar{X}^{*}$
injects into $\pi_{1}\widetilde{X^{*}}$, and is therefore trivial. ∎

We can now state and prove our main technical theorem.

######  Theorem 4.5.

Let $X^{*}=\langle X|\\{Y_{i}\\}\rangle$ be a cubical presentation that
satisfies the $C(9)$ condition. Let $\bar{X}^{*}$ be the reduced space of
$\widetilde{X}^{*}$. Then $\pi_{2}\bar{X}^{*}=0$.

###### Proof.

Let $\Sigma\hookrightarrow\bar{X}^{*}$ be a minimal complexity spherical
diagram, where the minimum is taken over all spherical diagrams representing a
fixed non-zero homotopy class in $\pi_{2}\bar{X}^{*}$. If $\Sigma$ has no
cone-cells, then it lies on the cubical part of $\bar{X}^{*}$ and it is
therefore contractible by Theorem 2.1. Thus, we may assume that $\Sigma$ has a
cone-cell.

Let $C_{\infty}$ be a cone-cell in $\Sigma$ and let $D_{0}$ be obtained from
$\Sigma$ by removing from it the interior of $C_{\infty}$. Note that $D_{0}$
is a disc diagram. Indeed, since $\Sigma$ has minimal complexity, then
$C_{\infty}$ embeds in $\Sigma$, as by Theorem 3.27 the cone of $\bar{X}^{*}$
containing $C_{\infty}$ embeds in $\bar{X}^{*}$, so a non-embedded cone-cell
in $\Sigma$ would have to be nullhomotopic, contradicting Condition 5 of
Definition 3.20. Moreover, $D_{0}$ has minimal complexity as any complexity
reduction in $D_{0}$ would lead to a complexity reduction in $\Sigma$.

By Theorem 3.26, either

  1. (1)

$D_{0}$ is a single vertex or a single cone-cell,

  2. (2)

$D_{0}$ has at least 2 shells and/or corners and/or spurs.

The first case would imply that $\Sigma$ consists exactly of two cone-cells
$C_{1}$ and $C_{2}$ with $\partial C_{1}=\partial C_{2}=\sigma$. Assume
$\partial C_{1}\rightarrow Y_{1}$ and $\partial C_{2}\rightarrow Y_{2}$. If
$Y_{1}=Y_{2}$, then $\Sigma\rightarrow\bar{X}^{*}$ factors as
$\Sigma\rightarrow C(Y_{1})\rightarrow\bar{X}^{*}$. But $C(Y_{1})$ is
contractible, because it is a cone, so $\Sigma$ is nullhomotopic in
$C(Y_{1})$, and therefore also in $\bar{X}^{*}$, which contradicts the choice
of $\Sigma$.

Otherwise, $Y_{1}\neq Y_{2}$. If $\sigma$ is essential in either $Y_{1}$ or
$Y_{2}$, then either $\sigma$ is a piece, and thus an essential path that is
the concatenation of $<9$ pieces, contradicting the $C(9)$ condition, or
$\sigma$ is not a piece, which implies that $Y_{1}$ and $Y_{2}$ are elevations
of the same $Y_{i}$ that differ by an element of
$Stab_{\pi_{1}X}(\widetilde{Y})$. In this case, $Y_{1}$ and $Y_{2}$ have the
same image, so $C(Y_{1})$ and $C(Y_{2})$ are identified in $\bar{X}^{*}$, and
thus $C_{1}=C_{2}$, again contradicting the minimal complexity of $\Sigma$. If
$\sigma$ is not essential in neither $Y_{1}$ nor $Y_{2}$, then there are disc
diagrams $D_{1}\rightarrow Y_{1},D_{2}\rightarrow Y_{2}$ with $\partial
D_{1}=\partial C_{1}$ and $\partial D_{2}=\partial C_{2}$, which together
bound a spherical diagram $\Sigma^{\prime}\rightarrow\hat{X}$. As $C(Y_{1})$
and $C(Y_{2})$ are contractible, $\Sigma^{\prime}$ and $\Sigma$ are homotopic.
Since $\hat{X}$ is aspherical – it is a covering space of a non-positively
curved cube complex, and thus also non-positively curved – then
$\Sigma^{\prime}$ is nullhomotopic in $\hat{X}$, and therefore also in
$\bar{X}^{*}$. This in turn implies that $\Sigma$ is nullhomotopic in
$\bar{X}^{*}$.

In the third case, the presence of spurs immediately implies that $D_{0}$ is
not a minimal complexity diagram. Likewise, the presence of a corner on
$\partial D_{0}$ implies the presence of a corner on $\partial C_{\infty}$,
contradicting that the spherical diagram $\Sigma$ has minimal complexity, as
such a corner could be absorbed into $C_{\infty}$, reducing the number of
squares in $\Sigma$ by Condition 2 of Definition 3.20. In particular, this
implies that $D_{0}$ cannot be a square disc diagram. We can therefore
conclude that $D_{0}$ is a disc diagram with at least two shells – in fact,
all we need to use now is that $D_{0}$ has at least one shell.

Let $C$ be a shell of $D_{0}$. Then the inner-path of $C$ is at most $4$
pieces, and the outerpath of $\partial C$ coincides with a subpath of
$\partial C_{\infty}$ in $\Sigma$. Thus, the outerpath of $C$ is a single
piece, contradicting the $C(9)$ condition. ∎

We can now deduce asphericity for the reduced space $\bar{X}^{*}$ associated
to a low-dimensional $C(9)$ cubical presentation, and in particular for
$X^{*}$ when the cubical presentation is minimal:

######  Corollary 4.6.

Let $X^{*}=\langle X|\\{Y_{i}\\}\rangle$ be a cubical presentation that
satisfies the $C(9)$ condition. If $dim(X)\leq 2$ and
$\max\\{cd(\pi_{1}Y_{i})\\}=1$, then $\bar{X}^{*}$ is contractible.

###### Proof.

As explained in Construction 5.1, the preimage $\hat{X}$ of $X$ in
$\widetilde{X^{*}}$ is a covering space of $X$, corresponding to
$ker(\pi_{1}X\rightarrow\pi_{1}X^{*})$. By Theorem 3.27, each $Y_{i}$ embeds
in $\widetilde{X^{*}}$, and thus also in the quotient $\bar{X}^{*}$. Viewing
$\bar{X}^{*}$ as a cubical presentation, $\bar{X}^{*}$ decomposes as the union
of its cubical part $\hat{X}$ and the cones over all elevations of $Y_{i}$’s
with distinct images in $\widetilde{X^{*}}$. To simplify notation, let

$\bigsqcup_{i}\bigsqcup_{gStab_{\pi_{1}(X,x_{0})}(\widetilde{Y}_{i})\in\pi_{1}X/Stab_{\pi_{1}(X,x_{0})}(\widetilde{Y}_{i})}gY_{i}:=\mathbf{Y}$

and

$\bigsqcup_{i}\bigsqcup_{gStab_{\pi_{1}(X,x_{0})}(\widetilde{Y}_{i})\in\pi_{1}X/Stab_{\pi_{1}(X,x_{0})}(\widetilde{Y}_{i})}gC(Y_{i}):=\mathbf{C(Y)},$

And note that

$\hat{X}\cap\mathbf{C(Y)}=\mathbf{Y}.$

We have the Mayer-Vietoris sequence:

${\cdots}$${H_{n}(\mathbf{Y})}$${H_{n}(\hat{X})\oplus
H_{n}(\mathbf{C(Y)})}$${H_{n}(\bar{X}^{*})}$
${H_{n-1}(\mathbf{Y})}$$\cdots\hfill\cdots$${H_{0}(\hat{X}^{*})}$${0}$

Since $\mathbf{Y}$ is homotopy equivalent to a graph, then
$H_{n}(\mathbf{Y})=0$ whenever $n\geq 2$, so we get isomorphisms
$H_{n}(\hat{X})\cong H_{n}(\bar{X}^{*})$ for each $n\geq 2$, since
$C(gY_{i}))$ is contractible for each $g\in\pi_{1}X^{*}$ and $i\in I$. Now,
$H_{2}(\bar{X}^{*})\cong\pi_{2}\bar{X}^{*}=0$ by Theorem 4.5 and Hurewicz’s
Theorem, and $H_{3}(\bar{X}^{*})=0$ since the sequence is exact and the terms
on the right and left of $H_{3}(\bar{X}^{*})$ are equal to zero. Since
$\bar{X}^{*}$ is $2$-connected, we may apply Hurewicz again to conclude that
$\pi_{3}\bar{X}^{*}=0$.

The proof is now finished: since by Theorem 4.5, $\pi_{1}\bar{X}^{*}=0$, and
$\bar{X}^{*}$ has no cells of dimension $\geq 3$, then
$H_{n}(\bar{X}^{*})\cong\pi_{n}\bar{X}^{*}=0$ for all $n\in\mathbb{N}$. ∎

The first of our two theorems is now established:

######  Theorem 4.7.

Let $X^{*}=\langle X|\\{Y_{i}\\}_{i\in I}\rangle$ be a minimal cubical
presentation that satisfies the $C(9)$ condition. Let $\pi_{1}X^{*}=G$. If
$dim(X)\leq 2$ and each $Y_{i}$ is homotopy equivalent to a graph, then
$X^{*}$ is a $K(G,1)$, so $G$ is torsion-free and $gd(G)\leq 2$.

###### Proof.

This is a direct consequence of Corollary 4.6, since then, by minimality,
$\bar{X}^{*}=\widetilde{X^{*}}$, so $\widetilde{X^{*}}$ is contractible. ∎

We point out that neither Theorem 4.5 nor Corollary 4.6 require the cubical
presentation $X^{*}=\langle X|\\{Y_{i}\\}_{i\in I}\rangle$ to have finitely
many relators, or for these relators to be compact. Thus, Theorem 4.7 holds
even if the $Y_{i}$ have infinitely-generated fundamental group, and even if
$|I|=\infty$. However, this is not the case for the results that follow: for
the remainder of this section, we adopt the following convention.

######  Convention 4.8.

Let $X^{*}=\langle X|\\{Y_{i}\\}_{I}\rangle$ be a cubical presentation. If
$X^{*}$ is _not_ minimal, then we always assume that $|I|<\infty$, and that
each $Y_{i}$ is compact.

######  Lemma 4.9.

Let $X^{*}=\langle X|\\{Y_{i}\\}_{i\in I}\rangle$ be a symmetric cubical
presentation. Then $\bar{X}^{*}$ is a $\pi_{1}X^{*}$-CW-complex, and if $v$ is
a cone-vertex of $\bar{X}^{*}$ corresponding to a cone over some $Y_{i}$, then
$Stab_{\pi_{1}X^{*}}(v)=Stab_{\pi_{1}X}(\widetilde{Y}_{i})/\pi_{1}{Y_{i}}$.

###### Proof.

The group $\pi_{1}X^{*}$ acts on $\widetilde{X^{*}}$ by permuting the cones.
The action is a covering space action, and in particular is free, and induces
an action on $\bar{X}^{*}$. For each cone-vertex $v$ in $\bar{X}^{*}$,
$Stab_{\pi_{1}X^{*}}(v)=Stab_{\pi_{1}X^{*}}(C(Im_{\hat{X}}(\widetilde{Y}_{i}))=\\{g_{\ell}\pi_{1}Y_{i}\\}$
where $C(Im_{\hat{X}}(\widetilde{Y}_{i}))$ is the cone with $v$ as its cone-
vertex, and $\\{g_{\ell}\pi_{1}Y_{i}\\}$ are left coset representatives of
$\pi_{1}(Y_{i},y_{i_{0}})$ in $Stab_{\pi_{1}(X,x_{0})}(\widetilde{Y}_{i})$.
This is exactly the quotient
$Stab_{\pi_{1}X}(\widetilde{Y}_{i})/\pi_{1}{Y_{i}}$. ∎

######  Lemma 4.10.

If $X^{*}=\langle X|\\{Y_{i}\\}\rangle$ is a symmetric cubical presentation,
then the action of $\pi_{1}X^{*}$ on $\bar{X}^{*}$ has finite cell-
stabilisers.

###### Proof.

All stabilisers of cells in $\hat{X}\subset\bar{X}^{*}$ are trivial; since the
cubical presentation is symmetric, then
$[Stab_{\pi_{1}X}(\widetilde{Y}):\pi_{1}Y]<\infty$, so Lemma 4.9 implies that
the stabilisers of cone-vertices are finite. ∎

So far we have shown that if $X^{*}=\langle X|\\{Y_{i}\\}\rangle$ satisfies
the $C(9)$ condition, then $\bar{X}^{*}$ is contractible, and that if, in
addition, $X^{*}$ is symmetric, then the action of $\pi_{1}X^{*}$ on
$\bar{X}^{*}$ has finite cell-stabilisers. To show that $\bar{X}^{*}$ is a
classifying space for proper actions for $\pi_{1}X^{*}$, we still must prove
that every finite subgroup $H<\pi_{1}X^{*}$ fixes a point in $\bar{X}^{*}$,
and that this point is unique. Note that since $\bar{X}^{*}$ is aspherical and
finite-dimensional, then the action of a finite subgroup $H$ on $\bar{X}^{*}$
cannot be free, as otherwise the quotient $\bar{X}^{*}/H$ would be a
classifying space for $H$. Since we will invoke it later on, we record this
observation as a lemma:

######  Lemma 4.11.

Let $Z$ be an aspherical, finite-dimensional cell-complex and let $G$ be a
group acting on $Z$ by combinatorial isometries. If $H<G$ is finite, then the
action of $H$ on $Z$ is not free.

We emphasize that this does not necessarily imply that such an action has a
global fixed point.

######  Lemma 4.12.

If a non-trivial subgroup $H<\pi_{1}X^{*}$ fixes a point in $\bar{X}^{*}$,
then $H$ is conjugate into $Stab_{\pi_{1}X^{*}}(ImY_{i})$ for some elevation
$Y_{i}\rightarrow\hat{X}$ of some $Y_{i}\rightarrow X$.

###### Proof.

Let $H$ be a non-trivial subgroup of $\pi_{1}X^{*}$. Since the action of $H$
on $\widetilde{X^{*}}$ is free and $\widetilde{X^{*}}$ coincides with
$\bar{X}^{*}$ outside of the cones, then a fixed point $\xi$ under the action
of $H$ on $\bar{X}^{*}$ must be a cone-vertex, which in turn, by Construction
5.1, corresponds to some left coset $gStab_{\pi_{1}X^{*}}(Y_{i})$ for some
elevation $Y_{i}\rightarrow\hat{X}$ of some $Y_{i}\rightarrow X$ and some
$g\in\pi_{1}X^{*}$. Thus,
$HgStab_{\pi_{1}X^{*}}(Y_{i})=gStab_{\pi_{1}X^{*}}(Y_{i})$, so
$g^{-1}HgStab_{\pi_{1}X^{*}}(Y_{i})=Stab_{\pi_{1}X^{*}}(Y_{i})$ and $H$ is
conjugate into $Stab_{\pi_{1}X^{*}}(Y_{i})$. ∎

######  Lemma 4.13.

If a non-trivial subgroup $H<\pi_{1}X^{*}$ fixes a point in $\bar{X}^{*}$,
then that point is unique, and in particular the fixed-point space
$(\bar{X}^{*})^{H}$ is contractible.

Before proving the lemma, we need an auxiliary result.

######  Proposition 4.14.

Let $X^{*}=\langle X|\\{Y_{i}\\}^{k}_{i=1}\rangle$ be a cubical presentation
that satisfies the $C(n)$ condition for some $n\geq 2$, and where each $Y_{i}$
is compact. Then there is a uniform upper-bound $\mathcal{L}\geq 0$ on the
size of pieces in $\widetilde{X^{*}}$.

###### Proof.

Since each $Y_{i}$ is compact, $\pi_{1}Y_{i}$ is finitely generated for each
$i\in\\{1,\ldots,k\\}$, and since the cubical presentation has finitely many
relations $Y_{i}\rightarrow X$, then finitely many elements of $\pi_{1}X$
suffice to generate all the $\pi_{1}Y_{i}$’s. The compactness of the $Y_{i}$’s
implies additionally that there is an uniform upper-bound $\ell_{i}$ for the
length of minimal-length paths representing the generators of $\pi_{1}Y_{i}$,
thus, there is a bound $\mathcal{L}=\max\\{\ell_{i}\\}$ for the size of pieces
arising in generators of $\pi_{1}Y_{1},\ldots,\pi_{1}Y_{k}$. Since $X^{*}$
satisfies the $C(n)$ condition for $n\geq 2$, then each path representing a
generator is a concatenation of at least $2$ pieces, and in particular any
such piece $p$ satisfies $|p|<\ell_{i}\leq\mathcal{L}$ whenever it arises in a
generator of a $\pi_{1}Y_{i}$.

Assume that there is an essential closed path $\sigma\rightarrow Y_{i}$ that
is a concatenation of pieces having a piece $p_{r}$ with
$|p_{r}|>\mathcal{L}$. By the discussion above, we can assume in particular
that $\sigma$ does not represents a generator of a $\pi_{1}Y_{i^{\prime}}$ for
any $i^{\prime}\in\\{1,\ldots,k\\}$. Note that any expression for $\sigma$ as
a concatenation of pieces can be further expressed as a concatenation of
pieces in generators of $\pi_{1}Y_{i}$. Write
$\sigma=\alpha_{1}\cdots\alpha_{m}=p_{1}\cdots p_{r}\cdots p_{n}$, where each
$\alpha_{j}$ is a (not necessarily distinct) generator of $\pi_{1}Y_{i}$.
Since $|p_{r}|>\mathcal{L}$, then there is an $\alpha_{j}$ with
$j\in\\{1,\ldots,m\\}$ for which $p_{r}\cap\alpha_{j}=\alpha_{j}$. Thus,
$\alpha_{j}$ can be expressed as a single piece $q_{r}\subset p_{r}$,
contradicting the discussion in the previous paragraph, and hence the $C(n)$
condition. ∎

The proof of Lemma 4.13 now combines Proposition 4.14 with the hypothesised
symmetry of the cubical presentation under consideration.

###### Proof of Lemma 4.13.

Let $H<\pi_{1}X^{*}$. Assume $\xi\neq\xi^{\prime}$ are cone-vertices of
$\bar{X}^{*}$ fixed by $H$, so $H\xi=\xi$ and $H\xi^{\prime}=\xi^{\prime}$.
Then Lemma 4.12 implies that $H\subset g^{-1}Stab_{\pi_{1}X^{*}}(Y_{i})g\cap
g^{\prime-1}Stab_{\pi_{1}X^{*}}(Y_{i^{\prime}})g^{\prime}$ for some
$i,i^{\prime}\in\\{1,\ldots,k\\}$ and $g,g^{\prime}\in\pi_{1}X^{*}$. We claim
that the collection $\\{Stab_{\pi_{1}X^{*}}(Y_{i})\\}$ is malnormal, and thus
$H$ is the trivial subgroup. To this end, we first show that the collection
$\\{Stab_{\pi_{1}X}(\widetilde{Y}_{i})\\}$ is malnormal in $\pi_{1}X$.

Suppose that the intersection
$Stab_{\pi_{1}X}(\widetilde{Y}_{i})^{\tilde{g}}\cap
Stab_{\pi_{1}X}(\widetilde{Y}_{j})^{\tilde{g}^{\prime}}$ is infinite for some
$i,i^{\prime}\in\\{1,\ldots,k\\}$ and
$\tilde{g},\tilde{g}^{\prime}\in\pi_{1}X$. Since $X^{*}$ is symmetric, then
$[Stab_{\pi_{1}X}(\widetilde{Y}_{i}):\pi_{1}Y_{i}]<\infty$ and
$[Stab_{\pi_{1}X}(\widetilde{Y}_{j}):\pi_{1}Y_{j}]<\infty$, so
$\pi_{1}Y_{i}^{\tilde{g}}\cap\pi_{1}Y_{j}^{\tilde{g}^{\prime}}$ has finite
index in $Stab_{\pi_{1}X}(\widetilde{Y}_{i})^{\tilde{g}}\cap
Stab_{\pi_{1}X}(\widetilde{Y}_{j})^{\tilde{g}^{\prime}}$. Thus,
$\pi_{1}Y_{i}^{\tilde{g}}\cap\pi_{1}Y_{j}^{\tilde{g}^{\prime}}$ is infinite,
and in particular contains an infinite order element $\tilde{h}$. Therefore,
the axis of $\tilde{h}$ in $\widetilde{X}$ is an unbounded piece between
elevations $\widetilde{Y}_{i}$ and $\widetilde{Y}_{j}$ of $Y_{i}$ and $Y_{j}$,
contradicting Proposition 4.14, and thus the $C(9)$ condition. We conclude
that $Stab_{\pi_{1}X}(\widetilde{Y}_{i})^{\tilde{g}}\cap
Stab_{\pi_{1}X}(\widetilde{Y}_{j})^{\tilde{g}^{\prime}}$ is finite; since
$\pi_{1}X$ is torsion-free, then in fact
$Stab_{\pi_{1}X}(\widetilde{Y}_{i})^{\tilde{g}}\cap
Stab_{\pi_{1}X}(\widetilde{Y}_{j})^{\tilde{g}^{\prime}}$ must be trivial, so
the collection $\\{Stab_{\pi_{1}X}(\widetilde{Y}_{i})\\}$ is malnormal in
$\pi_{1}X$.

To promote this to malnormality of the collection
$\\{Stab_{\pi_{1}X^{*}}(Y_{i})\\}$ in $\pi_{1}X^{*}$, let $h\in
Stab_{\pi_{1}X^{*}}(Y_{i})^{g}\cap Stab_{\pi_{1}X^{*}}(Y_{j})^{g^{\prime}}$,
and let $\tilde{h_{i}}\in Stab_{\pi_{1}X}(\widetilde{Y}_{i})^{\tilde{g}}$ and
$\tilde{h_{j}}\in Stab_{\pi_{1}X}(\widetilde{Y}_{j})^{\tilde{g}^{\prime}}$ be
preimages of $h$, so $\tilde{h_{i}}\tilde{h_{j}}^{-1}$ is an element of
$\pi_{1}\hat{X}$. Since $\tilde{h_{i}}\in
Stab_{\pi_{1}X}(\widetilde{Y}_{i})^{\tilde{g}}$ and $\tilde{h_{j}}\in
Stab_{\pi_{1}X}(\widetilde{Y}_{j})^{\tilde{g}^{\prime}}$ and as mentioned
before, by symmetry,
$[Stab_{\pi_{1}X}(\widetilde{Y}_{i}):\pi_{1}Y_{i}]<\infty$ and
$[Stab_{\pi_{1}X}(\widetilde{Y}_{j}):\pi_{1}Y_{j}]<\infty$, then there are
paths $\sigma_{i}\rightarrow Y_{i},\sigma_{j}\rightarrow Y_{j}$ such that for
some $k,k^{\prime}\in\mathbb{N}$, the paths
$\sigma_{i}^{k},\sigma_{j}^{k^{\prime}}$ bound disc diagrams $D_{i}\rightarrow
Y_{i},D_{j}\rightarrow Y_{j}$ in $\widetilde{X^{*}}$, each of which can be
assumed to have minimal complexity, and therefore consisting of a single cone-
cell. Moreover, by the discussion above, $\sigma_{i}$ and $\sigma_{j}$
represent lifts of conjugates of $\tilde{h_{i}}$ and $\tilde{h_{j}}$, so there
are paths $\sigma^{\prime}_{i},\sigma^{\prime}_{j}$ such that
$\sigma^{\prime}_{i}\sigma_{j}^{{}^{\prime}-1}$ is a closed path that
represents $\tilde{h_{i}}\tilde{h_{j}}^{-1}$. Thus,
$\sigma^{\prime}_{i}\sigma_{j}^{{}^{\prime}-1}$ bounds a disc diagram $D$ in
$\widetilde{X^{*}}$, and there are paths
$\ell_{i},\ell_{i}^{\prime},\ell_{j},\ell_{j}^{\prime}$ that are lifts of the
corresponding conjugating elements, so that
$\sigma_{i}\ell_{i}\sigma^{\prime}_{i}\ell^{\prime}_{i}$ and
$\sigma_{j}\ell_{j}\sigma^{\prime}_{j}\ell^{\prime}_{j}$ bound square-disc
diagrams $R_{i},R_{j}$ in $\widetilde{X^{*}}$. We can further assume that
$R_{i},R_{j}$ and $D$ are chosen to have minimal area and minimal complexity
amongst all possible diagrams with the same corresponding boundary. For
$R_{i}$ and $R_{j}$, this means in particular that there are no cornsquares on
any of $\sigma_{i},\sigma^{\prime}_{i},\sigma_{j},\sigma^{\prime}_{j}$, so
each of these paths determines a hyperplane carrier in the corresponding
diagram $R_{i}$ or $R_{j}$, and more generally that $R_{i}$ and $R_{j}$ are
_grids_ , in the sense that they are cubically isomorphic to a product of
combinatorial intervals $I_{n}\times I_{m}$ for suitable $m,n\in\mathbb{N}$.
We note that, of course, $R_{i}$ or $R_{j}$ could be degenerate square
diagrams, in which case $\sigma_{i}=\sigma^{\prime}_{i}$ or
$\sigma_{j}=\sigma^{\prime}_{j}$, and the argument in the next paragraph
becomes a bit simpler.

If $D$ is a square disc diagram, then $\sigma^{\prime}_{1}$ and
$\sigma^{\prime}_{j}$ are homotopic rel their endpoints in
$\widetilde{X^{*}}$, so $\tilde{h_{i}}=\tilde{h_{j}}$, contradicting
malnormality in $\pi_{1}X$. We claim that this is the only possibility for
$D$. Assume towards a contradiction that $D$ contains at least one cone-cell,
and observe that since any corners or spurs on $\partial D$ could be removed,
improving the choices made in the previous paragraph, that cone-cell must be a
shell $S$. Consider the disc diagram
$F=D_{i}\cup_{\sigma_{i}}R_{i}\cup_{\sigma^{\prime}_{i}}D\cup_{\sigma^{\prime}_{j}}R_{j}\cup_{\sigma_{j}}D_{j}$.
Then, since $S$ is a shell of $D$, its innerpath is a concatenation of at most
$4$ pieces, and either the outerpath of $S$ is a subpath of
$\sigma^{\prime}_{i}\subset R_{i}$, or a subpath of
$\sigma^{\prime}_{j}\subset R_{j}$, or a subpath of their concatenation. In
the first case, since $\partial S\cap R_{i}$ in $F$ is the outerpath of $S$ in
$D$, then $\partial S\cap R_{i}$ is a single piece (a wall piece if $R_{i}$ is
not a degenerate diagram, and a cone piece otherwise), so $\partial S$ is the
concatenation of at most $5$ pieces, contradicting the $C(9)$ condition.
Similarly if the outerpath of $S$ is a subpath of $\sigma^{\prime}_{j}$. In
the last case, $\partial S$ is the concatenation of at most $6$ pieces: the
$4$ pieces coming from its innerpath and a piece coming from each of
$\sigma^{\prime}_{i},\sigma^{\prime}_{j}$, contradicting again the $C(9)$
condition.

So $\\{Stab_{\pi_{1}X^{*}}(Y_{i})\\}$ is malnormal in $\pi_{1}X^{*}$, and
either $H=\\{1\\}$, or $\xi=\xi^{\prime}$. ∎

######  Remark 4.15.

In the proof above, we use that $\pi_{1}X$ is torsion-free, which follows
trivially from the assumption that $X$ is finite-dimensional, since $X$ is
aspherical, but is true also for infinite dimensional non-positively curved
cube complexes, and is a consequence of CAT(0) geometry [BH99, Lea13].

We can finally conclude that, in fact, the finite subgroups of $\pi_{1}X^{*}$
have non-empty fixed-point sets under their action on $\bar{X}^{*}$:

######  Corollary 4.16.

Every finite subgroup $H<\pi_{1}X^{*}$ fixes a point in $\bar{X}^{*}$.

###### Proof.

Let $H$ be a finite subgroup of $\pi_{1}X^{*}$. If $H$ is trivial, then it
fixes all of $\bar{X}^{*}$. Otherwise, since the action of $H$ on
$\bar{X}^{*}$ cannot be free by Lemma 4.11, for any non-trivial $h\in H$,
there exists $k\in\mathbb{N}$ for which $h^{k}$ fixes a point in
$\bar{X}^{*}$. This implies that each cyclic subgroup $\langle h\rangle$ fixes
a point, and in particular that every $h\in H$ fixes a point in $\bar{X}^{*}$.
Indeed, if $h^{k}v=v$ for some $k\in\mathbb{N}$ and $v\in\bar{X}^{*}$, then
$hv=h(h^{k}v)=h^{k}(hv)$, so $h^{k}$ fixes $hv$ and by Lemma 4.13 $hv=v$.

To see now that every element of $H$ fixes the same point, we use Burnside’s
Lemma: since $H$ is a finite group acting on a finite set $S$ (the union of
the orbits of the fixed points of non-trivial elements), and each non-trivial
element fixes a unique point, if $m$ is the number of orbits of elements, then
$m=\frac{|S|+|H|-1}{|H|}$, so $|S|=(m-1)|H|+1$ and in particular, as the size
of each orbit must divide $|H|$, at most one orbit of points can have size
$<|H|$. There must therefore be $m-1$ orbits of size $|H|$, and a single orbit
of size $1$, yielding a fixed point in $\bar{X}^{*}$. ∎

Putting together the previous results, we obtain:

######  Theorem 4.17.

Let $X^{*}=\langle X|\\{Y_{i}\\}^{k}_{i=1}\rangle$ be a symmetric cubical
presentation that satisfies the $C(9)$ condition. Let $\pi_{1}X^{*}=G$. If
$dim(X)\leq 2$ and each $Y_{i}$ is homotopy equivalent to a graph, then there
is a quotient $\bar{X}^{*}$ of $\widetilde{X^{*}}$ that is an
$\underline{E}G$, so $cd_{\mathbb{Q}}(G)\leq dim(\bar{X}^{*})\leq 2$.

###### Proof.

This follows from Corollary 4.6, Lemma 4.13, and Corollary 4.16. ∎

We can go a step further if we assume that $X$ has a finite regular cover
where every $Y_{i}\rightarrow X$ lifts to an embedding, as is observed in
[Wis21, 4.4] for $C^{\prime}(\frac{1}{20})$ cubical presentations.

######  Lemma 4.18.

Let $X^{*}=\langle X|\\{Y_{i}\\}_{i\in I}\rangle$ be a symmetric $C(9)$
cubical presentation where $dim(X)\leq 2$ and each $Y_{i}$ is homotopy
equivalent to a graph. If there exists a finite regular cover
$\leavevmode\hbox to9.47pt{\vbox to9.39pt{\pgfpicture\makeatletter\hbox{\hskip
4.53471pt\lower-3.41666pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-4.53471pt}{-3.41666pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$X$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} { {}}{}{{}{}}{
{}{}{}}{}{{}{}}{{}}{}{}{}{}{}{}{}
{}{{}{}{}}{}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-4.30415pt}{3.61665pt}\pgfsys@curveto{-4.30415pt}{3.61665pt}{-3.2606pt}{5.76942pt}{0.0pt}{5.76942pt}\pgfsys@curveto{3.2606pt}{5.76942pt}{4.73471pt}{3.61665pt}{4.73471pt}{3.61665pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\rightarrow X$ where
$Y_{i}\rightarrow X$ lifts to an embedding for each $i\in I$, then
$\pi_{1}X^{*}$ is virtually torsion-free.

###### Proof.

Consider the covering $\leavevmode\hbox to12.27pt{\vbox
to10.69pt{\pgfpicture\makeatletter\hbox{\hskip 5.93471pt\lower-4.06804pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-5.93471pt}{-4.06804pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$X^{*}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} { {}}{}{{}{}}{
{}{}{}}{}{{}{}}{{}}{}{}{}{}{}{}{}
{}{{}{}{}}{}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-5.70415pt}{4.26804pt}\pgfsys@curveto{-5.70415pt}{4.26804pt}{-4.27066pt}{6.4208pt}{0.0pt}{6.4208pt}\pgfsys@curveto{4.27066pt}{6.4208pt}{6.1347pt}{4.26804pt}{6.1347pt}{4.26804pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\rightarrow X^{*}$
induced by $\leavevmode\hbox to9.47pt{\vbox
to9.39pt{\pgfpicture\makeatletter\hbox{\hskip 4.53471pt\lower-3.41666pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-4.53471pt}{-3.41666pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$X$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} { {}}{}{{}{}}{
{}{}{}}{}{{}{}}{{}}{}{}{}{}{}{}{}
{}{{}{}{}}{}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-4.30415pt}{3.61665pt}\pgfsys@curveto{-4.30415pt}{3.61665pt}{-3.2606pt}{5.76942pt}{0.0pt}{5.76942pt}\pgfsys@curveto{3.2606pt}{5.76942pt}{4.73471pt}{3.61665pt}{4.73471pt}{3.61665pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\rightarrow X$, which
is a finite-degree covering because $\leavevmode\hbox to9.47pt{\vbox
to9.39pt{\pgfpicture\makeatletter\hbox{\hskip 4.53471pt\lower-3.41666pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-4.53471pt}{-3.41666pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$X$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} { {}}{}{{}{}}{
{}{}{}}{}{{}{}}{{}}{}{}{}{}{}{}{}
{}{{}{}{}}{}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-4.30415pt}{3.61665pt}\pgfsys@curveto{-4.30415pt}{3.61665pt}{-3.2606pt}{5.76942pt}{0.0pt}{5.76942pt}\pgfsys@curveto{3.2606pt}{5.76942pt}{4.73471pt}{3.61665pt}{4.73471pt}{3.61665pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\rightarrow X$ is. If
$g\in\pi_{1}X^{*}-\\{1\\}$ satisfies $g^{n}=1$ for some $n\in\mathbb{N}$, then
Corollary 4.16 implies that if $\sigma\rightarrow X^{*}$ is a closed path
representing $g$, then $\sigma^{n}$ is conjugate to a closed path in some
$Y_{i}$. Now since $g$ is non-trivial, $\sigma$ does not lift to a closed path
in $Y_{i}$. Since $Y_{i}\rightarrow X$ lifts to an embedding
$Y_{i}\rightarrow\leavevmode\hbox to9.47pt{\vbox
to9.39pt{\pgfpicture\makeatletter\hbox{\hskip 4.53471pt\lower-3.41666pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-4.53471pt}{-3.41666pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$X$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} { {}}{}{{}{}}{
{}{}{}}{}{{}{}}{{}}{}{}{}{}{}{}{}
{}{{}{}{}}{}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-4.30415pt}{3.61665pt}\pgfsys@curveto{-4.30415pt}{3.61665pt}{-3.2606pt}{5.76942pt}{0.0pt}{5.76942pt}\pgfsys@curveto{3.2606pt}{5.76942pt}{4.73471pt}{3.61665pt}{4.73471pt}{3.61665pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$, then $\sigma$
cannot lift to a closed path in $X$ either, so $\sigma$ is not closed in
$X^{*}$, and $g\notin\pi_{1}\leavevmode\hbox to12.27pt{\vbox
to10.69pt{\pgfpicture\makeatletter\hbox{\hskip 5.93471pt\lower-4.06804pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-5.93471pt}{-4.06804pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$X^{*}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} { {}}{}{{}{}}{
{}{}{}}{}{{}{}}{{}}{}{}{}{}{}{}{}
{}{{}{}{}}{}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-5.70415pt}{4.26804pt}\pgfsys@curveto{-5.70415pt}{4.26804pt}{-4.27066pt}{6.4208pt}{0.0pt}{6.4208pt}\pgfsys@curveto{4.27066pt}{6.4208pt}{6.1347pt}{4.26804pt}{6.1347pt}{4.26804pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$. ∎

If a group $G$ has finite virtual cohomological dimension, then $vcd(G)\leq
dim(\underline{E}G)$, since a torsion-free subgroup $G^{\prime}<G$ of finite
index has $cd(G)=vcd(G)$, and $dim(\underline{E}G)$ gives an upper bound for
the cohomological dimension of any torsion-free subgroup of $G$. In
particular:

######  Corollary 4.19.

Let $X^{*}=\langle X|\\{Y_{i}\\}_{i\in I}\rangle$ be a symmetric $C(9)$
cubical presentation with $G=\pi_{1}X^{*}$. If $dim(X)\leq 2$ and each $Y_{i}$
is homotopy equivalent to a graph, and there exists a finite regular cover
$\leavevmode\hbox to9.47pt{\vbox to9.39pt{\pgfpicture\makeatletter\hbox{\hskip
4.53471pt\lower-3.41666pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-4.53471pt}{-3.41666pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$X$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} { {}}{}{{}{}}{
{}{}{}}{}{{}{}}{{}}{}{}{}{}{}{}{}
{}{{}{}{}}{}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-4.30415pt}{3.61665pt}\pgfsys@curveto{-4.30415pt}{3.61665pt}{-3.2606pt}{5.76942pt}{0.0pt}{5.76942pt}\pgfsys@curveto{3.2606pt}{5.76942pt}{4.73471pt}{3.61665pt}{4.73471pt}{3.61665pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\rightarrow X$ where
$Y_{i}\rightarrow X$ lifts to an embedding for each $i\in I$, then $vcd(G)\leq
2$. ∎

##  5\. Examples

Using Theorems 4.7 and 4.17, we can provide classifying spaces, or classifying
spaces for proper actions, in the situations outlined below:

######  Example 5.1.

(Forcing small-cancellation by taking covers) If $X$ is a compact non-
positively curved cube complex with hyperbolic fundamental group, and
$H_{1},...,H_{k}$ are quasiconvex subgroups of $\pi_{1}X$ that form a
malnormal collection, then for each $n>0$ there are finite index subgroups
$H^{\prime}_{i}\subset H_{i}$ and local isometries $Y_{i}\rightarrow X$ with
$Y_{i}$ compact and $\pi_{1}Y_{i}=H^{\prime}_{i}$ such that $X^{*}=\langle
X|\\{Y_{i}\\}\rangle$ satisfies $C(n)$. How one achieves this condition is
explained in [Wis21, 3.51]. To guarantee that $X^{*}$ is symmetric, it
suffices to require that $H_{i}=C_{\pi_{1}X}(H_{i})$. When $n\geq 9$,
$dim(X)\leq 2$ and $H_{1},...,H_{k}$ are free, we conclude that $\bar{X}^{*}$
is a classifying space for proper actions for $\pi_{1}X^{*}$, and in
particular that $\pi_{1}X^{*}$ has rational cohomological dimension
$cd_{\mathbb{Q}}(\pi_{1}X^{*})\leq 2$.

######  Example 5.2 (Forcing small-cancellation by creating noise).

Let $X$ be a compact non-positively curved cube complex with non-elementary
hyperbolic fundamental group, as above. Then by [Are23a, 3.2], for all $k\geq
1$ there exist (infinitely many choices of) free non-abelian subgroups
$\\{H_{1},\ldots,H_{k}\\}$, and cyclic subgroups $z_{i}<H_{i}$ such that there
exist local isometries $Y_{i}\rightarrow X$ with $Y_{i}$ compact and
$\pi_{1}Y_{i}=z_{i}$, so that the cubical presentation $X^{*}=\langle
X|\\{Y_{i}\\}\rangle$ is minimal, and satisfies the $C(9)$ condition. Thus, if
$\dim(X)\leq 2$, then $X^{*}$ is a classifying space for all such examples.

We now move away slightly from the hyperbolic setting with a construction that
is described in [Wis21, 5.r].

######  Example 5.3 (Some 2-dimensional Artin groups).

Let $\Gamma$ be a finite simplicial graph, and let

$A_{\Gamma}=\langle v_{i}\in
V(\Gamma)|(v_{i},v_{j})^{m_{ij}}=(v_{i},v_{j})^{m_{ij}}\text{ if }i<j\rangle$
(1)

be an Artin group on $\Gamma$, where either $m_{ij}$ is an integer $\geq 2$ or
$m_{ij}=\infty$, the notation $(v_{i},v_{j})^{m_{ij}}$ denotes the first half
of the word $(v_{i}v_{j})^{m_{ij}}$, and we adopt the (standard) convention
that $(v_{i}v_{j})^{\infty}$ signifies that there is no relation between
$v_{i}$ and $v_{j}$.

Let $B_{\Gamma}$ denote the bouquet of $|V|$ circles, where each circle is
labelled by a $v_{i}\in V$. Consider the cubical presentation

$B^{*}_{\Gamma}=\langle B_{\Gamma}|Y_{ij}\rightarrow B_{\Gamma}\rangle$ (2)

where, for each $m_{ij}<\infty$, the complex $Y_{ij}$ is the Cayley graph of
the 2-generator Artin group $\langle
v_{i},v_{j}|(v_{i},v_{j})^{m_{ij}}=(v_{j},v_{i})^{m_{ij}}\rangle$ and
$Y_{ij}\rightarrow B_{\Gamma}$ is the obvious covering projection. Then
$\pi_{1}B^{*}_{\Gamma}=A_{\Gamma}$, so this is a “non-trivial” cubical
presentation for $A_{\Gamma}$.

Note that $B^{*}_{\Gamma}$ is a minimal cubical presentation, and that the
$C(9)$ condition is satisfied provided that $5\leq m_{ij}\leq\infty$ for all
$i<j$. Indeed, since $B^{*}_{\Gamma}$ is a bouquet of circles, and hence
1-dimensional, there are no non-trivial wall-pieces in $B^{*}_{\Gamma}$. The
cone-pieces correspond to intersections between distinct elevations of the
$Y_{ij}$’s, and for a pair of relations $Y_{ij},Y_{i^{\prime}j^{\prime}}$,
these intersections are (subpaths of) bi-infinite lines of the form
$v^{\infty}_{\ell}$ where one of $i,j$ and one of $i^{\prime}j^{\prime}$ are
equal to $\ell$. Since an essential cycle in a $Y_{ij}$ lifts to a path that
intersects $v^{\infty}_{\ell}$ in a single edge, and the girth of any such
cycle is $\geq 5$, then no essential cycle is the concatenation of $<10$
pieces.

It is a well-established result [CD95] that $2$-dimensional Artin groups are
exactly those where no triangle in the defining graph has labels
$(2,3,4),(2,2,n),(2,3,3)$ or $(2,3,5)$. This class contains in particular the
class of _extra large type Artin groups_ , which are precisely those with
labels $4\leq m_{ij}\leq\infty$ for all $i<j$. In light of the discussion
above, we can partially recover the result that extra large type Artin groups
are 2-dimensional as a corollary of Theorem 4.7:

######  Theorem 5.4.

Let $A_{\Gamma}$ be an Artin group on $\Gamma$ given by a presentation (1),
where $5\leq m_{ij}\leq\infty$ for all $i<j$. Then $B^{*}_{\Gamma}$ is a
$K(A_{\Gamma},1)$, and in particular, $A_{\Gamma}$ is torsion-free and
$cd(A_{\Gamma})\leq 2$.

We note that, while the cubical presentation (2) can be viewed as a graphical
presentation, asphericity of $B_{\Gamma}^{*}$ cannot be deduced from the
asphericity results in [Gro03, Oll06, Gru15], since these handle only
graphical presentations where the relators are cycles, rather than arbitrary
–possibly non-compact– graphs. While we state Theorem 5.4 for Artin groups
whose associated cubical presentations satisfy the cubical $C(9)$ condition,
we believe that the result can be recovered for _all_ Artin groups where
$3\leq m_{ij}\leq\infty$ for all $i<j$, since in that case the $C(6)$
condition is satisfied, and in the graphical case this condition should be
enough to recover a suitable form of Theorem 3.26, and thus of Theorem 4.7. We
have not, however, explored this possibility closely.

It is claimed in [Wis21, 5.70] that the cubical presentation (2) satisfies the
$C(6)$ condition whenever the defining graph has no triangles with labels of
the form $(2,3,4),(2,2,n),(2,3,3)$ or $(2,3,5)$, but no justification is
presented in that text. Nevertheless, if the claim is true, then all
$2$-dimensional Artin groups admit minimal $C(6)$ graphical presentations.

When $X$ is a square complex whose fundamental group is not hyperbolic,
general constructions of $C(9)$ cubical presentations are harder to come by,
but can be produced by hand in specific situations. For instance, if
$X_{\Gamma}$ is the Salvetti complex of a RAAG whose defining graph $\Gamma$
has no triangles, then $dim(X_{\Gamma})\leq 2$, and one can produce a wealth
of cubical presentations $X^{*}_{\Gamma}=\langle
X_{\Gamma}|\\{Y_{i}\\}\rangle$ that satisfy the $C(9)$ condition and are
minimal. This is outlined in [Wis21, 3.s (4)], and will be elaborated upon in
forthcoming work [Are].

Thus, Theorems 4.7 and 4.17 are widely applicable, even when one starts with a
non-positively curved cube complex $X$ whose fundamental group is far from
being hyperbolic.

## References

  * [AB20] Karim Adiprasito and Bruno Benedetti.  Collapsibility of ${\rm CAT}(0)$ spaces.  Geom. Dedicata, 206:181–199, 2020.
  * [Ago08] Ian Agol.  Criteria for virtual fibering.  J. Topol., 1(2):269–284, 2008.
  * [Ago13] Ian Agol.  The virtual Haken conjecture.  Doc. Math., 18:1045–1087, 2013.  With an appendix by Agol, Daniel Groves, and Jason Manning.
  * [AH22] Goulnara N. Arzhantseva and Mark F. Hagen.  Acylindrical hyperbolicity of cubical small cancellation groups.  Algebr. Geom. Topol., 22(5):2007–2078, 2022.
  * [Are] Macarena Arenas.  Asphericity of cubical presentations: the general case.  in preparation.
  * [Are23a] Macarena Arenas.  A cubical rips construction.  2023\.  To appear in Algebraic & Geometric Topology. Available at https://arxiv.org/pdf/2202.01048.pdf.
  * [Are23b] Macarena Arenas.  Cubical small-cancellation theory and large-dimensional hyperbolic groups.  PhD thesis, University of Cambridge, 2023.
  * [BH99] Martin R. Bridson and André Haefliger.  Metric spaces of non-positive curvature.  Springer-Verlag, Berlin, 1999.
  * [BSV14] Mladen Bestvina, Michah Sageev, and Karen Vogtmann, editors.  Geometric group theory, volume 21 of IAS/Park City Mathematics Series.  American Mathematical Society, Providence, RI; Institute for Advanced Study (IAS), Princeton, NJ, 2014.
  * [BW12] Nicolas Bergeron and Daniel T. Wise.  A boundary criterion for cubulation.  Amer. J. Math., 134(3):843–859, 2012.
  * [CD95] Ruth Charney and Michael W. Davis.  The $K(\pi,1)$-problem for hyperplane complements associated to infinite reflection groups.  J. Amer. Math. Soc., 8(3):597–627, 1995.
  * [Cou11] Rémi Coulon.  Asphericity and small cancellation theory for rotation families of groups.  Groups Geom. Dyn., 5(4):729–765, 2011.
  * [Fen83] Roger A. Fenn.  Techniques of geometric topology, volume 57 of London Mathematical Society Lecture Note Series.  Cambridge University Press, Cambridge, 1983.
  * [Gre60] Martin Greendlinger.  On Dehn’s algorithms for the conjugacy and word problems, with applications.  Comm. Pure Appl. Math., 13:641–677, 1960.
  * [Gro03] M. Gromov.  Random walk in random groups.  Geom. Funct. Anal., 13(1):73–146, 2003.
  * [Gru15] Dominik Gruber.  Groups with graphical $C(6)$ and $C(7)$ small cancellation presentations.  Trans. Amer. Math. Soc., 367(3):2051–2078, 2015.
  * [Hag08] Frédéric Haglund.  Finite index subgroups of graph products.  Geom. Dedicata, 135:167–209, 2008.
  * [Hat02] Allen Hatcher.  Algebraic topology.  Cambridge University Press, Cambridge, 2002.
  * [HW12] Frédéric Haglund and Daniel T. Wise.  A combination theorem for special cube complexes.  Ann. of Math. (2), 176(3):1427–1482, 2012.
  * [HW15] Tim Hsu and Daniel T. Wise.  Cubulating malnormal amalgams.  Invent. Math., 199:293–331, 2015.
  * [Jan16] Kasia Jankiewicz.  The fundamental theorem of cubical small cancellation theory.  Trans. Amer. Math. Soc., 2016.
  * [JW22] Kasia Jankiewicz and Daniel T. Wise.  Cubulating small cancellation free products.  Indiana Univ. Math. J., 71(4):1397–1409, 2022.
  * [L0̈5] Wolfgang Lück.  Survey on classifying spaces for families of subgroups.  In Infinite groups: geometric, combinatorial and dynamical aspects, volume 248 of Progr. Math., pages 269–322. Birkhäuser, Basel, 2005.
  * [Lea13] Ian J. Leary.  A metric Kan-Thurston theorem.  J. Topol., 6(1):251–284, 2013.
  * [Lyn66] Roger C. Lyndon.  On Dehn’s algorithm.  Math. Ann., 166:208–228, 1966.
  * [Oll06] Yann Ollivier.  On a small cancellation theorem of Gromov.  Bull. Belg. Math. Soc. Simon Stevin, 13(1):75–89, 2006.
  * [Sag95] Michah Sageev.  Ends of group pairs and non-positively curved cube complexes.  Proc. London Math. Soc. (3), 71(3):585–617, 1995.
  * [Whi39] J. H. C. Whitehead.  Simplicial Spaces, Nuclei and m-Groups.  Proc. London Math. Soc. (2), 45(4):243–327, 1939.
  * [Wis21] Daniel T. Wise.  The structure of groups with a quasiconvex hierarchy, volume 209 of Annals of Mathematics Studies.  Princeton University Press, Princeton, NJ, 2021.