# Surface groups among cubulated hyperbolic and one-relator groups

Henry Wilton

###### Abstract

Let $X$ be a non-positively curved cube complex with hyperbolic fundamental
group. We prove that $\pi_{1}(X)$ has a non-free subgroup of infinite index
unless $\pi_{1}(X)$ is either free or a surface group, answering a question of
Wise. A similar result for one-relator groups follows, answering a question
posed by several authors. The proof relies on a careful analysis of free and
cyclic splittings of cubulated groups.

Surface groups – the fundamental groups of closed, aspherical surfaces – and
free groups are often seen as close cousins. One way this can be made precise
is to look at their subgroup structures. The Nielsen–Schreier theorem asserts
that every subgroup of a free group is free, while Whitehead also proved that
every subgroup _of infinite index_ in a surface group is free. It is natural
to wonder whether this property characterises free and surface groups.
Conjectures and questions along these lines have been stated under various
hypotheses [4, 3, 20, 18, 55].

To the best of the author’s knowledge, the following very general question is
open.

######  Question 0.1.

Let $G$ be an infinite, finitely presented group, such that every subgroup of
infinite index is free. Must $G$ be isomorphic to either a free group or a
surface group?

A finitely generated, but infinitely presented, group with every proper
subgroup isomorphic to $\mathbb{Z}$ was constructed by Ol’shanskii [36].
Clearly, a group $G$ with a surface subgroup can only have every subgroup of
infinite index free if $G$ itself is virtually a surface group, so Question
0.1 can be thought of as a useful precursor to the famous problem of finding
surface subgroups.

We provide positive answers to Question 0.1 for two important classes of
groups. Recall that a group $G$ is _cubulated_ if $G=\pi_{1}(X)$, where $X$ is
a compact, non-positively curved cube complex. Cubulated groups have been a
major theme in geometric group theory for several decades, culminating in the
work of Agol and Wise on the virtual Haken conjecture [1, 56]. Our main
theorem gives a strong positive answer to Question 0.1 for cubulated
hyperbolic groups.

######  Theorem A.

Let $G$ be a cubulated hyperbolic group. Unless $G$ is free or a surface
group, $G$ has a one-ended, quasiconvex subgroup of infinite index.

In particular, Theorem A applies to $C^{\prime}(1/6)$ small-cancellation
groups [54].

The proof makes essential use of the cubulation hypothesis, and removing it
seems to be well beyond current technology. The hyperbolicity hypothesis, on
the other hand, can be relaxed to the assumption that $G$ is _virtually
special_ in the sense of Haglund–Wise [26].111Agol’s theorem asserts that
cubulated hyperbolic groups are virtually special, so this is indeed a
relaxation [1].

######  Corollary B.

Let $G$ be the fundamental group of a compact, virtually special cube complex.
Unless $G$ is free or a surface group, $G$ has a one-ended, cubically convex-
cocompact subgroup of infinite index.

###### Proof.

If $G$ is hyperbolic then the result follows from Theorem A. Otherwise,
Caprace–Haglund proved that $G$ contains a cubically convex-cocompact subgroup
isomorphic to $\mathbb{Z}^{2}$ [10, Corollary 4], and the result follows
unless $\mathbb{Z}^{2}$ has finite index. But, in the latter case, $G$ is a
surface group by Bieberbach’s theorem. ∎

This corollary answers a question asked by Wise in the 2014 ICM proceedings
[55, Question 13.50].

Mapping tori of free-group automorphisms are another appealing class of groups
to which Theorem A applies.

######  Corollary C.

Let $F$ be a finitely generated free group of rank at least 2. Any semidirect
product $G=F\rtimes\mathbb{Z}$ has a finitely generated, one-ended subgroup.

###### Proof.

Brinkmann proved that $G$ is hyperbolic unless $G$ contains a subgroup
isomorphic to $\mathbb{Z}^{2}$ [9]. Since $F$ has rank at least 2, such a
subgroup must have infinite index, so the result follows unless $G$ is
hyperbolic. In this case, $G$ is cubulated by the work of Hagen–Wise [23, 24].
Since $G$ is neither free nor a surface group, the result again follows, this
time by Theorem A. ∎

Historically, Question 0.1 has received the most attention when $G$ is a one-
relator group. In this context, the question was attributed to
Fine–Rosenberger–Wienke in the Kourovka notebook [18, Question 20.22], but it
has also received attention in many other works on one-relator groups,
sometimes with additional hypotheses [3, 4, 14, 19]. In this context, it is
sometimes referred to as the _surface group conjecture_. In [52], the author
proved the surface group conjecture for Sela’s limit groups, and also for
graphs of free groups with cyclic edge groups.222The latter class includes the
classes of _cyclically pinched_ and _conjugacy pinched_ one-relator groups
considered in [14, 19].

Most recently, the surface group conjecture was addressed by
Gardam–Kielak–Logan [20, Conjecture 1.4], who proved it in the case when $G$
has two generators. Combining their work with Theorem A and other recent
advances in our understanding of one-relator groups, the surface group
conjecture follows.

######  Theorem D.

Let $G$ be an infinite one-relator group. Unless $G$ is free or a surface
group, $G$ has a subgroup of infinite index that is not free.

The alert reader will notice that the conclusion in the case of one-relator
groups is slightly weaker than the conclusions for cubulated hyperbolic or
free-by-cyclic groups. While Theorem A produces a finitely generated (indeed,
quasiconvex) subgroup that is not free, the non-free subgroup produced by
Theorem D may be infinitely generated.

######  Remark 0.2.

In fact, the proof of Theorem D produces a finitely generated non-free
subgroup unless $G$ has two generators. In the two-generator case, the
stronger result is not true: every finitely generated subgroup of infinite
index in the $(1,2)$-Baumslag–Solitar group $BS(1,2)$ is either trivial or
isomorphic to $\mathbb{Z}$, but $BS(1,2)$ contains a subgroup isomorphic to
the infinitely generated non-free group $\mathbb{Q}_{2}$ of dyadic rationals.
(Cf. Question 5.4 below.)

The proof of Theorem A is inspired by the author’s proof of the corresponding
statement for graphs of free groups with cyclic edge groups [52]. That proof
relied on Whitehead theory to recognise relative free splittings of free
groups, so our first task is to generalise that machinery to the context of
non-positively curved cube complexes.

Let $X$ be a compact, non-positively curved cube complex. We consider compact,
convex subcomplexes $Y$ of the universal cover $\widetilde{X}$, and define a
_Whitehead complex_ $\mathrm{Wh}_{X}(Y)$. This is a simplicial complex that
encodes the intersections of the bounding hyperplanes of $Y$, and provides a
topological model for the complement $\widetilde{X}\smallsetminus Y$. We will
be especially interested in the minimal cardinality $k$ of a cut set of
vertices of $\mathrm{Wh}_{X}(Y)$: if there is such a cut set, then $Y$ is
called a _$k$ -cut_. An essential cube complex $X$ has a 0-cut if and only if
$\pi_{1}(X)$ has more than one end.

We analyse the behaviour of the Whitehead complex when $Y$ is cut along a
hyperplane. It turns out that $\mathrm{Wh}_{X}(Y)$ decomposes as a connect sum
in the natural sense for simplicial complexes, in a way that generalises the
work of Manning [34] and Cashen–Macura [13] in the classical context of
Whitehead graphs. Our key technical results analyse how cut sets of vertices
behave under the connect sum operation. This enables us to relate cuts to
links of vertices. The following criterion, a cubical analogue of Whitehead’s
cut-vertex lemma, is a sample application.

######  Theorem E.

Let $X$ be a compact, non-positively curved cube complex. If the link of every
vertex is connected and has no cut simplices then $\pi_{1}(X)$ has at most one
end.

While 0-cuts detect free splittings, 1-cuts turn out to obstruct analysing
splittings over non-trivial subgroups. Fortunately, using recent work of
Shepherd [42], as long as $\pi_{1}(X)$ is one-ended we may assume that $X$ has
no 1-cuts.

Assuming $\pi_{1}(X)$ is one-ended, our next task is to analyse splittings
over $\mathbb{Z}$, and to do this we restrict to the case when $\pi_{1}(X)$ is
hyperbolic. The approach generalises work of Cashen–Macura [13] and Cashen
[12], who showed that a line pattern in a free group does not split over
$\mathbb{Z}$ if and only if there is a bound on the diameters of cut pairs in
the Whitehead graph. We prove an analogous result, Theorem 3.11, which asserts
that $\pi_{1}(X)$ is cyclically indecomposable if and only if there is a
uniform bound on the widths of 2-cuts. The point is that cyclic splittings
correspond to _periodic_ 2-cuts, from which a pigeonhole argument implies a
bound on the width of a 2-cut if $\pi_{1}(X)$ is cyclically indecomposable.
Using the projection-complex machinery of Bestvina–Bromberg–Fujiwara [5], we
strengthen this to a uniform bound on the widths of $k$-cuts, independent of
$k$.

With these technical results in hand, we are ready to prove Theorem A. Using
the Grushko and JSJ decompositions, and since the case of graphs of free
groups with cyclic edge groups is already known, we may reduce to the
cyclically indecomposable case. By the above analysis, we may assume that $X$
has no 0- or 1-cuts, and that there is a uniform bound $C$ on the widths of
$k$-cuts. Agol’s theorem [1] implies that hyperplane stabilisers are
separable, so we may pass to a finite-sheeted cover $X_{0}\to X$ with the
property that, for some hyperplane $H$ of $\widetilde{X}$, every
$\pi_{1}(X_{0})$-translate of $H$ is at distance $>C$ from $H$. Cutting
$X_{0}$ along $H$ and passing to a connected component $X^{\prime}$ gives a
quasiconvex subgroup that is necessarily of infinite index. But now a $0$-cut
in $X^{\prime}$ would lead to a $k$-cut in $X$ of width greater than $C$,
which is a contradiction, so $\pi_{1}(X^{\prime})$ is not free, as required.

The paper is structured as follows. Section 1 contains preliminaries on non-
positively curved complexes, cube complexes and simplicial complexes. Experts
may wish to jump straight to Section 2, where we define Whitehead complexes
and use them to analyse free splittings, including proving Theorem E. In
Section 3 we extend the analysis to cyclic splittings in the hyperbolic case.
Section 4 contains the proofs of the main theorems, while some open questions
are listed in Section 5.

### Acknowledgements

I am grateful to Mark Hagen for useful conversations about this work, most
especially for suggesting Lemma 3.16, which greatly simplifies the argument.
Thanks also to Macarena Arenas, Jack Button, Chris Cashen, Mark Hagen again,
Marco Linton, Lars Louder and Sam Shepherd for comments on an early draft.

##  1 Preliminaries

Two kinds of cell complexes will be of particular importance in this paper. On
the one hand, our main objects of study are non-positively curved cube
complexes and their universal covers, which are CAT(0). On the other hand, we
will study them by constructing certain simplicial complexes, called
_Whitehead complexes_. In this section, we recall some basic facts about all
these classes of complexes. The reader is referred to the standard reference
[8] for background material on cube complexes and other metric polyhedral
complexes, and for details of much of what follows.

###  1.1 Non-positively curved cube complxes

A _cube complex_ $X$ is a cell complex constructed from a disjoint union of
finite-dimensional unit cubes by identifying faces homeomorphically [8,
Example I.7.40(4)].

The _link_ $\mathrm{Lk}_{X}(x)$ of a point $x$ in a metric polyhedral complex
$X$ is the space of directions at $x$, i.e. the space of germs of geodesics
emanating from $x$ [8, I.7.38], and admits a natural angular metric. When $X$
is a cube complex and $x$ is a vertex, $\mathrm{Lk}_{X}(x)$ has the natural
structure of a simplicial complex, where the simplices are the corners of the
cubes incident at $x$.

As long as a cube complex $X$ is finite-dimensional, $X$ admits a natural
complete geodesic metric in which the interior of every $k$-cube is locally
isometric to $\mathbb{R}^{k}$. We will refer to this metric as the _$\ell^{2}$
-metric_ on $X$. Gromov’s link condition asserts that a polyhedral complex is
non-positively curved – i.e. the metric is locally CAT(0) – if and only if the
angular metric on the link of any point is CAT(1) [22, 4.2.A]. Gromov further
proved that, when $X$ is a cube complex, the link condition is satisfied if
and only if the link of every vertex is _flag_ , meaning that, for every
$k\geq 2$, every $k$-clique in the 1-skeleton of $X$ is the boundary of a
$k$-simplex [22, 4.2.C].

The features of a non-positively curved cube complex $X$ can often be captured
by studying their _hyperplanes_. Let $C$ be a $k$-dimensional cube, identified
with $[-1/2,1/2]^{k}\subseteq\mathbb{R}^{k}$. A _hypercube_ of $C$ is an
intersection of $C$ with any coordinate hyperplane of $\mathbb{R}^{k}$. The
set of hypercubes of cubes in $X$ form a natural directed system under
inclusion, and gluing along these inclusions gives a cube complex with a
natural map to $X$. The path components of this cube complexes are the
_hyperplanes_ of $X$.

It is immediate from the definition that the natural map from a hyperplane $H$
to $X$ is a local isometry. Therefore, if $X$ is non-positively curved then
$H$ is too.

Since hyperplanes are constructed by gluing together hypercubes of $X$, each
hyperplane $H\to X$ admits a natural _(closed) cubical neighbourhood_ $N(H)$,
which is a bundle over $H$ with fibre a closed interval. Likewise, $N(H)$ has
a natural _interior_ $\mathring{N}(H)$, also a bundle over $H$ with fibre an
open interval. The hyperplane $H$ is said to be _two-sided_ when these bundles
are trivial.

If the map $H\to X$ is injective we say that $H$ is _embedded_. In this case,
the interior $\mathring{N}(H)$ is also naturally embedded in $X$. The
complement $X_{0}=X\smallsetminus\mathring{N}(H)$ is a locally convex
subcomplex of $X$, and is said to be obtained by _cutting $X$ along $H$_. In
this case, we write $X=X_{0}\cup_{H}$. When $H$ is two-sided, this expresses
$X$ as a _graph of spaces_ in the sense of Scott and Wall [39]. As long as $X$
itself is connected, the vertex space $X_{0}$ has at most two components, and
if there are are two then we write $X=X_{1}\cup_{H}X_{2}$ in the obvious way.

Finally, we can make use of the following important lemma [8, Proposition
II.4.14].

######  Lemma 1.1.

Let $X,Y$ be complete, connected, geodesic, metric spaces. If $X$ is non-
positively curved then any local isometry $Y\to X$ is injective on fundamental
groups.

Thus, the inclusion of a hyperplane is $\pi_{1}$-injective, so cutting along
$H$ also decomposes the fundamental group $G=\pi_{1}(X)$ as a graph of groups
in the sense of Serre [40], with a single edge group $\pi_{1}(H)$.

###  1.2 CAT(0) cube complexes

When $X$ is non-positively curved, the natural geodesic metric on the
universal cover $\widetilde{X}$ is CAT(0), by the Cartan–Hadamard theorem [22,
§4]. This gives us access to the standard tools of CAT(0) geometry detailed in
[8], which are especially useful when studying a closed, convex subspace
$Y\subseteq\widetilde{X}$. For instance, we may define the orthogonal
projection $\pi_{Y}:\widetilde{X}\to Y$ as in [8, Proposition II.2.4].

One important consequence of the uniqueness of geodesics in CAT(0) spaces is
that any local isometry between CAT(0) spaces is a global isometric embedding
[8, Proposition II.4.14]. Thus, the map from a hyperplane $H\to X$ lifts to a
convex embedding of the universal cover
$\widetilde{H}\hookrightarrow\widetilde{X}$, and so the hyperplanes of
$\widetilde{X}$ are the universal covers of the hyperplanes of $X$.

The natural map $N(H)\to X$ is also a local isometry. Thus, the map $N(H)\to
H$ also lifts to an embedding

$\widetilde{N(H)}=N(\widetilde{H})\hookrightarrow\widetilde{X}\,.$

The interval bundle $N(\widetilde{H})$ is necessarily trivial because
$\widetilde{H}$ is simply connected, so $\widetilde{H}$ is embedded and two-
sided, and cutting along $\widetilde{H}$ realises $\widetilde{X}$ as a graph
of spaces. Because $\widetilde{X}$ is simply connected it follows that
$\widetilde{H}$ separates it into two, and so we have

$\widetilde{X}=\widetilde{U}_{+}\cup_{\widetilde{H}}\widetilde{U}_{-}$

for two convex subcomplexes $\widetilde{U}_{\pm}$. These subcomplexes are
called the _half-spaces_ of $\widetilde{H}$.

It is usually important when studying CAT(0) cube complexes to rule out
‘uninteresting’ half-spaces that are within a bounded distance of the
corresponding hyperplane.

######  Definition 1.2.

A hyperplane $\widetilde{H}$ of a CAT(0) cube complex $\widetilde{X}$ is
called _inessential_ if some half-space of $\widetilde{H}$ is within a bounded
neighbourhood of $\widetilde{H}$. If $\widetilde{X}$ has no inessential
hyperplanes then $\widetilde{X}$ itself is called _essential_. We will call a
non-positively curved complex $X$ essential or inessential according to its
universal cover.

Fortunately, we may always assume that $X$ is essential by invoking a result
of Caprace–Sageev [11, Proposition 3.5].

######  Lemma 1.3.

Every compact, non-positively curved cube complex $X$ deformation retracts to
an essential, convex subcomplex of its first subdivision.

Hyperplanes also make it possible to define a notion of convex hull for a
subset of $\widetilde{X}$.

######  Definition 1.4.

For any subset $Y$ of a CAT(0) cube complex $\widetilde{X}$, the _cubical
convex hull_ $\operatorname{Hull}(Y)$ is defined to be the intersection of the
half-spaces of $\widetilde{X}$ that contain $Y$.

It is important to be able to control the geometry of convex hulls, and we can
do this using a fundamental theorem of Haglund. This theorem makes use of the
_$\ell^{1}$ -metric_ on (the vertices of) $\widetilde{X}$.

######  Definition 1.5.

Let $\widetilde{X}$ be a CAT(0) cube complex. The $\ell^{1}$-metric on the
1-skeleton $X^{(1)}$ declares the distance between two points $x$ and $y$ to
be equal to the length of the shortest path in $X^{(1)}$ between them.

Note that the $\ell^{1}$ metric on $X^{(1)}$ is also geodesic, and that the
$\ell^{1}$ distance between two points is exactly the number of hyperplanes
that separate them. The next lemma is useful for relating the $\ell^{1}$ and
$\ell^{2}$ distances.

######  Lemma 1.6.

Let $Y_{1},Y_{2}$ be convex subcomplexes of a CAT(0) cube complex
$\widetilde{X}$. Suppose that a geodesic $\gamma$ minimises the distance from
$Y_{1}$ to $Y_{2}$. A hyperplane $H$ separates $Y_{1}$ from $Y_{2}$ if and
only if $\gamma$ crosses $H$.

###### Proof.

The necessity is obvious, so it remains to prove that, if $\gamma$ crosses
$H$, then $H$ separates $Y_{1}$ from $Y_{2}$. Suppose that $\gamma(0)\in
Y_{1}$ and $\gamma$ crosses $H$ at $\gamma(t)$. If $H$ intersects $Y_{1}$ then
$\gamma(0)$, $\gamma(t)$ and $\pi_{H}(\gamma(0))$ forms a triangle with two
right angles, contradicting the hypothesis that $\widetilde{X}$ is CAT(0).
Therefore, $H$ is disjoint from $Y_{1}$, and likewise from $Y_{2}$. Since
$\gamma$ crosses $H$ once, it follows that $H$ separates $Y_{1}$ from $Y_{2}$,
as required. ∎

We can now state Haglund’s theorem [25, Theorem H]

######  Theorem 1.7 (Haglund).

Let $\widetilde{X}$ be a $d$-dimensional CAT(0) cube complex and
$Y\subseteq\widetilde{X}^{(0)}$ a set of vertices that is $\kappa$-quasiconvex
in the $\ell^{1}$-metric. There is $R=R(d,\kappa)$ such that every point of
$\operatorname{Hull}(Y)$ is at distance at most $R$ from a point of $Y$.

One useful consequence is to characterise geometrically well-behaved
subgroups. A subgroup $H$ of the fundamental group of a non-positively curved
cube complex $X$ is called _cubically convex-cocompact_ if $H$ acts
cocompactly on a subcomplex $Y\subseteq\widetilde{X}$ that is convex in the
$\ell^{2}$ metric.

######  Corollary 1.8.

Let $X$ be a compact, non-positively curved cube complex. A subgroup $H$ of
$\pi_{1}(X)$ is cubically convex-cocompact if and only if the $H$-orbit of
some vertex is quasiconvex in the $\ell^{1}$ metric.

###  1.3 Simplicial complexes

As the above discussion makes clear, our investigations of cube complexes will
often be metric. By contrast, when simplicial complexes arise, we will be
interested in their combinatorial structure, and especially interested in
collections of vertices and simplices that separate. Here, we introduce the
necessary terminology.

We already saw that, in a metric complex, the link of a point can be thought
of metrically as a space of directions. In a simplicial complex, it is
convenient to give an equivalent combinatorial definition, which also extends
to certain subcomplexes. A subcomplex $B$ of a simplicial complex $A$ is
called _full_ if, whenever all the vertices of a simplex $\sigma$ are in $B$,
then $\sigma$ itself is in $B$.

######  Definition 1.9 (Stars and links).

Let $A$ be a simplicial complex and $B\subseteq A$ a full subcomplex. The
_open star_ of $B$, $\mathrm{St}_{A}(B)$, is the union of the set of simplices
of $A$ that contain a simplex of $B$ as a face. The _link_ of $B$,
$\mathrm{Lk}_{A}(B)$, is the set of faces of simplices of $\mathrm{St}_{A}(B)$
that are disjoint from $B$.

Note that $\mathrm{St}_{A}(B)$ may not be a subcomplex of $A$, but
$\mathrm{Lk}_{A}(B)$ always is. Furthermore, after putting a suitable metric
on $A$, for any vertex $v$, $\mathrm{Lk}_{A}(v)$ coincides with the metric
definition of link given in §1.1.

As mentioned above, we are especially interested in the separation properties
of subcomplexes.

######  Definition 1.10 (Cut sets and connectivity).

Let $A$ be a simplicial complex and $B$ a full subcomplex. Write
$A_{B}:=A\smallsetminus\mathrm{St}(B)$. The subcomplex $B$ is said to
_separate_ $A$ if $A_{B}$ is disconnected. If $B$ is a simplex we say $B$ is a
_cut simplex_.

In the specific case where $B$ is a set of $k$ pairwise non-adjacent vertices,
we say that $B$ is a _cut set_ for $A$. If $\\#B=1$ we say $B$ is a _cut
vertex_ , and if $\\#B=2$ we say $B$ is a _cut pair_. If $A$ has no cut set of
cardinality at most $k$, then $A$ is said to be _$k$ -connected_.

##  2 Free splittings

In 1936, J.H.C. Whitehead explained how to determine whether an element $w$ of
a free group $F$ is primitive. For a fixed basis $B$ of $F$, his algorithm
uses a graph $\mathrm{Wh}_{B}(w)$ which is now known as the _Whitehead graph_
[51]. His algorithm can equally be interpreted as searching for free
splittings $F\cong\langle w\rangle*F^{\prime}$. Many elaborations and
generalisations of Whitehead’s argument have been given over the years by,
among others, Lyon [33], Stallings [46] and Stong [47]. Diao–Feighn, building
on work of Gersten [21], gave an algorithm to compute the Grushko
decomposition of a graph of free groups [15].333See the recent preprint of
Dicks for a comprehensive history of this topic [16, §2].

The power of this point of view is that _global_ information about a group $G$
– whether or not $G$ is free, or splits freely, or most generally the
structure of the Grushko decomposition of $G$ – is encoded in the _local_
information of a Whitehead graph, which can be read off a presentation. The
following theorem, which is the result of combining Whitehead’s algorithm with
a theorem of Shenitzer [41, Theorem 5], exemplifies this idea in the case of
one-relator groups.

######  Theorem 2.1.

Consider a one-relator group $G=F/\llangle w\rrangle$ and fix a basis $B$ for
$F$. If the Whitehead graph $\mathrm{Wh}_{B}(w)$ is connected without cut
vertices, then $G$ does not split freely.

Note that $\mathrm{Wh}_{B}(w)$ is exactly the link of the unique vertex in the
natural presentation complex for $G$. Our goal is to develop a similar theory
for non-positively curved cube complexes. To this end, our first task is to
generalise the notion of Whitehead graph to the context of non-positively
curved cube complexes.

###  2.1 Bounding hyperplanes and Whitehead complexes

Let $X$ be a non-positively curved cube complex, let $\widetilde{X}\to X$ be
its universal cover and let $Y\subseteq\widetilde{X}$ be a convex subcomplex.
Equivalently, we may think of $Y$ as a CAT(0) complex equipped with a locally
isometric combinatorial map $Y\to X$ – any such immersion lifts to an
embedding $Y\hookrightarrow\widetilde{X}$, and all such embeddings differ by a
deck transformation of $X$.

######  Definition 2.2.

A hyperplane $H$ of $\widetilde{X}$ _bounds_ $Y$ if it is disjoint from $Y$
but no other hyperplane separates $Y$ from $H$. Equivalently, $H$ and $Y$ are
disjoint, but $Y$ intersects the (closed) regular neighbourhood $N(H)\cong
H\times[0,1]$. We may orient the interval factor so that the intersection is
of the form $H_{Y}\times\\{0\\}$, where $H_{Y}$ is a convex subcomplex of $H$
called the _bounding hyperplane component of $Y$ in $H$_.

The Whitehead complex associated to such a $Y$ encodes the way that the
bounding hyperplanes of $Y$ intersect each other.

######  Definition 2.3.

Let $X$ be a non-positively curved cube complex, and $Y$ a convex subcomplex
of the universal cover $\widetilde{X}$. The _Whitehead complex_ of $Y$,
$\mathrm{Wh}_{X}(Y)$, is a simplicial complex defined as follows. The vertices
of $\mathrm{Wh}_{X}(Y)$ are the bounding hyperplanes of $Y$. A finite set of
bounding hyperplanes $\mathcal{H}=\\{H_{0},\ldots,H_{k}\\}$ spans a
$k$-simplex $\sigma_{\mathcal{H}}$ in $\mathrm{Wh}_{X}(Y)$ if and only if the
intersection $\bigcap_{i=0}^{k}H_{i}$ is non-empty.

######  Remark 2.4.

The Helly property for hyperplanes of CAT(0) cube complexes, proved by Sageev,
asserts that if a set of hyperplanes intersect pairwise then their
intersection is non-empty [38, Theorem 4.14]. Hence, Whitehead complexes are
flag.

The name is justified by the following example.

######  Example 2.5.

Let $F$ be a free group, thought of as the fundamental group of a bouquet of
circles $R$. (This is equivalent to choosing a basis $B$ for $F$.) The
conjugacy class of a non-trivial element $w\in F$ can be represented by an
immersion $w:S^{1}\to R$. The mapping cylinder $X$ of this immersion is
naturally a non-positively curved square complex. Let $*$ be the one-point
space and consider the map $*\to X$ with image equal to the unique vertex of
$R$. Then $\mathrm{Wh}_{X}(*)$ is the Whitehead graph $\mathrm{Wh}_{B}(w)$,
subdivided once.

More generally, in [13], Cashen and Macura defined _generalised Whitehead
graphs_ $\mathrm{Wh}_{B}(T)\\{w\\}$ associated to a word $w$ in a free group
$F$, for any subtree $T$ of the Cayley tree $\widetilde{R}$ of $F$. Our
complex $\mathrm{Wh}_{X}(T)$ is the graph obtained from
$\mathrm{Wh}_{B}(T)\\{w\\}$ by subdividing once.

The case when $Y$ is a vertex provides another important example.

######  Example 2.6.

Consider $\tilde{y}$ a vertex of $\widetilde{X}$ with image $y\in X$. The
bounding hyperplanes of $\tilde{y}$ are dual to the edges incident at $y$, and
a set of such hyperplanes cross exactly when the edges bound a cube with
corner $\tilde{y}$. Therefore,
$\mathrm{Wh}_{X}(\tilde{y})\cong\mathrm{Lk}_{X}(y)$ as simplicial complexes.

Whitehead complexes are useful because they provide a topological model for
the complement of $Y$: we shall see that $Wh_{X}(Y)$ is homotopy equivalent to
the complement $\widetilde{X}\smallsetminus Y$. In order to describe this
homotopy equivalence, it is useful to define the _link_ of $Y$, and to extend
our discussion of bounding hyperplanes to _codimension- $k$ hyperplanes_.

######  Definition 2.7.

Consider a set of $k$ distinct hyperplanes $H_{1},\ldots,H_{k}$ of $Y$ that
pairwise intersect. The intersection

$K=\bigcap_{i=1}^{k}H_{i}$

is called a _codimension- $k$ hyperplane_. It has a closed regular
neighbourhood $N(K)\cong K\times[0,1]^{k}$. If the $H_{i}$ are all bounding
hyperplanes of $Y$ then $K$ is called a _bounding codimension- $k$ hyperplane
of $Y$_. In this case, we may orient the interval factors so that

$N(K)\cap Y=K_{Y}\times\\{0\\}^{k}$

for some convex subcomplex $K_{Y}$, called the _bounding codimension- $k$
hyperplane component of $K$ in $Y$_.

The _link_ of $K_{Y}\times\\{0\\}^{k}$ in $K_{Y}\times[0,1]^{k}$ consists of
unit vectors orthogonal to $K_{Y}\times\\{0\\}^{k}$, and is naturally of the
form $K_{Y}\times\sigma_{K}$, where $\sigma_{K}$ is a $(k-1)$-simplex.

Whenever a codimension-$k$ hyperplane $K$ is contained in a codimension-$l$
hyperplane $L$ (necessarily with $l\geq k$), there are natural inclusions
$K_{Y}\subseteq L_{Y}$ and $\sigma_{L}\subseteq\sigma_{K}$, which in turn lead
to inclusions

$K_{Y}\times\sigma_{L}\subseteq
K_{Y}\times\sigma_{K}\,,\,K_{Y}\times\sigma_{L}\subseteq
L_{Y}\times\sigma_{L}\,.$ (1)

Gluing along these inclusions defines the _link_ of the subcomplex $Y$.

######  Definition 2.8.

The _link_ of a convex subcomplex $Y$ of a CAT(0) cube complex $\widetilde{X}$
is the cell complex $\mathrm{Lk}_{\widetilde{X}}(Y)$ defined from the disjoint
union

$\coprod_{K}K_{Y}\times\sigma_{K}$

over all bounding hyperplane components of any codimension, by gluing along
the natural inclusions (1). As usual, when $\widetilde{X}$ is the universal
cover of a non-positively curved cube complex $X$, we will write
$\mathrm{Lk}_{X}(Y)$.

######  Remark 2.9.

The link $\mathrm{Lk}_{\widetilde{X}}(Y)$ can also be described metrically as
the germs of geodesics in $\widetilde{X}$ that are orthogonal to $Y$. Thus,
orthogonal projection defines a natural map
$\hat{\pi}_{Y}:\widetilde{X}\smallsetminus
Y\to\mathrm{Lk}_{\widetilde{X}}(Y)$.

We can now prove that $\mathrm{Wh}_{X}(Y)$ is homotopy equivalent to the
complement of $Y$ in two stages.

######  Lemma 2.10.

For any convex subcomplex $Y$ of a CAT(0) cube complex $\widetilde{X}$, the
orthogonal projection $\hat{\pi}_{Y}:\widetilde{X}\smallsetminus
Y\to\mathrm{Lk}_{\widetilde{X}}(Y)$ is a homotopy equivalence.

###### Proof.

As in the remark, we identify the link $\mathrm{Lk}_{\widetilde{X}}(Y)$ with
the set of germs of geodesics orthogonal to $Y$. Since $Y$ is a subcomplex,
every such germ extends uniquely to a geodesic of length $1$. Therefore,
$\hat{\pi}_{Y}$ has a well-defined right-inverse
$\rho:\mathrm{Lk}_{\widetilde{X}}(Y)\to\widetilde{X}\smallsetminus Y$ sending
the germ of $\gamma$ to $\gamma(1)$. For any $x\notin Y$, let $\gamma_{x}$
denote the unique geodesic from $\hat{\pi}_{Y}(x)$ to $x$. Then

$(x,t)\mapsto\gamma_{x}((1-t)d(x,Y)+t)$

defines a homotopy equivalence between the identity and
$\rho\circ\hat{\pi}_{Y}$, which completes the proof. ∎

Finally, we relate links to Whitehead complexes by contracting bounding
hyperplane components.

######  Lemma 2.11.

For any convex subcomplex $Y$ of a CAT(0) cube complex $\widetilde{X}$, there
is a homotopy equivalence
$\mathrm{Lk}_{\widetilde{X}}(Y)\simeq\mathrm{Wh}_{\widetilde{X}}(Y)$.

###### Proof.

By definition, $\mathrm{Wh}_{\widetilde{X}}(Y)$ is the complex obtained by
contracting each $K_{Y}$ factor to a point in the definition of
$\mathrm{Lk}_{\widetilde{X}}(Y)$. Since $\mathrm{Lk}_{\widetilde{X}}(Y)$ is a
cell complex and each $K_{Y}$ is contractible, the result follows by, for
example, [27, Proposition 0.16]. ∎

The previous two lemmas combine to give the following fundamental result.

######  Lemma 2.12.

For any convex subcomplex $Y$ of a CAT(0) cube complex $\widetilde{X}$, there
is a homotopy equivalence $\widetilde{X}\smallsetminus
Y\simeq\mathrm{Wh}_{\widetilde{X}}(Y)$.

Finally, it will be useful to understand the links of vertices in Whitehead
complexes. (Note that here, in the setting of simplicial complexes, we mean
link in the sense of Definition 1.9.)

######  Lemma 2.13.

Let $Y$ be a convex subcomplex of a CAT(0) cube complex $\widetilde{X}$ and
let $H$ be a bounding hyperplane of $Y$. Then

$\mathrm{Lk}_{\mathrm{Wh}_{\widetilde{X}}(Y)}(H)\cong\mathrm{Wh}_{H}(H_{Y})\,.$

###### Proof.

Since both simplicial complexes are flag, it suffices to define an isomorphism
at the level of 1-skeleta.

The vertices of $\mathrm{Lk}_{\mathrm{Wh}_{\widetilde{X}}(Y)}(H)$ are, by
definition, the bounding hyperplanes of $Y$ that cross $H$. For every such
hyperplane $K$, the intersection $H\cap K$ is a bounding hyperplane of $H_{Y}$
in $H$. Conversely, every bounding hyperplane of $H_{Y}$ in $H$ extends
uniquely to a bounding hyperplane of $Y$ that crosses $H$. Thus, $K\mapsto
H\cap K$ defines the required bijection on the 0-skeleta. The bijection
extends to an isomorphism of 1-skeleta, because bounding codimension-three
hyperplanes of $Y$ contained in $H$ (which correspond to edges of
$\mathrm{Lk}_{\mathrm{Wh}_{\widetilde{X}}(Y)}(H)$) are the same thing as
bounding codimension-2 hyperplanes of $H_{Y}$ in $H$ (which correspond to
edges of $\mathrm{Wh}_{H}(H_{Y})$). ∎

###  2.2 Ends and splittings

Invoking a famous theorem of Stallings, the number of _(Freudenthal) ends_ of
the universal cover of a non-positively curved cube complex $X$ detects the
free splittings of the fundamental group [45]. (Note that $\pi_{1}(X)$ is
necessarily torsion-free.)

######  Theorem 2.14 (Stallings).

Let $X$ be a compact, non-positively curved cube complex.

  1. (i)

$\mathrm{Ends}(\widetilde{X})=0$ if and only if $\pi_{1}(X)$ is trivial.

  2. (ii)

$\mathrm{Ends}(\widetilde{X})>1$ if and only if $\pi_{1}(X)$ splits
freely.444A non-trivial free splitting corresponds to an action on a
simplicial tree without a global fixed point and with trivial edge
stabilisers. Note that $\mathbb{Z}\cong 1*_{1}$ splits freely.

We will see that the number of ends can be conveniently encoded in the
connectivity properties of Whitehead graphs. The next result is prototypical,
and deals with the case when $X$ is simply connected.

######  Proposition 2.15.

Let $X$ be a compact, non-positively curved cube complex. There is a compact
CAT(0) cube complex $Y$ mapping to $X$ with $\mathrm{Wh}_{X}(Y)=\varnothing$
if and only if $\pi_{1}(X)$ is trivial.

###### Proof.

The Whitehead graph $\mathrm{Wh}_{X}(Y)$ is empty if and only if
$Y=\widetilde{X}$. Thus such a compact $Y$ exists if and only if
$\widetilde{X}$ is compact, meaning that $\pi_{1}(X)$ is finite, i.e. trivial.
∎

In §1.2, we saw that is is often convenient to assume that our cube complexes
are essential. In fact, when studying free splittings, we will only need a
weaker notion.

######  Definition 2.16.

A CAT(0) cube complex $\widetilde{X}$ is called _freely inessential_ if some
(necessarily compact) hyperplane $H\subseteq\widetilde{X}$ has the property
that one of the associated half-spaces is compact. Otherwise, $\widetilde{X}$
is called _freely essential_.

######  Remark 2.17.

Any compact CAT(0) cube complex is freely inessential, unless it is a point,
and any essential cube complex is freely essential.

The next result explains how Whitehead complexes encode free splittings of the
fundamental group.

######  Proposition 2.18.

Let $\widetilde{X}$ be a freely essential CAT(0) cube complex. There is a
compact, convex $Y\subseteq\widetilde{X}$ such that
$\mathrm{Wh}_{\widetilde{X}}(Y)$ is non-empty and disconnected if and only if
$\widetilde{X}$ has more than one end.

###### Proof.

Suppose such a $Y$ exists. Since $\mathrm{Wh}_{\widetilde{X}}(Y)$ is
disconnected, Lemma 2.12 implies that $\widetilde{X}\smallsetminus Y$ is also
disconnected, so $Y$ separates $\widetilde{X}$ into at least two components.
If one of these components is bounded then any bounding hyperplane in the
corresponding component of $\mathrm{Wh}_{\widetilde{X}}(Y)$ has a compact half
space, so $\widetilde{X}$ is freely inessential. Otherwise, all components are
unbounded so $\widetilde{X}$ has more than one end.

For the converse direction, suppose that $\widetilde{X}$ has more than one
end, so there is a compact subcomplex $Y\subseteq\widetilde{X}$ such that
$\widetilde{X}\smallsetminus Y$ has at least two unbounded components. By
Theorem 1.7, we may replace $Y$ by its convex hull and assume that $Y$ is
convex. Since $\widetilde{X}\smallsetminus Y$ has at least two components, so
does $\mathrm{Wh}_{\widetilde{X}}(Y)$ by Lemma 2.12. ∎

The proposition connects Whitehead complexes and free splittings, but the
resulting criterion is ineffective because there is no bound on the possible
size of a $Y$ for which $\mathrm{Wh}_{X}(Y)$ is disconnected. To make the
criterion effective, we need to understand how to build up Whitehead complexes
inductively by gluing, generalising the _splicing_ construction of Manning
[34] used by Cashen–Macura [13].

###  2.3 Connected sums of simplicial complexes

The definition of the connected sum of two simplicial complexes is directly
analogous to the definition for manifolds.

######  Definition 2.19 (Connected sum of complexes).

Let $A,B$ be simplicial complexes, let $a\in A$, $b\in B$ be vertices, and let
$\phi:\mathrm{Lk}_{A}(a)\to\mathrm{Lk}_{B}(b)$ be an isomorphism. The complex

$A\\#_{\phi}B:=A_{a}\cup_{\phi}B_{b}$

is the _connected sum of $A$ to $B$ along $\phi$_. (Often, we will have a
natural identification $v=a=b$, and $\phi$ will be implicit. In this case, we
may write $A\\#_{v}B$ instead.)

######  Remark 2.20.

Definition 2.19 generalises the _splicing_ construction used by Manning to
analyse Whitehead graphs in finite-index subgroups [34], and exploited further
by Cashen and Macura [12, 13]. Up to taking first subdivisions, Manning’s
definition coincides with Definition 2.19 when restricted to graphs. (Recall
from Example 2.5 that the classical Whitehead graph is the first subdivision
of our Whitehead complex.)

The next result explains how cutting $Y$ along a hyperplane decomposes the
Whitehead complex as a connected sum. When a convex subcomplex
$Y\subseteq\widetilde{X}$ intersects a hyperplane $H$ of $\widetilde{X}$, we
will abuse notation by identifying $H$ with $H\cap Y$. Cutting $Y$ along $H$
then leads to the decomposition

$Y=Y_{1}\cup_{H}Y_{2}\,;$

note that $H$ is then a bounding hyperplane of both $Y_{1}$ and $Y_{2}$.

######  Proposition 2.21 (Cutting and connected sum).

Let $Y$ be a convex subcomplex of a CAT(0) cube complex $\widetilde{X}$. If

$Y=Y_{1}\cup_{H}Y_{2}$

for some hyperplane $H$ of $\widetilde{X}$ then

$\mathrm{Wh}_{X}(Y)\cong\mathrm{Wh}_{X}(Y_{1})\\#_{H}\mathrm{Wh}_{X}(Y_{2})\,.$

###### Proof.

For notational convenience, write $W_{i}=\mathrm{Wh}_{X}(Y_{i})$ for $i=1,2$,
and $W=\mathrm{Wh}_{X}(Y)$. Our first task is to define the isomorphism
$\mathrm{Lk}_{W_{1}}(H)\cong\mathrm{Lk}_{W_{2}}(H)$ needed to define the
connected sum.

A $(k-1)$-simplex $\sigma_{K}$ of $W_{i}$ corresponds to a bounding
codimension-$k$ hyperplane $K$ of $Y_{i}$. This simplex $\sigma_{K}$ is
contained in $\mathrm{Lk}_{W_{i}}(H)$ exactly when $K$ crosses $H$. A
codimension-$k$ hyperplane that crosses $H$ bounds $Y_{1}$ if and only it
bounds $Y_{2}$, so $\mathrm{Lk}_{W_{1}}(H)$ is naturally identified with
$\mathrm{Lk}_{W_{2}}(H)$ as required. Let $L$ denote the natural copy of
$\mathrm{Lk}_{W_{1}}(H)\equiv\mathrm{Lk}_{W_{2}}(H)$ in $W_{1}\\#_{H}W_{2}$.

To see that $W\cong W_{1}\\#_{H}W_{2}$, note that the $(k-1$)-simplices of
$W_{1}\\#_{H}W_{2}$ correspond bijectively to bounding codimension-$k$
hyperplanes of $Y$. Indeed, the simplices of $W_{1}\\#_{H}W_{2}$ are of three
kinds:

  1. (i)

simplices $\sigma_{K}$ in $L$ correspond to bounding codimension-$k$
hyperplanes of $Y$ that cross $H$;

  2. (ii)

simplices $\sigma_{K}$ of $W_{1}$ that are not adjacent to $H$ correspond to
bounding codimension-$k$ hyperplanes of $Y_{1}$ disjoint from $H$;

  3. (iii)

simplices $\sigma_{K}$ of $W_{2}$ that are not adjacent to $H$ correspond to
bounding codimension-$k$ hyperplanes of $Y_{2}$ disjoint from $H$.

Since this list also covers all the codimension-$k$ hyperplanes of $Y$, and
inclusions are preserved, the result follows. ∎

###  2.4 Cut-set lemmas in connnected sums

To understand splittings of fundamental groups of non-positively curved cube
complexes, the key idea is to combine Proposition 2.21 with an understanding
of how the connectivity properties of simplicial complexes, in the sense of
Definition 1.10, change under connected sum. The first lemma is prototypical.

######  Lemma 2.22.

Suppose that $A$ and $B$ are simplicial complexes. If $B$ is connected but
$A\\#_{v}B$ is disconnected, then $v$ is a cut vertex in $A$.

###### Proof.

Let $L$ denote the link of $v$ and $C(L)$ the cone on $L$. Then
$B=B_{v}\cup_{L}C(L)$ so, because $C(L)$ is contractible, the Mayer–Vietoris
sequence for reduced homology gives that

$\tilde{H}_{0}(L)\to\tilde{H}_{0}(B_{v})\to\tilde{H}_{0}(B)\to 0$

is exact. Therefore, $\tilde{H}_{0}(L)$ surjects $\tilde{H}_{0}(B_{v})$
because $B$ is connected.

On the other hand, a second application of Mayer–Vietoris to
$A\\#_{v}B=A_{a}\cup_{L}B_{b}$ gives that

$\tilde{H}_{0}(L)\to\tilde{H}_{0}(A_{v})\oplus\tilde{H}_{0}(B_{v})\to\tilde{H}_{0}(A\\#_{v}B)\to
0$

is exact, so $\tilde{H}_{0}(L)$ does _not_ surject
$\tilde{H}_{0}(A_{v})\oplus\tilde{H}_{0}(B_{v})$, because $A\\#_{v}B$ is
disconnected. Hence $\tilde{H}_{0}(A_{v})\neq 0$, i.e. $v$ is a cut vertex in
$A$. ∎

Since we will need to apply the lemma inductively, we also need to understand
when cut vertices can arise, and more generally cut simplices.

######  Lemma 2.23.

Suppose that $A$ and $B$ are connected and a simplex $\sigma$ of $A$ is a cut
simplex of $A\\#_{v}B$. Then one of the following holds:

  1. (i)

$v$ is a cut vertex in $B$;

  2. (ii)

$\sigma$ is a cut simplex in either $A$ or $B$; or

  3. (iii)

$\sigma$ is contained in the link of $v$, and the simplex
$\tau=\langle\sigma,v\rangle$ spanned by $\sigma$ and $v$ together is a cut
simplex in both $A$ and $B$.

In particular, if $A\\#_{v}B$ has a cut simplex then either $A$ or $B$ has a
cut simplex.

###### Proof.

Again, let $C=A\\#_{v}B$ for notational convenience. The hypothesis that
$\sigma$ is a cut simplex of $C$ means that $C_{\sigma}$ is disconnected. The
proof now divides into cases, according to whether $\sigma$ is in $A$ but not
in $B$, or $\sigma$ is contained in the canonical copy of the link of $v$.

In the first case, when $\sigma$ is in $A$ but not in $B$, we have

$C_{\sigma}=A_{\sigma}\\#_{v}B\,,$

so Lemma 2.22 implies that $v$ is a cut vertex in $B$, as in item (i).

In the second case, if $\sigma$ is in the link of $v$ then

$C_{\sigma}=A_{\sigma}\\#_{v}B_{\sigma}\,,$

so Lemma 2.22 leads to two further subcases: either one of $A_{\sigma}$ or
$B_{\sigma}$ is disconnected, or $v$ is a cut vertex in both $A_{\sigma}$ and
$B_{\sigma}$. The former corresponds to item (ii), while the latter case gives
item (iii). ∎

The above results will help us to understand free splittings of $\pi_{1}(X)$.
More generally, we will need to understand how arbitrary cut sets decompose.

######  Lemma 2.24.

Suppose that $U$ is a full set of vertices of $A$ and $V$ is a full set of
vertices of $B$, both non-adjacent to a common vertex $v$. If $U\cup V$ is a
cut set of $A\\#_{v}B$ then: either

  1. (i)

$V$ is a cut set in $B$; or

  2. (ii)

$U\cup\\{v\\}$ is a cut set in $A$.

###### Proof.

Again, let $C=A\\#_{v}B$ for notational convenience. Because $v$ is adjacent
to neither $U$ nor $V$,

$C_{U\cup V}=A_{U}\\#_{v}B_{V}$

which is disconnected by hypothesis. Therefore, by Lemma 2.22, either $B_{V}$
is disconnected, giving item (i), or $v$ is a cut vertex in $A_{U}$, giving
item (ii). ∎

###  2.5 Cuts and the generalised Whitehead’s lemma

In this section, we combine the results developed so far to prove a sufficient
condition for the fundamental group of a compact, non-positively curved cube
complex $X$ to be one-ended. These conditions are phrased in terms of _$k$
-cuts_.

######  Definition 2.25 ($k$-cuts).

Consider a finite CAT(0) cube complex equipped with an immersion $Y\to X$. If
$\mathrm{Wh}_{X}(Y)$ has a finite cut set then $Y$ is said to be a _cut_. More
precisely, if $\mathrm{Wh}_{X}(Y)$ is $(k-1)$-connected but not $k$-connected
– i.e. if the smallest cut set for $\mathrm{Wh}_{X}(Y)$ has cardinality $k$ –
then $Y$ is said to be a _$k$ -cut_. If $\\{H_{1},\ldots,H_{k}\\}$ is a
minimal cut set for a $k$-cut $Y$ then we say that $Y$ is a cut _between
$H_{1},\ldots,H_{k}$_.

Our analysis of splittings of fundamental groups of non-positively curved cube
complexes will rest on understanding cuts. We start with 0-cuts, which
characterise free splittings. The results of §2.4 explain how 0-cuts
decompose.

######  Lemma 2.26.

Consider a finite CAT(0) cube complex equipped with a locally isometric
immersion $Y\to X$, and let $Y=Y_{1}\cup_{H}Y_{2}$ for some hyperplane $H$ of
$Y$. If $Y$ is a 0-cut then either one of $Y_{1}$ or $Y_{2}$ is a 0-cut, or
both $Y_{1}$ and $Y_{2}$ are 1-cuts.

###### Proof.

The immersion lifts to an embedding of $Y$ as a convex subcomplex of the
universal cover $\widetilde{X}$, and the hyperplane $H$ of $Y$ then extends
uniquely to a hyperplane of $\widetilde{X}$, still denoted by $H$. Proposition
2.21 implies that

$\mathrm{Wh}_{X}(Y)\cong\mathrm{Wh}_{X}(Y_{1})\\#_{H}\mathrm{Wh}_{X}(Y_{2})\,,$

so the result follows from Lemma 2.22. ∎

To iterate the argument, we also need to analyse cut simplices.

######  Lemma 2.27.

Consider a finite CAT(0) cube complex equipped with a locally isometric
immersion $Y\to X$, and let $Y=Y_{1}\cup_{H}Y_{2}$ for some hyperplane $H$ of
$Y$. If $\mathrm{Wh}_{X}(Y)$ has a cut simplex, then one of
$\mathrm{Wh}_{X}(Y_{1})$ or $\mathrm{Wh}_{X}(Y_{2})$ has a cut simplex.

###### Proof.

This follows in the same way as Lemma 2.26, using Proposition 2.21 and Lemma
2.23. ∎

We now have all the tools needed to prove Theorem E, which we think of as a
version of Whitehead’s lemma for non-positively curved cube complexes.

###### Proof of Theorem E.

Let $Y$ be a CAT(0) cube complex, locally isometrically immersed in $X$. We
prove that $\mathrm{Wh}_{X}(Y)$ is connected without cut simplices, by
induction on the number of vertices of $Y$.

In the base case, $Y$ is a single vertex $y$, so
$\mathrm{Wh}_{X}(Y)\cong\mathrm{Lk}_{X}(y)$ by Example 2.6 and there is
nothing to prove. If $Y$ has more than one vertex, we may choose a
(necessarily separating) hyperplane $H$ of $Y$ and write
$Y=Y_{1}\cup_{H}Y_{2}$. By the inductive hypothesis, both $Y_{1}$ and $Y_{2}$
are connected without cut simplices, and therefore so is $Y$ by Lemma 2.27.

In conclusion, $X$ has no 0-cuts, so Proposition 2.18 implies that
$\pi_{1}(X)$ has at most one end. ∎

Theorem E provides a convenient certificate that $\pi_{1}(X)$ does not split
freely. However, the necessary condition that it provides is not sufficient:
there are non-positively curved square complexes in which the link of a vertex
contains a cut vertex, and yet the universal cover is one-ended. The following
example is based on [28, Example 2.1].

######  Example 2.28.

Consider the rank-two free group $F=\langle a,b\rangle$, thought of as the
fundamental group of the wedge of two circles $\Gamma=S^{1}\vee S^{1}$.
Consider the element $w=abab^{2}ab^{3}$ of $F$ and realise the corresponding
cyclic word by an immersion of a circle $\gamma:S^{1}\to\Gamma$. Let $X$ be
the non-positively curved square complex obtained by gluing either end of a
cylinder $S^{1}\times[0,1]$ to two copies of $\Gamma$, using $\gamma$ as the
attaching map at each end. The links of the two vertices of $X$ are both
isomorphic to the Whitehead graph of $w$, which has a cut vertex.

The fundamental group of $X$ is the double

$\pi_{1}(X)=F*_{\langle w\rangle}F$

by the Seifert–van Kampen theorem. Applying the automorphism $\phi$ of $F_{2}$
setting $\phi(a)=ab^{-2}$ and $\phi(b)=b$ sends $w\mapsto
w^{\prime}=ab^{-1}a^{2}b$. We may realise $w^{\prime}$ by an immersion
$\gamma^{\prime}:S^{1}\to\Gamma$ and glue along a cylinder as before to obtain
a new square complex $X^{\prime}$ with $\pi_{1}(X^{\prime})\cong\pi_{1}(X)$.
However, the links of the two vertices of $X^{\prime}$ are isomorphic to the
Whitehead graph of $w^{\prime}$, which consists of two triangles glued along
an edge, and in particular is connected with no cut vertices. Therefore, by
Theorem E, $\pi(X)\cong\pi_{1}(X^{\prime})$ is one-ended, even though the
links of $X$ have cut vertices.

In the two-dimensional case, a square complex $X$ can always be modified so
that the sufficient condition of Theorem E can be applied. A map of square
complexes $\phi:X^{\prime}\to X$ is called an _unfolding of $X$_ if (i) the
restriction of $\phi$ to 1-skeleta is a combinatorial homotopy equivalence of
graphs; and (ii) $\phi$ is a homeomorphism on the complements of the
1-skeleta.

######  Proposition 2.29.

Let $X$ be a compact, non-positively curved square complex. There is an
unfolding $\phi:X^{\prime}\to X$ such that

$X^{\prime}=\Gamma\vee X_{1}\vee\ldots\vee X_{m}\vee S_{1}\vee\ldots\vee
S_{n}$

is also a compact, non-positively curved square complex and, furthermore:

  1. (i)

$\Gamma$ is a graph;

  2. (ii)

each $X_{i}$ has the property that every link is connected without cut
vertices or cut edges;

  3. (iii)

each $S_{j}$ is a square.

###### Proof.

The existence of $X^{\prime}$ is [53, Lemma 4.6] stated for square complexes
(see also [31, §3]); it remains to prove that $X^{\prime}$ is non-positively
curved. To do this, it suffices to check Gromov’s link condition, that any
cycle $\gamma$ in a link of a vertex of $X^{\prime}$ should have length at
least $2\pi$ in the angular metric [8, Lemma II.5.6]. Because $\phi$ is a
homeomorphism on the complement of the 1-skeleton, the induced maps on links
are injective on edges and hence immersions. Thus, $\gamma$ is also an
immersed cycle in a link of $X$, so has length at least $2\pi$ by the link
condition for $X$. ∎

The square complexes $X_{i}$ provided by the proposition satisfy the
hypotheses of Theorem E. Thus, the wedge decomposition $X^{\prime}$ actually
gives the Grushko decomposition of $X$:

$\pi_{1}(X)\cong\pi_{1}(\Gamma)*\pi_{1}(X_{1})*\ldots*\pi_{1}(X_{m})\,,$

and each $\pi_{1}(X_{i})$ is non-trivial and freely indecomposable.
Furthermore, the unfolding procedure that produces $X^{\prime}$ from $X$ is
algorithmic: one simply unfolds an edge of the 1-skeleton whenever the link of
a vertex contains a cut vertex. In summary, Theorem E and Proposition 2.29
together provide an effective, geometric method to determine the Grushko
decomposition of the fundamental group of a non-positively curved square
complex.

###  2.6 Shepherd’s theorem and $1$-cuts

The results of §2.5 show that 0-cuts detect when an essential cube complex has
more than one end, but they also illustrate the fact that the existence of
1-cuts makes it difficult to determine whether or not $\pi_{1}(X)$ splits.
Therefore, for one-ended $\pi_{1}(X)$, we would like to be able to choose $X$
to have no 1-cuts. In the 2-dimensional case, this can be done by _unfolding_
the 1-skeleton as in Proposition 2.29. To handle the general case, however, we
can use a recent deep theorem of Shepherd.

A _quarter-space_ of a CAT(0) cube complex is the intersection of two half-
spaces. A quarter-space is said to be _deep_ if it is not contained in a
bounded neighbourhood of the intersection of its two defining hyperplanes. The
next result is [42, Theorem 1.5].

######  Theorem 2.30 (Shepherd’s theorem).

If $G$ is one-ended and the fundamental group of a non-positively curved cube
complex, then $G=\pi_{1}(X)$ for an essential non-positively curved cube
complex $X$, such that the universal cover $\widetilde{X}$ has the property
that all half-spaces of $\widetilde{X}$ are one-ended and all quarter-spaces
are deep.

The following lemma connects Shepherd’s theorem to 1-cuts.

######  Lemma 2.31.

Let $\widetilde{X}$ be a one-ended, essential CAT(0) cube complex in which all
quarter-spaces are deep. If $\widetilde{X}$ has a 1-cut, then some half-space
of $\widetilde{X}$ has more than one end.

###### Proof.

Let $Y\subseteq\widetilde{X}$ be a 1-cut, i.e. $Y$ is compact and convex and
$\mathrm{Wh}_{\widetilde{X}}(Y)$ has a cut vertex $H$. Let $U$ be the half-
space of $H$ that contains $Y$.

We claim that $U$ is freely essential. Indeed, suppose that some hyperplane
$K$ of $U$ had a compact half-space $V$. If $K$ were disjoint from $H$ then
$K$ would also be an inessential hyperplane in $\widetilde{X}$, contradicting
the hypothesis that $\widetilde{X}$ is essential. If $K$ crossed $H$ then $V$
would be a compact quarter-space in $\widetilde{X}$, contradicting the
hypothesis that all quarter-spaces are deep.

Now, $\mathrm{Wh}_{U}(Y)$ is obtained from $\mathrm{Wh}_{\widetilde{X}}(Y)$ by
deleting the star of $H$, so is disconnected. Therefore, $U$ has more than one
end by Proposition 2.18, as required. ∎

Combining Shepherd’s theorem with the lemma, we may always assume that one-
ended cube complexes have no 1-cuts.

######  Corollary 2.32.

If $G$ is the fundamental group of a compact, non-positively curved cube
complex and $G$ is one-ended, then $G=\pi_{1}(X)$, where $X$ is a compact,
essential, non-positively curved cube complex without 0-cuts or 1-cuts.

##  3 Cyclic splittings

Having understood the free splittings of $\pi_{1}(X)$, we may henceforth
assume that $\pi_{1}(X)$ is one-ended. Our next task is to adapt the analysis
of the previous section to understand splittings of $\pi_{1}(X)$ over
$\mathbb{Z}$. We restrict attention to the case where $\pi_{1}(X)$ is
hyperbolic and, for brevity, we will say _hyperbolic cube complex_ to mean a
compact, connected, non-positively curved cube complex $X$ with
$\widetilde{X}$ (and hence $\pi_{1}(X)$) Gromov-hyperbolic. Let $\delta$ be a
hyperbolicity constant for $\widetilde{X}$.

Cashen–Macura studied line patterns in free groups, i.e. relatively hyperbolic
structures on free groups with cyclic parabolics [13]. They explained how to
recognise cut pairs in Otal’s decomposition space – i.e. the Bowditch boundary
– by analysing cut pairs in generalised Whitehead graphs. Cashen went on to
use this analysis to recognise relative cyclic splittings [12]. Barrett used
similar ideas to describe an algorithm that computes the Bowditch JSJ
decomposition of any hyperbolic group [2].

Our first task is to adapt their ideas to the setting of hyperbolic cube
complexes.

###  3.1 Periodic $2$-cuts

Suppose that $\pi_{1}(X)$ is one-ended. If every splitting of $\pi_{1}(X)$
with cyclic edge groups is trivial, then $\pi_{1}(X)$ is said to be
_cyclically indecomposable_. The main theorem of this section is a
combinatorial characterisation of cyclic indecomposibility. Just as we
understood free splittings of $\pi(X)$ in terms of 0-cuts and 1-cuts of $X$,
so cyclic splittings can be understood using 2-cuts.

As we will see, cyclic splittings give rise to 2-cuts, but the converse is not
quite true: in order to construct a cyclic splitting from a 2-cut, we also
need to consider the combinatorial types of the bounding hyperplane
components.

######  Definition 3.1 (Abstract hyperplane component).

Let $X$ be compact, non-positively curved cube complex. An _abstract
hyperplane component_ over $X$ consists of the following data:

  1. (i)

a compact, convex subcomplex $Z$ of a hyperplane $H\subseteq\widetilde{X}$;

  2. (ii)

a non-trivial partition of $\pi_{0}(\mathrm{Wh}_{H}(Z))$.

Two abstract hyperplane components $Z\subseteq H$ and $Z^{\prime}\subseteq
H^{\prime}$ are of _isomorphic type_ if there is a deck transformation
$\phi\in\pi_{1}(X)$ such that $\phi(Z)=Z^{\prime}$, inducing an isomorphism
$\mathrm{Wh}_{H}(Z)\cong\mathrm{Wh}_{H^{\prime}}(Z^{\prime})$ that preserves
the chosen partitions of $\pi_{0}$.

An _orientation_ of an abstract hyperplane component consists of the following
extra datum:

  1. (iii)

a choice of one of the two half spaces of the hyperplane $H$.

Two oriented abstract hyperplane components $Z\subseteq H$ and
$Z^{\prime}\subseteq H^{\prime}$ are of _opposite type_ if they are of
isomorphic type and, furthermore, the isomorphism $\phi$ can be chosen to
reverse the chosen orientations.

This definition enables us to define a _periodic_ 2-cut.

######  Definition 3.2 (Periodic 2-cut).

Let $X$ be a compact, non-positively curved cube complex. Consider a 2-cut $Y$
in $X$ between hyperplanes $H,H^{\prime}$. Define an oriented abstract
hyperplane component corresponding to $H$ as follows:

  1. (i)

$Z=H_{Y}$;

  2. (ii)

the partition on
$\pi_{0}(\mathrm{Wh}_{H}(Z))=\pi_{0}(\mathrm{Lk}_{\mathrm{Wh}_{X}(Y)})(H)$ is
induced by connectivity in $\mathrm{Wh}_{X}(Y)_{\\{H,H^{\prime}\\}}$;

  3. (iii)

the orientation is determined by the half-space of $H$ that contains $Y$.

Define an oriented abstract hyperplane component corresponding to $H^{\prime}$
in the same way. The 2-cut $Y$ is said to be _periodic_ if the two oriented
abstract hyperplane components are of opposite type. In this case, the element
$\phi\in\pi_{1}(X)$ sending $H\to H^{\prime}$ is called the _associated
covering transformation of $Y$_.

We shall see that periodic 2-cuts characterise cyclic splittings. However,
first we need to relate cyclic splittings to Whitehead complexes.

######  Lemma 3.3.

Let $X$ be an essential hyperbolic cube complex with $\pi_{1}(X)$ one-ended.
Then $\pi_{1}(X)$ splits over $\mathbb{Z}$ if and only if some infinite cyclic
subgroup acts cocompactly on some isometrically embedded subcomplex
$\widetilde{Z}$ with $\mathrm{Wh}_{X}(\widetilde{Z})$ disconnected.

###### Proof.

By the algebraic annulus theorem of Dunwoody–Swenson [17], and noting that
$\pi_{1}(X)$ is torsion-free, $\pi_{1}(X)$ splits over a $\mathbb{Z}$ subgroup
if and only if some subgroup $C\cong\mathbb{Z}$ satisfies that the number of
relative ends $e(G,C)>1$. Because $\pi_{1}(X)$ is hyperbolic, $C$ is
quasiconvex.

Choosing a base vertex $*$ in $\widetilde{X}$, the condition that $e(G,C)>1$
means that, for some $R$, the $R$-neighbourhood of the orbit $C*$ separates
$\widetilde{X}$ into at least two infinite components. Let $\widetilde{Z}$ be
the cubical convex hull of the $R$-neighbourhood of $C*$, which still
separates $\widetilde{X}$ into at least two infinite components. By Haglund’s
theorem (Theorem 1.7), $C$ acts cocompactly on $\widetilde{Z}$. Because
$\widetilde{Z}$ separates $\widetilde{X}$ into two components,
$\mathrm{Wh}_{X}(\widetilde{Z})$ is disconnected by Lemma 2.12, as required.

Conversely, suppose that some $C\cong\mathbb{Z}$ acts cocompactly on some
convex subcomplex $\widetilde{Z}$ with $\mathrm{Wh}_{X}(\widetilde{Z})$
disconnected, so $\widetilde{Z}$ separates $\widetilde{X}$ into two components
by Lemma 2.12. Either some complementary component remains within a bounded
distance of $\widetilde{Z}$, contradicting the hypothesis that $X$ is
essential, or all complementary components are unbounded away from
$\widetilde{Z}$, meaning that $e(G,C)>1$, as required. ∎

Lemma 3.3 makes it easy to prove that periodic 2-cuts characterise cyclic
splittings.

######  Theorem 3.4 (Cyclic splittings and periodic 2-cuts).

Let $X$ be an essential hyperbolic cube complex. Suppose that $\pi_{1}(X)$ is
one-ended, and suppose furthermore that $X$ has no 0- or 1-cuts. Then
$\pi_{1}(X)$ is cyclically decomposable if and only if there is a periodic
2-cut $Y\to X$.

###### Proof.

By Lemma 3.3, $\pi_{1}(X)$ is cyclically decomposable if and only if there is
a convex $\mathbb{Z}$-periodic subcomplex
$\widetilde{Z}\subseteq\widetilde{X}$ that separates $\widetilde{X}$. By Lemma
2.12, $\widetilde{Z}$ separates $\widetilde{X}$ if and only if
$\mathrm{Wh}_{X}(\widetilde{Z})$ is disconnected.

Suppose that there is a periodic 2-cut $Y\subseteq\widetilde{X}$ between
hyperplanes $H,H^{\prime}$. By definition, $H,H^{\prime}$ define a cut pair in
$\mathrm{Wh}_{X}(Y)$, and the corresponding oriented abstract hyperplane
components have opposite type. Therefore, we may define the amalgamated
subcomplex

$Y\cup_{\phi(H)}\phi(Y)\,.$

Iterating using all powers of $\phi$, we may set $\widetilde{Z}$ to be the
infinite amalgamation

$\cdots\cup_{\phi^{-1}(H)}\phi^{-1}(Y)\cup_{H}Y\cup_{\phi(H)}\phi(Y)\cup_{\phi^{2}(H)}\cdots\,,$

which is $\langle\phi\rangle$-invariant and cocompact. By Proposition 2.21,
$\mathrm{Wh}_{X}(\widetilde{Z})$ is constructed by iteratively splicing copies
of $\mathrm{Wh}_{X}(Y)$ along copies of $H$, meaning that
$\mathrm{Wh}_{X}(\widetilde{Z})$ is the infinite union

$\cdots\cup_{L}\mathrm{Wh}_{X}(Y)_{\\{H,H^{\prime}\\}}\cup_{L}\mathrm{Wh}_{X}(Y)_{\\{H,H^{\prime}\\}}\cup_{L}\mathrm{Wh}_{X}(Y)_{\\{H,H^{\prime}\\}}\cup_{L}\cdots\,,$

where $L=\mathrm{Lk}_{\mathrm{Wh}_{X}(Y)}(H)\cong\mathrm{Wh}_{H}(H_{Y})$ by
Lemma 2.13. The partitions that are part of the data of an abstract hyperplane
component imply that there is a class $c\in\tilde{H}_{0}(L)$ with the property
that $c$ and $\phi_{*}(c)$ have the same image in
$\tilde{H}_{0}(\mathrm{Wh}_{X}(Y)_{\\{H,H^{\prime}\\}})$. Applying
Mayer–Vietoris and using that homology commutes with direct limits, it follows
that $\tilde{H}_{0}(\mathrm{Wh}_{X}(\widetilde{Z}))$ is non-trivial, so
$\mathrm{Wh}_{X}(\widetilde{Z})$ is indeed disconnected as required.

Conversely, suppose that there is a convex $\mathbb{Z}$-periodic subcomplex
$\widetilde{Z}\subseteq\widetilde{X}$ with $\mathrm{Wh}_{X}(\widetilde{Z})$
disconnected. The compact quotient $Z=\mathbb{Z}\backslash\widetilde{Z}$ has
fundamental group $\mathbb{Z}$, and therefore $H^{1}(Z)\cong\mathbb{Z}$. Since
the first cohomology is generated by cocycles dual to hyperplanes, some
hyperplane $H$ of $Z$ is crossed by any loop representing the generator $\phi$
of $\mathbb{Z}$. Since $\pi_{1}(H)$ is contained in the kernel of the dual map

$\mathbb{Z}\cong\pi_{1}(Z)\to\mathbb{Z}$

it follows that $H$ is simply connected, and so lifts to a hyperplane (still
denoted by $H$) of $\widetilde{Z}$. By [11, Lemma 2.3], after replacing $\phi$
by a proper power we may assume that $\phi(H)$ is disjoint from $H$.

Let $\widetilde{Z}_{\pm}$ be the two half-spaces of $H$, so

$\widetilde{Z}=\widetilde{Z}_{-}\cup_{H}\widetilde{Z}_{+}\,.$

We claim that $\mathrm{Wh}_{X}(\widetilde{Z}_{+})$ and
$\mathrm{Wh}_{X}(\widetilde{Z}_{-})$ are connected. Indeed,
$\widetilde{Z}_{+}$ may be written as the ascending union of compact, convex
subcomplexes $Y_{n}$ where

$Y_{n}=Y\cup_{\phi(H)}\phi(Y)\cup_{\phi^{2}(H)}\ldots\cup_{\phi^{n-1}(H)}\phi^{n-1}(Y)$

and, correspondingly, the Whitehead complex of $\widetilde{Z}_{+}$ may may be
written as an ascending union

$\mathrm{Wh}_{X}(\widetilde{Z}_{+})=\bigcup_{n\geq
1}\mathrm{Wh}_{X}(Y_{n})_{\phi^{n}(H)}\,.$

Thus, if $\mathrm{Wh}_{X}(\widetilde{Z}_{+})$ were disconnected then it would
follow that $\mathrm{Wh}_{X}(Y_{n})_{\phi^{n}(H)}$ were disconnected for some
$n$. If so then $Y_{n}$ would be a 1-cut in $X$, contradicting the assumption.
Thus, $\mathrm{Wh}_{X}(\widetilde{Z}_{+})$ is connected and, symmetrically, so
is $\mathrm{Wh}_{X}(\widetilde{Z}_{-})$. Since

$\mathrm{Wh}_{X}(\widetilde{Z})=\mathrm{Wh}_{X}(\widetilde{Z}_{-})\\#_{H}\mathrm{Wh}_{X}(\widetilde{Z}_{+})$

is disconnected, it follows that $H$ is a cut vertex in both
$\mathrm{Wh}_{X}(\widetilde{Z}_{-})$ and $\mathrm{Wh}_{X}(\widetilde{Z}_{+})$
by Lemma 2.22.

Now, $H^{\prime}=\phi(H)$ is a hyperplane of $\widetilde{Z}_{+}$ disjoint from
$H$, and cutting along $H^{\prime}$ decomposes $\widetilde{Z}_{+}$ as

$\widetilde{Z}_{+}=Y\cup_{H^{\prime}}\phi(\widetilde{Z}_{+})$

where $Y=\widetilde{Z}_{+}\cap\phi(\widetilde{Z}_{-})$; note that $H$ is a
bounding hyperplane of $Y$. Hence,

$\mathrm{Wh}_{X}(\widetilde{Z}_{+})=\mathrm{Wh}_{X}(Y)\\#_{H^{\prime}}\mathrm{Wh}_{X}(\phi(\widetilde{Z}_{+}))$

and so, since we saw above that
$\mathrm{Wh}_{X}(\phi(\widetilde{Z}_{+}))\cong\mathrm{Wh}_{X}(\widetilde{Z}_{+})$
is connected, $Y$ is a 2-cut between $H$ and $H^{\prime}$ by Lemma 2.24.
Finally, note that $Y$ is a periodic 2-cut by construction. ∎

###  3.2 A bound on the widths of $2$-cuts

Although Theorem 3.4 provides a useful characterisation of the existence of
cyclic splittings, in order to make it effective we need an _a priori_ bound
on the complexity of the 2-cuts that arise. Such a bound is the main result of
this section, and is proved in Theorem 3.11 below. It is phrased in terms of
the _width_ of a 2-cut.

######  Definition 3.5 (Width of a $k$-cut).

Let $X$ be a compact, non-positively curved cube complex, and consider a
$k$-cut $Y\to X$ between bounding hyperplanes $\\{H_{1},\ldots,H_{k}\\}$ for
$k\geq 2$. The _width_ of $Y$ is the shortest $\ell^{2}$-distance between any
distinct $H_{i}$ and $H_{j}$.

The next lemma provides some geometric control on cuts, assuming that there
are no 1-cuts.

######  Lemma 3.6.

Let $\widetilde{X}$ be a one-ended a CAT(0) cube complex without 0- or 1-cuts.
Consider a $k$-cut $Y$ between bounding hyperplanes $H_{1},\ldots,H_{k}$. Then
$Y\cap\mathrm{Hull}\left(\bigcup_{i=1}^{k}H_{i}\right)$ is also a $k$-cut
between $H_{1},\ldots,H_{k}$.

###### Proof.

Let $K_{1},\ldots K_{l}$ be the hyperplanes of $Y$ that have a half-space
$U_{j}$ such that $H_{i}\subseteq U_{j}$ for all $i$. Write
$Y_{n}=Y\cap\bigcap_{j=1}^{n}U_{j}$ for each $0\leq n\leq l$. Note that

$Y_{l}=Y\cap\mathrm{Hull}\left(\bigcup_{i=1}^{k}H_{i}\right)\,.$

To prove the lemma, we show by induction on $n$ that each $Y_{n}$ is a
$k$-cut.

In the base case, $Y_{0}=Y$ and there is nothing to prove. For the inductive
step, let $n\geq 1$. If $K_{n}$ is disjoint from $Y_{n-1}$ then
$Y_{n-1}=Y_{n}$ and there is, again, nothing to prove. Otherwise, $K_{n}$
decomposes $Y_{n-1}$ as

$Y_{n-1}=Y_{n}\cup_{K_{n}}Y^{\prime}_{n}$

where $Y_{n}=Y_{n-1}\cap U_{n}$. Then $H_{1},\ldots,H_{k}$ all bound $Y_{n}$
and not $Y^{\prime}_{n}$ (since they are disjoint from $K_{n}$, and $K_{n}$
cannot separate $Y$ from any $H_{i}$). By Proposition 2.21

$\mathrm{Wh}_{X}(Y_{n-1})=\mathrm{Wh}_{X}(Y_{n})\\#_{K_{n}}\mathrm{Wh}_{X}(Y^{\prime}_{n})\,,$

and $K_{n}$ is not a cut vertex in $\mathrm{Wh}_{X}(Y^{\prime}_{n})$ because
$\widetilde{X}$ has no 1-cuts. Therefore $\\{H_{1},\ldots,H_{k}\\}$ is a cut
set in $\mathrm{Wh}_{X}(Y_{n})$ by Lemma 2.24, so $Y_{n}$ is also a $k$-cut
between $H_{1},\ldots,H_{k}$. Taking $n=l$, the result follows. ∎

We now turn our attention to some CAT(0) geometry of hyperplanes. Consider a
2-cut $Y$ between hyperplanes $H_{1}$ and $H_{2}$, and let $\gamma$ be an
$\ell^{2}$ geodesic in between $H_{1}$ and $H_{2}$. In order to prove the
desired bound, we need to find long chains of mutually disjoint hyperplanes
that cross $\gamma$ at an angle that isn’t too acute. A hyperplane $K$ is said
to be _$\epsilon$ -almost orthogonal_ to $\gamma$ if $\gamma$ crosses $K$ at
an angle of at least $\epsilon$.

The next lemma shows that many of the hyperplanes crossed by $\gamma$ will be
almost orthogonal, for a suitable value of $\epsilon$. The proof uses
elementary Euclidean trigonometry.

######  Lemma 3.7.

Let $d\geq 2$, and let $\gamma$ be an $\ell^{2}$ geodesic in a $d$-dimensional
CAT(0) cube complex $\widetilde{X}$. Suppose that $\gamma$ crosses two
hyperplanes $K_{1}$ and $K_{2}$ consecutively. Then at least one of $K_{1}$
and $K_{2}$ is $\epsilon_{d}$-almost orthogonal to $\gamma$, where
$\epsilon_{d}=\tan^{-1}\left(\frac{1}{2\sqrt{d-1}}\right)$.

###### Proof.

If $\gamma$ crosses $K_{1}$ and $K_{2}$ simultaneously then the result is
immediate, because $K_{1}$ and $K_{2}$ are orthogonal to each other and
$\epsilon_{d}<\pi/4$.

Otherwise, suppose that $\gamma(0)\in K_{1}$ and $\gamma$ crosses $K_{1}$ at
an angle $\theta<\epsilon_{d}$. Because the (closed) regular neighbourhood
$N(K_{1})$ of $K_{1}$ is isometric to $K_{1}\times[-1/2,1/2]$, we have
$\gamma(t)\in N(K_{1})$ whenever $|t|\sin\theta\leq 1/2$. In particular, if
$\gamma(t_{0})$ is in the boundary of $N(K_{1})$ then
$|t_{0}|=1/(2\sin\theta)$. Let $p$ be the orthogonal projection of
$\gamma(t_{0})$ to $K_{1}$. The three points $\gamma(0)$, $\gamma(t_{0})$, $p$
together form a right-angled Euclidean triangle, so

$d(\gamma(0),p)=|t_{0}|\cos\theta=\frac{1}{2}\cot\theta\,.$

Because $\theta<\epsilon_{d}$, it follows that

$d(\gamma(0),p)>\frac{1}{2}\cot\epsilon_{d}=\sqrt{d-1}$

which implies that $\gamma(0)$ and $p$ are separated by a hyperplane in the
$(d-1)$-dimensional cube complex $K_{1}$. Since every such hyperplane extends
uniquely to a hyperplane in $\widetilde{X}$ orthogonal to $K_{1}$, it follows
that $\gamma$ crosses a hyperplane in $N(K_{1})$ orthogonal to $K_{1}$.

The first such hyperplane is by assumption $K_{2}$. Let $\gamma(t_{1})\in
K_{2}$ and let $q$ be the point of intersection between $K_{1}$ and $K_{2}$.
Then the three points $\gamma(0)$, $\gamma(t_{1})$, $q$ together span a right-
angled Euclidean triangle, whence $\gamma$ makes an angle

$\pi/2-\theta>\pi-\epsilon_{d}>\pi/4>\epsilon_{d}$

with $K_{2}$, as required. ∎

The next lemma guarantees that a sufficiently distant pair of hyperplanes will
be separated by many mutually disjoint hyperplanes, all themselves almost
orthogonal to a geodesic.

######  Lemma 3.8.

Let $H_{1}$ and $H_{2}$ be hyperplanes in a $d$-dimensional CAT(0) cube
complex $\widetilde{X}$, and let $\gamma$ be an $\ell^{2}$-geodesic between
them. For every $n$ there is a $C$ such that, if the length of $\gamma$ is
greater than $C$, then there exists a sequence of mutually disjoint
hyperplanes

$K_{1},\ldots,K_{n}$

that each separate $H_{1}$ from $H_{2}$, all of which are
$\epsilon_{d}$-almost orthogonal to $\gamma$.

###### Proof.

Let $N$ be the Ramsey number $R(n,d+1)$. Recall that the $\ell^{1}$ distance
between $H_{1}$ and $H_{2}$ is equal to the number of hyperplanes crossed by
$\gamma$. Since the $\ell^{1}$ and $\ell^{2}$ metrics are quasi-isometric,
there is a $C$ such that, if $\gamma$ has $\ell^{2}$ length at least $C$, then
it crosses at least $2N$ hyperplanes. By Lemma 3.7, at least $N$ of these
hyperplanes are $\epsilon_{d}$-almost orthogonal to $\gamma$. Since no set of
$d+1$ hyperplanes of $\widetilde{X}$ pairwise intersect, it follows from the
definition of $N$ that there is a pairwise disjoint set

$\\{K_{1},\ldots,K_{n}\\}$

of $n$ hyperplanes that intersect $\gamma$ $\epsilon_{d}$-almost orthogonally.

Finally, note that any hyperplane $K$ that crosses $H_{i}$ meets it
orthogonally, and therefore if $\gamma$ also crossed $K$ there would be a
compact geodesic triangle in $\widetilde{X}$ with two right angles,
contradicting the CAT(0) condition. Therefore, each $K_{j}$ is disjoint from
$H_{1}$ and $H_{2}$, and therefore separates them from each other. ∎

We needed to find almost orthogonal hyperplanes because of the following
lemma, which will enable us to bound the diameter of a hyperplane intersecting
a suitably chosen 2-cut.

######  Lemma 3.9.

Suppose that an $\ell^{2}$ geodesic $\gamma$ intersects a hyperplane $H$ at an
angle $\theta$. Let $x$ be the point of intersection and let $y$ be a point in
$H$. Then

$d(x,y)\sin\theta\leq d(y,\gamma)\,.$

In particular, if $H$ is $\epsilon$-almost orthogonal to $\gamma$, $d(x,y)\leq
d(y,\gamma)/\sin\epsilon$.

###### Proof.

Let $z$ be the closest point on $\gamma$ to $y$, and consider the geodesic
triangle with vertices $x$, $y$ and $z$. Let $\bar{\theta}$ be the angle in
the Euclidean comparison triangle at the vertex corresponding to $x$, which is
at least $\theta$ by [8, Proposition 1.7(4)]. Now, the Euclidean sine rule
gives

$d(y,z)\geq d(x,y)\sin\bar{\theta}\geq d(x,y)\sin\theta$

as required. ∎

The proof of the claimed bound on the widths of 2-cuts uses the pigeonhole
principle, applied to isomorphism types of abstract hyperplane components of a
given diameter.

######  Remark 3.10.

Let $X$ be a compact, non-positively curved cube complex, and let $R\geq 0$.
The set of isomorphism types of abstract bounding hyperplane components $Z$
over $X$ of diameter at most $R$ is finite.

This finiteness is the basis for our argument using the pigeonhole principle.

######  Theorem 3.11.

Let $X$ be an essential, $d$-dimensional, hyperbolic cube complex without
0-cuts or 1-cuts. Then $\pi_{1}(X)$ is cyclically indecomposable if and only
if there is a bound on the widths of 2-cuts in $X$.

###### Proof.

The ‘only if’ direction is similar to the proof of Theorem 3.4. If
$\pi_{1}(X)$ has a cyclic splitting then, by Lemma 3.3, there is an infinite
cyclic subgroup $\langle\phi\rangle$ acting cocompactly on some isometrically
embedded subcomplex $\widetilde{Z}\subseteq\widetilde{X}$ with
$\mathrm{Wh}_{X}(\widetilde{Z})$ disconnected. As in the proof of Theorem 3.4,
we can find a hyperplane $H$ of $\widetilde{Z}$ such that, for any $n$,
cutting $\widetilde{Z}$ along $H$ and $\phi^{n}(H)$ gives a 2-cut between $H$
and $\phi^{n}(H)$. Since $d(H,\phi^{n}(H))\to\infty$ with $n$, it follows that
there is no bound on the widths of 2-cuts, as required.

For the converse, let $Y$ be a 2-cut between hyperplanes $H_{1}$ and $H_{2}$.
By Lemma 3.6, we may replace $Y$ by its intersection with
$\operatorname{Hull}(H_{1}\cup H_{2})$, and so assume that $Y$ is contained in
$\operatorname{Hull}(H_{1}\cup H_{2})$. Let $\gamma$ be an $\ell^{2}$ geodesic
between $H_{1}$ and $H_{2}$. It is easy to see that $H_{1}\cup\gamma\cup
H_{2}$ is $2\delta$-quasiconvex in the $\ell^{2}$ metric and therefore, by
Theorem 1.7, $\operatorname{Hull}(H_{1}\cup H_{2})$ is contained in the
$R$-neighbourhood of $H_{1}\cup\gamma\cup H_{2}$, for some uniform $R$.

Let $M$ be the number of isomorphism types of oriented abstract hyperplane
components over $X$ of diameter at most $2R\sqrt{4d-3}$ (which is finite by
Remark 3.10). Let $M^{\prime}$ be such that any geodesic crossing $M^{\prime}$
hyperplanes has an $\ell^{2}$ length of at least $(1+\sqrt{4d-3})R$, and let
$N=M+2M^{\prime}$. By Lemma 3.8, if the width of $Y$ is greater than a
constant $C$, then we can find a sequence of $M+1$ hyperplanes

$K_{0},K_{1},\ldots,K_{M}\,,$

all separating $H_{1}$ from $H_{2}$, all $\epsilon_{d}$-almost orthogonal to
$\gamma$, and such that each $K_{i}$ intersects $\gamma$ at distance greater
than $(1+\sqrt{4d-3})R$ from both $H_{1}$ and $H_{2}$.

From each $K_{i}$ we define an oriented abstract hyperplane component $Z_{i}$
as follows. First, set $Z_{i}=K_{i}\cap Y$; we next need to partition
$\pi_{0}(\mathrm{Wh}_{K_{i}}(Z_{i}))$. Cutting $Y$ along $K_{i}$ decomposes it
as

$Y=Y_{1}\cup_{K_{i}}Y_{2}\,,$

and Proposition 2.21 then tells us that

$\mathrm{Wh}_{X}(Y)=\mathrm{Wh}_{X}(Y_{1})\\#_{K_{i}}\mathrm{Wh}_{X}(Y_{2})\,.$

In particular, $\mathrm{Wh}_{X}(Y)$ contains a copy of
$\mathrm{Wh}_{K_{i}}(Z_{i})$, which is canonically isomorphic to the links of
$K_{i}$ in both $\mathrm{Wh}_{X}(Y_{i})$ by Lemma 2.13. The partition on
$\pi_{0}(\mathrm{Wh}_{K_{i}}(Z_{i}))$ is now induced by connectivity in
$\mathrm{Wh}_{X}(Y)\smallsetminus\\{H_{1},H_{2}\\}$. Furthermore, each $Z_{i}$
is naturally oriented from $H_{1}$ to $H_{2}$.

We next prove a bound on the diameter of any $Z_{i}$. Indeed, let $x$ be the
point at which $K_{i}$ intersects $\gamma$. Since $Y$ is contained in the
closed $R$-neighbourhood of $H_{1}\cup\gamma\cup H_{2}$, $Z_{i}$ is the union
of two closed subsets:

$Z_{i}=P\cup Q$

where $P=Z_{i}\cap(N_{R}(H_{1})\cup N_{R}(H_{2}))$ and $Q=Z_{i}\cap
N_{R}(\gamma)$.

By Lemma 3.9, the distance from any $y\in Q$ to $x$ is bounded above by

$\frac{R}{\sin\epsilon_{d}}=\frac{R}{\sin\cot^{-1}(2\sqrt{d-1})}=R\sqrt{4d-3}\,.$

We next argue that $P$ is empty. Indeed, if not then there is $y\in P\cap Q$,
since $Z_{i}$ is connected and $P$ and $Q$ are both closed. But such a $y$ is
within distance $R$ of $H_{1}\cup H_{2}$ and within $R\sqrt{4d-3}$ of $x$, so
$K_{i}$ intersects $\gamma$ at distance less than $R(1+\sqrt{4d-3})$ from
either $H_{1}$ or $H_{2}$, contradicting the construction of the $K_{i}$
above. Thus, $P$ is indeed empty, so $Z_{i}=Q$ and
$\operatorname{diam}(Z_{i})=\operatorname{diam}(Q)\leq 2R\sqrt{4d-3}$.

Therefore, by the pigeonhole principle and the choice of $M$, there exist
distinct $i<j$ such that $Z_{i}$ and $Z_{j}$ are isomorphic oriented abstract
hyperplane components. Cutting $Y$ along $K_{i}$ and $K_{j}$ decomposes it as

$Y=Y_{1}\cup_{K_{i}}Y^{\prime}\cup_{K_{j}}Y_{2}\,.$

Applying Lemma 2.24 twice, $Y^{\prime}$ is a 2-cut between $K_{i}$ and
$K_{j}$, while $Z_{i}$ and $Z_{j}$ are the corresponding bounding hyperplane
components. Therefore, $Y^{\prime}$ is in fact a periodic 2-cut between
$K_{i}$ and $K_{j}$, as required. ∎

###  3.3 From $2$-cuts to $k$-cuts

In the previous section, we proved a uniform bound on the widths of 2-cuts. In
this section, we will upgrade this to a uniform bound on the widths of
$k$-cuts, where the bound is independent of $k$. The key result is Lemma 3.16,
which encapsulates the fact that a finite collection of hyperplanes coarsely
has the structure of a quasi-tree. I am grateful to Mark Hagen for pointing
out that this lemma should be true, and that it would simplify the proof my
argument.

The proof of Lemma 3.16 is inspired by the projection-complex machinery of
Bestvina–Bromberg–Fujiwara [5]. Consider a collection $\mathbb{V}=\\{V_{i}\\}$
of convex subcomplexes of a hyperbolic, CAT(0) cube complex $\widetilde{X}$.
(In [5] this collection would typically be infinite. For us it will be finite,
but of unbounded cardinality.) For each $U\in\mathbb{V}$, we consider the
closest-point projection $\pi_{U}:\widetilde{X}\to U$. For each $U$,
Bestvina–Bromberg–Fujiwara use this to define a _distance function_ on pairs
of elements of $\mathbb{V}\smallsetminus\\{U\\}$ via

$d^{\pi}_{U}(V,W):=\operatorname{diam}(\pi_{U}(V)\cup\pi_{U}(W))\,.$

They go on to state _projection complex axioms_ for the distance functions
$d^{\pi}_{\bullet}$, with respect to a fixed constant $\theta>0$; see [6,
§1.3] for a concise statement of the axioms. The next proposition asserts
that, as long as the elements of $\mathbb{V}$ are pairwise sufficiently
distant from each other, the projection complex axioms are satisfied.

######  Proposition 3.12.

Let $\widetilde{X}$ be a $\delta$-hyperbolic CAT(0) cube complex and let
$\mathbb{V}$ be a finite set of convex subcomplexes of $\widetilde{X}$. There
is a constant $C$, depending only on $\widetilde{X}$, such that, as long as
the elements of $\mathbb{V}$ are pairwise at distance at least $C$, there is
$\theta>0$ such that the following hold for all distinct
$U,V,W,Y\in\mathbb{V}$:

  1. (i)

$d^{\pi}_{U}(V,W)=d^{\pi}_{U}(W,V)$;

  2. (ii)

$d^{\prime\pi}_{U}(V,W)+d^{\pi}_{U}(W,Y)\geq d^{\pi}_{U}(V,Y)$;

  3. (iii)

$d^{\pi}_{U}(V,V)\leq\theta$;

  4. (iv)

if $d^{\pi}_{U}(V,W)>\theta$ then $d_{V}(U,W)\leq\theta$;

  5. (v)

$\\#\\{Z\in\mathbb{V}\mid d^{\pi}_{Z}(V,W)>\theta\\}<\infty$

###### Proof.

Properties (i) and (ii) are obvious, while property (v) is immediate by
hypothesis since $\mathbb{V}$ is assumed to be finite. Property (iii) is known
as the _contracting property_ for convex subspaces of $\delta$-hyperbolic
spaces, and a standard exercise shows that it holds for any $\theta\geq
8\delta$ as long as $C\geq 2\delta$. Finally, property (iv) is known as the
_Behrstock inequality_. A sketch is given in [44, p. 133]. ∎

In order to construct a quasitree, it is necessary to adjust the distance
functions $d^{\pi}_{\bullet}$ to strengthen the Behrstock inequality. For
this, we follow Bestvina–Bromberg–Fujiwara–Sisto [6]. From
$d^{\pi}_{\bullet}$, they construct a new family of distance functions
$d_{\bullet}$ with improved properties.

######  Proposition 3.13.

For $d^{\pi}_{\bullet}$ and $\theta$ as above, there are functions
$d_{\bullet}$ such that

$|d^{\pi}_{\bullet}-d_{\bullet}|\leq 2\theta\,,$

and that satisfy the following axioms for all distinct $U,V,W,Y\in\mathbb{V}$:

  1. (i)

$d_{U}(V,W)=d_{U}(W,V)$;

  2. (ii)

$d_{U}(V,W)+d_{U}(W,Y)\geq d_{U}(V,Y)$;

  3. (iii)

$d_{U}(V,V)\leq 11\theta$;

  4. (iv)

if $d_{U}(V,W)>11\theta$ then $d_{W}(V,Y)=d_{W}(U,Y)$ unless $Y=W$;

  5. (v)

$\\#\\{Z\in\mathbb{V}\mid d_{Z}(V,W)>\theta\\}<\infty$.

###### Proof.

This is [6, Proposition 4.14]. ∎

######  Remark 3.14.

Proposition 3.13(iv) should be thought of as a strengthened version of the
Behrstock inequality. Indeed, taking $Y=V$ in Proposition 3.13(iv) shows that,
if $d_{U}(V,W)>11\theta$, then $d_{W}(U,V)=d_{W}(V,V)\leq 11\theta$ by
Proposition 3.13(iii).

Bestvina–Bromberg–Fujiwara–Sisto use the strengthened projection complex
axioms to construct a quasitree. Instead, we will use them to find
$U\in\mathbb{V}$ onto which the diameter of the projection of all the other
elements of $\mathbb{V}$ is bounded.

######  Lemma 3.15.

Let $\widetilde{X}$ be a $\delta$-hyperbolic CAT(0) cube complex and let
$\mathbb{V}$ be a finite set of unbounded convex subcomplexes of
$\widetilde{X}$. There is a constant $C$, depending only on $\widetilde{X}$,
such that, as long as the elements of $\mathbb{V}$ are pairwise at distance at
least $C$, there is $U\in\mathbb{Y}$ such that

$\operatorname{diam}\left(\bigcup_{V\in\mathbb{V}\smallsetminus
U}\pi_{U}(V)\right)\leq 13\theta\,.$

###### Proof.

Take $C$ as in Proposition 3.12. Since $d_{\bullet}$ is within $2\theta$ of
$d^{\pi}_{\bullet}$, it suffices to prove that there is $U$ such that
$d_{U}(V,W)\leq 11\theta$ for all $V,W\neq U$. We prove this by induction on
the cardinality of $\mathbb{V}$.

The base cases when $\\#\mathbb{V}\leq 2$ are trivial. For the inductive step,
fix $W_{0}\in\mathbb{V}$ and let
$\mathbb{V}^{\prime}:=\mathbb{V}\smallsetminus\\{W_{0}\\}$. By induction,
there is $U^{\prime}\in\mathbb{V}^{\prime}$ such that
$d_{U^{\prime}}(V^{\prime},W^{\prime})\leq 11\theta$ for all
$V^{\prime},W^{\prime}\in\mathbb{V}^{\prime}\smallsetminus\\{U^{\prime}\\}$.

If $d_{U^{\prime}}(V^{\prime},W_{0})\leq 11\theta$ for all
$V^{\prime}\in\mathbb{V}^{\prime}$ then we are done, taking $U=U^{\prime}$.
Suppose, therefore, that $d_{U^{\prime}}(V^{\prime},W_{0})>11\theta$ for some
$V^{\prime}\in\mathbb{V}^{\prime}\smallsetminus\\{U^{\prime}\\}$. In this
case, we set $U:=W_{0}$, and we need to show that $d_{U}(V,W)\leq 11\theta$
for any $V,W\in\mathbb{V}^{\prime}$.

Proposition 3.13(iv) implies that

$d_{U^{\prime}}(V,U)=d_{U^{\prime}}(V,W_{0})=d_{U^{\prime}}(V^{\prime},W_{0})>11\theta$

for any $V\neq U^{\prime}$, while $d_{U^{\prime}}(U^{\prime},U)>11\theta$
since $U^{\prime}$ is unbounded. Applying Proposition 3.13(iv) again, it
follows that $d_{U}(V,W)=d_{U}(V,U^{\prime})$ for any $W\neq U$. But
$d_{U}(V,U^{\prime})\leq 11\theta$ by Remark 3.14, so $d_{U}(V,W)\leq
11\theta$ for any $V,W\neq U$, as required. ∎

The geometric lemma that we need can now be easily deduced.

######  Lemma 3.16.

Let $\widetilde{X}$ be a $d$-dimensional, $\delta$-hyperbolic CAT(0) cube
complex. There is a constant $D$ with the following property. Suppose that
$\mathbb{V}$ is any finite, non-empty set of unbounded convex subcomplexes of
$\widetilde{X}$. If $d(U,V)>D$ for every distinct $U,V\in\mathbb{V}$, then
there is a hyperplane $H$ and $U\in\mathbb{V}$ such that $H$ separates $U$
from every $V\in\mathbb{V}\smallsetminus\\{U\\}$.

###### Proof.

Let $C$ and $\theta$ be as in Lemma 3.15. By Theorem 1.7, there is $r$ such
that, for any subset $Z\subseteq\widetilde{X}$ of diameter at most $13\theta$,
the diameter of $\operatorname{Hull}(Z)$ is at most $r$.

For each $U\in\mathbb{V}$, let $\widehat{U}$ be the convex hull of the
$(\sqrt{d}+r)$-neighbourhood of $U$, which is contained in the
$R$-neighbourhood of $U$, for some uniform $R$, by Theorem 1.7. Let
$\widehat{\mathbb{V}}:=\\{\widehat{U}\mid U\in\mathbb{V}\\}$ and let $D=C+2R$.

As long as the elements of $\mathbb{V}$ are at distance at least $D$ from each
other, it follows that the elements of $\widehat{\mathbb{V}}$ are at distance
at least $C$ from each other. Hence, by Lemma 3.15, there is $\widehat{U}$
such that

$\operatorname{diam}\left(\bigcup_{\widehat{V}\in\widehat{\mathbb{V}}\smallsetminus\widehat{U}}\pi_{\widehat{U}}(\widehat{V})\right)\leq
13\theta\,,$

whence

$\operatorname{diam}\operatorname{Hull}\left(\bigcup_{\widehat{V}\in\widehat{\mathbb{V}}\smallsetminus\widehat{U}}\pi_{\widehat{U}}(\widehat{V})\right)\leq
r$

by the definition of $r$.

This hull contains a point on the frontier of $\widehat{U}$, so is at distance
at least $\sqrt{d}$ from $U$ itself. Therefore, any shortest path from $U$ to
the hull crosses some hyperplane $H$. Thus, $H$ separates $U$ from
$\pi_{\widehat{U}}(V)$, for every $V\neq U$. It follows that $H$ separates $U$
from $V$, for all $V\neq U$, as required. ∎

We can now upgrade our bound on the widths of 2-cuts to a bound on the widths
of $k$-cuts.

######  Theorem 3.17.

Let $X$ be an essential, $d$-dimensional, hyperbolic cube complex without
0-cuts or 1-cuts. Then $\pi_{1}(X)$ is cyclically indecomposable if and only
if there is a uniform bound, independent of $k$, on the widths of all $k$-cuts
with $k\geq 2$.

###### Proof.

If there is a bound on the widths of $k$-cuts for all $k\geq 2$ then, in
particular, there is a bound on the widths of all $2$-cuts. Therefore, $X$ is
cyclically essential and $\pi_{1}(X)$ is cyclically indecomposable by Theorem
3.11.

For the converse implication, let $C$ be the bound from Theorem 3.11. By
Theorem 1.7, there is $R$ such that the hull of the $C$-neighbourhood of any
convex subset is contained in the $R$ neighbourhood of that subset. Let $D$ be
the constant from Lemma 3.16.

Let $Y$ be a $k$-cut between a collection of $k\geq 2$ hyperplanes
$\\{H_{1},\ldots H_{k}\\}$, and suppose that the width of $Y$ is greater than
$2R+D$. Suppose, furthermore, that $k$ is minimal such that there exists a
$k$-cut of width greater than $2R+D$.

For each $i$, let $U_{i}$ be the convex hull of the $C$-neighbourhood of
$H_{i}$.Then the $U_{i}$ are pairwise at distance at least $D$.

By Lemma 3.16, there is a hyperplane $K$ that, without loss of generality,
separates $U_{1}$ from the remaining $H_{i}$. In particular, $K$ is at
distance at least $C$ from $H_{1}$ and separates all the other $H_{i}$ from
$H_{1}$. Thus, cutting $Y$ along $K$ gives

$Y=Y_{1}\cup_{K}Y_{2}$

where $H_{1}$ bounds $Y_{1}$ and the remaining $H_{i}$ all bound $Y_{2}$.

By the minimality of $k$, $\\{H_{2},\ldots,H_{k}\\}$ is not a cut set in
$\mathrm{Wh}_{X}(Y_{2})$. Therefore, Lemma 2.24 implies that $Y_{1}$ is a
2-cut between $H_{1}$ and $K$, which is of width at least $C$ by construction.
By the choice of $C$, it follows that $\pi_{1}(X)$ is cyclically decomposable,
as required. ∎

##  4 One-ended subgroups

In this section, we prove both our main theorem about cubulated hyperbolic
groups and its corollary about one-relator groups.

###  4.1 Cubulated hyperbolic groups

With the uniform bound on the widths of cuts in hand, it is easy to prove the
main theorem in the cyclically indecomposable case.

######  Theorem 4.1.

Let $X$ be a hyperbolic cube complex. If $\pi_{1}(X)$ is one-ended and
cyclically indecomposable then it has a one-ended quasiconvex subgroup of
infinite index.

###### Proof.

By Corollary 2.32, we may without loss of generality assume that $X$ is
essential, without 0- or 1-cuts. Let $C$ be the uniform bound on widths of
$k$-cuts given by Theorem 3.17.

Choose any hyperplane $H$ of $X$, and fix an elevation $\widetilde{H}$ to the
universal cover $\widetilde{X}$. Because $\pi_{1}H$ is separable in
$G=\pi_{1}X$ by the work of Agol [1] and Haglund–Wise [26], there is a finite-
index subgroup $G_{0}$ of $G$ such that

$d(\widetilde{H},g\widetilde{H})>C$

for any $g\in G_{0}\smallsetminus\operatorname{Stab}(\widetilde{H})$.

Let $X_{0}$ denote the quotient $G_{0}\backslash\widetilde{X}$ and let $H_{0}$
be the image of $\widetilde{H}$, which is a hyperplane of $X_{0}$. Let
$X^{\prime}$ be (a component of) the result of cutting $X_{0}$ along $H_{0}$.
Since $X^{\prime}$ is finite and maps locally isometrically to $X$,
$\pi_{1}(X^{\prime})$ is a quasiconvex subgroup of $G$. Furthermore, since
$\pi_{1}(X^{\prime})$ is a vertex-group of a non-trivial graph-of groups
decomposition for $G_{0}$, it has infinite index in $G_{0}$, and hence in $G$.
It remains to show that $\pi_{1}(X^{\prime})$ is one-ended.

With a view to a contradiction, suppose that $\pi_{1}(X^{\prime})$ is freely
decomposable. By Proposition 2.18, there is a 0-cut $Y\to X^{\prime}$. But
$\mathrm{Wh}_{X^{\prime}}(Y)$ is obtained from $\mathrm{Wh}_{X}(Y)$ by
deleting the $G_{0}$-translates of $\widetilde{H}$, so some finite set of
translates

$g_{1}\widetilde{H},\ldots,g_{k}\widetilde{H}$

(with $g_{i}\in G_{0}$) define a cut set in $\mathrm{Wh}_{X}(Y)$. Since
$\mathrm{Wh}_{X}(Y)$ has no 0-cut or 1-cuts, $k\geq 2$. Therefore $Y$ is a
$k$-cut in $\widetilde{X}$ of width greater than $C$, contradicting the choice
of $C$. ∎

By passing to the Grushko and Bowditch JSJ decompositions, the main theorem
follows.

###### Proof of Theorem A.

Since $\pi_{1}(X)$ is a finitely generated, torsion-free group, it is freely
indecomposable if and only if it is one-ended, by Stallings’ theorem [45]. If
$\pi_{1}(X)$ is freely decomposable, Grushko’s theorem implies that either
$\pi_{1}(X)$ is free or it has a one-ended free factor, which is necessarily a
quasiconvex subgroup of infinite index. Therefore, we may assume that
$\pi_{1}(X)$ is itself one-ended.

Consider the Bowditch JSJ decomposition of $\pi_{1}(X)$ [7]. If the
decomposition has only one vertex and no edges, then that vertex is either of
surface or of rigid type. In the former case, $\pi_{1}(X)$ is a surface group.
In the latter case, $\pi_{1}(X)$ is cyclically indecomposable, so it has a
one-ended, quasiconvex subgroup of infinite index by Theorem 4.1.

Now consider the case where the JSJ decomposition is non-trivial. Every vertex
group $V$ is quasiconvex [7, Proposition 1.2], and in particular finitely
generated and hyperbolic. If some vertex group $V$ is not itself free, then by
Grushko’s theorem it has a one-ended free factor $V^{\prime}$, which provides
a one-ended, quasiconvex subgroup, necessarily of infinite index in
$\pi_{1}(X)$. On the other hand, if every vertex group is free, then
$\pi_{1}(X)$ is either a surface group or contains a one-ended, quasiconvex
subgroup by [52, Theorem 3]. ∎

###  4.2 One-relator groups

Combining the result for cubulated hyperbolic groups with recent joint work of
the author with Louder [30, 31], and also work of Linton [29], we can also
answer Question 0.1 for one-relator groups. The key is Puder’s notion of
_primitivity rank_ [37].

######  Definition 4.2 (Primitivity rank).

Let $w$ be a non-trivial element of a free group $F$. The _primitivity rank_
$\pi(w)$ is the minimal rank of a subgroup $w\in H\subseteq F$ such that $w$
is not a primitive element of $H$.

By convention, $\min\varnothing=\infty$, so $\pi(w)=\infty$ if and only if $w$
is primitive in $F$ (and hence in every subgroup containing it). As a result
of this convention, $\pi(w)\geq 1$ for all $w$. (It can be convenient to
extend the definition of $\pi$ to the trivial element $1\in F$ and set
$\pi(1)=0$.)

It has recently been discovered that $\pi(w)$ determines several features of
the one-relator group $G=F/\llangle w\rrangle$. These results can be
summarised in the following theorem.

######  Theorem 4.3.

Let $F$ be free, $w$ a non-trivial element of $F$ and $G=F/\llangle
w\rrangle$. Then:

  1. (i)

$\pi(w)=\infty$ if and only if $G$ is free;

  2. (ii)

$\pi(w)=1$ if and only if $G$ has torsion;

  3. (iii)

if $\pi(w)=2$ then $G$ has a two-generator, one-relator subgroup $P$ that is
not free;

  4. (iv)

if $\pi(w)>2$ then $G$ is hyperbolic and cubulated.

###### Proof.

Item (i) is a result of the classical theorem of Whitehead that, for non-
trivial $w$, $G$ is free if and only if $w$ is primitive [32, Proposition
5.10]. It is easy to see that $\pi(w)=1$ if and only if $w$ is a proper power
in $F$; a classical theorem of Karrass–Magnus–Solitar asserts that this is
true exactly when $G$ has torsion [32, Proposition 5.17]. Item (iii) is a
theorem of Louder and the author [30, Theorem 1.5], while item (iv) is a
theorem of Linton [29, Theorem 8.2] (using deep work of Wise [56], as well as
further work of Louder and the author [31]). ∎

######  Remark 4.4.

The one-relator group $G$ is also hyperbolic and cubulated when $\pi(w)=1$, by
work of Newman [35] and Wise [56]. However, we will not need either of these
facts here.

The final piece of the jigsaw is a recent result of Gardam–Kielak–Logan, who
handled the two-generator, one-relator case of Question 0.1 [20, Theorem 1.5].

######  Theorem 4.5 (Gardam–Kielak–Logan).

Let $G$ be a two-generator, one-relator group. If every subgroup of infinite
index in $G$ is free, then $G$ is either free or the fundamental group of a
closed surface.

Combining these results with Theorem A, we obtain the desired result for all
one-relator groups.

###### Proof of Theorem D.

A one-relator group $G=F/\llangle w\rrangle$ is free if $\pi(w)=0$ or
$\pi(w)=\infty$. Since $G$ is infinite and every subgroup of infinite index is
free, $G$ is torsion-free, so $\pi(w)>1$. If $\pi(w)>2$ then $G$ is hyperbolic
and cubulated by Theorem 4.3(iv), so the result follows from Theorem A. Thus,
it remains to handle the case when $\pi(w)=2$.

In this case, $G$ has a two-generator, one-relator subgroup $P$ that is not
free by Theorem 4.3(iii). The index of $P$ in $G$ is finite by hypothesis, and
every subgroup of infinite index in $P$ is free. By Theorem 4.5, $P$ is either
$\mathbb{Z}^{2}$ or $\mathbb{Z}\rtimes\mathbb{Z}$, so the same is true of $G$
by Bieberbach’s theorem. ∎

##  5 Questions

This final section contains some open questions that are suggested by the
results of this paper.

Strebel’s theorem asserts that every subgroup of infinite index in an
$n$-dimensional Poincaré duality group has cohomological dimension at most
$n-1$ [48]. Since groups of cohomological dimension one are precisely free
groups [49], Theorems A and D can be thought of as converses to Strebel’s
theorem for dimension $n=2$, in the contexts of cubulated hyperbolic and one-
relator groups. This is the whole story for one-relator groups, which are
always virtually of cohomological dimension 2. However, for cubulated
hyperbolic groups, it makes sense to ask whether the converses of the higher-
dimensional instances of Strebel’s theorem are also true.

######  Question 5.1.

Let $X$ be a hyperbolic cube complex with $\operatorname{cd}(\pi_{1}(X))=n>2$.
If the cohomological dimension of every subgroup of infinite index in
$\pi_{1}(X)$ is less than $n$, does it follow that $\pi_{1}(X)$ must be a
$PD_{n}$ group?

As with Theorem A, it is also conceivable that the geometric hypotheses of
Question 5.1 can be relaxed.

The proof of Theorem A made essential use of the separability of hyperplane
stabilisers, which is closely related to the fact that $X$ is virtually
special. Removing this condition would necessitate a different approach.

######  Question 5.2.

Let $X$ be a compact, non-positively curved cube complex with every subgroup
of infinite index free. Must $\pi_{1}(X)$ be either a free group or a surface
group?

An answer to Question 5.2 in the case of lattices in products of trees would
be interesting. The well known flat-closing conjecture would imply a positive
answer.

In a similar vein, removing the cubulation hypothesis from Theorem A would be
a major breakthrough.

######  Question 5.3.

Let $G$ be an infinite hyperbolic group. If every subgroup of infinite index
in $G$ is free, must $G$ be either a free group or a surface group?

Of course, if every one-ended hyperbolic group had a surface subgroup, then
Question 5.3 would have an affirmative answer.

The next question addresses the slight mismatch between the conclusions of our
main theorems.

######  Question 5.4.

Let $G$ be a two-generator, one-relator group such that every finitely
generated subgroup of infinite index is free. Must $G$ be isomorphic to either
$\mathbb{Z}$ or a solvable Baumslag–Solitar group?

(Note that the two two-generator surface groups, $\mathbb{Z}^{2}$ and
$\mathbb{Z}\rtimes\mathbb{Z}$, are $BS(1,1)$ and $BS(1,-1)$ respectively.)

As explained in §2.5, any non-positively curved _square_ complex can be
‘unfolded’ until Theorem E applies, providing an effective geometric
computation of the Grushko decomposition. I would like to extend this method
to higher-dimensional cube complexes.555Touikan’s algorithm provides a general
computation of the Grushko decomposition for any torsion-free group with
solvable word problem [50]. However, a cubical method would be preferable.

The first step is to recognise whether or not the cube complex $X$ is
essential. Sam Shepherd has explained to me that being essential can be
recognised algorithmically: the key observation is that an inessential CAT(0)
cube complex contains a half-space that is minimal with respect to inclusion,
and the bounding hyperplane of a minimal half-space can be recognised [43].

Given essential $X$, the next step is to determine whether or not $\pi_{1}(X)$
splits freely. The following question asks whether or not the sufficient
condition of Theorem E can always be applied.

######  Question 5.5.

Let $G$ be a one-ended cubulable group. Does $G=\pi_{1}(X)$, where $X$ is
essential and no vertex of $X$ has a link with a separating simplex?

Shepherd’s theorem (Theorem 2.30) implies that there is an $X$ without
separating vertices in the links, but does not provide information about
higher-dimensional simplices. Regardless of the answer to Question 5.5, it is
also important to find an algorithmic condition to recognise whether or not
$X$ admits 0- or 1-cuts.

Assuming an understanding of free splittings, one may pose similar questions
about splittings over $\mathbb{Z}$. It is not clear if there is an analogue of
Theorem E for cyclic splittings.

######  Question 5.6.

Let $X$ be a compact, non-positively curved cube complex with $\pi_{1}(X)$
one-ended. Is there a condition on the links of vertices of $X$ that
guarantees that $\pi_{1}(X)$ does not split over $\mathbb{Z}$?

For instance, it seems probable that the techniques of this article can be
used to prove that, if $X$ is a hyperbolic square complex and the links of
vertices of $X$ are connected without cut vertices or cut pairs, then
$\pi_{1}(X)$ does not split over $\mathbb{Z}$. But it would be better still to
provide a local criterion for higher-dimensional cube complexes, or outside
the hyperbolic setting.

As a step towards understanding cyclic splittings outside the hyperbolic
setting, the next question asks whether the hyperbolicity hypothesis can be
removed from Theorem 3.4.

######  Question 5.7.

Let $X$ be a compact, essential, non-positively curved cube complex without 0-
or 1-cuts. If $\pi_{1}(X)$ splits over $\mathbb{Z}$, must $X$ admit a periodic
2-cut?

The converse statement, that $\pi_{1}(X)$ splits over $\mathbb{Z}$ if $X$
admits a periodic 2-cut, follows from the algebraic annulus theorem of
Dunwoody–Swenson [17].

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