# Higher dimensional digraphs from cube complexes and their spectral theory

Nadia S. Larsen  Department of Mathematics
University of Oslo
PO BOX 1053 Blindern
N-0316 Oslo
Norway<EMAIL_ADDRESS>and  Alina Vdovina  Department of Mathematics
City College of New York
CUNY
160 Convent avenue, New York, NY, 10031
USA<EMAIL_ADDRESS>
(Date: 3 November 2022)

###### Abstract.

We define $k$-dimensional digraphs and initiate a study of their spectral
theory. The $k$-dimensional digraphs can be viewed as generating graphs for
small categories called $k$-graphs. Guided by geometric insight, we obtain
several new series of $k$-graphs using cube complexes covered by Cartesian
products of trees, for $k\geq 2$. These $k$-graphs can not be presented as
virtual products, and constitute novel models of such small categories. The
constructions yield rank-$k$ Cuntz-Krieger algebras for all $k\geq 2$. We
introduce Ramanujan $k$-graphs satisfying optimal spectral gap property, and
show explicitly how to construct the underlying $k$-digraphs.

##  1\. Introduction

The study of higher rank graphs and their $C^{*}$-algebras originates in work
of Robertson and Steger in [28] and expanded into a very active direction of
research in operator algebras following the work of Kumjian and Pask [13],
where the term $k$-graph was formalised. A higher rank graph, or $k$-graph,
with $k\geq 1$, is a small category with a functor into the monoid
${\mathbb{N}}^{k}$ that enjoys a unique factorisation property. While many
important structural results about higher rank graph $C^{*}$-algebras have
been obtained, the supply of examples of $k$-graphs for $k\geq 3$ is limited.
The main contribution of the present paper is to provide new, infinite series
of examples of higher rank graphs. Our main technical innovation is the
concept of a $k$-dimensional digraph, where $k\geq 2$. Using input from
geometric group theory, we obtain several infinite series of explicit
constructions of $k$-dimensional digraphs for $k\geq 3$, and from these we
obtain novel examples of higher rank graphs with rank at least $3$ which are
not of product type, in particular are not skew-products, as will be
explained.

A $k$-dimensional digraph is a generalisations of a directed graph, or
digraph. It is known that $1$-graphs are free categories defined by directed
graphs, see e.g. [13] or [16, Proposition 3.12]. Our definition of
$k$-dimensional digraph is made so that the natural category associated to it
will be a $k$-graph. This definition is as follows:

######  Definition 1.1.

Let $k\geq 2$ be a positive integer. A $k$-dimensional digraph
$\operatorname{DG}=(V,E,o,t,k,\phi)$ is a directed graph with $V$ a finite set
of vertices, $E$ finite set of edges, maps $o,t:E\to V$ which determine the
origin and terminus of each $x\in E$, and with the property that the edge set
decomposes as a disjoint union $E=E_{1}\sqcup E_{2}\sqcup\dots\sqcup E_{k}$
with $E_{i}$ for $i=1,\dots,k$ regarded as edges of colour $i$, such that
there is a bijection $\phi:Y\to Y$ on the set $Y:=Y_{\operatorname{DG}}$ of
all directed paths of length two formed of edges of colours given by ordered
pairs $(i,j)$ with $i\neq j$ in $\\{1,2,\dots,k\\}$, satisfying the following
properties:

  1. (F1)

If $xy$ is a path of length two with $x$ of colour $i$ and $y$ of colour $j$,
then $\phi(xy)=y^{\prime}x^{\prime}$ for a unique pair
$(y^{\prime},x^{\prime})$ where $y^{\prime}$ has colour $j$, $x^{\prime}$ has
colour $i$ and the origin and terminus vertices of the paths $xy$ and
$y^{\prime}x^{\prime}$ coincide. We write this as $xy\sim
y^{\prime}x^{\prime}$ and note that $\phi^{2}=1$.

  2. (F2)

For all $x\in E_{i}$, $y\in E_{j}$ and $z\in E_{l}$ so that $xyz$ is a path on
$E$, where $i,j,l$ are distinct colours, if $x_{1},x_{2},x^{2}\in E_{i}$,
$y_{1},y_{2},y^{2}\in E_{j}$ and $z_{1},z_{2},z^{2}\in E_{l}$ satisfy

$xy\sim y^{1}x^{1},x^{1}z\sim z^{1}x^{2},y^{1}z^{1}\sim z^{2}y^{2}$

and

$yz\sim z_{1}y_{1},xz_{1}\sim z_{2}x_{1},x_{1}y_{1}\sim y_{2}x_{2},$

it follows that $x_{2}=x^{2},y_{2}=y^{2}$ and $z_{2}=z^{2}$.

One of the main motivations for this definition is to have a purely
combinatorial finite set of data to deal with as input to defining a
$C^{*}$-algebra. The definition of a $k$-graph in [13] involves an infinite
category. There are combinatorial conditions describing $k$-coloured graphs
with complete and associative set of squares in [10, p. 577] leading to a
$k$-graph, but these too are formulated in terms of category theory and the
$k$-coloured graph consists of infinite data. While the complete and
associative collection of squares contains the same combinatorial information
as our $k$-dimensional digraph, the advantage of our concept of
$k$-dimensional digraph is two-fold, as it involves only a finite number of
conditions to be checked without reference to graph morphisms in an infinite
category, and because it allows for providing a rich supply of examples from
$k$-cube complexes via geometric group theory.

If we seek for an analogy from topology, we may talk about a CW-complex having
a simply-connected universal cover, so our $k$-dimensional digraph is an
analogue of the complex and the $k$-graph in the existing literature is an
analogue of the universal cover.

The constructions in [28] provided generalisation of the Cuntz-Krieger
algebras from topological Markov shifts introduced in [4]. They were motivated
by an observation of Spielberg [33] clarifying that a free group $\Gamma$ on
finitely many generators, viewed as the fundamental group of a finite
connected graph, acts on the boundary $\Omega$ of its Cayley graph such that
in the associated crossed product $C(\Omega)\rtimes\Gamma$ one finds
generating partial isometries for an ordinary Cuntz-Krieger algebra
$\mathcal{A}$ associated to a matrix $M$ that records incidence in the
universal covering graph. This provided the basis for defining a _higher rank
Cuntz-Krieger_ algebra $\mathcal{A}$ in [28]: the input consists of a finite
alphabet and a family of commuting $(0,1)$-matrices $M_{1},M_{2},\dots,M_{k}$,
with $k\geq 1$, having entries in the alphabet, satisfying a number of
conditions, and controlling the formation of words of
${\mathbb{N}}^{k}$-valued shape. Shortly after [28], Kumjian and Pask [13]
defined a $k$-graph $\Lambda$ as a small category with a
${\mathbb{N}}^{k}$-valued degree of morphisms modelling the formation of words
from [28] and with an associated $C^{*}$-algebra $C^{*}(\Lambda)$ generated by
partial isometries subject to Cuntz-Krieger type relations, recovering [28].

The explicit examples in [28] feature $k=2$. There, the foundation for the
construction of the $C^{*}$-algebra is a group action on the boundary of an
affine $\tilde{A}_{2}$ building, from which one extracts a suitable alphabet
and defines two commuting matrices with the properties (H0)-(H3) specified in
[28], see also [12]. Now, even though $k$-graphs were defined some time ago,
not many explicit examples are known for $k\geq 3$, see though [19].

Here we construct infinite series of examples of $k$-graphs, with $k\geq 3$,
from groups acting freely and transitively on products of $k$ regular trees of
constant valencies, as were explicitly found by Rungtanapirom, Stix and
Vdovina in [26]. One of our key tools is Theorem 3.1 establishing that there
is a $k$-graph, unique in the appropriate sense, associated to every
$k$-dimensional digraph.

######  Theorem 1.2.

(Cf. Theorem 3.1) Suppose that $\operatorname{DG}=(V,E,o,t,k,\phi)$ is a
$k$-dimensional digraph, $k\geq 2$. Then there exist a small category
$\Lambda_{\operatorname{DG}}$ with $V$ as set of objects and a functor
$d:\Lambda_{\operatorname{DG}}\to{\mathbb{N}}^{k}$ which assigns value
(degree) $e_{i}$ to morphisms determined by edges in $E_{i}$, for all $1\leq
i\leq k$. Moreover, the functor $d$ has the unique factorisation property and
in particular, $(\Lambda_{\operatorname{DG}},d)$ is a higher rank graph.

The other main tool shows that every $k$-cube complex gives rise to a
$k$-dimensional digraph, as follows:

######  Theorem 1.3.

(Cf. Theorem 3.3) For any complex $\mathcal{X}$ with universal cover equal to
the product of $k$ regular trees, where $k\geq 2$, there is a $k$-dimensional
digraph $\operatorname{DG}(\mathcal{X})$ defined by sending a vertex of
$\mathcal{X}$ to a vertex in $\operatorname{DG}(\mathcal{X})$, and by sending
each geometric edge in $\mathcal{X}$ to two edges in the edge set of
$\operatorname{DG}(\mathcal{X})$, of same colour and opposite orientations.

The $k$-graphs resulting from our constructions are not skew-products of a
$k$-graph by some finite group, because such a setup would involve a
fundamental group with torsion elements, while all fundamental groups of
complexes in our construction are infinite torsion-free groups, since they are
fundamental groups of CAT(0) spaces.

The examples of $k$-graphs we obtain by Corollary 3.4 and Theorem 4.7 differ
from, for example, the $k$-graphs constructed in [12], [27], [28], [29], since
these papers only considered the case $k=2$. Moreover, our examples are new
for $k=2$, see Remark 3.9. Our construction also differs from the more recent
[18] when $k=2$ and [19] for $k\geq 3$, both of which emulate [28] and [12].
The core idea in these references is as follows: Given a cell complex
$\mathcal{X}$, each $k$-dimensional cell in $\mathcal{X}$ becomes a vertex in
a $k$-graph $\Lambda(\mathcal{X})$, and for two such cells there is an edge in
$\Lambda(\mathcal{X})$ if the given cells are adjacent via a $(k-1)$-cell, for
$k\geq 2$. In this construction one may take pointed cells as vertices, or
unpointed. Another way to distinguish our higher rank graphs from the ones in
[28] is to compute the products of coordinate matrices. The products of
coordinate matrices in [28] have to be $(0,1)$-matrices, but this is not
necessarily the case in our construction (see Example 5.10).

To place our definition in the context of similar developments, we recall that
a recipe for constructing $2$-graphs was proposed in [13, Section 6], starting
from two distinct directed graphs on the same set of vertices with commuting
vertex matrices. A step forward was taken in [7], see their Remark 2.3, where
a certain associativity type condition was identified as sufficient. In [10],
the authors distilled these earlier attempts at constructing higher rank
graphs and landed on a prescription requiring a skeleton, or a $k$-coloured
graph (where the colouring refers to edges and employs $k$ distinct colours)
and a collection of building blocks termed squares that satisfy compatibility
requirements, see [10, Theorem 4.4]. The squares here are certain coloured-
graph morphisms.

A further simplification of the prescription of a $k$-graph from its skeleton,
seen as a $k$-coloured graph, has been employed in concrete examples such as
[14, Example 7.7 ] and [16, Section 8.2]. In fact, this last example
articulates the requirements on the coloured graph that inspired our
conditions (F1) and (F2) in Definition 1.1. It is interesting to note that the
validity of the associativity condition (F2), for $k\geq 3$, is a priori
highly nontrivial. There are connections to finding solutions of the Yang-
Baxter equation, see for example [37] and [34].

There is a strong connection between geometry of CW-complexes, groups and
semigroup actions, higher rank graphs and the theory of $C^{*}$-algebras. The
difficulty is that there are many ways to associate $C^{*}$-algebras to
groups, semigroups and CW-complexes, and this can lead to both isomorphic and
non-isomorphic $C^{*}$-algebras. For the higher rank graphs, there is a
canonical way to associate a $C^{*}$-algebra, cf [13], but it happens that
non-isomorphic $k$-rank graphs lead to the same $C^{*}$-algebra. This
conclusion is often achieved through computation of K-theory and applications
of the powerful Kirchberg-Phillips classification machinery for purely
infinite simple unital nuclear $C^{*}$-algebras.

One important question is what is a genuine higher rank? Meaning that our
$k$-rank graph can not be obtained by some standard procedure from graphs of
smaller ranks. We address this question by introducing the spectral theory of
combinatorial higher rank graphs. So far the spectrum of strongly connected
higher rank graphs was considered in [11] and [14], through Perron-Frobenius
theory, which lead to new explicit constructions of von Neumann factors. We
generalise results of [14] by constructing infinite series of IIIλ factors for
any $k$, and infinitely many values of $\lambda$.

We want to stress the following simple but important point about our
construction of $k$-graphs: Recall that for an undirected graph with (vertex)
adjacency matrix $A$, we have $A(v,w)=1=A(w,v)$ if vertices $v,w$ are
connected by an edge. Thus the adjacency matrix is _symmetric_ and the
eigenvalues are real. In a $k$-graph, we have directed edges in the various
colours in its $1$-skeleton. There is no reason why the adjacency matrix
should be symmetric. What can be said in general about a $k$-graph is that, if
it is strongly connected, then its associated coordinate matrices jointly
admit a unimodular Perron-Frobenius eigenvector, [11]. Now, in our
construction of the $k$-graph $\Lambda(P)$ from a $k$-cube complex $P$, the
procedure is such that it assigns to each undirected edge in the 1-skeleton of
$P$ a pair of morphisms (arrows) with opposite orientation in $\Lambda(P)$. As
a consequence, the adjacency matrix for the complex in direction $i$ is the
same as the coordinate matrix $M_{i}$ of $\Lambda(P)$ in colour $i$, for all
$i=1,\dots,k$. Thus all our constructions of $k$-graphs have symmetric
matrices.

With this in mind, we suggest a new class of higher rank graphs, which we call
Ramanujan $k$-rank graphs, see Section 5.2. Their coordinate matrices are
symmetric, so all eigenvalues are real and it makes sense to consider the
spectral gap. We show that our $k$-graphs satisfy the optimal spectral gap
condition, which distinguishes them from the examples that have appeared in
the literature so far.

The structure of the paper is the following: In a preliminary section 2 we
collect conventions and results about categories, groups acting on products of
trees, $k$-cube complexes, in particular one-vertex $k$-cube complexes from
$k$-cube groups, $k$-graphs and their $C^{*}$-algebras. Section 3 starts with
one of our main results, Theorem 3.1, which prescribes the construction of a
$k$-graph from a given $k$-dimensional digraph. We then associate a
$k$-dimensional digraph to any complex covered by a product of $k$ trees, see
Theorem 3.3. The ensuing Corollary 3.4 describes the new family of $k$-graphs
from $k$-cube complexes. In the case of a one-vertex complex $P$, or
equivalently a $k$-cube group with complex $P$, we show that the resulting
$k$-graphs are rigid in the sense of [15], in particular they are aperiodic
and yield classifiable $C^{*}$-algebras in the sense of the Kirchberg-Phillips
classification [22]. In Section 4 we construct $k$-cube complexes on $N$
vertices from $N$-covers of one-vertex complexes, with $N\geq 2$, and prove
that the $C^{*}$-algebras of the associated $k$-graphs with $N$ vertices are
covered by the Kirchberg-Phillips classification theory. In Corollary 5.3 we
expand the scope of the constructions of $2$-graphs in [14, Example 7.7]
leading to factors of type III1/2 and give an explicit infinite family giving
type III${}_{{1}/{(2L)^{2}}}$ factors, with $L$ an arbitrary integer. In
section 5.2 we introduce the notion of Ramanujan $k$-graphs and show that
there are infinite families of such $k$-graphs, see Theorem 5.9. Example 5.10
details an explicit Ramanujan $3$-graph on $25$ vertices with optimal spectral
gap. We compute the associated $25\times 25$ incidence matrices and estimate
the joint spectral gap of the $3$-dimensional digraph with the help of MAGMA.
The $3$-graph moreover features the interesting property that while it arises
from an infinite $3$-cube group $\Gamma_{1}$ which is also an irreducible
lattice, in the cover with $25$ sheets each of the three distinct alphabets in
$\Gamma_{1}$ generates a finite group of order $25$.

Acknowledgment. We thank Mark Lawson and Aidan Sims for useful comments. We
are grateful to the anonymous referee for an extensive list of suggestions
towards improving the presentation and several proofs. This research was
initiated during a visit by A.V. to Oslo in connection to the Master Class on
”Equilibrium states in semigroup theory, K-theory and number theory”, 4-6
November, 2019, supported by the Trond Mohn Foundation through the project
”Pure Mathematics in Norway”. It was carried out while N.L. was a Research
Fellow in the Cluster of Excellence Mathematics Münster at WWU, Germany. She
thanks for warm hospitality and excellent working conditions in the Cluster.

##  2\. Preliminaries

###  2.1. Categories.

We follow the principles laid out in [17], but also keep in mind the
interpretation in [5, Chapter II, §1.1 and §1.2 ]. A category $\mathcal{C}$
consists of objects $\operatorname{Obj}(\mathcal{C})$ and morphisms
$\operatorname{Hom}(\mathcal{C})$. We often blur the distinction between
$\mathcal{C}$ and $\operatorname{Hom}(\mathcal{C})$, and refer to the latter
as the _elements_ or _arrows_ of $\mathcal{C}$. To each $f\in\mathcal{C}$
there are two objects associated, its _origin_ and _terminus_ , and the
category is seen as a collection of elements endowed with a partial product
governed by compatibility of objects. Of interest to us are categories
associated to directed graphs, where concatenation of edges on the graph will
determine composition of arrows in the category, upon natural reversal of
origin and terminus.

###  2.2. Directed graphs

By a directed graph $D$ we mean a set $D^{0}$ of vertices, a set $D^{1}$ of
edges and maps $o,t:D^{1}\to D^{0}$ determining the origin and terminus of
edges. Edges whose origin and terminus coincide, also called loops, will be
allowed. We shall assume that $D^{0},D^{1}$ are finite. We form a path $ef$
when $t(e)=o(f)$ for $e,f\in D^{1}$, and extend this to finite directed paths
$f_{1}f_{2}\dots f_{m}$ on $D$, and likewise in the case of a finite number of
graphs on the same vertex set, see Section 3.

###  2.3. Complexes covered by products of trees

We start by introducing our definition of $k$-cube complex. Then we expand on
the case of one-vertex $k$-cube complexes, for which we follow the notation
and approach of [34] and [19]. We refer to [36], [3], and especially [25,
Section 1.2] for an introduction to square complexes and $(2m,2l)$ groups,
$m,l\geq 1$. We refer to [30] for details on $CAT(0)$ complexes and to [9] for
the basic theory of CW complexes. We use the letter $T$ for an arbitrary
regular tree, and $\mathcal{T}_{l}$ for the regular $l$-valent tree, where
$l\geq 1$.

######  Definition 2.1.

Let $k\geq 1$ be a positive integer. A CW complex $\mathcal{X}$ is a
$k$-dimensional cube complex, or $k$-cube complex, if its universal cover is a
Cartesian product of $k$ trees
$\operatorname{T}_{1}\times\operatorname{T}_{2}\times\dots\times\operatorname{T}_{k}$,
each of which has finite constant valency.

In the case of a one-vertex $k$-cube complex, for which we use the letter $P$,
an equivalent definition is as the quotient space $P=Z\backslash G$ of a group
$G$ with a free and transitive action on a product
$Z=\operatorname{T}_{1}\times\operatorname{T}_{2}\times\dots\times\operatorname{T}_{k}$
of $k$ trees. Such $G$ are called $k$-cube groups, see Definition 2.3. For
general $k$-cube complexes with more than one vertex, the similar definition
as a quotient space $\mathcal{X}=Z\backslash G$ can be enforced upon replacing
transitive action with cocompact action.

We leave the case of trees with possibly non-constant and/or infinite valency
for future discussion.

To describe a $k$-cube complex for $k\geq 2$ it is useful to recall the
formalism of $2$-complexes (or square complexes) covered by products of two
trees, see e.g. [34]. We use the letter $S$ to denote a generic $2$-complex. A
square complex $S$ is a $2$-dimensional combinatorial cell complex with
$1$-skeleton consisting of a graph $\mathcal{G}(S)=(V(S),E(S))$ with set of
vertices $V(S)$, and set of oriented edges $E(S)$, and with $2$-cells arising
from a set of squares that are combinatorially glued to the graph
$\mathcal{G}(S)$. More precisely, let $e\mapsto e^{-1}$ denote orientation
reversal of an edge $e\in E(S)$, and suppose that $(e_{1},e_{2},e_{3},e_{4})$
is a $4$-tuple of oriented edges in $E(S)$ with the origin of $e_{i+1}$ equal
to the terminus of $e_{i}$ (for $i$ modulo $4$). A square
$\square=(e_{1},e_{2},e_{3},e_{4})$ is the orbit of
$(e_{1},e_{2},e_{3},e_{4})$ under the dihedral action generated by cyclically
permuting the edges $e_{i}$ and by the reverse orientation map

(2.1)
$(e_{1},e_{2},e_{3},e_{4})\mapsto(e_{4}^{-1},e_{3}^{-1},e_{2}^{-1},e_{1}^{-1}).$

As customary, we think of a square-shaped $2$-cell glued to the (topological
realization of the) respective edges of the graph $\mathcal{G}(S)$.

A vertical/horizontal structure (in short, a VH-structure) on a square complex
is given by a bipartite structure of the set of unoriented edges
$\overline{E(S)}=E_{V}\sqcup E_{H}$ such that for every vertex $v$ in $V(S)$
the link at $v$ is the complete bipartite graph on the resulting partition
$E(v)=E(v)_{V}\sqcup E(v)_{H}$, with $E(v)$ denoting the set of oriented edges
with origin $v$. Torsion-free cocompact lattices $\Gamma$ in
$\mathbb{A}ut(\mathcal{T}_{m})\times\mathbb{A}ut(\mathcal{T}_{l})$ with
$m,l\geq 1$, not interchanging the factors and considered up to conjugation,
correspond uniquely to finite square complexes $S$ with a VH-structure of
partition size $(2m,2l)$ up to isomorphism. Simply transitive torsion-free
lattices not interchanging the factors correspond to finite square complexes
with one vertex and a VH-structure, necessarily of constant partition size.

###  2.4. One-vertex $k$-cube complexes

We first look at the case when $k=2$. Let $S$ be a square complex with one
vertex $v\in S$ and a VH-structure $\overline{E(S)}=E_{V}\sqcup E_{H}$.
Pictorially, this consists of a collection of squares, each of which has four
vertices labelled $v$. Passing from the origin to the terminus of an oriented
edge $e$ in a square corresponds to a fixed point free involution $e\to
e^{-1}$ on $E(v)_{V}$ and on $E(v)_{H}$. Thus the partition size is
necessarily a tuple $(2m,2l)$ of even integers, $m,l\geq 1$. The lattice
identified with $\pi_{1}(S,v)$ admits a description in terms of two generating
subsets $A,B$, see [34, Definition 5].

######  Definition 2.2.

A vertical/horizontal structure, or VH-structure, in a group $G$ is an ordered
pair $(A,B)$ of finite subsets $A,B\subseteq G$ such that the following hold.

  1. (1)

Taking inverses induces fixed point free involutions on $A$ and $B$.

  2. (2)

The union $A\cup B$ generates $G$.

  3. (3)

The product sets $AB$ and $BA$ have size $\\#A\cdot\\#B$ and satisfy $AB=BA$.

  4. (4)

The sets $AB$ and $BA$ do not contain $2$-torsion.

The tuple $(\\#A,\\#B)$ is called the valency vector of the VH-structure in
$G$.

If a group $G$ admits a VH-structure $(A,B)$ of valency vector $(\\#A,\\#B)$,
then by [3, Section 6.1], there is a square complex $S_{A,B}$ with one vertex
and a VH-structure in the sense of subsection 2.3. The set of oriented edges
of $S_{A,B}$ is the disjoint union $E(S_{A,B})=A\sqcup B$, with the
orientation reversion map given by $e\mapsto e^{-1}$, and with $A,B$ labelling
the edges in vertical and horizontal direction, respectively. The link of
$S_{A,B}$ in $v$ is the complete bipartite graph with vertices labelled by $A$
and $B$, see [34, Lemma 1], and [1, Theorem C] implies that the universal
cover of $S_{A,B}$ is a product of trees. Conversely, given a square complex
$S$ with a VH-structure $(A,B)$ and a single vertex, its fundamental group
(i.e. the fundamental group of its topological realisation) admits a VH-
structure of valency $(\\#A,\\#B)$, see [26, Proposition 5.7]. We refer to it
as a $(\\#A,\\#B)$-group. Example 3.8 shows a $(4,4)$-complex with associated
$(4,4)$-group.

To describe the geometric squares of $S_{A,B}$, note that a relation
$ab=b^{\prime}a^{\prime}$ in $G$ with $a,a^{\prime}\in A$ and $b,b^{\prime}\in
B$ (not necessarily pairwise distinct), as prescribed by Definition 2.2, leads
to four algebraic relations obtained upon cyclic permutation and inversion,
namely

(2.2) $ab=b^{\prime}a^{\prime},\quad a^{-1}b^{\prime}=ba^{\prime-1},\quad
a^{\prime-1}b^{\prime-1}=b^{-1}a^{-1}\quad\text{ and
}a^{\prime}b^{-1}=b^{\prime-1}a.$

This leads to the definition of a geometric square as a tuple of four
Euclidean squares. All four vertices in each square coalesce into the single
vertex $v$ of $S_{A,B}$ when we glue the edges according to labels and
orientation. Before we introduce our convention, we recall briefly two other
(equivalent) conventions for describing geometric squares.

###  2.5. One convention - see e.g. Rattaggi

The formalism of a geometric square seen as a $4$-tuple of squares in
Euclidean space is well-known. In [25, Figure 4.1, page 182], for example, the
group relation $ab=b^{\prime}a^{\prime}$ is reflected by the $4$-tuple of
squares having edges labelled according to a one-way cyclic permutation in
counterclockwise direction, see below:

(2.3)
$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{S_{O}}$$\scriptstyle{b^{\prime}}$$\textstyle{{\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a^{\prime}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{S_{R}}$$\scriptstyle{b}$$\textstyle{{\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{S_{H}}$$\scriptstyle{a^{\prime}}$$\textstyle{{\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{b^{\prime}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{S_{V}}$$\scriptstyle{a}$$\textstyle{{\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{b}$$\textstyle{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{{\bullet}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$$\textstyle{{\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{b}$$\textstyle{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{{\bullet}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a^{\prime}}$$\textstyle{{\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{b^{\prime}}$$\textstyle{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{{\bullet}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{b}$$\textstyle{{\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$$\textstyle{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{{\bullet}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{b^{\prime}}$$\textstyle{{\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a^{\prime}}$

The notation means that if $S_{O}$ is regarded as a reference square, then
$S_{H}$ is obtained by reflection in the horizontal direction (about $b$),
$S_{V}$ by reflection in the vertical direction (about $a$) and, finally,
$S_{R}$ arises from rotation counterclockwise by $\pi$. Our use of
$S_{O},S_{R},S_{H},S_{V}$ as notation for the squares is inspired by [19,
Section 2].

The geometric square associated to $ab=b^{\prime}a^{\prime}$ in [25] and
visualised in (2.3) is given by

$\\{(a,b,a^{\prime},b^{\prime}),\,(a^{\prime},b^{\prime},a,b),\,(a^{-1},{b^{\prime}}^{-1},{a^{\prime}}^{-1},b^{-1}),\,({a^{\prime}}^{-1},b^{-1},a^{-1},{b^{\prime}}^{-1})\\}.$

###  2.6. A second convention - see Kimberley-Robertson

In [12], Kimberley-Robertson adopted a two-direction labelling of their
squares which to a group relation $ab=b^{\prime}a^{\prime}$ assigns a
$4$-tuple of squares according to the convention below:

(2.4)
$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{S_{O}}$$\scriptstyle{a^{\prime}}$$\textstyle{{\bullet}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{S_{R}}$$\scriptstyle{b}$$\textstyle{{\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$$\scriptstyle{b^{\prime}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\bullet}}$$\scriptstyle{S_{H}}$$\textstyle{{\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a^{\prime}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{S_{V}}$$\scriptstyle{a}$$\scriptstyle{b^{\prime}}$$\textstyle{{\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{b}$$\textstyle{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{{\bullet}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{b^{\prime}}$$\scriptstyle{a}$$\textstyle{{\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{b}$$\textstyle{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{{\bullet}}}}$$\textstyle{{\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a^{\prime}}$$\textstyle{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{{\bullet}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{b}$$\textstyle{{\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{b^{\prime}}$$\scriptstyle{a}$$\textstyle{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{{\bullet}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a^{\prime}}$$\textstyle{{\bullet}}$

The geometric square associated to $ab=b^{\prime}a^{\prime}$ and visualised in
(2.4) is given by

$\\{(a,b,b^{\prime},a^{\prime}),\,({a^{\prime}}^{-1},{b^{\prime}}^{-1},b^{-1},a^{-1}),\,(a^{-1},b^{\prime},b,{a^{\prime}}^{-1}),\,(a^{\prime},b^{-1},{b^{\prime}}^{-1},a)\\}.$

###  2.7. Our convention

We choose a convention that will facilitate our constructions of $k$-graphs
later on, and in particular we swap the letters for vertical and horizontal
directions, as follows: we keep the cyclic permutation in counterclockwise
direction from [25] but choose labelling of edges as ”starting” at one vertex
by going out in both horizontal and vertical direction, similar to [12].

(2.5)
$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{S_{O}}$$\scriptstyle{b}$$\textstyle{{\bullet}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{S_{R}}$$\scriptstyle{a^{\prime}}$$\textstyle{{\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{b^{\prime}}$$\scriptstyle{a}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\bullet}}$$\scriptstyle{S_{H}}$$\textstyle{{\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{b}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{S_{V}}$$\scriptstyle{b^{\prime}}$$\scriptstyle{a}$$\textstyle{{\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a^{\prime}}$$\textstyle{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{{\bullet}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$$\scriptstyle{b^{\prime}}$$\textstyle{{\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a^{\prime}}$$\textstyle{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{{\bullet}}}}$$\textstyle{{\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{b}$$\textstyle{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{{\bullet}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a^{\prime}}$$\textstyle{{\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$$\scriptstyle{b^{\prime}}$$\textstyle{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{{\bullet}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{b}$$\textstyle{{\bullet}}$

Explicitly, we define a _geometric square_ as visualised in (2.5) to be a
tuple

(2.6)
$\\{({a},{b},{a}^{\prime-1},{b}^{\prime-1}),\,({a}^{\prime-1},{b}^{\prime-1},{a},{b}),\,({a}^{\prime},{b}^{-1},{a}^{-1},{b}^{\prime}),\,({a}^{-1},{b}^{\prime},{a}^{\prime},{b}^{-1})\\},$

where any two squares are seen as equivalent.

Since for our purposes it will be important to keep track of how such squares
arise, we introduce the following more precise notation: for $a\in A$ and
$b\in B$, we let

(2.7) $S_{O}^{a,b}:=(a,b,a^{\prime-1},b^{\prime-1}),$

where $a^{\prime},b^{\prime}$ are the unique elements in $A$ and $B$,
respectively, such that $ab=b^{\prime}a^{\prime}$. We refer to $ab$ as the
_vertical-horizontal pair of edges in $S_{O}^{a,b}$_ and to
$b^{\prime}a^{\prime}$ as the _horizontal-vertical pair of edges in
$S_{O}^{a,b}$_.

In [34], the last named author generalised VH-structure to the $k$-dimensional
case, as follows.

######  Definition 2.3.

(See [34, Definition 7]) A $k$-cube structure in a group $G$ is an ordered
$k$-tuple $(A_{1},\ldots,A_{k})$ of finite subsets $A_{i}\subseteq G$ such
that the following hold for all $i,j=1,\dots,k$, $i\neq j$:

  1. (1)

Taking inverses induces fixed point free involutions on $A_{i}$.

  2. (2)

The union $\cup A_{i}$ generates $G$.

  3. (3)

The product sets $A_{i}A_{j}$ and $A_{j}A_{i}$ have size
$\\#A_{i}\cdot\\#A_{j}$ and $A_{i}A_{j}=A_{j}A_{i}$.

  4. (4)

The sets $A_{i}A_{j}$ and $A_{j}A_{i}$ do not contain $2$-torsion.

  5. (5)

The group $G$ acts simply transitively on a Cartesian product of $k$ trees.

The tuple $(\\#A_{1},\ldots,\\#A_{k})$ is the valency vector of the $k$-cube
structure in $G$, and $A_{1},\ldots,A_{k}$ are generating sets of $G$.

We note that each pair $(A_{i},A_{j})\subseteq G$ for $i,j=1,\dots,k$ with
$i\neq j$ forms a subgroup $G_{i,j}$ of $G$ equipped with a VH-structure. This
observation can be used to show that to a given $k$-cube group $G$ with
generating family $(A_{1},\ldots,A_{k})$ there is an associated $k$-cube
complex $P_{(A_{1},\ldots,A_{k})}$. Its $2$-dimensional cells are prescribed
by the square complexes $S_{A_{i},A_{j}}$ obtained from each
$(\\#A_{i},\\#A_{j})$ group $G_{i,j}$ for $i\neq j$.

######  Remark 2.4.

In studying $k$-cube groups and one-vertex $k$-cube complexes, we will move
freely between two equivalent interpretations. Starting from a $k$-cube group
$G$ defined algebraically through properties (1)-(5) in Definition 2.3, the
associated quotient space $Z\backslash G$ with $Z$ a product of $k$ trees is a
$k$-cube complex with one vertex. Its construction as a CW complex from
$j$-cells for $0\leq j\leq k$ is detailed in [19, Definition 2.3]. Conversely,
one may define a $k$-cube group $G$ from combinatorial data by starting with
$k$ finite sets of even cardinalities (encoding edges), and building up by
induction (on dimension of cells) a $k$-dimensional complex with the necessary
compatibility to yield generating sets $A_{1},A_{2},\dots,A_{k}$ for $G$, see
[19, Definition 2.4].

We next introduce some notation for cubes in a one-vertex $k$-cube complex $P$
for $k\geq 2$, with motivation and inspiration drawn from [19, Section 2]. We
let $E$ be the set of edges. Because $P$ is the complex associated to a
$k$-cube group, $E$ partitions into $k$ subsets as $E=E_{1}\sqcup\dots\sqcup
E_{k}$ in such a way that if an edge $e$ is in $E_{j}$ then $e^{-1}\in E_{j}$
for $j=1,\dots,k$. We refer to $E_{i}$ as the subset of edges of _colour_ $i$,
for $i=1,\dots,k$, where the $k$ colours are assumed distinct. The $2$-cells
of $P$ are geometric squares of the form
$S_{O}=(a,b,a^{\prime-1},b^{\prime-1})$ regarded as the equivalence class
$\\{S_{O},S_{R},S_{H},S_{V}\\}$ described in (2.6), where $a,a^{\prime}\in
E_{i}$ and $b,b^{\prime}\in E_{j}$ for $i\neq j$. By a _geometric square_ we
mean any square in $\\{S_{O},S_{R},S_{H},S_{V}\\}$. Similar to [19], for
distinct colours $i\neq j$ we let

$F(i,j)=\\{S=(a,b,a^{\prime-1},b^{\prime-1})\mid S\text{ is a geometric square
with }a,a^{\prime}\in E_{i},b,b^{\prime}\in E_{j}\\},$

and we denote by $S^{ij}$ a generic square in $F(i,j)$ for all $i\neq j$ in
$\\{1,\dots,k\\}$. The $3$-cells are determined by _geometric cubes_ , all
whose $6$ faces are geometric squares, see figure 1 for a (generic) such cube.

Figure 1. A generic $3$-cube

More precisely, the $6$ faces of the cube are geometric squares
$(S_{1}^{ij},S_{2}^{il},S_{3}^{jl},S_{4}^{jl},S_{5}^{il},S_{6}^{ij})$, with

$\displaystyle S_{1}^{ij}$
$\displaystyle=(a_{1},b_{1},a_{2}^{-1},b_{2}^{-1}),\text{ front face }$
$\displaystyle S_{2}^{il}$
$\displaystyle=(a_{2},c_{3},a_{3}^{-1},c_{2}^{-1}),\text{ right face}$
$\displaystyle S_{3}^{jl}$
$\displaystyle=(b_{2},c_{2},b_{3}^{-1},c_{1}^{-1}),\text{ bottom face}$
$\displaystyle S_{4}^{jl}$
$\displaystyle=(b_{1},c_{3},b_{4}^{-1},c_{4}^{-1}),\text{ top face}$
$\displaystyle S_{5}^{il}$
$\displaystyle=(a_{1},c_{4},a_{4}^{-1},c_{1}^{-1}),\text{ left face}$
$\displaystyle S_{6}^{ij}$
$\displaystyle=(a_{4},b_{4},a_{3}^{-1},b_{3}^{-1}),\text{ back face}.$

In particular, any one of the $6$ geometric squares in this list is given
subject to the equivalence relation (2.6), and the geometric cube can be
equivalently presented with any one of the $8$ vertices in the bottom-left
position. We stress that the labels $a_{1},\dots,a_{4}$, $b_{1},\dots,b_{4}$,
$c_{1},\dots,c_{4}$ here are formal symbols that keep track of how the cubes
are glued in the complex. As already mentioned, if $e$ is a label for an edge,
then $e^{-1}$ is the label recording orientation reversal.

For $4\leq l\leq k$, the $l$-dimensional cells are $l$-cubes, see [19].

A given $k$-cell in a $k$-cube complex $P$ has a topological realisation as
the product of intervals $[0,1]^{k}$. Denoting by $\varepsilon_{i}$ the
standard basis elements in $\mathbb{R}^{k}$, for $i=1,\dots,k$, we view a
geometric edge in $P$ as having _degree $\varepsilon_{i}$_ if it lies in the
span of the generator $\varepsilon_{i}$ in its topological realisation. This
agrees with the degree of paths in higher rank graphs in section 3.

###  2.8. Examples of cube groups

The cube groups in these section were introduced in [34]. They contain as a
particular case arithmetic lattices and non-residually finite CAT(0) groups
constructed in [26] . We refer to them as _RSV-groups_. They are _the first
explicit examples_ of arithmetic groups acting freely and transitively on
products of $k$ trees of constant valencies, for $k\geq 3$, as well as non-
residually finite CAT(0) groups of dimensions $k\geq 3$. RSV groups are
irreducible in the sense that they can not be presented as virtual products of
group actions on products of smaller number of trees.

We recall here a construction of an explicit series of RSV lattices, which is
infinite in several parameters, $k$, $q$ and $\delta$. The significance of the
irreducibility of these groups is that the associated $k$-graphs can not
presented as virtual products, so are entirely new.

For $q$ an odd prime, let $\delta\in{\mathbb{F}}_{q^{2}}^{\times}$ be a
generator of the multiplicative group of the field with $q^{2}$ elements. If
$i,j\in{\mathbb{Z}}/(q^{2}-1){\mathbb{Z}}$ satisfy $i\not\equiv j\pmod{q-1}$,
then $1+\delta^{j-i}\not=0$, and it follows that there is a unique
$x_{i,j}\in{\mathbb{Z}}/(q^{2}-1){\mathbb{Z}}$ with
$\delta^{x_{i,j}}=1+\delta^{j-i}.$

Set $y_{i,j}:=x_{i,j}+i-j$, and note that
$\delta^{y_{i,j}}=\delta^{x_{i,j}+i-j}=(1+\delta^{j-i})\cdot\delta^{i-j}=1+\delta^{i-j}$.
Define

$\displaystyle l(i,j)$ $\displaystyle:=i-x_{i,j}(q-1),$ $\displaystyle k(i,j)$
$\displaystyle:=j-y_{i,j}(q-1),$

and further let $M\subseteq{\mathbb{Z}}/(q^{2}-1){\mathbb{Z}}$ be a union of
cosets under $(q-1){\mathbb{Z}}/(q^{2}-1){\mathbb{Z}}$ with $\\#M=k$.

If $q$ is odd, it was shown in [26] that the following groups act freely and
transitively on a product of $k$ trees:

$\Gamma_{M,\delta}=\left\langle a_{i},i\in M\
\left|\begin{array}[]{c}a_{i+(q^{2}-1)/2}a_{i}=1\text{ for all $i\in M$},\\\
a_{i}a_{j}=a_{k(i,j)}a_{l(i,j)}\text{ for all $i,j\in M$ with $i\not\equiv
j\pmod{q-1}$}\end{array}\right.\right\rangle$

######  Example 2.5.

The smallest example in dimension $k=3$ arises with $q=5$ and $M$ equal to the
collection of cosets $i\in{\mathbb{Z}}/{24}{\mathbb{Z}}$ with $i$ not dividing
$4$. This group, denoted $\Gamma_{1}$, acts vertex transitively on the product
of three regular trees
$\mathcal{T}_{6}\times\mathcal{T}_{6}\times\mathcal{T}_{6}$ and has the
presentation

$\Gamma_{1}=\left\langle\begin{array}[]{c}a_{1},a_{5},a_{9},a_{13},a_{17},a_{21},\\\
b_{2},b_{6},b_{10},b_{14},b_{18},b_{22},\\\
c_{3},c_{7},c_{11},c_{15},c_{19},c_{23}\end{array}\
\left|\begin{array}[]{c}a_{i}a_{i+12}=b_{i}b_{i+12}=c_{i}c_{i+12}=1\ \text{
for all $i$ },\\\ a_{1}b_{2}a_{17}b_{22},\ a_{1}b_{6}a_{9}b_{10},\
a_{1}b_{10}a_{9}b_{6},\ a_{1}b_{14}a_{21}b_{14},\ a_{1}b_{18}a_{5}b_{18},\\\
a_{1}b_{22}a_{17}b_{2},\ a_{5}b_{2}a_{21}b_{6},\ a_{5}b_{6}a_{21}b_{2},\
a_{5}b_{22}a_{9}b_{22},\\\ a_{1}c_{3}a_{17}c_{3},\ a_{1}c_{7}a_{13}c_{19},\
a_{1}c_{11}a_{9}c_{11},\ a_{1}c_{15}a_{1}c_{23},\ a_{5}c_{3}a_{5}c_{19},\\\
a_{5}c_{7}a_{21}c_{7},\ a_{5}c_{11}a_{17}c_{23},\ a_{9}c_{3}a_{21}c_{15},\
a_{9}c_{7}a_{9}c_{23},\\\ b_{2}c_{3}b_{18}c_{23},\ b_{2}c_{7}b_{10}c_{11},\
b_{2}c_{11}b_{10}c_{7},\ b_{2}c_{15}b_{22}c_{15},\ b_{2}c_{19}b_{6}c_{19},\\\
b_{2}c_{23}b_{18}c_{3},\ b_{6}c_{3}b_{22}c_{7},\ b_{6}c_{7}b_{22}c_{3},\
b_{6}c_{23}b_{10}c_{23}.\end{array}\right.\right\rangle$

Thus $\Gamma_{1}$ is a $3$-cube group with
$A_{1}=\\{a_{1},a_{5},a_{9},a_{13},a_{17},a_{21}\\}$ and similar descriptions
for $A_{2}$ and $A_{3}$. It is an arithmetic group, so it is residually
finite. Of interest to us is the fact that it admits quotients of order
$5^{l},l\in\mathbb{N}$, see Example 5.10.

In [26], the authors also constructed $k$-cube groups acting on a product of
trees of distinct constant valencies. Explicitly, for any set of size $k$ of
distinct odd primes $p_{1},\ldots,p_{k}$, there is a group acting simply
transitively on a product of $k$ trees of valencies $p_{1}+1,\ldots,p_{k}+1$,
obtained using Hamiltonian quaternion algebras.

######  Example 2.6.

For $p_{1}=3,p_{2}=5,p_{3}=7$, there is an explicit presentation of a group
acting simply transitively on a product of three trees
$\mathcal{T}_{4}\times\mathcal{T}_{6}\times\mathcal{T}_{8}$, see [26]. Indeed,
with $\mathbf{i},\mathbf{j},\mathbf{k}$ denoting the quaternions, let

$\displaystyle a_{1}=1+\mathbf{j}+\mathbf{k},\ a_{2}=1+\mathbf{j}-\mathbf{k},\
a_{3}=1-\mathbf{j}-\mathbf{k},\ a_{4}=1-\mathbf{j}+\mathbf{k},$ $\displaystyle
b_{1}=1+2\mathbf{i},\ b_{2}=1+2\mathbf{j},\ b_{3}=1+2\mathbf{k},\
b_{4}=1-2\mathbf{i},\ b_{5}=1-2\mathbf{j},\ b_{6}=1-2\mathbf{k},$
$\displaystyle c_{1}=1+2\mathbf{i}+\mathbf{j}+\mathbf{k},\
c_{2}=1-2\mathbf{i}+\mathbf{j}+\mathbf{k},\
c_{3}=1+2\mathbf{i}-\mathbf{j}+\mathbf{k},\
c_{4}=1+2\mathbf{i}+\mathbf{j}-\mathbf{k},$ $\displaystyle
c_{5}=1-2\mathbf{i}-\mathbf{j}-\mathbf{k},\
c_{6}=1+2\mathbf{i}-\mathbf{j}-\mathbf{k},\
c_{7}=1-2\mathbf{i}+\mathbf{j}-\mathbf{k},\
c_{8}=1-2\mathbf{i}-\mathbf{j}+\mathbf{k}.$

Then $a_{l}^{-1}=a_{l+2}$, $b_{l}^{-1}=b_{l+3}$, and $c_{l}^{-1}=c_{l+4}$. The
required group is given by

$\Gamma_{2}=\left\langle\begin{array}[]{c}a_{1},\dots a_{4}\\\
b_{1},\dots,b_{6}\\\ c_{1},\dots,c_{8}\end{array}\
\left|\begin{array}[]{c}a_{1}b_{1}a_{4}b_{2},\ a_{1}b_{2}a_{4}b_{4},\
a_{1}b_{3}a_{2}b_{1},\ a_{1}b_{4}a_{2}b_{3},\ a_{1}b_{5}a_{1}b_{6},\
a_{2}b_{2}a_{2}b_{6}\\\ a_{1}c_{1}a_{2}c_{8},\ a_{1}c_{2}a_{4}c_{4},\
a_{1}c_{3}a_{2}c_{2},\ a_{1}c_{4}a_{3}c_{3},\\\ a_{1}c_{5}a_{1}c_{6},\
a_{1}c_{7}a_{4}c_{1},\ a_{2}c_{1}a_{4}c_{6},\ a_{2}c_{4}a_{2}c_{7}\\\
b_{1}c_{1}b_{5}c_{4},\ b_{1}c_{2}b_{1}c_{5},\ b_{1}c_{3}b_{6}c_{1},\
b_{1}c_{4}b_{3}c_{6},\ b_{1}c_{6}b_{2}c_{3},\ b_{1}c_{7}b_{1}c_{8},\\\
b_{2}c_{1}b_{3}c_{2},\ b_{2}c_{2}b_{5}c_{5},\ b_{2}c_{4}b_{5}c_{3},\
b_{2}c_{7}b_{6}c_{4},\ b_{3}c_{1}b_{6}c_{6},\
b_{3}c_{4}b_{6}c_{3}\end{array}\right.\right\rangle.$

This is also denoted $\Gamma_{3,5,7}$, see [26].

###  2.9. Higher rank graphs

We recall the definition of a $k$-graph due to Kumjian and Pask [13]. For an
integer $k\geq 1$, we view ${\mathbb{N}}^{k}$ as a monoid under pointwise
addition. A $k$-graph is a countable small category $\Lambda$ together with an
assignment of a _degree_ $d(\mu)\in{\mathbb{N}}^{k}$ to every morphism
$\mu\in\Lambda$ such that for all $\mu,\nu,\pi\in\Lambda$ the following hold

  1. (1)

$d(\mu\nu)=d(\mu)+d(\nu)$; and

  2. (2)

whenever $d(\pi)=m+n$ for $m,n\in{\mathbb{N}}^{k}$, there is a unique
factorisation $\pi=\mu\nu$ such that $d(\mu)=m$ and $d(\nu)=n$.

Condition (2) is known as the _factorisation property_ in the $k$-graph. The
composition in $\mu\nu$ is understood in the sense of morphisms, thus the
source $s(\mu)$ of $\mu$ equals the range $r(\nu)$ of $\nu$. Note that the
morphisms of degree $0$ (in ${\mathbb{N}}^{k}$) are the identity morphisms in
the category. Denote this set by $\Lambda^{0}$, and refer to its elements as
_vertices_ of $\Lambda$. With $e_{1},\dots,e_{k}$ denoting the generators of
${\mathbb{N}}^{k}$, the set $\Lambda^{e_{i}}=\\{\lambda\in\Lambda\mid
d(\lambda)=e_{i}\\}$ consists of edges (or morphisms) of degree $e_{i}$, for
$i=1,\dots,k$. We write $v\Lambda^{n}$ for the set of morphisms of degree
$n\in{\mathbb{N}}^{k}$ with range $v$.

Throughout this paper we are concerned with $k$-graphs where $\Lambda^{0}$ and
all $\Lambda^{e_{i}}$, $i=1,\dots,k$, are finite. A $k$-graph $\Lambda$ so
that $0<\\#v\Lambda^{n}<\infty$ for all $v\in\Lambda^{0}$ and all
$n\in{\mathbb{N}}^{k}$ is source free and row-finite. Following [11], a finite
$k$-graph $\Lambda$ is _strongly connected_ if $v\Lambda w\not=\emptyset$ for
all vertices $v,w\in\Lambda^{0}$.

The _coordinate matrices_
$M_{1},\dots,M_{k}\in\operatorname{Mat}_{\Lambda^{0}}({\mathbb{N}})$ of
$\Lambda$ are $\Lambda^{0}\times\Lambda^{0}$ matrices with

$M_{i}(v,w)=|v\Lambda^{e_{i}}w|.$

By the factorisation property, the matrices $M_{i}$ pairwise commute for
$i=1,\dots,k$. For ${n}=(n_{i})_{i=1,\dots,k}\in{\mathbb{N}}^{k}$, we define

$\textstyle M^{{n}}:=\prod^{k}_{i=1}M_{i}^{n_{i}}.$

We denote the spectral radius of a square matrix $B$ by $\rho(B)$, and we let

$\rho(\Lambda):=(\rho(M_{1}),\rho(M_{2}),\dots,\rho(M_{k}))\in[0,\infty)^{k}.$

For ${m}\in{\mathbb{Z}}^{k}$ we write $\rho(\Lambda)^{{m}}$ for the product
$\prod^{k}_{i=1}\rho(M_{i})^{m_{i}}$.

Given a row-finite, source free $k$-graph $\Lambda$, its associated
$C^{*}$-algebra $C^{*}(\Lambda)$ is the universal $C^{*}$-algebra generated by
a family $\\{\mathbf{s}_{\mu}\mid\mu\in\Lambda\\}$ of partial isometries
satisfying

  * (CK1)

$\\{\mathbf{s}_{v}\mid v\in\Lambda^{0}\\}$ is a family of mutually orthogonal
projections;

  * (CK2)

$\mathbf{s}_{\mu}\mathbf{s}_{\nu}=\mathbf{s}_{\mu\nu}$ whenever
$s(\mu)=r(\nu)$;

  * (CK3)

$\mathbf{s}_{\mu}^{*}\mathbf{s}_{\mu}=\mathbf{s}_{s(\mu)}$ for all $\mu$;

  * (CK4)

$\mathbf{s}_{v}=\sum_{\mu\in
v\Lambda^{{n}}}\mathbf{s}_{\mu}\mathbf{s}^{*}_{\mu}$ for all
$v\in\Lambda^{0},{n}\in{\mathbb{N}}^{k}$.

##  3\. Construction of $k$-graphs from $k$-cube groups: the one-vertex case

In this section we construct $k$-graphs with one vertex, for $k\geq 2$. There
are two main steps. The first is a general procedure by which we associate a
category to a $k$-dimensional digraph as introduced in Definition 1.1, and
prove that the conditions (F1) and (F2) inherent to the $k$-dimensional
digraph imply the existence of a degree functor from the category to
${\mathbb{N}}^{k}$ that satisfies the required factorisation property. This
step will be reminiscent of the construction of Artin monoids as quotients of
free monoids by an equivalence relation on the collection of positive words
identifying braid strings, see [2]. The degree functor will be similar to the
degree map on an Artin monoid, cf. [31]. The second step is to provide
$k$-dimensional digraphs, and here we shall use one-vertex cube complexes
associated to $k$-cube groups as a source from which to construct such
digraphs.

In Section 4 we use the results of this section combined with concrete
constructions of covering maps in the context of complexes to produce higher
rank graphs with more than one vertex. We stress that our constructions are
performed on the complexes, which depend on finite combinatorial data, and not
on the $k$-graphs, which are categories with additional structure.

The next result is the abstract construction of the category associated to a
$k$-dimensional digraph.

######  Theorem 3.1.

Suppose that $\operatorname{DG}=(V,E,o,t,k,\phi)$ is a $k$-dimensional
digraph, $k\geq 2$. Then there exist a small category
$\Lambda_{\operatorname{DG}}$ with $V$ as set of objects and a functor
$d:\Lambda_{\operatorname{DG}}\to{\mathbb{N}}^{k}$ which assigns value
(degree) $e_{i}$ to morphisms determined by edges in $E_{i}$, for all $1\leq
i\leq k$. Moreover, the functor $d$ has the unique factorisation property and
in particular, $(\Lambda_{\operatorname{DG}},d)$ is a higher rank graph.

###### Proof.

First, let ${\mathcal{C}}:={\mathcal{C}}_{\operatorname{DG}}$ be the free
category associated to the directed graph $\operatorname{DG}=(V,E,o,t)$, where
we disregard the colouring of the edges, see e.g. [17, Theorem 1, page 49].
The object set of ${\mathcal{C}}$ is $V$ and the arrows are given by finite
strings, or paths, consisting of finite sequences $v_{1},\dots,v_{m}$ of
objects connected by $m-1$ arrows $x_{s}:v_{s}\to v_{s+1}$ with the
compatibility of objects $o(x_{s+1})=t(x_{s})$ for $1\leq s\leq m-1$, where
$m>1$ is arbitrary.

To conform later with the conventions of higher rank graphs, we assign length
$0$ to identity morphisms, or arrows $\langle v_{1}\rangle$, as opposed to
length $1$ in [17]. An arrow $\langle v_{1},x_{1},v_{2}\rangle$ of length $1$
will be called an _elementary_ arrow, where we again shift the value of the
length by $-1$ compared to [17]. For $m>1$, an arrow $A:=\langle
v_{1},x_{1},v_{2},\cdots,v_{m-1},x_{m-1},v_{m}\rangle$ is determined by the
finite path $x_{1}x_{2}\cdots x_{m-1}$ of length $m-1$ on the graph, and we
shall often regard it as such, by disregarding the contribution of the objects
in the notation. In the category, the same arrow is equal to a composition

(3.1) $A=A_{m-1}\circ A_{m-2}\circ\cdots\circ A_{2}\circ A_{1}$

of $m-1$ elementary arrows $A_{s}=\langle v_{s},x_{s},v_{s+1}\rangle$, $1\leq
s\leq m$. For short, we write it as $x_{m-1}\circ\cdots\circ x_{2}\circ
x_{1}.$ The category underlying the $k$-graph will be obtained as a quotient
category of $C$, as in [17, Proposition 1, page 51].

We let ${\mathcal{C}}_{Y}$ be the collection of arrows in ${\mathcal{C}}$
arising from paths on the digraph $DG$ belonging to $Y$,

(3.2) ${\mathcal{C}}_{Y}=\\{\langle v,x,u,y,w\rangle\mid v,u,w\in V,x\in
E_{i},y\in E_{j},1\leq i\neq j\leq k\\}.$

We refer to elements in ${\mathcal{C}}_{Y}$ as _bicoloured_ arrows.

For objects $v,w$ in ${\mathcal{C}}$, we let ${\mathcal{C}}(v,w)$ be the set
of morphisms in ${\mathcal{C}}$ which are arrows from $v$ to $w$. For each
pair of objects $v,w\in{\mathcal{C}}$, we will define a binary relation
$\mathcal{R}_{v,w}$ on the set of morphisms ${\mathcal{C}}(v,w)$. For this
purpose, we introduce the following terminology: Given two arrows
$A,A^{\prime}$ in ${\mathcal{C}}$, we say that $A^{\prime}$ is obtained from
$A$ by _diverting in ${\mathcal{C}}_{Y}$ over $\phi$_ if we have

(3.3) $A:=\langle
v_{1},x_{1},v_{2},\cdots,v_{s},x_{s},v_{s+1},x_{s+1},v_{s+2},\dots,v_{m-1},x_{m-1},v_{m}\rangle$

and

(3.4) $A^{\prime}:=\langle
v_{1},x_{1},v_{2},\cdots,v_{s},x_{s+1}^{\prime},v_{s+1}^{\prime},x_{s}^{\prime},v_{s+2},\dots,v_{m-1},x_{m-1},v_{m}\rangle$

with $\langle v_{s},x_{s},v_{s+1},x_{s+1},v_{s+2}\rangle,\langle
v_{s},x_{s+1}^{\prime},v_{s+1}^{\prime},x_{s}^{\prime},v_{s+2}\rangle\in{\mathcal{C}}_{Y}$
and $x_{s+1}^{\prime}x_{s}^{\prime}=\phi(x_{s}x_{s+1})$, for some $1\leq s\leq
m-1$ and $m>1$. Note that this makes sense on arrows of length at least $2$.
Note also that diverting $A$ to $A^{\prime}$ over $\phi$ does not change the
number of edges of given colour in $A$. Further, since $\phi^{2}=1$, if
$A^{\prime}$ is obtained from $A$ by diverting over $\phi$, then also $A$ is
obtained from $A^{\prime}$ by diverting in ${\mathcal{C}}_{Y}$ over $\phi$:
simply use that $x_{s}x_{s+1}=\phi(x_{s+1}^{\prime}x_{s}^{\prime})$.

Let $v,w\in{\mathcal{C}}$. For $A,A^{\prime}\in{\mathcal{C}}(v,w)$ we let

$A\mathcal{R}_{v,w}A^{\prime}$

if $A^{\prime}=A$ or there is a finite sequence
$A_{0}=A,A_{1},\dots,A_{n}=A^{\prime}$ of elements in ${\mathcal{C}}$ so that
$A_{p+1}$ for $1\leq p\leq n-1$ is obtained from $A_{p}$ by diverting over
$\phi$. The number of edges of same colour stays the same in each $A_{p}$.

We claim that $R_{v,w}$ is an equivalence relation. The reflexivity
$A\mathcal{R}_{v,w}A$ is clear for each $A$.

To see that the relation is symmetric, let
$A_{0}=A,A_{1},\dots,A_{n}=A^{\prime}$ be a sequence of morphisms in $C$
implementing the relation $A\mathcal{R}_{v,w}A^{\prime}$. Then the sequence
$A_{0}^{\prime}=A^{\prime},A_{1}^{\prime}=A_{n-1},\dots,A_{n-1}^{\prime}=A_{1},A_{n}^{\prime}=A_{0}$
will implement $A^{\prime}\mathcal{R}_{v,w}A$.

For transitivity, suppose that $A\mathcal{R}_{v,w}A^{\prime}$ and
$A^{\prime}\mathcal{R}_{v,w}A^{\prime\prime}$, which we may assume are
represented by the sequences

$A_{0}=A,A_{1},\dots,A_{n}=A^{\prime},B_{0}=A^{\prime},B_{1},\dots,B_{p}=A^{\prime\prime}.$

Then it is clear that
$A_{0}=A,A_{1},\dots,A_{n}=A^{\prime},A_{n+1}=B_{1},A_{n+2}=B_{2},\dots,A_{n+p}=B_{p}$
implements $A\mathcal{R}_{v,w}A^{\prime\prime}.$

We next claim that the equivalence relation $\mathcal{R}$ is preserved by
composition of morphisms. Suppose that $B\in{\mathcal{C}}(v^{\prime},v)$ and
$A\mathcal{R}_{v,w}A^{\prime}$. Let $A_{0}=A,A_{1},\dots,A_{n}=A^{\prime}$
implement the equivalence between $A,A^{\prime}$, then $C_{0}=A_{0}\circ
B,C_{1}=A_{1}\circ B,\dots,C_{n}=A_{n}\circ B$ implements the relation

$(A\circ B)\mathcal{R}_{v,w}(A^{\prime}\circ B).$

If now $D\in{\mathcal{C}}(w,w^{\prime})$, then similarly we have $(D\circ
A)\mathcal{R}_{v,w^{\prime}}(D\circ A^{\prime})$. In all, $\mathcal{R}$ is a
congruence in the sense of [17, Page 52]. Hence there is a category
${\mathcal{C}}/\mathcal{R}$ with object set $V$, the object set of
${\mathcal{C}}$, and set of morphisms $({\mathcal{C}}/\mathcal{R})(v,w)$ equal
to the quotient ${\mathcal{C}}(v,w)/\mathcal{R}_{v,w}$, for $v,w\in V$. We
denote the class in ${\mathcal{C}}(v,w)/\mathcal{R}_{v,w}$ of a morphism
$A\in{\mathcal{C}}(v,w)$ by $\dot{A}$.

Next we show existence of the functor
$d:{\mathcal{C}}/\mathcal{R}\to{\mathbb{N}}^{k}$. For an object $v$ in
${\mathcal{C}}/\mathcal{R}$ we let $d(v)=0$. For an elementary arrow ${A}$ in
${\mathcal{C}}$ with $A\in{\mathcal{C}}(v,w)$ for $v,w\in V$, there is a
unique colour $i\in\\{1,\dots,k\\}$ of the edge underlying the arrow. We set
$d({A}):=e_{i}$.

We extend this to an arbitrary arrow ${A}$ in ${\mathcal{C}}$ by

$d({A}):=e_{i_{1}}+e_{i_{2}}+\cdots+e_{i_{n}}\text{ if }A=\langle
v_{1},x_{1},v_{2},\cdots,x_{n},v_{n+1}\rangle,x_{s}\in E_{i_{s}},1\leq s\leq
n,$

where $i_{s}$ are not necessarily distinct. If two arrows $A,A^{\prime}$ are
given as in (3.3) and (3.4) with $A\mathcal{R}_{v,w}A^{\prime}$, we have that
$d(A)=d(A^{\prime})$ because at the vertex $v_{s}$ where the path in $A$ is
diverted we have $e_{i_{s}}+e_{i_{s+1}}=e_{i_{s+1}}+e_{i_{s}}$. Let

(3.5) $n_{K}=\begin{cases}\\#\\{s\mid 1\leq s\leq n,i_{s}=K\\}&\text{ if
}K\in\\{i_{1},i_{2},\dots,i_{n}\\}\\\ 0&\text{ if
}K\not\in\\{i_{1},i_{2},\dots,i_{n}\\}\end{cases}$

for $K=1,\dots,k$. Then

$d(\dot{A}):=(n_{1},n_{2},\dots,n_{k})\in{\mathbb{N}}^{k}$

is well-defined. Moreover, it satisfies
$d(\dot{A}\dot{B})=d(\dot{A})+d(\dot{B})$ for
$\dot{A}\in{\mathcal{C}}(v,w)/\mathcal{R}_{v,w}$ and
$\dot{B}\in{\mathcal{C}}(v^{\prime},v)/\mathcal{R}_{v^{\prime},v}$. Likewise
for composition on the right. This defines the functor
$d:{\mathcal{C}}/\mathcal{R}\to{\mathbb{N}}^{k}$. We will show that $d$ enjoys
the unique factorisation property required of a higher rank graph.

For this we fix an ordering of the colours and show, as an intermediate step,
that every morphism $\dot{A}$ has a representative $A$ in ${\mathcal{C}}$ with
all edges of the same colour grouped together. For simplicity of notation, we
assume that the colours appear in the order $\\{1,2,\dots,k\\}$. We claim that
each class $\dot{A}$ contains a representative

(3.6) $A=\langle
v_{1},x_{1},\cdots,x_{n_{1}},v_{n_{1}+1},x_{n_{1}+1},\cdots,x_{n_{1}+n_{2}},\cdots,v_{m}\rangle,$

where $x_{1},\dots,x_{n_{1}}$ are edges of colour $i_{1}$, followed by edges
$x_{n_{1}+1},\dots,x_{n_{1}+n_{2}}$ of colour $i_{2}$, and so on, with $n_{K}$
edges of colour $i_{K}$ at the end, where $i_{1}>i_{2}>\cdots>i_{K}$ are
distinct colours in $\\{1,\dots,k\\}$. Thus, the largest colour appears
nearest to the origin of the path on the digraph that determines the arrow,
and the colours appear in decreasing order along the path towards it terminus.

Note that (3.6) is trivially satisfied if $\dot{A}$ is the class of an arrow
$A$ on the digraph that only follows one colour. Assume next that there are
only two colours $i>j$ in a representative $A$ for $\dot{A}$. Let $v_{s}$ be
the first vertex at which we have a bicoloured path $\langle
v_{s},x_{s},v_{s+1},x_{s+1},v_{s+2}\rangle$ with $x_{s}\in E_{j},x_{s+1}\in
E_{i}$. Then we divert this into the equivalent path $\langle
v_{s},x_{s+1}^{\prime},v_{s+1}^{\prime},x_{s}^{\prime},v_{s+2}\rangle$ for
unique $x_{s}^{\prime}\in E_{j}$ and $x_{s+1}^{\prime}\in E_{i}$. If there are
no more edges from $E_{i}$ past the vertex $v_{s+2}$, we are done. If not,
continue the process until we are left with a representative for $\dot{A}$ of
the form required in (3.6).

Assume now that there are three distinct colours $l>j>i$ in a representative
$A$ for $\dot{A}$. If all three colours appear in decreasing order $l,j,i$ in
the path supporting $A$, there is nothing to prove. If two colours appear in
reverse order at a time, we reduce to the previous case. Assume now that a
tricoloured path appears in increasing order of colours. For simplicity of
notation, we may assume that this is an arrow in ${\mathcal{C}}$ of the form

$\langle v_{1},x,v_{2},y,v_{3},z,v_{4}\rangle,v_{1},\dots,v_{4}\in V,x\in
E_{i},y\in E_{j},z\in E_{l}.$

This is the composition $\langle v_{3},z,v_{4}\rangle\circ\langle
v_{2},y,v_{3}\rangle\circ\langle v_{1},x,v_{2}\rangle$ (or $z\circ y\circ x$)
of elementary arrows in colours $l>j>i$, and we claim that it is the same in
${\mathcal{C}}/\mathcal{R}$ as a (unique) arrow of the form

(3.7) $\langle
v_{1},\bar{z},v_{2}^{\prime},\bar{y},v_{3}^{\prime},\bar{x},v_{4}\rangle,\bar{z}\in
E_{l},\bar{y}\in E_{j},\bar{x}\in E_{i},$

where $v_{2}^{\prime},v_{3}^{\prime}\in V$. By successive application of this
reversing of order, it will follow that $\dot{A}$ admits a representative with
the colours appearing as in (3.6).

Since $\langle
v_{1},x,v_{2},y,v_{3}\rangle\in{\mathcal{C}}(v,w)\cap{\mathcal{C}}_{Y}$, there
are unique $x^{1}\in E_{i},y^{1}\in E_{j}$ such that

$\langle v_{1},x,v_{2},y,v_{3}\rangle\mathcal{R}_{v_{1},v_{3}}\langle
v_{1},y^{1},u_{1},x^{1},v_{3}\rangle,$

with $u_{1}=t(y^{1})=o(x^{1})$. For simplicity, write this as $(y\circ
x)\mathcal{R}_{v_{1},v_{3}}(x^{1}\circ y^{1})$. Continuing this way, the
bijection $\phi$ prescribes edges $x_{1},x_{2},x^{2}\in E_{i}$,
$y_{1},y_{2},y^{2}\in E_{j}$ and $z_{1},z^{1},z_{2},z^{2}\in E_{l}$ such that

$\displaystyle(z\circ x^{1})$
$\displaystyle\mathcal{R}_{u_{1},v_{4}}\,(x^{2}\circ z^{1}),\quad\text{ with
}o(x^{2})=t(z^{1})=u_{2}$ $\displaystyle(z^{1}\circ y^{1})\,$
$\displaystyle\mathcal{R}_{v_{1},u_{2}}\,(y^{2}\circ z^{2}),\quad\text{ with
}o(y^{2})=t(z^{2})=u_{3}$ $\displaystyle(z\circ y)\,$
$\displaystyle\mathcal{R}_{v_{2},v_{4}}\,(y_{1}\circ z_{1}),\quad\text{ with
}o(y_{1})=t(z_{1})=u_{4}$ $\displaystyle(z_{1}\circ x)\,$
$\displaystyle\mathcal{R}_{v_{1},u_{4}}\,(x_{1}\circ z_{2}),\quad\text{ with
}o(x_{1})=t(z_{3})=u_{5}$ $\displaystyle(y_{1}\circ x_{1})\,$
$\displaystyle\mathcal{R}_{u_{3},v_{4}}\,(x_{2}\circ y_{2}),\quad\text{ with
}o(x_{2})=t(y_{2})=u_{6},$

where $u_{1}=o(z^{1})$ and $u_{4}=t(x_{1})$. By our assumption (F2) on
$\operatorname{DG}$, we have $u_{2}=u_{6}$ and $u_{3}=u_{5}$ and

$\displaystyle\bar{x}:=x_{2}=x^{2},$ $\displaystyle\bar{y}:=y_{2}=y^{2},$
$\displaystyle\bar{z}:=z_{2}=z^{2}.$

With $v_{2}^{\prime}:=u_{3}$ and $v_{3}^{\prime}:=u_{2}$, this gives the
claimed representative in (3.7), where we have used that the relation
$\mathcal{R}$ preserves composition of morphisms. Successive applications of
this reversing of order in a tricoloured path show that $\dot{A}$ admits a
representative as in (3.6). Note at the same time that the class in
${\mathcal{C}}/\mathcal{R}$ of the morphism $\bar{x}\circ\bar{y}\circ\bar{z}$
decomposes uniquely as products of two morphisms along any choice in
$\mathcal{R}_{v_{1},v_{4}}$ which involves tricoloured paths. More precisely,
the decompositions are

(3.8) $\displaystyle z\circ(y\circ x)$ $\displaystyle\text{ corresponding to
}e_{l}+(e_{j}+e_{i});$ (3.9) $\displaystyle z\circ(x^{1}\circ y^{1})$
$\displaystyle\text{ corresponding to }e_{l}+(e_{i}+e_{j});$ (3.10)
$\displaystyle\bar{x}\circ(z^{1}\circ y^{1})$ $\displaystyle\text{
corresponding to }e_{i}+(e_{l}+e_{j});$ (3.11)
$\displaystyle\bar{x}\circ(\bar{y}\circ\bar{z})$ $\displaystyle\text{
corresponding to }e_{i}+(e_{j}+e_{l});$ (3.12) $\displaystyle
y_{1}\circ(x_{1}\circ\bar{z})$ $\displaystyle\text{ corresponding to
}e_{j}+(e_{i}+e_{l});$ (3.13) $\displaystyle y_{1}\circ(z_{1}\circ x)$
$\displaystyle\text{ corresponding to }e_{j}+(e_{l}+e_{i}).$

If more than three colours appear in a representative $A$ for $\dot{A}$, say
$i_{1}<i_{2}<\cdots<i_{K}$ with $K\geq 4$, and if a path supporting $A$ has
colours in increasing order then we resort to the previous cases. Thus, if
$K=4$ and a path appears with colours in the order $i_{1}<i_{2}<i_{3}<i_{4}$,
we first reverse the path onto colours $i_{1},i_{4},i_{3},i_{2}$, working from
the terminus of the path (source of its arrow in the category) to the left
towards its origin. Then we move the edges of colour $i_{1}$ past the ones of
colours $i_{4},i_{3},i_{2}$, using the previous cases. Similarly for $K>4$. In
all, (3.6) follows.

Now we are ready to prove the factorisation property of $d$. Suppose that
$\dot{A}$ is in ${\mathcal{C}}/\mathcal{R}$ with
$d(\dot{A})=(n_{1},n_{2},\dots,n_{k})\in N^{k}$. We must show that whenever
$(n_{1},n_{2},\dots,n_{k})=(m_{1},m_{2},\dots,m_{k})+(p_{1},p_{2},\dots,p_{k})$
in ${\mathbb{N}}^{k}$ there are unique morphisms $\dot{B},\dot{C}$ in
${\mathcal{C}}/\mathcal{R}$ so that

$\dot{A}=\dot{B}\dot{C},d(\dot{B})=(m_{1},m_{2},\dots,m_{k})\text{ and
}d(\dot{C})=(p_{1},p_{2},\dots,p_{k}).$

The proof is structured into cases determined by the number of non-zero
entries $n_{s}$, that is, by the number of colours that appear in a morphism
in the class $\dot{A}$.

Case 1: single colour. Thus $n_{i}>0$ for a unique $1\leq i\leq k$. If
$n_{i}=1$, then by our construction of ${\mathcal{C}}/\mathcal{R}$ and $d$ we
know that $\dot{A}$ is the class of an elementary arrow $A=\langle
v_{1},x_{1},v_{2}\rangle$ with $x_{1}\in E_{i}$ and only the trivial
decomposition involving identities at $v_{1},v_{2}$ is possible. If $n_{i}>1$,
then a representative for $A$ consists of a path of length $n_{i}$ along edges
in $E_{i}$, and so we can decompose $\dot{A}=\dot{B}\dot{C}$ with
$d(\dot{B})=m_{i},d(\dot{C})=p_{i}$ for any choice $n_{i}=m_{i}+p_{i}$ in
${\mathbb{N}}$.

Case 2: two colours. By our earlier claim (3.6), we may assume that
$d(\dot{A})=(n_{1},n_{2},\dots,n_{k})$ with $n_{i}\geq 1,n_{j}\geq 1$ at $i>j$
in $\\{1,\dots,k\\}$ and $n_{l}=0$ at all other entries.

Case 2.1: $n_{i}=n_{j}=1$. By definition of ${\mathcal{C}}/\mathcal{R}$, the
morphism $\dot{A}$ is the class of a bicoloured arrow $\langle
v_{1},x_{1},v_{2},x_{2},v_{3}\rangle$, with $x_{1}\in E_{i}$ and $x_{2}\in
E_{j}$. Let $x_{2}^{\prime}\in E_{j}$ and $x_{1}^{\prime}\in E_{i}$ so that
$\phi(x_{1}x_{2})=x_{2}^{\prime}x_{1}^{\prime}$, and put
$v_{2}^{\prime}=t(x_{2}^{\prime})=o(x_{1}^{\prime})$. Then

$A=\langle v_{2}^{\prime},x_{1}^{\prime},v_{3}\rangle\langle
v_{1},x_{2}^{\prime},v_{2}^{\prime}\rangle=\langle
v_{2},x_{2},v_{3}\rangle\langle v_{1},x_{1},v_{2}\rangle$

give the unique decompositions $\dot{A}=\dot{B}\dot{C}$ according to the
decompositions $e_{i}+e_{j}$ and $e_{j}+e_{i}$ of $d(\dot{A})$.

Case 2.2: $n_{i}>1,n_{j}\geq 1$. Without loss of generality, we may assume
that $n_{i}\geq n_{j}$.

The given morphism $\dot{A}$ is the class of an arrow following a path with
$n_{i}$ edges of colour $i$ and $n_{j}$ edges of colour $n_{j}$. Let
$n_{i}=m_{i}+p_{i}$ and $n_{j}=m_{j}+p_{j}$, and consider first the case that
$m_{i},p_{j}\geq 1$. We then divert the path in $A$ after $p_{i}+m_{i}-1$
edges of colour $i$ by applying case 2.1 to obtain a representative for
$\dot{A}$ in the form of a path with $p_{i}+m_{i}-1$ edges of colour $i$, then
one edge of colour $j$ followed by an edge of colour $i$, and finally with
$p_{j}+m_{j}-1$ edges of colour $j$ at the end. If $p_{i}+m_{i}-2=0$ and
$p_{j}+m_{j}-2=0$, we are done, having recovered case 2.1. Otherwise, if for
example $p_{i}+m_{i}-2\geq 1$, then apply case 2.1 to divert an edge of colour
$j$ onto one of colour $i$, thus obtaining a representative with
$p_{i}+m_{i}-2$ edges of colour $i$, then one edge of colour $j$ followed by
two of colour $i$, and finally $m_{j}+p_{j}-1$ edges of colour $j$. If also
$m_{j}+p_{j}-2\geq 1$, we divert another edge of colour $j$ successively past
the two of colour $i$ to get $p_{i}+m_{i}-2$ of colour $i$, two of colour $j$,
two of colour $i$, and finally $p_{j}+m_{j}-2$ of colour $j$. Continuing this
process, we divert all $m_{i}$ edges of colour $i$ followed by the $p_{j}$
edges of colour $j$ into a path where there are $p_{j}$ edges of colour $j$
first, followed by $m_{i}$ edges of colour $i$. This determines the required
decomposition of $\dot{A}=\dot{B}\dot{C}$, where
$d(\dot{B})=m_{i}e_{i}+m_{j}e_{j}$ and $d(\dot{C})=p_{i}e_{i}+p_{j}e_{j}$.

The remaining cases where only two colours are present are treated similarly.

Case 3: three colours. By our earlier claim, we may assume that
$d(\dot{A})=(n_{1},n_{2},\dots,n_{k})$ with $n_{i}\geq 1,n_{j}\geq 1,n_{l}\geq
1$ at $i>j>l$ in $\\{1,\dots,k\\}$ and $n_{h}=0$ at all other entries
$h\in\\{1,\dots,k\\}$. Assuming $n_{i}=m_{i}+p_{i}$, $n_{j}=m_{j}+p_{j}$ and
$n_{l}=m_{l}+p_{l}$, then by considering cases as before and applying the
factorisations (3.8)-(3.13) accordingly results in a factorisation
$\dot{A}=\dot{B}\dot{C}$ with $d(\dot{B})=m_{i}e_{i}+m_{j}e_{j}+m_{l}e_{l}$
and $d(\dot{C})=p_{i}e_{i}+p_{j}e_{j}+p_{l}e_{l}$. We omit the details.

In the general case where $d(\dot{A})$ has nonzero entries in more than three
distinct colours $i_{1}>i_{2}>\cdots>i_{K}$, $K\geq 4$, the problem reduces to
decomposing along any expression of $(n_{1},n_{2},\dots,n_{K})$ where at least
one of the entries $n_{i_{s}}$, $s=1,\dots,K$, is expressed as a sum
$m_{i_{s}}+p_{i_{s}}$ for $m_{i_{s}},p_{i_{s}}\geq 1$. This reduces to cases 2
and 3. ∎

We remark that an alternative argument to construct a higher rank graph from
the input of a $k$-dimensional digraph could be obtained by appealing to [10].
The main step is to identify a complete collection of squares that is
associative based on $\operatorname{DG}$ and then an application of [10,
Theorem 4.4] provides the desired $k$-graph.

######  Remark 3.2.

It is possible to express the bijection $\phi$ using notation from [13]. If
$C$ and $D$ are directed $1$-graphs with common set of vertices
$V=C^{0}=D^{0}$, distinct sets of edges $C^{1}$, $D^{1}$, and commuting vertex
matrices, let

$C^{1}\ast D^{1}=\\{(x,y)\in C^{1}\times D^{1}\mid t(x)=o(y)\\}.$

Then the bijection $\phi$ in (F1) is given by its restrictions
$\phi_{i,j}:E_{i}^{1}\ast E_{j}^{1}\to E_{j}^{1}\ast E_{i}^{1}$, for all
$i\neq j$ in $\\{1,2,\dots,k\\}$. This construction is reminiscent of an older
idea of a product of two (possibly directed) graphs, as described in Ore’s
monograph [21].

We note that condition (F2) in Definition 1.1 is vacuous when $k=2$. We shall
refer to $E=E_{1}\sqcup E_{2}\sqcup\dots\sqcup E_{k}$ as the _$1$ -skeleton_
of $\Lambda_{\operatorname{DG}}$.

######  Theorem 3.3.

For any complex $\mathcal{X}$ with universal cover equal to the product of $k$
regular trees, where $k\geq 2$, there is a $k$-dimensional digraph
$\operatorname{DG}(\mathcal{X})$ defined by sending a vertex of $\mathcal{X}$
to a vertex in $\operatorname{DG}(\mathcal{X})$, and by sending each geometric
edge in $\mathcal{X}$ to two edges in the edge set of
$\operatorname{DG}(\mathcal{X})$, of same colour and opposite orientations.

We prove this theorem in stages. First we prove the one-vertex case for $k=2$,
where the statement is a consequence of the description of a one-vertex
$2$-complex as $S_{A,B}$ for a VH-structure $A,B$. Then we prove the case
$k=3$ by employing geometric cubes. First we record a key consequence whose
proof is immediate from Theorem 3.1 and Theorem 3.3.

######  Corollary 3.4.

Given $\mathcal{X}$ a $k$-cube complex with $k\geq 2$, there is a $k$-graph
$\Lambda_{\operatorname{DG}(\mathcal{X})}$ with vertex set equal to the vertex
set of $\mathcal{X}$ and whose $1$-skeleton contains two edges for each
geometric edge of $\mathcal{X}$, in a colour preserving and orientation
reversing assignment. Further, $\Lambda_{\operatorname{DG}(\mathcal{X})}$ is
row-finite.

######  Definition 3.5.

Given a $k$-cube complex $\mathcal{X}$, its associated $C^{*}$-algebra is the
higher rank graph $C^{*}$-algebra
$C^{*}(\Lambda_{\operatorname{DG}(\mathcal{X})})$.

To simplify the notation we write $\Lambda(\mathcal{X})$ in place of
$\Lambda_{\operatorname{DG}(\mathcal{X})}$. Continuing our convention from the
proof of Theorem 3.1, we use letters such as $x,y$ to denote both generic
elements in the edge set $E$ of $\operatorname{DG}(\mathcal{X})$ and their
corresponding morphisms in $\Lambda(\mathcal{X})$ associated to elementary
arrows. In particular, a bicoloured path $xy$ on the digraph with $x\in
E_{i},y\in E_{j}$ for distinct colours $i\neq j$ will be $y\circ x$ as
composition as morphisms in the $k$-graph.

As another point of notation, in the proof of the next result and that of
Theorem 3.3 in the one-vertex case, we shall distinguish between labels such
as $a$ in a generating subset $A$ of a $k$-cube group $G$ and the label of the
edge it defines in the associated $k$-digraph, for which we reserve notation
of the form $\mathfrak{a}$. In the group we have $aa^{-1}=1=1_{G}$, so in the
one-vertex complex $a^{-1}$ means orientation reversal of the edge labelled
with $a$, while in the $k$-dimensional digraph the edge labelled $a$ is sent
into distinct directed edges $\mathfrak{a}_{1}$, $\mathfrak{a}_{2}$ (with no
cancellation inherited from the group).

######  Lemma 3.6.

Assume that $S_{A,B}$ is a one-vertex square complex with associated group $G$
given by a VH-structure $(A,B)$ with $\\#A$ and $\\#B$ both even positive
integers. Suppose that

(3.14) $A=\\{a_{1},\dots,a_{L},a_{L+1},\dots,a_{2L}\\}\text{ and
}B=\\{b_{1},\dots,b_{K},b_{K+1},\dots,b_{2K}\\},$

with $a_{r}a_{L+r}=1$ in $G$ for all $r=1,\dots,L$ and $b_{s}b_{K+s}=1$ in $G$
for all $s=1,\dots,K$, with $K,L\geq 1$. In particular, for each
$r=1,\dots,L$, we have that $a_{r}$ and $a_{L+r}$ label the same geometric
edge in $S_{A,B}$, but with opposite orientations. Similarly for $b_{s}$,
$b_{K+s}$ with $s=1,\dots,K$.

Then there is a $2$-dimensional digraph in the sense of Definition 1.1 with
edge set $E(S_{A,B})=E_{1}\sqcup E_{2}$ obtained by associating to each
$a_{r}$ for $r=1,\dots,2L$ a directed edge $\mathfrak{a}_{r}$ in $E_{1}$, and
to each $b_{s}$ for $s=1,\dots,2K$ a directed edge $\mathfrak{b}_{s}$ in
$E_{2}$.

###### Proof.

We have that for each $a_{r}\in A$ and $b_{s}\in B$, with $r=1,\dots,2L$ ,
$s=1,\dots,2K$, there are unique $a_{l(r,s)}\in A$ and $b_{m(r,s)}\in B$, with
$l(r,s)\in\\{1,\dots,2L\\}$ and $m(r,s)\in\\{1,\dots,2K\\}$, such that

$a_{r}b_{s}=b_{m(r,s)}a_{l(r,s)}.$

In particular, $a_{r}b_{s}$ is contained as a vertical-horizontal pair of
edges in a unique geometric square in the family

$S_{*}^{a_{r},b_{s}},r=1,\dots,2L,s=1,\dots,2K,*=O,H,V,R,$

with $b_{m(r,s)}a_{l(r,s)}$ forming the horizontal-vertical pair of edges in
$S_{*}^{a_{r},b_{s}}$ starting and ending at the same vertices as $a_{r}b_{s}$
(which we recall coalesce to the single vertex $v$ of $S_{A,B}$).

Let $xy$ be a path of length two with $x\in E_{1}$ and $y\in E_{2}$. Then $xy$
is uniquely determined by $x=\mathfrak{a}_{r}$ for some $r=1,\dots,2L$ and
$y=\mathfrak{b}_{s}$ for some $s=1,\dots,2K$. Let now $a_{r},b_{s},a_{l(r,s)}$
and $b_{m(r,s)}$ be as above. Then $y^{\prime}:=\mathfrak{b}_{m(r,s)}\in
E_{2}$ and $x^{\prime}:=\mathfrak{a}_{l(r,s)}\in E_{1}$ determine a unique
path of length two $y^{\prime}x^{\prime}$ so that $xy\sim
y^{\prime}x^{\prime}$.

This defines the required bijection $\phi:Y\to Y$ with
$\phi(xy)=y^{\prime}x^{\prime}$ on the set $Y$ of all paths of length two of
distinct colours. ∎

######  Example 3.7.

A simple construction of a $2$-graph based on the procedure of Corollary 3.4
recovers a known example, see [23], [37] and [15, Example 11.1(1)], where
$\theta(i,j)=(i,j)$ is the identity permutation of the set
$\\{1,2\\}\times\\{1,2\\}$. Consider the $(2,2)$-group
$G={\mathbb{Z}}\times{\mathbb{Z}}$ with generating sets $A=\\{a,a^{-1}\\}$
corresponding to the first copy of ${\mathbb{Z}}$ and $B=\\{b,b^{-1}\\}$ for
the second copy. We have the commutation relation $ab=ba$ as the basis for a
geometric square $S_{O}^{a,b}$. The one-vertex complex has two loops. An
application of Lemma 3.6 yields a $2$-dimensional digraph with edge set a
disjoint union of $E_{1}=\\{\mathfrak{a}_{1},\mathfrak{a}_{2}\\}$ and
$E_{2}=\\{\mathfrak{b}_{1},\mathfrak{b}_{2}\\}$, thus four loops, with the
bijection $\phi$ on the set of paths of length two of distinct colours read
off from $S_{O}^{a,b},S_{H}^{a,b},S_{V}^{a,b}$ and $S_{R}^{a,b}$ as follows:

$\mathfrak{a}_{1}\mathfrak{b}_{1}\sim\mathfrak{b}_{1}\mathfrak{a}_{1},\mathfrak{a}_{1}\mathfrak{b}_{2}\sim\mathfrak{b}_{2}\mathfrak{a}_{1},\mathfrak{a}_{2}\mathfrak{b}_{1}\sim\mathfrak{b}_{1}\mathfrak{a}_{2}\text{
and }\mathfrak{a}_{2}\mathfrak{b}_{2}\sim\mathfrak{b}_{2}\mathfrak{a}_{2}.$

######  Example 3.8.

We now present a $2$-graph from this recipe where the group $G$ is not of
product type. As we will explain, figure 2 shows an example of a $(4,4)$-group
$G$, cf. [34].

Figure 2. A concrete example of $(4,4)$-group from four squares

The four squares are geometric squares representing the $2$-cells of an
associated complex $S_{A,B}$, where $A=\\{a_{1},a_{2},a_{3},a_{4}\\}$ for
$a_{3}=a_{1}^{-1}$ and $a_{4}=a_{2}^{-1}$, and
$B=\\{b_{1},b_{2},b_{3},b_{4}\\}$ for $b_{3}=b_{1}^{-1}$ and
$b_{4}=b_{2}^{-1}$. Here $L=K=2$, cf. Lemma 3.6. With our convention in (2.7)
we have, from left to right, $S_{O}^{a_{1},b_{1}}$, $S_{O}^{a_{1},b_{3}}$,
$S_{O}^{a_{1},b_{4}}$ and $S_{O}^{a_{2},b_{1}}$.

The associated $2$-graph $\Lambda(S_{A,B})$ from Corollary 3.4 has
$1$-skeleton determined by the $2$-dimensional digraph whose edges are given
by the disjoint union of
$E_{1}=\\{\mathfrak{a}_{1},\mathfrak{a}_{2},\mathfrak{a}_{3},\mathfrak{a}_{4}\\}$
and
$E_{2}=\\{\mathfrak{b}_{1},\mathfrak{b}_{2},\mathfrak{b}_{3},\mathfrak{b}_{4}\\}$,
see Lemma 3.6. Let us now describe explicitly the bijection $\phi:Y\to Y$.
Note that the digraph has 4 loops of one colour (red) and 4 loops of the
second colour (blue).

We have 16 paths of length two of the form $xy$, where $x\in E_{1}$ and $y\in
E_{2}$, given by all the possible $\mathfrak{a}_{i}\mathfrak{b}_{j}$ with
$i,j=1,\dots,4$. Correspondingly, we have all possible vertical-horizontal
pairs of edges $a_{i}b_{j}$ in the collection of geometric squares

$S_{*}^{a_{j},b_{j}},i,j=1,\dots,4,*=O,V,R,H.$

Pick for each $a_{i}b_{j}$ the unique $a_{l(i,j)}\in A$ and $b_{m(i,j)}\in B$
such that $b_{m(i,j)}a_{l(i,j)}$ is the corresponding horizontal-vertical pair
of edges in the same square, and let $y^{\prime}=\mathfrak{b}_{m(i,j)}$,
$x^{\prime}=\mathfrak{a}_{l(i,j)}$ as prescribed by the proof of Lemma 3.6.

Explicitly, corresponding to the horizontal-vertical pairs of edges in the
geometric square

$\\{S_{O}^{a_{1},b_{1}},S_{V}^{a_{1},b_{1}},S_{R}^{a_{1},b_{1}},S_{H}^{a_{1},b_{1}}\\},$

it is seen that
$\phi(\mathfrak{a}_{1}\mathfrak{b}_{1})=\mathfrak{b}_{4}\mathfrak{a}_{3}$,
$\phi(\mathfrak{a}_{3}\mathfrak{b}_{4})=\mathfrak{b}_{1}\mathfrak{a}_{1}$,
$\phi(\mathfrak{a}_{1}\mathfrak{b}_{2})=\mathfrak{b}_{3}\mathfrak{a}_{3}$ and
$\phi(\mathfrak{a}_{3}\mathfrak{b}_{3})=\mathfrak{b}_{2}\mathfrak{a}_{1}$.

Similarly, from the geometric square

$\\{S_{O}^{a_{1},b_{3}},S_{V}^{a_{1},b_{3}},S_{R}^{a_{1},b_{3}},S_{H}^{a_{1},b_{3}}\\}$

we get
$\phi(\mathfrak{a}_{1}\mathfrak{b}_{3})=\mathfrak{b}_{1}\mathfrak{a}_{4}$,
$\phi(\mathfrak{a}_{3}\mathfrak{b}_{1})=\mathfrak{b}_{3}\mathfrak{a}_{2}$,
$\phi(\mathfrak{a}_{2}\mathfrak{b}_{3})=\mathfrak{b}_{1}\mathfrak{a}_{3}$ and
$\phi(\mathfrak{a}_{4}\mathfrak{b}_{1})=\mathfrak{b}_{3}\mathfrak{a}_{1}$;
from

$\\{S_{O}^{a_{1},b_{4}},S_{V}^{a_{1},b_{4}},S_{R}^{a_{1},b_{4}},S_{H}^{a_{1},b_{4}}\\}$

we get
$\phi(\mathfrak{a}_{1}\mathfrak{b}_{4})=\mathfrak{b}_{2}\mathfrak{a}_{2}$,
$\phi(\mathfrak{a}_{3}\mathfrak{b}_{2})=\mathfrak{b}_{4}\mathfrak{a}_{4}$,
$\phi(\mathfrak{a}_{4}\mathfrak{b}_{4})=\mathfrak{b}_{2}\mathfrak{a}_{3}$ and
$\phi(\mathfrak{a}_{2}\mathfrak{b}_{2})=\mathfrak{b}_{4}\mathfrak{a}_{1}$;
finally, from the geometric square

$\\{S_{O}^{a_{2},b_{1}},S_{V}^{a_{2},b_{1}},S_{R}^{a_{2},b_{1}},S_{H}^{a_{2},b_{1}}\\}$

we get
$\phi(\mathfrak{a}_{2}\mathfrak{b}_{1})=\mathfrak{b}_{2}\mathfrak{a}_{4}$,
$\phi(\mathfrak{a}_{4}\mathfrak{b}_{2})=\mathfrak{b}_{1}\mathfrak{a}_{2}$,
$\phi(\mathfrak{a}_{2}\mathfrak{b}_{4})=\mathfrak{b}_{3}\mathfrak{a}_{4}$ and
$\phi(\mathfrak{a}_{4}\mathfrak{b}_{3})=\mathfrak{b}_{4}\mathfrak{a}_{2}$;
this describes the bijection $\phi$ completely.

The link of $S_{A,B}$ at its vertex $v$ is the complete bipartite graph of
type $(4,4)$, see Figure 3.

Figure 3. The link of the complex

It follows that $G:=\pi_{1}(S_{A,B},v)$ is a $(4,4)$-group. In fact, $G$ is
the fundamental group of a CAT(0) complex with Gromov link condition, see [8].
We remind that every edge of the complex belongs to four squares, see figure 4
for a fragment of the universal cover of the complex showing the edge $a_{1}$
belonging to four squares (in the universal cover).

Figure 4. Fragment of the universal cover showing the edge $a_{1}$

######  Remark 3.9.

The group with the same VH-structure as in Figure 2 appears also in [12],
Section 7, as the group $2\times 2.37$ in their list. However, their $2$-graph
is different from the one in Example 3.8, since, if we translate the notions
of [12] into higher rank graphs, the $2$-graph corresponding to the group
$2\times 2.37$ would have sixteen vertices. In general, the $2$-graphs of [12]
corresponding to $(2m,2n)$ groups give $2$-graphs with $4mn$ vertices,
$4(m-1)mn$ edges of one colour and $4mn(n-1)$ edges of another colour. The
$2$-graphs of [27], [29] have $3(q^{2}+q+1)$ vertices and $3(q^{2}+q+1)q$
edges of each colour for $q$ being a prime power different from $3$.

###### Proof.

(Proof of Theorem 3.3, the general case.) Fix $k\geq 3$. Assume first that we
have a one-vertex $k$-cube complex $P$, thus we may take it of form
$P_{A_{1},\dots,A_{k}}$ associated with a $k$-cube group $G$ with underlying
structure determined by the ordered tuple $(A_{1},\dots,A_{k})$. As explained,
the edges of the complex are labelled by the generators of $G$. For each
$i=1,\dots,k$ we write

$A_{i}=\\{a_{1}^{i},\dots,a_{L_{i}}^{i},a_{L_{i}+1}^{i},\dots,a_{2L_{i}}^{i}\\},$

with the convention that $a_{r}^{i}a_{L_{i}+r}^{i}=1$ in $G$ for $1\leq r\leq
L_{i}$. Define a digraph with edge set $E=E_{1}\sqcup E_{2}\sqcup\dots\sqcup
E_{k}$ by the assignment that to each $a_{r}^{i}$ corresponds a directed edge
$\mathfrak{a}_{r}^{i}$ in $E_{i}$, with $i=1,\dots,k$ and $r=1,\dots,2L_{i}$.
We must identify a bijection $\phi$ on the set of bicoloured edges and
establish conditions (F1) and (F2) of Definition 1.1.

Suppose that $xy$ is a path of length two in $E$ with $x\in E_{i}$ and $y\in
E_{j}$ for distinct $i,j$ in $\\{1,\dots,k\\}$. Then there are $a_{r}^{i}\in
A_{i}$ and $b_{s}^{j}\in A_{j}$ for unique $r=1,\dots,2L_{i}$ and
$s=1,\dots,2L_{j}$, such that $x=\mathfrak{a}_{r}^{i}$ and
$y=\mathfrak{a}_{s}^{j}$. Since we have a cube complex, there is an associated
square complex $P_{A_{i},A_{j}}$ with corresponding group $G_{A_{i},A_{j}}$,
where we choose the convention that $A_{i}$ is vertical and $A_{j}$ horizontal
direction. Lemma 3.6 implies that there is a unique path of length two
$y^{\prime}x^{\prime}$ with $y^{\prime}\in A_{j}$ and $x^{\prime}\in A_{i}$,
corresponding to a unique square with vertical-horizontal and horizontal-
vertical pairs given by

$a_{r}^{i}a_{s}^{j}=a_{m(r,s)}^{j}a_{l(r,s)}^{i}$

such that $xy\sim y^{\prime}x^{\prime}$. Here $a_{m(r,s)}^{j}\in A_{j}$ and
$a_{l(r,s)}^{i}\in A_{i}$ are uniquely determined. This provides the desired
bijection $\phi$ and settles requirement (F1).

Next suppose that we are given a path $xyz$ with $x\in E_{i},y\in E_{j},z\in
E_{l}$ for distinct colours $i,j,l$ in $\\{1,\dots,k\\}$. The key ingredient
is that by condition (5) in Definition 2.3, each directed cube, in the sense
of (F2), arises as a directed copy of one of the $3$-dimensional cubes of the
complex. We now identify a directed cube satisfying the hypotheses of (F2) and
an associated geometric $3$-cube.

First, there is a unique square $S_{1}^{ij}$ which contains a vertical-
horizontal pair $a_{r}^{i}a_{s}^{j}$ with $a_{r}^{i}\in A_{i},a_{s}^{j}\in
A_{j}$ so that $x=\mathfrak{a}_{r}^{i}$ and $y=\mathfrak{a}_{s}^{j}$. Upon
completing the square $S_{1}^{ij}$ to
$a_{r}^{i}a_{s}^{j}=a_{s^{1}}^{j}a_{r^{1}}^{i}$, as in the beginning of the
proof, for unique $r^{1}\in\\{1,\dots,2L_{i}\\}$ and
$s^{1}\in\\{1,\dots,2L_{j}\\}$, we have

$xy\sim y^{1}x^{1}\text{ for }x^{1}=\mathfrak{a}_{r^{1}}^{i}\text{ and
}y^{1}=\mathfrak{a}_{s^{1}}^{j}.$

Next we use $x^{1}$ and $z$ to extract a geometric square $S_{2}^{il}$,
determined by $a_{r^{1}}^{i}a_{t}^{l}=a_{t^{1}}^{l}a_{r^{2}}^{i}$, for unique
$t^{1}\in\\{1,\dots,2L_{l}\\}$ and $r^{2}\in\\{1,\dots,2L_{i}\\}$, so that

$x^{1}z\sim z^{1}x^{2}\text{ for
}z=\mathfrak{a}_{t}^{l},z^{1}=\mathfrak{a}_{t^{1}}^{l}\text{ and
}x^{2}=\mathfrak{a}_{r^{2}}^{i}.$

Finally, by using $y^{1},z^{1}$ we extract a geometric square $S_{3}^{jl}$,
determined by $a_{s^{1}}^{j}a_{t^{1}}^{l}=a_{t^{2}}^{l}a_{s^{2}}^{j}$, for
unique $s^{2}\in\\{1,\dots,2L_{j}\\}$ and $t^{2}\in\\{1,\dots,2L_{l}\\}$, so
that

$y^{1}z^{1}\sim z^{2}y^{2}\text{ for }y^{2}=\mathfrak{a}_{s^{2}}^{j}\text{ and
}z^{2}=\mathfrak{a}_{t^{2}}^{l}.$

There is a unique geometric cube containing the squares
$S_{1}^{ij},S_{2}^{il}$ and $S_{3}^{jl}$, where the notation follows the
convention after figure 1, and $a_{r}^{i}a_{s}^{j}a_{t}^{l}$ is a path joining
two vertices in the cube at longest possible distance due to condition (5) in
Definition 2.3. Thus, in this geometric cube we have also obtained the path
$a_{t^{2}}^{l}a_{s^{2}}^{j}a_{r^{2}}^{i}$ opposite to
$a_{r}^{i}a_{s}^{j}a_{t}^{l}$ and joining the same vertices in the cube.

If we perform the same argument starting with $y,z$ to obtain $yz\sim
z_{1}y_{1}$, followed by $x,z_{1}$ to obtain $xz_{1}\sim z_{2}x_{1}$ and
finally $x_{1},y_{1}$ to get $x_{1}y_{1}\sim y_{2}x_{2}$, we find unique
squares $S_{4}^{jl},S_{5}^{il},S_{6}^{ij}$ determined by
$a^{j}_{s}a^{l}_{t}=a^{l}_{t_{1}}a^{j}_{s_{1}}$,
$a^{i}_{r}a^{l}_{t_{1}}=a^{l}_{t_{2}}a^{i}_{r_{1}}$ and
$a^{i}_{r_{1}}a^{j}_{s_{1}}=a^{j}_{s_{2}}a^{i}_{r_{2}}$, respectively, for
$r_{2}\in\\{1,\dots,2L_{i}\\},s_{2}\in\\{1,\dots,2L_{j}\\}$ and
$t_{2}\in\\{1,\dots,2L_{l}\\}$. So

$x_{2}=\mathfrak{a}_{r_{2}}^{i},\,y_{2}=\mathfrak{a}_{s_{2}}^{j}\,z_{2}=\mathfrak{a}_{t_{2}}^{l}.$

Since $a_{r}^{i}a_{s}^{j}a_{t}^{l}$ is a common $ijl$-path, the squares
$S_{1}^{ij},S_{2}^{il},S_{3}^{jl},S_{4}^{jl},S_{5}^{il},S_{6}^{ij}$ determine
the same geometric cube. We have that
$a_{t_{2}}^{l}a_{s_{2}}^{j}a_{r_{2}}^{i}$ is another path in this
$3$-dimensional cube opposite to $a_{r}^{i}a_{s}^{j}a_{t}^{l}$ and joining
vertices at longest possible distance. Since there only is one path of longest
distance opposite to $a_{r}^{i}a_{s}^{j}a_{t}^{l}$ in a $3$-dimensional cube,
we must have

$a_{s^{2}}^{j}=a_{s_{2}}^{j},\,a_{t^{2}}^{l}=a_{t_{2}}^{l}\text{ and
}a_{r^{2}}^{i}=a_{r_{2}}^{i}.$

Then $x_{2}=x^{2}$, $y_{2}=y^{2}$ and $z_{2}=z^{2}$, as required to fulfill
condition (F2).

We now assume that $\mathcal{X}$ has $N$ vertices, with $N\geq 2$, and we
declare these to be the vertices of $\operatorname{DG}(\mathcal{X})$. As
prescribed, each geometric edge in the complex is sent into two edges with
opposite orientation in the edge set $E$ of $\operatorname{DG}(\mathcal{X})$.
For every path of length two $xy$ with $x\in E_{i}$ and $y\in E_{j}$ for
$i\neq j$ so that $o(y)=t(x)$, there is a unique geometric square $S_{1}$ in
$\mathcal{X}$ in which $xy$ corresponds to a vertical-horizontal pair of
edges. Then the corresponding horizontal-vertical pair of edges gives rise to
$x^{\prime}\in E_{i}$ and $y^{\prime}\in E_{j}$ such that $xy\sim
y^{\prime}x^{\prime}$, and this defines uniquely the bijection $\phi$ on the
set $Y$ of paths of length two of distinct colours required in (F1). The
condition (F2) holds by the same argument as the one-vertex case because any
$ijl$-coloured path $xyz$ will be contained in a unique $3$-cube in
$\mathcal{X}$, determined through unique squares
$S_{1}^{ij},S_{2}^{il},S_{3}^{jl},S_{4}^{jl},S_{5}^{il},S_{6}^{ij}$. The
difference is that since the complex has more than one vertex, we do not have
a labelling of the edges by elements of the group $G$ acting cocompactly on
$\mathcal{X}$, but this does not affect the existence of the squares and of
the $3$-cube in which $xyz$ determines a path with unique opposite path at
longest distance between the same vertices of the $3$-dimensional cube. ∎

To visualise the argument in the proof of Theorem 3.3, we refer to the generic
geometric cube in figure 1. Let $x=a_{1}$ (or, for consistency, $x$ is a
directed edge in $E_{i}$ labelled with $a_{1}\in A_{i}$), $y=b_{1}$ and
$z=c_{3}$. The argument produces the path
$a_{t^{2}}^{l}a_{j^{2}}^{j}a_{r^{2}}^{i}$ given by $c_{1}b_{3}a_{3}$ following
the squares $S_{1}^{ij}=(a_{1},b_{1},a_{2}^{-1},b_{2}^{-1})$,
$S_{2}^{il}=(a_{2},c_{3},a_{3}^{-1},c_{2}^{-1})$ and
$S_{3}^{jl}=(b_{2},c_{2},b_{3}^{-1},c_{1}^{-1})$. Alternatively, it produces
the path $a_{t_{2}}^{l}a_{j_{2}}^{j}a_{r_{2}}^{i}$ following the squares
$(b_{1},c_{3},b_{4}^{-1},c_{4}^{-1})$, $(a_{1},c_{4},a_{4}^{-1},c_{1}^{-1})$
and $(a_{4},b_{4},a_{3}^{-1},b_{3}^{-1})$.

######  Example 3.10.

In figure 5 we present a geometric cube which is part of the data of the
$3$-cube group $\Gamma_{2}$ from Section 2.8.

Figure 5. A geometric cube for the $\Gamma_{2}$ group

The generating sets of $\Gamma_{2}$ are
$A_{1}=\\{a_{1},a_{2},a_{1}^{-1},a_{2}^{-1}\\}$,
$A_{2}=\\{b_{1},b_{2},b_{3},b_{1}^{-1},b_{2}^{-1},b_{3}^{-1}\\}$ and
$A_{3}=\\{c_{1},c_{2},c_{3},c_{4},c_{1}^{-1},c_{2}^{-1},c_{3}^{-1},c_{4}^{-1}\\}$.
There are $(|A_{1}|\cdot|A_{2}|\cdot|A_{3})|/2^{3}=24$ cubes in total, where
the factor $2^{3}$ in the denominator corresponds to the fact that there are
$8$ vertices in the cube, and we can complete the cube starting with three
edges of distinct colours from any one of them.

The cube in figure 5 is obtained from the triple $(a_{1},b_{1},c_{2})$ of
edges in the three alphabets by completing its faces with geometric squares.
With the notation of figure 1, the faces $S_{1}^{12}$, $S_{2}^{13}$ and
$S_{3}^{23}$ arise, respectively, from the group relations
$a_{1}b_{1}a_{4}b_{2}$, $a_{2}c_{2}a_{1}c_{3}$ and $b_{2}c_{4}b_{5}c_{3}$
(identified with $b_{2}^{-1}c_{3}^{-1}b_{2}c_{4}^{-1}$). The remaining three
faces correspond to the geometric squares $b_{1}c_{2}b_{1}c_{5}$,
$a_{1}c_{1}a_{2}c_{8}$ (identified with $a_{1}c_{1}a_{2}c_{4}^{-1}$) and
$a_{4}b_{4}a_{1}b_{2}$ (identified with $a_{2}^{-1}b_{1}^{-1}a_{1}b_{2}$).

###  3.1. Aperiodicity

In $C^{*}$-algebra theory, the classification of purely infinite, simple,
unital, nuclear, $C^{*}$-algebras is a landmark result by Kirchberg-Phillips,
see [22]. Aperiodicity of a higher rank graph is an important property,
because together with cofinality it implies simplicity of the associated
$C^{*}$-algebra, and further implies pure infiniteness if every vertex can be
reached from a loop with an entrance. We next investigate aperiodicity of
$\Lambda(P)$ from Corollary 3.4.

We recall the necessary facts and notation from [13]. Let $\Lambda$ be a
$k$-graph. If ${m}=(m_{i})_{i},{q}=(q_{i})_{i}\in{\mathbb{N}}^{k}$, we write
${m}\leq{q}$ if $m_{i}\leq q_{i}$ for all $i=1,\dots,k$. By $\Omega_{k}$ we
denote the $k$-graph with vertex set $\Omega_{k}^{0}={\mathbb{N}}^{k}$ and set
of elements (morphisms) consisting of pairs
$(m,n)\in{\mathbb{N}}^{k}\times{\mathbb{N}}^{k}$ with $m\leq n$ and
$d(m,n)=n-m$. The set $\Lambda^{\infty}$ of infinite paths consists of degree
preserving functors $\omega:\Omega_{k}\to\Lambda$. An infinite path $\omega$
is _aperiodic_ provided that for every $q\in{\mathbb{N}}^{k}$ and all
$p\in{\mathbb{Z}}^{k}\setminus\\{0\\}$, there is $(m,n)\in\Omega_{k}$ such
that $m+p\geq 0$ and $\omega(m+p+q,n+p+q)\neq\omega(m+p,n+p)$. The $k$-graph
$\Lambda$ satisfies the _aperiodicity condition (A)_ provided that for every
$v\in\Lambda^{0}$ there is an aperiodic path $\omega$ with $r(\omega)=v$.

In our case, the existence of an aperiodic infinite path will be provided by
the theory of rigid $k$-monoids from [15]. Guided by the work of Lawson-
Vdovina, we first extend the notion of left and right rigid to $k$-dimensional
digraphs with one vertex. We note that the idea of rigidity below appeared in
a first form in [35, Definition on page 3, items(2) and (3)].

######  Definition 3.11.

Let $\operatorname{DG}$ be a $k$-dimensional digraph with one vertex and edge
set $E=E_{1}\sqcup E_{2}\sqcup\dots\sqcup E_{k}$ for $k\geq 2$. We say that

  1. (1)

$\operatorname{DG}$ is _right rigid_ if for every ${x}^{\prime}\in
E_{i},{y}^{\prime}\in E_{j}$ with $i\neq j$ there are unique $x\in E_{i},y\in
E_{j}$ such that $x{y}^{\prime}\sim y{x}^{\prime}$.

  2. (2)

$\operatorname{DG}$ is _left rigid_ if for every $x\in E_{i},y\in E_{j}$,
$i\neq j$, there are unique ${x}^{\prime}\in E_{i},{y}^{\prime}\in E_{j}$ such
that $x{y}^{\prime}\sim y{x}^{\prime}$.

######  Lemma 3.12.

Suppose that $P$ is a one-vertex $k$-cube complex with underlying structure
$(A_{1},\dots,A_{k})$ for $i=1,\dots,k$, where each $A_{i}$ is of the form
$\\{a_{1}^{i},\dots,a_{2L_{i}}^{i}\\}$ and $a_{r}^{i}a_{L_{i}+r}^{i}=1$ in the
associated group for all $1\leq r\leq L_{i}$. Let $\operatorname{DG}(P)$ be
the associated $k$-dimensional digraph from Theorem 3.3. Then
$\operatorname{DG}(P)$ is left and right rigid.

###### Proof.

The two properties of being rigid above arise from the fact that the link of
the vertex $v$ in $P$ has no multiple edges. Therefore, every top-left corner
and every bottom-right corner appear exactly once in a geometric square. The
formal proof is below.

Suppose that $x^{\prime}\in E_{i}$ and $y^{\prime}\in E_{j}$ for $i\neq j$ in
$\\{1,\dots,k\\}$. Since we have a cube complex, there is an associated square
complex $P_{A_{i},A_{j}}$. By our construction of $\operatorname{DG}(P)$,
there are $a_{r}^{i}\in A_{i}$ and $b_{s}^{j}\in A_{j}$ for unique
$r=1,\dots,2L_{i}$ and $s=1,\dots,2L_{j}$ such that
$x^{\prime}=\mathfrak{a}_{r}^{i}$ and $y^{\prime}=\mathfrak{b}_{s}^{j}$. In
the square complex $P_{A_{i},A_{j}}$ there is a unique square of the form

$S_{O}=(a_{r}^{i},(b_{s}^{j})^{-1},(a_{g}^{i})^{-1},b_{h}^{j})$

for $g\in\\{1,\dots,2L_{i}\\}$ and $h\in\\{1,\dots,2L_{j}\\}$. The associated
$S_{H}$ is $(a_{g}^{i},b_{s}^{j},(a_{r}^{i})^{-1},(b_{h}^{j})^{-1})$, and thus
$a_{g}^{i}b_{s}^{j}=b_{h}^{j}a_{r}^{i}$ in $G_{A_{i},A_{j}}$. Letting
$x=\mathfrak{a}_{g}^{i}$ and $y=\mathfrak{b}_{h}^{j}$ gives $xy^{\prime}\sim
yx^{\prime}$ in $\operatorname{DG}(P)$, as claimed.

Right rigidity is similar. Starting this time with $x\in E_{i}$ and $y\in
E_{j}$ for distinct $i,j$ in $\\{1,\dots,k\\}$, we find $a_{r}^{i}\in A_{i}$
and $b_{s}^{j}\in A_{j}$ for unique $r\in\\{1,\dots,2L_{i}\\}$ and
$s\in\\{1,\dots,2L_{j}\\}$ so that $x=\mathfrak{a}_{r}^{i}$ and
$y=\mathfrak{b}_{s}^{j}$. Consider the unique square

$S_{O}=((a_{r}^{i})^{-1},b_{s}^{j},a_{m}^{i},(b_{n}^{j})^{-1})$

in $P_{A_{i},A_{j}}$, and form its associated $S_{V}$, which is
$(a_{r}^{i},b_{n}^{j},(a_{m}^{i})^{-1},(b_{s}^{j})^{-1})$. Then
$a_{r}^{i}b_{n}^{j}=b_{s}^{j}a_{m}^{i}$ in $G_{A_{i},A_{j}}$, so letting
$x^{\prime}=\mathfrak{a}_{m}^{i}$ and $y^{\prime}=\mathfrak{b}_{n}^{j}$ leads
to $xy^{\prime}\sim yx^{\prime}$ in the digraph, as claimed.

∎

Given a one-vertex $k$-cube complex $P$, the associated $k$-graph $\Lambda(P)$
is a monoid, being a category with a single object. It is a $k$-monoid in the
sense of [15], with alphabets
$E_{i}=\\{\mathfrak{a}_{1}^{i},\dots,\mathfrak{a}_{2L_{i}}^{i}\\}$ for
$i=1,\dots,k$ where each $L_{i}\geq 1$.

######  Corollary 3.13.

Given a one-vertex $k$-cube complex $P$, the graph $\Lambda(P)$ is left and
right rigid, and satisfies the aperiodicity condition. In particular,
$C^{*}(\Lambda(P))$ is simple.

###### Proof.

Let $P$ be a one-vertex $k$-complex, which we may assume as in the hypothesis
of Lemma 3.12. Let $\Lambda(P)=\Lambda(\operatorname{DG}(P))$ be the
associated $k$-graph from Corollary 3.4. The right rigidity of the digraph
implies that for any choice of elements $y^{\prime},x^{\prime}$ with
$y^{\prime}$ in the alphabet $E_{j}$ and $x^{\prime}$ in $E_{i}$, where $i\neq
j$, there are unique elements $x$ and $y$ in the alphabets $E_{i}$ and
$E_{j}$, respectively, so that $y^{\prime}\circ x=x^{\prime}\circ y$. This
means that $\Lambda(P)$ is right rigid. Left rigid follows in a similar way.
We conclude from [15, Corollary 11.10 and Lemma 4.15] that $\Lambda(P)$ is
effective and hence admits an aperiodic infinite path. As there is only one
vertex, the aperiodicity condition is satisfied. Since $\Lambda(P)$ is also
cofinal, $C^{*}(\Lambda(P))$ is simple by [13, Proposition 4.8]. ∎

The constructions of [28] produce a purely infinite simple rank two Cuntz-
Krieger algebra $\mathcal{A}$. This uses in a crucial way the fact that every
word $w$ of a given shape $m=(m_{1},m_{2})\in{\mathbb{N}}^{2}$ admits at least
two distinct extensions $w^{\prime},w^{\prime\prime}$, in the sense that the
origin of $w^{\prime},w^{\prime\prime}$ (with suitable interpretation) equals
the terminus of $w$, and both have same shape $e_{j}$ for all $j=1,2$.

For a row-finite and source free $k$-graph $\Lambda$, [13, Proposition 4.9]
put forward conditions that would imply $C^{*}(\Lambda)$ is purely infinite
simple. A correct version of these conditions was identified in [32,
Proposition 8.8], which we present here (writing _cycle_ instead of _loop_):
given a finitely aligned $k$-graph $\Lambda$, a morphism
$\mu\in\Lambda\setminus\Lambda^{0}$ is a _cycle with an entrance_ if
$s(\mu)=r(\mu)$ and there exists $\alpha\in s(\mu)\Lambda$ having
$d(\alpha)\leq d(\mu)$ and being distinct from the initial segment of $\mu$ of
degree $d(\alpha)$. Thus for some factorisation $\mu=\mu_{1}\mu_{2}$ where
$d(\mu_{1})=n\leq d(\mu)$, there exists $\alpha\neq\mu_{1}$ with $d(\alpha)=n$
and $r(\alpha)=r(\mu_{1})$. Therefore, upon interpreting concatenation of
edges on the digraph $\operatorname{DG}(P)$ as composition of morphisms in the
associated $\Lambda(P)$, see (3.1), and by interpreting the constructions of
[28] in terms of higher rank graphs, the existence of a cycle with an entry
requires that for a given $\mu_{2}$ there are two distinct extensions, with
the additional property that the origin of $\mu_{2}$ is the terminus of one of
the extensions. As we will show below, our $k$-graphs satisfy the stronger
aperiodicity condition used in [28].

######  Proposition 3.14.

Let $P$ be a one-vertex $k$-complex as in the hypothesis of Lemma 3.12 for
$k\geq 2$ and let $\Lambda(P)$ be the associated one-vertex $k$-graph. If
$L_{i}\geq 2$ for $i=1,\dots,k$, then the vertex in $\Lambda(P)$ supports at
least two distinct cycles of length two in colour $i$, hence
$C^{*}(\Lambda(P))$ is purely infinite. Furthermore, $C^{*}(\Lambda(P))$ falls
under the Kirchberg-Phillips classification theory and is thus determined by
its K-theory.

###### Proof.

Fix $i\in\\{1,\dots,k\\}$ with $L_{i}\geq 2$. Then we can form the length-two
cycles $\mu=\mathfrak{a}_{1}^{i}\mathfrak{a}_{L_{i}+1}^{i}$ and
$\nu=\mathfrak{a}_{2}^{i}\mathfrak{a}_{L_{i}+2}^{i}$ based at $v$ with
$d(\mu)_{i}=2=d(\nu)_{i}$. Now $\mathfrak{a}_{2}^{i}$ provides an edge
$\alpha$ with nontrivial degree $d(\alpha)\leq d(\mu)$ which is an entry to
$\mu$ not already contained in $\mu$. We conclude that $C^{*}(\Lambda(P))$ is
purely infinite. The last claim follows by [32, Corollary 8.15], see also [6,
Remark 5.2], by appealing to the classification result in [22]. ∎

##  4\. Construction of $k$-graphs with several vertices

In this section we present our construction of $k$-graphs with several
vertices, for $k\geq 2$, and provide examples and applications.

Towards this aim we need a procedure to get $k$-cube complexes with several
vertices. It is known that in a complex with several vertices one cannot
consistently identify the label of an edge with the label as generator in the
fundamental group. We come around this challenge by introducing additional
layers of labels, corresponding to covers with $N$ sheets, similar to what is
done for $N=2$ in [16, Section 8.1]. In general, this is a hard problem, since
there exist complexes without non-trivial finite covers, cf. [3]. Even when
the complexes are known to admit $N$-covers, corresponding to subgroups of
index $N$ of the fundamental group, see e.g. [9, Theorem 1.38], it is
difficult to construct covers explicitly. One challenge is that the subgroups
can be defined in many different ways. For us a cover will be defined by
picture, meaning that it is explicitly defined by the images of vertices,
edges, faces and so on. In all our pictures, the covering map amounts to
forgetting the upper indexes, and we can explicitly see that we have a local
homeomorphism at each point.

Recall that in a one-vertex $k$-cube complex with generating structure
$A_{1},\dots,A_{k}$ we view edges as being coloured, with each $A_{i}$ for
$i=1,\dots,k$ endowed with a distinct colour. For a complex
$\mathcal{X}=\tilde{P}$ obtained as an $N$-cover of a one-vertex $k$-cube
complex $P_{A_{1},\dots,A_{k}}$, the coloured edges are given by
$p^{-1}(A_{i})$ for $i=1,\dots,k$, where $p:\mathcal{X}=\tilde{P}\to
P_{A_{1},\dots,A_{k}}$ is the covering map. A $k$-complex, when viewed as
undirected, is always connected.

######  Proposition 4.1.

Suppose that $G$ is a $k$-cube group with associated $k$-cube complex
$P_{A_{1},\dots,A_{k}}$. Then $P_{A_{1},\dots,A_{k}}$ admits a double cover
$p:\tilde{P}\to P_{A_{1},\dots,A_{k}}$ with $\tilde{P}$ a complex with $2$
vertices.

We establish this proposition by writing down an explicit double cover, which
will be prescribed ”by picture” on $2$-cells and $3$-cells of the complexes
under consideration, see Lemmas 4.3 and 4.4. Before presenting the proof we
point out a consequence.

######  Corollary 4.2.

Each $k$-cube group $G$ admits a subgroup of index $2$.

###### Proof.

By e.g. [9, Theorem 1.38], for a given path-connected, locally path-connected,
and semilocally simply connected space $X$ there is a bijection between the
set of basepoint preserving isomorphism classes of path-connected covering
spaces $p:(\tilde{X},\tilde{x_{0}})\to(X,x_{0})$ and the set of subgroups of
$\pi_{1}(X,x_{0})$. The correspondence associates the subgroup
$p_{\ast}(\pi_{1}(\tilde{X},\tilde{x_{0}}))$ to the covering
$(\tilde{X},\tilde{x_{0}})$, and the number of sheets of the covering equals
the index of $p_{\ast}(\pi_{1}(\tilde{X},\tilde{x_{0}}))$ in
$\pi_{1}(X,x_{0})$, see [9, Proposition 1.32]. Applying this to the $2$-cover
from Proposition 4.1 yields existence of a subgroup of $G$ of index $2$. ∎

In order to motivate our constructions of coverings, we review a construction
of a $2$-cover of the complex associated to the torus $\mathbb{T}^{2}$.

Figure 6. A double cover of the torus

Recall from, for example [9, Page 14], that $\mathbb{T}^{2}$ is obtained from
a $2$-cell given by a square with pairs of opposite edges having same
orientation and label $a$ (vertically) or $b$ (horizontally), by gluing it
onto the wedge of two circles. A double cover with _two vertices_ arises from
two squares with oriented edges having distinct labels $a^{1}$, $a^{2}$,
$b^{1}$ and $b^{2}$, with the upper index $1$ or $2$ indicating the source
vertex, as shown in figure 6. The complex is obtained by attaching the two
squares (the $2$-cells) to the graph with vertices 1 and 2 in Figure 6. If $P$
is the complex associated to the $(2,2)$-group in Example 3.7, then the cover
$\tilde{P}$ just described will lead to a $2$-graph with two vertices, see
Theorem 4.7, see also Figure 2 in [16, Section 8.1].

The proof of Proposition 4.1 relies on a two lemmas, with the first one
detailing an explicit double cover for the $2$-cells of the given $k$-cube
complex.

######  Lemma 4.3.

Suppose that we have a one-vertex square complex $S=S_{A,B}$ with VH-structure
$(A,B)$ and vertex $v$. Then there is a $2$-cover $p:\tilde{S}\to S$ given by
a square complex $\tilde{S}$ with two vertices $v_{1}$ and $v_{2}$ whose
squares are given by the prescription: The inverse image of a geometric square
$S_{O}^{a,b}=(a,b,c^{-1},d^{-1})$ in $S_{A,B}$ consists of two geometric
squares, $S_{1}=(a^{1},b^{2},(c^{2})^{-1},(d^{1})^{-1})$ and
$S_{2}=(a^{2},b^{1},(c^{1})^{-1},(d^{2})^{-1})$ in $\tilde{S}$, and the
covering map is determined by

$p(*^{1})=p(*^{2})=*\text{ for }*=a,b,c,d.$

Equivalently, if we denote $v_{1}=1$ and $v_{2}=2$, the covering $p$ is
depicted on the squares by

(4.1)
$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{S_{1}}$$\scriptstyle{b^{2}}$$\textstyle{{1}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{S_{2}}$$\scriptstyle{b^{1}}$$\textstyle{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{2}}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{S_{O}^{a,b}}$$\scriptstyle{b}$$\textstyle{1}$$\textstyle{{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a^{1}}$$\scriptstyle{d^{1}}$$\textstyle{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{c^{2}}$$\textstyle{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a^{2}}$$\scriptstyle{d^{2}}$$\textstyle{{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{c^{1}}$$\textstyle{{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$$\scriptstyle{d}$$\textstyle{{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{c}$

Note that in the $2$-cover there are two geometric edges for each geometric
edge in $S_{A,B}$, so for example to $a\in S_{A,B}$ having origin and terminus
the vertex $1$ (identified with $v$) there will correspond $a^{1}$ and $a^{2}$
in $\tilde{S}$, with $a^{1}$ having origin $1$ and terminus $2$, and $a^{2}$
with origin $2$ and terminus $1$.

###### Proof.

We need only observe that
$p(S_{*}^{a^{1},b^{2}})=S_{*}^{a,b}=p(S_{*}^{a^{2},b^{1}})$ for $*=O,H,V,R$.
Thus the square complex $\tilde{S}$ is well-defined. The map $p$ is a local
homeomorphism because it is defined on the cells, and sends edges to edges and
vertices to vertices. ∎

######  Lemma 4.4.

Let $P_{A_{1},\dots,A_{k}}$ be a $k$-cube complex associated with a $k$-cube
group $G$ with underlying structure determined by the ordered tuple
$(A_{1},\dots,A_{k})$, with $\\#A_{i}=2L_{i}$ for every $i=1,\dots,k$. There
is a $2$-cover $\tilde{P}$ of $P_{A_{1},\dots,A_{k}}$ determined as follows:
On each $2$-dimensional cell, the covering $p$ is defined in Lemma 4.3. On a
$3$-dimensional geometric cube $C$, such as is described in figure 1 where we
assume $a_{r}\in A_{i}$, $b_{s}\in A_{j}$ and $c_{t}\in A_{l}$, for
$r,s,t=1,\dots,4$ and $i,j,l\in\\{1,\dots,k\\}$, the cover $\tilde{C}$ of $C$
consists of two geometric cubes, see figure 7, with labelling of edges
$\\{a_{r}^{\epsilon}\\}$, $\\{b_{s}^{\epsilon}\\}$ and
$\\{c_{t}^{\epsilon}\\}$ for $\epsilon=1,2$, and with the covering map given
by

(4.2) $p:\tilde{C}\to C,p(a_{r}^{1})=p(a_{r}^{2})=a_{r},r=1,\dots,4,$

and similarly for $p(b_{s}^{\epsilon})$ and $p(c_{t}^{\epsilon})$. For $4\leq
l\leq k$, the map $p$ is defined on an arbitrary $l$-cube by its prescription
on the underlying $3$-cubes.

Figure 7. A two cover of a generic geometric cube

###### Proof.

At $k=3$, it suffices to verify that $p$ on $\tilde{C}$ is well-defined. But
this is clear from the construction of the map $p$ in (4.2). The map $p$ is
constructed recursively on higher dimensional cubes: if $C$ is an $l$-cube for
$4\leq l\leq k$, then $p$ is prescribed consistently on all
$(l-1)$-dimensional faces of $C$, similarly to how (4.2) is obtained on
$3$-cubes from its prescription in (4.1) on squares. ∎

###### Proof of Proposition 4.1.

This follows by applying Lemmas 4.3 and 4.4. ∎

To obtain $k$-complexes with $N$ vertices for $N>2$, there are several ways to
use $k$-cube groups. Many of the $k$-cube groups are residually finite, so,
because of the 1-to-1 correspondence between the subgroups of index $N$ and
$N$-covers of the corresponding $k$-cube complex, we can get infinitely many
$k$-cube complexes with $N$ vertices. In principle, different subgroups of the
same index $N$ can lead to different coverings. If $N\geq 3$, then the
labelling of vertices is hard to sort out and we do not know of an explicit
prescription similar to (4.2). We shall use the following general procedure.

######  Proposition 4.5.

Suppose that $G$ is a residually finite $k$-cube group with $k\geq 2$. To each
normal subgroup $H$ of $G$ of finite index $N$ there is a $k$-complex
$\mathcal{X}$ with $N$ vertices obtained by the following prescription: let
$Q:G\to S_{N}$ the homomorphism obtained by composing the embedding of $G/H$
into the symmetric group on $N$ letters $S_{N}$ given by Cayley’s theorem with
the quotient map $q:G\to G/H$. For $a\in G$, the permutation $Q(a)$ in $S_{N}$
encodes the edges in the complex, with $a^{n}$ labelling an edge from the
vertex $n$ to the vertex $n^{\prime}=Q(a)(n)$ for $n=1,\dots,N$.

###### Proof.

Let $P$ be the one-vertex complex determined by $G$. Suppose that $H$ is a
subgroup of $G$ so that $(G:H)=N$. The complex $\mathcal{X}$ is constructed by
associating to each square $S_{O}=(a,b,(a^{\prime})^{-1},(b^{\prime})^{-1})$
in $P$ a total of $N$ squares

$(Q(a),Q(b),Q((a^{\prime})^{-1}),Q((b^{\prime})^{-1}))$

in the new, $N$-vertex complex, with vertices labelled by elements in
$\\{1,2,\dots,N\\}$. Indeed, applying $Q$ to the relation
$ab=b^{\prime}a^{\prime}$ in $G$ gives the identity
$Q(a)Q(b)=Q(b^{\prime})Q(a^{\prime})$ in $S_{N}$, which in turn yields $N$
squares in the complex $\mathcal{X}$ determined by

$Q(a)Q(b)(n)=Q(b^{\prime})Q(a^{\prime})(n),\text{ for all }n=1,\dots,N.$

More precisely, for each $n$, let
$m=Q(a)Q(b)(n)=Q(b^{\prime})Q(a^{\prime})(n)$ and consider the square with
labelling $(a^{\prime})^{n}(b^{\prime})^{s}$ in vertical-horizontal direction
from vertex $n$ to vertex $m$ via vertex $s$, and with labelling $b^{n}a^{r}$
in horizontal-vertical direction via vertex $r$, where $m=Q(a)(r)$,
$r=Q(b)(n)$, $Q(a^{\prime})(n)=s$ and $Q(b^{\prime})(s)=m$, see the figure,
where $1$ in the right square denotes the vertex in the complex of $G$:

(4.3)
$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{s}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{S}$$\scriptstyle{(b^{\prime})^{s}}$$\textstyle{{m}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{S_{O}^{a,b}}$$\scriptstyle{b}$$\textstyle{1}$$\textstyle{{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{b^{n}}$$\scriptstyle{(a^{\prime})^{n}}$$\textstyle{{r}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a^{r}}$$\textstyle{{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$$\scriptstyle{b^{\prime}}$$\textstyle{{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a^{\prime}}$

The covering map $p:\mathcal{X}\to P$ is defined by collapsing all squares of
the form $S$ onto the given square $S_{O}^{a,b}$ in $P$, for each square in
$P$. ∎

We extend the notions of left and right rigid to $k$-dimensional digraphs and
$k$-graphs with more than one vertex in the natural way. The idea is that
being rigid means that if two edges can form a corner (either bottom-left or
top-right), then they do form a unique corner.

######  Definition 4.6.

(a) Suppose that $\operatorname{DG}$ is a $k$-dimensional digraph. Then
$\operatorname{DG}$ is right rigid if for $x\in E_{i}$ and $y\in E_{j}$ edges
of distinct colours $i\neq j$ so that $o(x)=o(y)$, there are unique
$x^{\prime}\in E_{i}$ and $y^{\prime}\in E_{j}$ with $xy^{\prime}\sim
yx^{\prime}$. Left rigid is defined in a similar way.

(b) Suppose that $(\Lambda,d)$ is a $k$-graph, and let
$E_{i}=d^{-1}(\\{e_{i}\\})$ be its alphabets, for $1\leq i\leq k$. We say that
$\Lambda$ is right rigid if for $x\in E_{i}$ and $y\in E_{j}$ with the same
origin, for $i\neq j$, there are unique $x^{\prime}\in E_{i}$ and
$y^{\prime}\in E_{j}$ with the property that $y^{\prime}\circ
x=x^{\prime}\circ y$. Left rigidity of $\Lambda$ is defined in a similar
manner.

######  Theorem 4.7.

Suppose that $P$ is a one-vertex $k$-cube complex that admits an $N$-cover
$p:\tilde{P}\to P$ as in Proposition 4.5, with $\mathcal{X}=\tilde{P}$ the
associated $k$-cube complex with $N$ vertices, for $k\geq 2$ and $N\geq 2$.
The following are valid.

(a) The $k$-dimensional digraph $\operatorname{DG}(\mathcal{X})$ determined in
Theorem 3.3 is left and right rigid.

(b) The $k$-graph
$\Lambda({\mathcal{X}}):=\Lambda(\operatorname{DG}({\mathcal{X}}))$ associated
to $\operatorname{DG}(\mathcal{X})$ by Corollary 3.4 is strongly connected,
left rigid and right rigid.

###### Proof.

Part (a) follows from Theorem 3.3. Turning to part (b), to show that
$\Lambda(\mathcal{X})$ is strongly connected let $v,w$ be distinct vertices.
By the construction of cover in Proposition 4.5, there is an element of the
$k$-cube group whose action on $v$ gives $w$. Associated to this element there
is a path $y_{1}y_{2}\dots y_{m}$ in the $1$-skeleton of the $k$-complex $P$,
and this has a unique lift to a path in $\mathcal{X}$ from $v=o(y_{1})$ to
$w=t(y_{m})$. In particular, $v_{s}=t(y_{s}),v_{s+1}=o(y_{s})$ are adjacent
vertices in the complex for each $1\leq s\leq m-1$ (identifying $v=v_{1}$).
Our definition of the $k$-dimensional digraph gives two directed edges, with
opposite orientation, having source $v_{s}$ and terminus $v_{s+1}$,
respectively the opposite, for each $s=1,\dots,m-1$. This allows to form
directed paths in $\operatorname{DG}(\mathcal{X})$, hence in
$\Lambda(\mathcal{X})$, from $v$ to $w$ and from $w$ to $v$, as needed.

To see that $\Lambda(\mathcal{X})$ is rigid, it suffices to note that every
vertex in the cover $\tilde{P}$ has the same link as the one vertex of $P$,
and in particular its link contains no multiple edges. Therefore the proof of
Lemma 3.12 carries through. ∎

######  Corollary 4.8.

The $k$-graph $\Lambda(\mathcal{X})$ from Theorem 4.7 satisfies the
aperiodicity condition.

###### Proof.

Since $\Lambda(\mathcal{X})$ is rigid, the existence of an aperiodic path in
$\Lambda(\mathcal{X})$ based at a given vertex is guaranteed as in the one-
vertex case, see [15, Lemma 4.15 and Corollary 11.10]. ∎

######  Proposition 4.9.

Assume the hypotheses of Theorem 4.7, where $P=P_{A_{1},\dots,A_{k}}$ with
$A_{i}$ given as in Lemma 3.12 for $i=1,\dots,k$. If $|A_{i}|\geq 2$ for all
$i=1,\dots,k$, then every vertex $\Lambda(\mathcal{X})$ supports at least two
cycles of each colour. In particular, $C^{*}(\Lambda(\mathcal{X}))$ is purely
infinite and therefore classifiable by the Kirchberg-Phillips classification
theory.

###### Proof.

Let $\Lambda(\mathcal{X})^{0}$ denote the vertices, or identities, in our
$k$-graph. Since every geometric edge in $P$ gives rise to $N$ geometric edges
in $\tilde{P}$, we have that for each colour $i\in\\{1,\dots,k\\}$, every
vertex $v\in\Lambda(\mathcal{X})^{0}$ admits $N|A_{i}|$ incident edges, namely
edges with origin or terminus $v$. Furthermore, by our construction of
$\Lambda(\mathcal{X})$ we also know that each edge is contained in a length-
two cycle. Thus, for a given $v\in\Lambda(\mathcal{X})^{0}$, there are at
least two cycles $\mu=x_{2}\circ x_{1}$ and $\nu=x_{4}\circ x_{3}$ based at
$v$ and consisting of edges of colour $i$, with the terminus of $x_{1}$
possibly distinct from the terminus of $x_{3}$. Then $x_{4}$ is an entry to
$\mu$ of smaller degree and not already contained in $\mu$. In this
consideration the vertex $v$ already supports a cycle with an entrance, but
since our $k$-graph is strongly connected we could have chosen a cycle $\mu$
based at a different vertex $w$ and apply the same consideration. Now [32,
Proposition 8.8 and Corollary 8.15] apply to give the claimed conclusion. ∎

We next illustrate our construction of $k$-graphs with more than one vertex
with an explicit example of an infinite family of $k$-graphs with two
vertices, for all $k\geq 2$. The construction was partly outlined in [16,
Section 8.1], as corresponding to the uniform labelling $l_{u}$, and explicit
factorisation rules of the $2$-vertex graph were given in the case of the
mixed labelling $l_{m}$. Here we describe completely the case $l_{u}$ as an
application of Theorem 4.7.

######  Proposition 4.10.

For $k\geq 2$ and any $k$-tuple $(L_{1},\dots,L_{k})$ of positive integers
there exists an aperiodic strongly connected $2$-vertex $k$-rank graph
$\Lambda$ with $|v\Lambda^{e_{i}}w|=2L_{i}$, where $v$ and $w$ are the
vertices in $\Lambda$ and $i=1,\dots,k$.

###### Proof.

Fix $k\geq 2$ and for each $i=1,\dots,k$ let $\mathcal{L}_{i}$ be an alphabet
with $L_{i}$ letters. Let $\mathbb{F}_{i}=\mathbb{F}_{L_{i}}$ be the free
group generated by $\mathcal{L}_{i}$ for each $i=1,\dots,k$. The product group
$\mathbb{F}_{1}\times\cdots\times\mathbb{F}_{k}$ acts simply and transitively
on the product of trees
$\mathcal{T}_{2L_{1}}\times\dots\times\mathcal{T}_{2L_{k}}$, and yields in the
quotient a complex $P$ with one vertex and skeleton a wedge of
$\sum_{i=1}^{k}L_{i}$ circles. The $2$-cells in $P$ arise from pairs
$a\in\mathcal{L}_{i}$, $b\in\mathcal{L}_{j}$ for $i\neq j$ with the
commutation relation $ab=ba$ as in Example 3.7; for each such pair, there is a
torus glued to the wedge of circles.

Let $\tilde{P}$ be the associated $2$-cover from Proposition 4.1. By Theorem
4.7 and Corollary 4.8 there is a $k$-graph $\Lambda:=\Lambda(\tilde{P})$ with
the desired property: for each geometric edge in $P$, say having label
$a\in\mathcal{L}_{i}$, there are two geometric edges labelled $a^{1}$ and
$a^{2}$ in $\tilde{P}$, and each of these gives exactly one edge in the
associated $k$-graph $\Lambda$ between the two vertices. ∎

######  Example 4.11.

To illustrate Proposition 4.9 and Proposition 4.10, suppose that $k=2$ and
$L_{1}=L_{2}=1$. The associated $\Lambda$ has $1$-skeleton the graph with two
vertices in figure 6, where we identify $v$ as vertex $1$ and $w$ as vertex
$2$. If we view the coloured edges in direction $e_{1}\in{\mathbb{N}}^{2}$ as
labelled by $\mathcal{L}_{1}$ and in direction $e_{2}\in{\mathbb{N}}^{2}$ to
be labelled by $\mathcal{L}_{2}$, then by Proposition 4.10 we have

$w\Lambda^{e_{1}}v=\\{a^{1},\bar{a^{2}}\\},\,w\Lambda^{e_{2}}v=\\{b^{1},\bar{b^{2}}\\},\,v\Lambda^{e_{1}}w=\\{a^{2},\bar{a^{1}}\\}\text{
and }v\Lambda^{e_{1}}w=\\{b^{2},\bar{b^{1}}\\}.$

The 8 factorisation rules are:

$\displaystyle a^{1}b^{2}=b^{1}a^{2},$ $\displaystyle
a^{2}\bar{b^{2}}=\bar{b^{1}}a^{1},\bar{a^{1}}b^{1}=b^{2}\bar{a^{2}},\bar{a^{2}}\bar{b^{2}}=\bar{b^{1}}\bar{a^{1}}$
$\displaystyle a^{2}b^{1}=b^{2}a^{1},$ $\displaystyle
a^{1}\bar{b^{1}}=\bar{b^{2}}a^{2},\bar{a^{2}}b^{2}=b^{1}\bar{a^{1}},\bar{a^{1}}\bar{b^{1}}=\bar{b^{2}}\bar{a^{2}}.$

If $k=2$ and $L_{1}=L_{2}=2$, then the corresponding $2$-graph on two vertices
has the same 1-skeleton, and for example
$|v\Lambda^{e_{1}}w|=|w\Lambda^{e_{1}}v|=4$, and similarly in colour $e_{2}$.

##  5\. Applications

###  5.1. Von Neumann algebras from strongly connected $k$-graphs

We now present a large supply of von Neumann type $\mathrm{III}_{\lambda}$
factors from $k$-graphs as in [14], for infinitely many values of $\lambda$ in
$(0,1]$. We start with some preparation.

We refer to [26, Section 6] for the notion of adjacency operator in
$i$-direction for a $k$-cube complex, where $i\in\\{1,\dots,k\\}$. The basic
ingredients are as follows: Let $X$ be a $k$-cube complex with vertex set (of
its $1$-skeleton) denoted $X_{0}$ and with universal cover a product
$\operatorname{T}_{1}\times\operatorname{T}_{2}\times\dots\times\operatorname{T}_{k}$
of regular trees. For each $i=1,\dots,k$ and $V,W\in X_{0}$, we write
$V\sim_{i}W$ if the two vertices in the complex are adjacent in the
$i$-direction of $X$. The adjacency operator $\mathcal{A}_{i}$ in
$i$-direction is defined on $L^{2}(X_{0})$ by

$\mathcal{A}_{i}(f)(V)=\sum_{W\sim_{i}V}f(W).$

Since all complexes considered here are locally finite in a strong sense,
meaning that at every vertex there are finitely many edges in each direction
$i$, or of each colour $i$, for $i\in\\{1,\dots,k\\}$, the operators
$\mathcal{A}_{i}$ become $|X_{0}|\times|X_{0}|$ matrices. It was further
observed in [26, Remark 6.4] that whenever each pair of edges starting at a
vertex of $X$ in direction ${i,j}$, with $i\neq j$, belong to a unique square
in $X$, then $\mathcal{A}_{i}$ and $\mathcal{A}_{j}$ commute.

######  Proposition 5.1.

Let $\mathcal{X}$ be a $k$-cube complex with $N$ vertices covered by a
cartesian product of $k$ trees with valencies
$n_{1},n_{2},\dots,n_{k}\in\mathbb{Z}^{+}$, respectively, where $k\geq 2$ and
$N\geq 1$. Let $\Lambda(\mathcal{X})$ be the associated $k$-graph as in
Theorem 4.7. Then, with the notation of subsection 2.9, we have

(5.1) $\rho(\Lambda(\mathcal{X}))=(n_{1},n_{2},\dots,n_{k}).$

###### Proof.

The assumption on $\mathcal{X}$ says that there are $n_{i}$ edges
(disregarding orientation) of colour $i$ for each $i=1,\dots,k$. The graph
$\Lambda(\mathcal{X})$ is constructed by assigning two edges in its skeleton,
of opposite orientation, for each geometric edge in $\mathcal{}X$. Therefore,
if $M_{i}$ denotes the coordinate matrix of $\Lambda(\mathcal{X})$ in colour
$i$, we have that $M_{i}$ is the same as the adjacency operator in
$i$-direction $\mathcal{A}_{i}$. Thus it is a symmetric matrix with largest
positive eigenvalue equal to the valency of the tree in colour $i$. Hence,
$\rho(M_{i})=n_{i}$ for each $i=1,\dots,k$, as claimed. ∎

######  Remark 5.2.

The graph of Proposition 4.10 satisfies $\rho(\Lambda)=(2L_{1},\dots,2L_{k})$.

Given a strongly connected $k$-graph $\Lambda$, it was shown in [11, Corollary
4.6] that $C^{*}(\Lambda)$ admits KMS states at inverse temperature $\beta=1$
for the one-parameter action
$\alpha\colon\mathbb{R}\to\operatorname{Aut}C^{*}(\Lambda)$, the so-called
preferred dynamics, characterised by
$\alpha_{t}(\mathbf{s}_{\mu})=e^{it\log\rho(\Lambda)\cdot
d(\mu)}\mathbf{s}_{\mu}$, $t\in\mathbb{R}$, $\mu\in\Lambda$. Following [14],
define
$\mathcal{S}:=\\{\rho(\Lambda)^{d(\mu)-d(\nu)}\mid\mu,\nu\in\Lambda\text{ are
cycles}\\}$ and let $\lambda:=\sup\\{s\in\mathcal{S}\mid s<1\\}$. By the main
result of [14], Theorem 3.1, we have $\lambda\in(0,1]$ and the von Neumann
algebra generated by the image of $C^{*}(\Lambda)$ in the GNS representation
$\pi_{\varphi}$ corresponding to an extremal KMS state $\varphi$ is the
injective type $\mathrm{III}_{\lambda}$ factor.

Our application here is motivated by [14, Example 7.7], see also [20] and
[38]. It consists of producing an infinite family of von Neumann factors
$(\pi_{\varphi}(C^{*}(\Lambda)))^{\prime\prime}$ of type
$\mathrm{III}_{\lambda}$ associated to $k$-graphs in this fashion. Recall from
[14, Section 6] that the group of periods of a strongly connected graph
$\Lambda$ is defined as
$\mathcal{P}_{\Lambda}=\mathcal{P}_{v}^{+}-\mathcal{P}_{v}^{+}$, where for an
arbitrary vertex $v\in\Lambda^{0}$, $\mathcal{P}_{v}^{+}$ is the subsemigroup
$d(v\Lambda v)$ of ${\mathbb{N}}^{k}$. Equivalently, $\mathcal{P}_{\Lambda}$
is the subgroup of ${\mathbb{Z}}^{k}$ determined as
$\\{d(\mu)-d(\nu)\mid\mu,\nu\text{ are cycles in $\Lambda$}\\}$. As shown in
[14, Theorem 7.3], the set $\mathcal{S}$ above is the closure inside the
positive real half-line of the set $\\{\rho(\Lambda)^{g}\mid
g\in\mathcal{P}_{\Lambda}\\}$.

######  Corollary 5.3.

For $k\geq 2$ and any $k$-tuple $(L_{1},\dots,L_{k})$ of positive integers,
let $\Lambda$ be the $k$-graph with two vertices from Proposition 4.10. There
is a $\mathrm{III}_{\lambda}$ von Neumann factor
$(\pi_{\varphi}(C^{*}(\Lambda)))^{\prime\prime}$, where $\pi_{\varphi}$ is the
GNS representation of $C^{*}(\Lambda)$ corresponding to an extremal KMS1 state
$\phi$, and the type is determined as

$\lambda=\operatorname{sup}\\{(2L_{1})^{m_{1}}(2L_{2})^{m_{2}}\dots(2L_{k})^{m_{k}}\mid(m_{1},m_{2},\dots,m_{k})\in\mathcal{P}_{\Lambda}\\}\cap(0,1].$

In particular, if $L_{1}=\cdots=L_{k}=L$, then $\lambda=(2L)^{-2}$.

###### Proof.

Fix $k\geq 2$ and positive integers $L_{1},\dots,L_{k}$, and let $\Lambda$ be
as specified. Then $\Lambda$ is strongly connected, and we may apply Theorem
3.1 of [14] to obtain the claimed von Neumann factors.

The remaining task is to compute $\mathcal{P}_{\Lambda}$. By our construction
of $\Lambda$ it is not hard to see that $\mathcal{P}_{\Lambda}$ is generated
by ${m}\in{\mathbb{Z}}^{k}$ where either ${m}_{i}=2$ for a unique
$i\in\\{1,\dots,k\\}$ while ${m}_{l}=0$ at $l\neq i$, or ${m}_{i}={m}_{j}=1$
for some $i\neq j$ in $\\{1,\dots,k\\}$ and ${m}_{l}=0$ for
$l\notin\\{i,j\\}$. If $L_{1}=L_{2}=\dots=L_{k}=L$, then
$\rho(\Lambda)^{{m}}=(2L)^{\sum_{i=1}^{k}m_{i}}$ with
${m}\in\mathcal{P}_{\Lambda}$, and because $\sum_{i=1}^{k}m_{i}\in
2{\mathbb{Z}}$, the required type is attained as $\lambda=(2L)^{-2}$.

∎

It was pointed out in [14, Remark 7.6] that the type of the von Neumann
factors arising from extremal KMS1 states depends only on the skeleton of the
$k$-graph, and not on its factorisation rules. In our examples in Corollary
5.3, this means that the type of the von Neumann factor depends only on the
complex $\tilde{P}$ built up as a 2-cover of the one-vertex complex
$(\mathcal{T}_{2L_{1}}\times\dots\times\mathcal{T}_{2L_{k}})\backslash(\mathbb{F}_{1}\times\cdots\times\mathbb{F}_{k})$.

###  5.2. Spectral theory of $k$-graphs

Alon and Boppana prove that asymptotically in families of finite
$(q+1)$-regular graphs $X_{n}$ with diameter tending to $\infty$ the largest
absolute value of a non-trivial eigenvalue $\lambda(X_{n})$ of the adjacency
operator $A_{X_{n}}$ has limes inferior
$\varliminf_{n\to\infty}\lambda(X_{n})\geq 2\sqrt{q}.$

Now, instead of graphs we may consider cube complexes covered by products of
trees $\rm{T}_{1}\times\dots\times\rm{T}_{k}$, such that $\rm{T}_{i}$ has
valency $q_{i}$ and look at adjacency operators $\mathcal{A}_{i}$ in direction
$i$ corresponding to an individual tree $\rm{T}_{i}$.

######  Definition 5.4.

Let $X$ be a finite $k$-cube complex that has constant valency $q_{i}+1$ in
all directions $i=1,\dots,k$. Then $X$ is a cubical Ramanujan complex, if for
each $i\in\\{1,\ldots,k\\}$, the eigenvalues $\lambda$ of $\mathcal{A}_{i}$
either satisfy the equality $\lambda=\pm(q_{i}+1)$ or the bound

$\lambda\leq 2\sqrt{q_{i}}.$

Each such complex yields a $k$-graph $\Delta$ such that
$\rho(\Delta)=(q_{1}+1,q_{2}+1,...,q_{k}+1)$.

There are explicit constructions of Ramanujan cube complexes for several
infinite families in [26]. We consider next the complexes from [26],
corresponding to congruence subgroups of arithmetic lattices. We reformulate
some results of [26] in the light of the present paper.

######  Theorem 5.5.

(Cf. [26, Section 6]) For $p$ a prime, $l$ a positive integer, and $N\geq 2$,
there are infinitely many $k$-cube complexes with $N$ vertices covered by
products of $k$ trees, where $k\leq p-1$ and each tree is of valency