1 Introduction
Since their introduction [Reference Kumjian and Pask26], higher-rank graphs, or k-graphs, have been a source of interesting new higher-dimensional phenomena: in algebra [Reference Aranda Pino, Clark, an Huef and Raeburn2, Reference Clark, Flynn and an Huef13, Reference Rosjanuardi40], dynamics [Reference Kakariadis22, Reference Pask, Raeburn and Weaver34, Reference Skalski and Zacharias43, Reference Spielberg44],
$C^*$
-algebras [Reference Armstrong and Brownlowe3, Reference Clark, an Huef and Sims12, Reference Ruiz, Sims and Sørensen41], K-theory [Reference Evans16, Reference Gillaspy18, Reference Pask, Rennie and Sims35], topology [Reference Kaliszewski, Kumjian, Quigg and Sims23, Reference Kumjian, Pask, Sims and Whittaker28, Reference Pask, Raeburn and Quigg32, Reference Pask, Raeburn and Quigg33] and geometry [Reference Konter and Vdovina24, Reference Robertson and Steger38, Reference Vdovina45]. However, many natural questions about their structure theory remain difficult to unravel.
One such question, and the primary motivation for this paper, is: when can a k-graph
$C^*$
-algebra be realised, modulo Rieffel–Morita equivalences, as a crossed product of a commutative
$C^*$
-algebra? For
$1$
-graphs, the answer is ‘always’: given a row-finite source-free directed graph E, the middle two authors showed [Reference Kumjian and Pask25] that the
$C^*$
-algebra of its universal cover F is Rieffel–Morita equivalent to a commutative approximately finite-dimensional (AF) algebra, and there is an action of the fundamental group
$\pi _1(E, v)$
on
$C^*(F)$
whose crossed product is Rieffel–Morita equivalent to
$C^*(E)$
. For k-graphs, the answer is more nuanced and is related to two other intriguing structural questions: when does a k-graph embed in its fundamental groupoid and when is the boundary of its universal cover Hausdorff?
Our main
$C^*$
-algebraic theorem, Theorem 4.1, clarifies the relationships between these questions: the
$C^*$
-algebra
$C^*(\Lambda )$
of any connected row-finite source-free k-graph is a crossed product of an AF algebra
$C^*(\Sigma )$
by the fundamental group of
$\Lambda $
; if
$\Lambda $
embeds in its fundamental groupoid, then the AF algebra
$C^*(\Sigma )$
is a Fell algebra; and if, additionally, a naturally arising sub-1-graph of
$\Sigma $
is simply connected, then the boundary of
$\Sigma $
is Hausdorff, and
$C^*(\Sigma )$
is Rieffel–Morita equivalent to a commutative AF algebra. The point is that the first part of the program of [Reference Kumjian and Pask25] above goes through smoothly for k-graphs: every connected k-graph
$\Lambda $
has a fundamental group
$\pi (\Lambda , v)$
[Reference Pask, Raeburn and Quigg32] and a universal cover
$\Sigma $
[Reference Pask, Raeburn and Quigg33] that carries an action of
$\pi (\Lambda , v)$
, and when
$\Lambda $
is row-finite and source-free, the resulting crossed product is Rieffel–Morita equivalent to
$C^*(\Lambda )$
[Reference Kumjian and Pask26]. Our main contribution is the analysis of
$C^*(\Sigma )$
.
Motivated by this, we study the question of when a k-graph
$\Lambda $
embeds in its fundamental groupoid. Many k-graphs do not embed: we give three examples in Section 3.1; and any k-graph containing a copy of one of these (of which there are many) also fails to embed. So we focus on checkable sufficient conditions. We show that singly connected k-graphs always embed (Proposition 3.9(ii)), and highlight a surprising difference between k-graphs and
$1$
-graphs: universal covers of k-graphs need not be singly connected. We include a proof that
$1$
-graphs always embed (Theorem 3.14). We then show that many standard k-graph constructions preserve embeddability: coverings (Proposition 3.13), affine pullbacks, Cartesian products, crossed-products and skew-products (Corollary 3.15), and action graphs (Corollary 3.18). The workhorse in this is Theorem 3.8, which exploits the universal properties of the fundamental groupoid and fundamental group. In Proposition 3.25, we reduce the embeddability of a connected k-graph to group-embeddability of the subsemigroup based at any vertex. Using Dilian Yang’s work [Reference Yang47] on k-graphs and Yang–Baxter solutions, we show that there are many embeddable k-graphs for all k (Lemma 3.22). We are far from a complete answer to the embeddability question. Johnstone’s general results [Reference Johnstone21] characterise groupoid-embeddability of categories, but the hypotheses seem uncheckable: we gleaned no practical conditions—either necessary or sufficient—from Johnstone’s work, beyond the neat result of Lawson and Vdovina [Reference Lawson and Vdovina29, Theorem 11.14] presented in Remark 3.24. One might hope for help from Ore’s theorem [Reference Dehornoy, Digne, Godelle, Krammer and Michel15, Proposition II.3.11], but by the factorisation property, no interesting k-graphs are Ore. Remarks 4.17 and 4.18 indicate how much we still do not know about embeddability.
Finally, as definitive general results about embeddability are still beyond reach, we present a class of examples arising from the combinatorial objects used by Roberston and Steger to construct higher-rank Cuntz–Krieger
$C^*$
-algebras in [Reference Robertson and Steger38, Reference Robertson and Steger39], which first inspired the middle two authors to develop the concept of a higher-rank graph. We show in Theorem 5.17 and Proposition 5.20 that every
$\tilde {A_2}$
-group
$\Gamma _{\mathcal {T}}$
yields
$2$
-graphs
$\Lambda _{\mathcal {T}}$
and
$\Sigma _{\mathcal {T}}$
, the latter being a cover of the former, and in Corollary 5.19 that
$\Lambda _{\mathcal {T}}$
embeds in its fundamental groupoid—Proposition 3.13 then shows that
$\Sigma _{\mathcal {T}}$
embeds as well. We also prove that
$\Sigma _{\mathcal {T}}$
is singly connected and deduce that its
$C^*$
-algebra is type I
$_0$
. The construction of
$\Lambda _{\mathcal {T}}$
is related to a number of existing constructions. It is directly inspired by [Reference Robertson and Steger38, pages 135–136]. As discussed in [Reference Cartwright, Mantero, Steger and Zappa8, Reference Robertson and Steger38], a thick
$\tilde {A_2}$
-building
$\mathscr {B}$
carrying a vertex-transitive action of an
$\tilde {A_2}$
-group
$\Gamma _{\mathcal {T}}$
arises from a finite projective plane
$(P, L)$
, a bijection between P and L, and a compatible triangle presentation
$\mathcal {T}$
on P, the points of the projective plane, arising from the local structure of the building (see [Reference Cartwright, Mantero, Steger and Zappa8, Section 3]). The
$\tilde {A_2}$
-group
$\Gamma _{\mathcal {T}}$
is generated by a set indexed by P subject to the relations encoded in
$\mathcal {T}$
. The
$\tilde {A_2}$
-building
$\mathscr {B}$
is constructed as an augmented Cayley graph of
$\Gamma _{\mathcal {T}}$
with 2-simplices given by
$\mathcal {T}$
. Our
$\Lambda _{\mathcal {T}}$
is isomorphic to the
$2$
-graph obtained from [Reference Kumjian and Pask26, Example 1.7(iv)] from the
$0$
–
$1$
matrices
$M_i$
of [Reference Robertson and Steger38, page 135] (see Remark 5.18). Geometric considerations suggest both that
$\Sigma _{\mathcal {T}}$
should be simply connected, and therefore equal to the universal cover of
$\Lambda _{\mathcal {T}}$
, and that its topological realisation should coincide with that of
$\mathscr {B}$
, so it should have Hausdorff boundary; we leave this for future work. Our construction is also related to the construction of k-graphs from groups in [Reference Mutter, Radu and Vdovina31], but cannot be recovered from it: the covering
$2$
-graphs in [Reference Mutter, Radu and Vdovina31] are products of trees rather than
$\tilde {A_2}$
-buildings.
2 Background and preliminary results
2.1 Higher-rank graphs
We write
$\mathbb {N}$
for the additive monoid
$\{ 0, 1 ,\ldots \}$
. We denote the standard generators of
$\mathbb {N}^k \subset \mathbb {Z}^k$
by
$\varepsilon _1, \ldots ,\varepsilon _k$
, and we write
$n_i$
for the ith coordinate of
$n \in \mathbb {N}^k$
. We write
$\mathbf {1}_k$
or just
$\mathbf {1}$
for
$(1, \ldots , 1) \in \mathbb {N}^k$
.
A k-graph is a small category
$\Lambda $
equipped with a functor
$d :\Lambda \to \mathbb {N}^k$
satisfying the factorisation property: whenever
$d(\lambda ) = m+n$
, there exist unique
$\mu ,\nu \in \Lambda $
such that
$d(\mu )=m$
,
$d(\nu )=n$
and
$\lambda =\mu \nu $
. This implies that
$\Lambda $
is cancellative. We write
${\Lambda ^n := d^{-1} (n)}$
for
$n \in \mathbb {N}^k$
. When
$d(\lambda )=n$
, we say
$\lambda $
has degree n. The factorisation property implies that
$\Lambda ^0$
is the set of identity morphisms, which we call vertices. Elements of
$\bigcup _i \Lambda ^{\varepsilon _i}$
are called edges. For
$u,v\in \Lambda ^0$
, we write
$u\Lambda := r^{-1}(u)$
,
$\Lambda v:=s^{-1}(v)$
and
${u\Lambda v:=u\Lambda \cap \Lambda v}$
.
Notation 2.1. For
$\lambda \in \Lambda $
and
$0 \le m \le n \le d(\lambda )$
, we write
$\lambda (m,n)$
for the unique element of
$\Lambda $
such that
$\lambda \in \Lambda ^m \lambda (m,n)\Lambda ^{d(\lambda ) -n}$
. We define
$\lambda (n) := \lambda (n,n) = s(\lambda (0,n))$
.
Definition 2.2. The k-graph
$\Lambda $
is connected if the equivalence relation
$\sim $
on
$\Lambda ^0$
generated by
$\{(u,v)\mid u\Lambda v\ne \emptyset \}$
is
$\Lambda ^0\times \Lambda ^0$
. A k-graph is strongly connected if
$u \Lambda v \neq \emptyset $
for all
$u,v \in \Lambda ^0$
.
A morphism
$\phi : \Omega \to \Lambda $
between k-graphs is a functor such that
$d_\Lambda ( \phi ( \lambda ) ) = d_\Omega ( \lambda )$
for all
$\lambda \in \Omega $
. A quasi-morphism from a k-graph
$( \Omega , d_\Omega )$
to an
$\ell $
-graph
$( \Lambda , d_\Lambda )$
is a pair
$( \phi , f)$
consisting of a functor
$\phi : \Omega \to \Lambda $
and a homomorphism
$f: \mathbb {N}^k \to \mathbb {N}^\ell $
such that
$d_\Lambda \circ \phi = f \circ d_\Omega $
. If
$\Lambda $
is a k-graph, then
$\Lambda ^{\mathbb {N}\mathbf {1}} := \{\lambda \in \Lambda : d(\lambda ) \in \mathbb {N}\mathbf { 1}\}$
is a
$1$
-graph and the natural inclusion
$\Lambda ^{\mathbb {N}\mathbf {1}} \hookrightarrow \Lambda $
together with the map
$f: \mathbb {N} \to \mathbb {N}^k$
given by
$f(n) := n\mathbf {1}$
is a quasimorphism.
Examples 2.3.
-
(i) Let
$B_n$
be the directed graph with
$B_n^{0} = \{u\}$
and
$B^1_n = \{f_1 , \ldots , f_n\}$
. Its path category
$B_n^*$
is a
$1$
-graph and coincides with the free semigroup
$\mathbb {F}_n^+$
on n generators. -
(ii) Let
$\Delta _k = \{ (m,n) \in \mathbb {Z}^k \times \mathbb {Z}^k : m \le n \}$
. Define
$r,s : \Delta _k \to \operatorname {Obj} \Delta _k$
by
$r (m,n) = m$
,
$s (m,n) = n$
, and for
$m \le m \le p \in \mathbb {Z}^k$
, define
$(m,n)(n,p)=(m,p)$
and
$d (m,n) = n-m$
. Then
$( \Delta _k , d )$
is a k-graph where
$\operatorname {Obj} \Delta _k$
is identified with
$\{ (m,m) : m \in \mathbb {Z}^k \} \subset \operatorname {Mor} \Delta _k$
. -
(iii) Similarly,
$\Omega _k = \{(m,n) \in \mathbb {N}^k \times \mathbb {N}^k : m \le n\}$
is a sub-k-graph of
$\Delta _k$
.
Example 2.4 (Skew-product graphs).
Let
$\Lambda $
be a k-graph, G a group and
$c: \Lambda \to G$
a
$1$
-cocycle (functor). Then the set
$G \times _c \Lambda := \{ (g, \lambda ) : g \in G, \lambda \in \Lambda \}$
, under the structure maps
$$ \begin{align*} s (g, \lambda ) &= ( g c ( \lambda ), s ( \lambda ) ), \quad r (g, \lambda ) = (g, r(\lambda )), \nonumber\\ (g, \lambda ) \cdot (g c (\lambda), \mu) &= (g, \lambda \mu ), \qquad\quad\kern0.8pt d (g , \lambda ) = d ( \lambda ) \end{align*} $$
is a k-graph called the skew-product graph [Reference Kumjian and Pask26, Definition 5.1]. Left translation by G on the first coordinate of
$G \times _c \Lambda $
is an action of G by k-graph automorphisms.
There are two equivalent conventions for skew-product graphs in the literature: the other is [Reference Pask, Raeburn and Quigg33, Definition 6.3] (see also [Reference Bédos, Kaliszewski and Quigg5, Definition 3.5]). In [Reference Pask, Raeburn and Quigg33, Definition 6.3],
$$ \begin{align*}\Lambda \times _c G := \{ (\lambda , g) : \lambda \in \Lambda , g \in G \}\end{align*} $$
with structure maps
$$ \begin{align*} s ( \lambda , g ) &= ( s ( \lambda ) , g ), \quad r ( \lambda , g ) = ( r(\lambda ) , c( \lambda)g ), \\ ( \lambda , c( \lambda)g ) \cdot ( \mu , g ) &= ( \lambda \mu , g ), \quad\kern1pt\ d ( \lambda , g ) = d ( \lambda ). \end{align*} $$
It is simple to check that
$\phi (g, \lambda ) = (\lambda , c( \lambda )^{-1}g^{-1})$
yields an isomorphism
${\phi : G \times _c \Lambda \to \Lambda \times _c G}$
.
Example 2.5 (Monoidal
$2$
-graphs).
The following class of 2-graphs was introduced in [Reference Kumjian and Pask26, Section 6] and later studied extensively by Yang et al. [Reference Davidson, Power and Yang14, Reference Yang46, Reference Yang47]. Fix
${n_1 , n_2 \ge 1}$
. Let
$[n_i ]= \{ 1 , \ldots , n_i \}$
for
$i=1,2$
. Let
$\theta : [n_1 ]\times [n_2] \to [n_2] \times [n_1]$
be a bijection. The monoidal
$2$
-graph
$\mathbb {F}_\theta ^+$
is the unique
$2$
-graph such that
$(\mathbb {F}_\theta ^+)^0 = \{v\}$
,
$(\mathbb {F}_\theta ^+)^{\varepsilon _1} = \{ e_{1}, \ldots , e_{n_2} \}$
,
$(\mathbb {F}_\theta ^+)^{\varepsilon _2} = \{ f_{1}, \ldots , f_{n_2} \}$
and
$$ \begin{align*} e_i f_j = f_{j'} e_{i'} \quad\text{whenever } \theta (i,j) = (j',i'). \end{align*} $$
Remark 2.6. In their early papers, Yang et al. define
$\mathbb {F}_\theta ^+$
in terms of a bijection
${\theta : [n_1 ]\times [n_2] \to [n_1] \times [n_2]}$
rather than
$[n_1 ]\times [n_2] \to [n_2] \times [n_1]$
.
An affine map
$f : \mathbb {N}^\ell \to \mathbb {N}^k$
is a map of the form
$f(n) = An+p$
for
$A \in M_{\ell ,k} ( \mathbb {N} )$
and
$p \in \mathbb {N}^\ell $
. The next proposition unifies the pullback construction of [Reference Kumjian and Pask26, Definition 1.9] (case
$p=0$
) and the p-dual graph of [Reference Allen, Pask and Sims1, Definition 3.2] (case
$A=I$
).
Proposition 2.7 (Affine pullbacks).
Let
$(\Lambda ,d)$
be a k-graph and let
$f : \mathbb {N}^\ell \to \mathbb {N}^k$
be an affine map with
$f(0)=p \in \mathbb {N}^k$
. Set
$f^* (\Lambda ) = \{ ( \lambda , n ) : d ( \lambda ) = f(n) \} \subseteq \Lambda \times \mathbb {N}^\ell $
. Then
$f^*(\Lambda )$
is an
$\ell $
-graph, with
$r (\lambda , n ) = [ \lambda (0,p) , 0]$
,
$s ( \lambda , n ) = [\lambda ( d(\lambda ) -p , d(\lambda )),0]$
,
$$ \begin{align*} (\lambda , m ) \circ (\mu , n ) = ( \lambda ( 0 , d(\lambda)-p) \mu , m+n) \text{ if }s (\lambda , m ) = r ( \mu , n ), \end{align*} $$
and
$d_{f^*(\Lambda )} (\lambda ,n) = n$
. We have
$f^*(\Lambda )^0 = \{ ( \lambda , 0 ) : \lambda \in \Lambda ^p \}$
.
Proof. As in [Reference Kumjian and Pask26, Definition 1.9], the pullback
$A^*(\Lambda )$
of
$\Lambda $
by the homomorphism
$A : \mathbb {N}^\ell \to \mathbb {N}^k$
is an
$\ell $
-graph. By [Reference Allen, Pask and Sims1, Proposition 3.2], its dual
$p(A^*(\Lambda ))$
is also an
$\ell $
-graph. As sets,
Direct calculations show that this identification intertwines the structure maps above with those of
$p(A^*(\Lambda ))$
.
Example 2.8 (Crossed-product graph).
Let
$\alpha : \mathbb {Z}^\ell \to \operatorname {Aut} \Lambda $
be an action of
$\mathbb {Z}^\ell $
on a k-graph
$\Lambda $
. Then the set
$\Lambda \times \mathbb {N}^\ell $
with the structure maps
$$\begin{align*}\begin{array}{lr@{\ }lr@{\ }l} &r ( \lambda , m ) &= ( r ( \lambda ) , 0 ) , &s ( \lambda , m ) &= ( \alpha^{-m} (s (\lambda ) ) , 0 ) , \\[5pt] &( \lambda , m )( \mu , n )&= ( \lambda \alpha^m ( \mu ) , m+n ), & d ( \lambda , m ) &= ( d ( \lambda) , m ) \end{array} \end{align*}$$
is a
$(k + \ell )$
-graph, called the crossed-product graph
$\Lambda \times _\alpha \mathbb {N}^\ell $
(see [Reference Farthing, Pask and Sims17]).
2.2 Fundamental groupoids, fundamental groups and universal covers
Every k-graph
$\Lambda $
has a fundamental groupoid, defined as follows (see [Reference Schubert42, Section 19.1] or [Reference Pask, Raeburn and Quigg32, Section 3]).
Definition 2.9. Let
$\Lambda $
be a k-graph. There exists a groupoid
$\Pi (\Lambda )$
and a functor
$i : \Lambda \to \Pi (\Lambda )$
such that
$i(\Lambda ^0)=\Pi (\Lambda )^0$
, with the following universal property: for every functor F from
$\Lambda $
into a groupoid
$\mathcal {G}$
, there exists a unique groupoid homomorphism
$\tilde {F} : \Pi (\Lambda ) \to \mathcal {G}$
such that
$\tilde {F} \circ i = F$
. The pair
$(\Pi (\Lambda ), i)$
is unique up to canonical isomorphism, so we refer to any such groupoid
$\Pi (\Lambda )$
as the fundamental groupoid of
$\Lambda $
.
Remark 2.10. The assignment
$\Lambda \mapsto \Pi (\Lambda )$
is a functor from k-graphs to groupoids. The restriction of
$i : \Lambda \to \Pi (\Lambda )$
to
$\Lambda ^0$
is injective with range
$\Pi (\Lambda )^0$
and thus we identify
$\Lambda ^0$
with
$\Pi (\Lambda )^0$
. Note that
$\Pi (\Lambda )$
is denoted
$\mathcal {G} (\Lambda )$
in [Reference Pask, Raeburn and Quigg32], but this clashes with the notation for path groupoids in Section 2.4 and [Reference Kumjian and Pask26].
Each component of a k-graph also has a fundamental group.
Definition 2.11. Let
$\Lambda $
be a k-graph. The fundamental group
$\pi _1(\Lambda , v)$
of
$\Lambda $
at
$v \in \Lambda ^0$
is the isotropy group
$\pi _1 (\Lambda , v) := v \Pi ( \Lambda ) v$
of
$\Pi (\Lambda )$
at v.
Definition 2.12. For
$X \not = \emptyset $
, the pair groupoid of X is
$T(X) := X \times X$
, the simple transitive groupoid with unit space
$\{(x,x) : x \in X\}$
identified with X; it has structure maps
$$ \begin{align*} r(x, y) := x, \quad s(x, y) := y, \quad (x, y)(y, z) := (x, z), \quad (x, y)^{-1} := (y, x). \end{align*} $$
Remark 2.13. Suppose that
$\Lambda $
is connected. Then for every
$u, v \in \Lambda ^0$
,
$\pi _1 (\Lambda , u) \cong \pi _1 (\Lambda , v)$
, but the isomorphism
$\pi _1(\Lambda , u) \to \pi _1(\Lambda , v)$
is noncanonical.
Let
$v \in \Lambda ^0$
. Then there exists a function
$w \mapsto \gamma _w$
from
$\Lambda ^0$
to
$\Pi (\Lambda )v$
such that
$\gamma _v = v$
and
$r(\gamma _w) = w$
for all w. Any such function
$\gamma $
determines a
$1$
-cocycle
$\kappa = \kappa _\gamma : \Lambda \to \pi _1 (\Lambda , v)$
, given by
$$ \begin{align*} \kappa_\gamma(\lambda) := \gamma_{r(\lambda)}^{-1} i(\lambda)\gamma_{s(\lambda)} \quad \text{for } \lambda \in \Lambda, \end{align*} $$
and an isomorphism
$\phi _\gamma : \Pi (\Lambda ) \to \pi _1 (\Lambda , v) \times T(\Lambda ^0)$
(see [Reference Pask, Raeburn and Quigg33, Corollary 6.5]) given for
$g \in \Pi (\Lambda )$
by
$$ \begin{align*} \phi_\gamma(g) := (\gamma_{r(g)}^{-1} g\gamma_{s(g)}, (r(g), s(g))). \end{align*} $$
Thus, we have
$(\phi _\gamma \circ i)(\lambda ) = (\kappa _\gamma (\lambda ), (r(\lambda ), s(\lambda )))$
for all
$\lambda \in \Lambda $
.
The following definitions appear in [Reference Pask, Raeburn and Quigg33]. We include them for completeness.
Definition 2.14. Let
$\Lambda , \Sigma , \Gamma $
be k-graphs.
-
(i) A surjective k-graph morphism
$p : \Sigma \to \Lambda $
is a covering if for all
$v \in \Sigma ^0$
, p restricts to bijections
$\Sigma v \to \Lambda p(v)$
and
$v \Sigma \to p(v) \Lambda $
. -
(ii) A covering
$p:\Sigma \to \Lambda $
is said to be connected if
$\Sigma $
(and hence
$\Lambda $
) is connected. -
(iii) If
$p: \Sigma \to \Lambda $
and
$q : \Gamma \to \Lambda $
are coverings, a morphism from
$(\Sigma ,p)$
to
$(\Gamma , q)$
is a k-graph morphism
$\phi : \Sigma \to \Gamma $
such that
$q \circ \phi = p$
. -
(iv) A covering
$p : \Sigma \to \Lambda $
is universal if it is connected in the sense of part (ii), and for every connected covering
$q : \Gamma \to \Lambda $
, there is a unique morphism
${\phi : (\Sigma ,p)\to (\Gamma ,q)}$
in the sense of part (iii).
Example 2.15. Let
$\Lambda $
be a k-graph, G a group,
$c: \Lambda \to G$
a
$1$
-cocycle and
$G \times _c \Lambda $
the skew product. There is a covering
$p : G \times _c \Lambda \to \Lambda $
given by
$p(g, \lambda ) = \lambda $
[Reference Pask, Raeburn and Quigg33, Proposition 6.3]. The quotient
$G \backslash (G \times _c \Lambda )$
by translation in G is a k-graph, and p descends to an isomorphism
$\tilde {p} : G \backslash (G \times _c \Lambda ) \to \Lambda $
.
Theorem 2.16 [Reference Bridson and Haefliger6, Proposition A.19], [Reference Pask, Raeburn and Quigg33, Theorem 2.7].
Every connected k-graph
$\Lambda $
has a universal covering. A connected covering
$p :\Sigma \to \Lambda $
is universal if and only if the induced homomorphism
$p_*: \pi _1(\Sigma ,v) \to \pi _1 (\Lambda , p(w))$
given by
$p_* ( [\gamma ] ) = [p(\gamma )]$
is the trivial homomorphism for some, and hence every,
$v\in \Sigma ^0$
.
2.3 Simply connected k-graphs
Definition 2.17. A k-graph
$\Lambda $
is simply connected if
$\pi _1(\Lambda , v)$
is trivial for every
${v \in \Lambda ^0}$
.
Theorem 2.18 [Reference Pask, Raeburn and Quigg33, Corollaries 5.5 and 6.5].
Let
$\Lambda $
be a connected k-graph.
-
(i) A connected covering
$p : \Sigma \to \Lambda $
is universal if and only if
$\,\Sigma $
is simply connected. -
(ii) Given
$v \in \Lambda ^0$
, there exists a cocycle
$\eta : \Lambda \to \pi _1 (\Lambda , v)$
for which the skew-product covering
$p : \pi _1(\Lambda , v) \times _\eta \Lambda \to \Lambda $
of Example 2.15 is a universal covering.
We can characterise simply connected k-graphs using either fundamental groupoids or
$1$
-cocycles.
Lemma 2.19. Let
$\Lambda $
be a connected k-graph. Then the following are equivalent:
-
(i)
$\Lambda $
is simply connected; -
(ii)
$\gamma \mapsto (r(\gamma ), s(\gamma ))$
is an isomorphism
$\Pi (\Lambda ) \cong T(\Lambda ^0)$
; and -
(iii) for every group G, every
$1$
-cocycle
$c : \Lambda \to G$
is a coboundary in the sense that there is a function
$b : \Lambda ^0 \to G$
such that
$b(r(\lambda )) c(\lambda ) = b(s(\lambda ))$
.
Proof. (i)
$\,\Rightarrow \,$
(ii). If
$\Lambda $
is simply connected, then by definition,
$\pi _1(\Lambda , v)$
is trivial for all v, so Remark 2.13 gives item (ii).
(ii)
$\,\Rightarrow \,$
(iii). Suppose that
$\Pi (\Lambda ) = T(\Lambda ^0)$
and fix a
$1$
-cocycle
$c : \Lambda \to G$
. By the universal property of
$\Pi (\Lambda )$
, there is a homomorphism
$\tilde {c} : T(\Lambda ^0) \to G$
that extends c (that is,
$c = \tilde {c}\circ {i}$
). Fix
$v \in \Lambda ^0$
. Define
$b : \Lambda ^0 \to G$
by
$b(w) = \tilde {c}(v,w)$
. For each
$\lambda \in \Lambda $
,
$$ \begin{align*} c(\lambda) = \tilde{c}(r(\lambda), s(\lambda)) = \tilde{c}((r(\lambda), v) (v, s(\lambda))) = b(r(\lambda))^{-1} b(s(\lambda)), \end{align*} $$
giving
$b(r(\lambda )) c(\lambda ) = b(s(\lambda ))$
.
(iii)
$\,\Rightarrow \,$
(i). Suppose that every
$1$
-cocycle on
$\Lambda $
is a coboundary. Fix
$v \in \Lambda ^0$
. As in Remark 2.13, for each
$w \in \Lambda ^0 \setminus \{v\}$
, fix
$\gamma _w \in \Pi (\Lambda )^w_v$
, put
$\gamma _v = v$
and define
$\kappa : \Lambda \to \pi _1(\Lambda , v)$
by
$\kappa (\lambda ) = \gamma _{r(\lambda )}^{-1} i(\lambda )\gamma _{s(\lambda )}$
. Then
$\kappa $
is a
$1$
-cocycle so there is a map
${b : \Lambda ^0 \to \pi (\Lambda , v)}$
such that
$\kappa (\lambda ) = b(r(\lambda ))^{-1}b(s(\lambda ))$
for all
$\lambda $
. By the universal property of the fundamental groupoid,
$\kappa $
extends uniquely to a
$1$
-cocycle
$\tilde {\kappa } : \Pi (\Lambda ) \to \pi _1(\Lambda , v)$
(that is,
$\kappa = \tilde {\kappa }\circ {i}$
). By uniqueness, it follows that for all
$\gamma \in \Pi (\Lambda )$
, we have
$$ \begin{align*} \gamma_{r(\gamma)}^{-1} \gamma\gamma_{s(\gamma)} = \tilde{\kappa}(\gamma) = b(r(\gamma))^{-1}b(s(\gamma)). \end{align*} $$
The first equation implies that the restriction of
$\tilde {\kappa }$
to
$ \pi _1(\Lambda , v)$
is the identity map and by the second equation, the restriction is trivial. Hence,
$ \pi _1(\Lambda , v)$
is trivial and so
$\Lambda $
is simply connected.
2.4 The path groupoid
$\mathcal {G}_\Lambda $
and the
$C^*$
-algebra
$C^*(\Lambda )$
Let
$\Lambda $
be a row-finite source-free k-graph. The infinite path space
$\Lambda ^\infty $
of
$\Lambda $
is the space of k-graph morphisms
$x : \Omega _k \to \Lambda $
under the locally compact Hausdorff topology with basic compact open sets
$Z(\lambda ) := \{ x \in \Lambda ^\infty : \lambda = x(0, d(\lambda )) \}$
, indexed by
$\lambda \in \Lambda $
. For
$p \in \mathbb {N}^k$
, the shift map
$\sigma ^p: \Lambda ^\infty \to \Lambda ^\infty $
is defined by
$\sigma ^px(m, n) = x(m+p, n+p)$
for
$x \in \Lambda ^\infty $
and
$(m, n) \in \Omega _k$
; and
$p \mapsto \sigma ^p$
is an action of
$\mathbb {N}^k$
by local homeomorphisms.
Elements
$x, y \kern1.3pt{\in}\kern1.3pt \Lambda ^\infty $
are shift equivalent, written
$x \kern1.4pt{\simeq}\kern1.4pt y$
, if
$\sigma ^px \kern1.3pt{=}\kern1.3pt \sigma ^qy$
for some
${p, q \kern1.3pt{\in}\kern1.3pt \mathbb {N}^k}$
. The path groupoid
$\mathcal {G}_{\Lambda} $
is the Deaconu–Renault groupoid of the action
$p \mapsto \sigma ^p$
.
Definition 2.20 [Reference Kumjian and Pask26, Definition 2.7].
The path groupoid is
$$ \begin{align*} \mathcal{G}_\Lambda := \{ (x, n, y) \in \Lambda^\infty \times \mathbb{Z} \times \Lambda^\infty : \sigma^\ell x = \sigma^my, n = \ell -m \}, \end{align*} $$
with unit space
$\mathcal {G}_\Lambda ^0 = \{(x, 0, x) : x \in \Lambda ^\infty \}$
identified with
$\Lambda ^\infty $
, with structure maps
$$ \begin{align*} r(x, n, y) = x, \quad s(x, n, y) = y, \quad (x, n, y)(y, \ell, z) = (x, n+\ell, z), \quad (x, n, y)^{-1} = (y, -n, x), \end{align*} $$
and under the topology with basic open sets
$$ \begin{align*} Z(\mu ,\nu ) = \{(\mu x, d(\mu ) - d(\nu ), \nu x) : x \in Z(s(\mu ))\} \end{align*} $$
indexed by pairs
$(\mu ,\nu ) \in \Lambda \times \Lambda $
such that
$s(\mu ) = s(\nu )$
.
The
$C^*$
-algebra of
$\Lambda $
is defined via generators and relations.
Definition 2.21 [Reference Kumjian and Pask26, Definition 1.5].
A family of partial isometries
$\{ s_\lambda : \lambda \in \Lambda \}$
is a Cuntz–Krieger
$\Lambda $
-family if:
-
(CK1)
$\{ s_v : v \in \Lambda ^0\}$
is a collection of mutually orthogonal projections; -
(CK2)
$s_{\lambda \mu } = s_{\lambda }s_\mu $
for all
$\lambda , \mu \in \Lambda $
such that
$s(\lambda ) = r(\mu )$
; -
(CK3)
$s_{\lambda }^*s_{\lambda } = s_{s(\lambda )}$
for all
$\lambda \in \Lambda $
; and -
(CK4) for all
$v \in \Lambda ^0$
and
$n \in \mathbb {N}^k$
, we have
$s_v = \sum _{\lambda \in v\Lambda ^n} s_{\lambda }s_{\lambda }^*$
.
We write
$C^*(\Lambda )$
for the universal
$C^*$
-algebra generated by a Cuntz–Krieger
$\Lambda $
-family
$\{ s_\lambda : \lambda \in \Lambda \}$
.
The groupoid
$\mathcal {G}_\Lambda $
is étale [Reference Kumjian and Pask26, Proposition 2.8], and [Reference Kumjian and Pask26, Corollary 3.5(i)] says that
$C^*(\Lambda) \cong C^*(\mathcal{G}_\Lambda)$
.
3 Embedding results for higher-rank graphs
In this section, we develop tools for determining when a k-graph
$\Lambda $
embeds in
$\Pi (\Lambda )$
, and describe classes of examples that do embed; we also present three examples—one from [Reference Pask, Raeburn and Quigg32], one due to Ben Steinberg and one that is new—that do not embed.
3.1 Nonembeddings
Even a fairly elementary monoidal
$2$
-graph
$\Lambda $
need not embed in
$\Pi (\Lambda )$
.
Example 3.1 [Reference Pask, Raeburn and Quigg32, Example 7.1].
Let
$\Lambda $
be the 1-vertex
$2$
-graph with
$\Lambda ^{\varepsilon _1} = \{ d, e \}$
and
$\Lambda ^{\varepsilon _2} = \{ a,b,c \}$
such that
$$ \begin{align} da = ad,\quad db = be,\quad dc = ae,\quad ea = cd,\quad eb = ce,\quad ec = bd. \end{align} $$
Using the first four relations from (3-1) and that the map
$i : \Lambda \to \Pi (\Lambda )$
is a morphism, we obtain
$$ \begin{align*} i(a) = i(d) i(a) i(d)^{-1} &= i(d)i(e)^{-1}i(c) = i(d)i(b)i(e)^{-1} = i(b), \end{align*} $$
so
$i(a)=i(b)$
in
$\Pi ( \Lambda )$
. The fifth equation in (3-1) gives
$i(d)=i(e)$
, so equations two and five give
$i(b)=i(c)$
. Hence,
$i(a)=i(b) = i(c)$
and
$i(d) = i(e)$
. The degree map descends to an isomorphism
$\tilde {d} : \Pi ( \Lambda ) \to \mathbb {Z}^2$
; so the universal cover of
$\Lambda $
is isomorphic to
${\mathbb {Z}^2 \times _d \Lambda }$
.
The next example, shown to us by Ben Steinberg, who attributes the idea to Mal’cev [Reference Mal’cev30, §2] (see also [Reference Lawson and Vdovina29, Example 11.13]), is a monoidal
$2$
-graph that does not embed even though its edge-set does.
Example 3.2 (Steinberg, private communication).
Let
$\Lambda $
be the unique
$1$
-vertex
$2$
-graph with
$\Lambda ^{\varepsilon _1} = \{ e_1, e_2, e_3, e_4 \}$
and
$\Lambda ^{\varepsilon _2} = \{ f_1, f_2, f_3, f_4 \}$
, and such that
$$ \begin{align} e_af_b := \begin{cases} f_be_a & \text{if } (a, b) = (1, 4), (4, 1); \\ f_ae_b & \text{otherwise.} \end{cases} \end{align} $$
Since
$i : \Lambda \to \Pi (\Lambda )$
is a functor,
$i(e_a)^{-1}i(f_a) = i(f_b)i(e_b)^{-1}$
for
$(a, b) = (1,2), (2,3), (3,4)$
, so
$$ \begin{align*} i( e_1 )^{-1} i ( f_1) = i( f_2 ) i ( e_2 )^{-1} = i( e_3 )^{-1} i ( f_3) = i ( f_4 ) i (e_4 )^{-1}, \end{align*} $$
and then rearranging the outer terms gives
$$ \begin{align*} i(f_1e_4) = i(f_1) i(e_4) = i(f_4) i (e_1) = i(f_4e_1). \end{align*} $$
Uniqueness of factorisations in
$\Lambda $
shows that
$f_1e_4 \not = f_4e_1$
, so i is not injective.
We show that i is injective on
$\Lambda ^{\varepsilon _1} \cup \Lambda ^{\varepsilon _2}$
. For this, define
$c : \Lambda ^{\varepsilon _1} \cup \Lambda ^{\varepsilon _2} \to \mathbb {Z}$
by
$c(e_j) = c(f_j) = j$
for
$j=1, \ldots , 4$
. Since c respects (3-2), it extends to a functor
${c : \Lambda \to \mathbb {Z}}$
. By Definition 2.9, there is a functor
$\tilde {c} : \Pi (\Lambda ) \to \mathbb {Z}$
such that
$\tilde {c} \circ i = c$
. In particular,
$\tilde {c}(i(e_j)) = j = \tilde {c}(i(f_j))$
for all j. Hence,
$(\tilde {c} \times \tilde {d}) \circ i : (\Lambda ^{\varepsilon _1} \cup \Lambda ^{\varepsilon _2}) \to \mathbb {Z}^2$
is injective. Thus, i is injective on
$\Lambda ^{\varepsilon _1} \cup \Lambda ^{\varepsilon _2}$
.
Example 3.3. For readers looking at this paper in monochrome, in the following example we refer to solid edges as blue, dashed edges as red, and dot-dashed edges as green. By [Reference Hazlewood, Raeburn, Sims and Webster19, Theorems 4.4 and 4.5], there is a unique
$3$
-graph
$\Gamma $
with the skeleton and factorisation rules below (there are no 3-coloured paths, so the associativity condition is vacuous).
By Proposition 3.12(ii) below, the 2-coloured sub-
$2$
-graphs of
$\Gamma $
are all embeddable: the
$\mathbb {Z}$
-valued cocycle on the blue–red graph carrying
$\{f_i : i \le 4\}$
to
$1$
and all other edges to
$0$
is essential (see Definition 3.10); the
$\mathbb {Z}$
-valued cocycle on the red–green graph carrying
$\{f_1\}\cup \{f^{\prime }_i : i \ge 2\}$
to
$1$
and all other edges to
$0$
is essential; and the trivial cocycle on the blue–green graph is essential.
However,
$\Gamma $
does not embed in
$\Pi (\Gamma )$
: writing
$[x]$
for
$i(x) \in \Pi (\Lambda )$
, we calculate:
$$ \begin{align*}[f_1'] [g_1]^{-1} &= [g_3]^{-1}[f_3] = [f_2] [g_2]^{-1} = [f_2] [e_2]^{-1} [e_2] [g_2]^{-1} = [e_3]^{-1}[f_4] [g_4]^{-1}[e_4] \\ &= [e_3]^{-1}[f_4][e_1][e_1]^{-1} [g_4]^{-1}[e_4]\\ &= [e_3]^{-1}[e_3][f_1][g_1]^{-1} [e_4]^{-1}[e_4] = [f_1] [g_1]^{-1}. \end{align*} $$
So cancellation gives
$[f_1'] = [f_1]$
. We then have
$[f^{\prime }_3] = [g_3][f_1][g_1]^{-1} = [g_3][f^{\prime }_1] [g_1]^{-1} = [f_3]$
and
$[f^{\prime }_4] = [e_3][f^{\prime }_1][e_1]^{-1} = [e_3][f_1][e_1]^{-1} = [f_4]$
, and then also
${[f^{\prime }_2] = [e_3]^{-1} [f^{\prime }_4][e_2] = [e_3]^{-1} [f_4][e_2] = [f_2]}$
.
Motivated by these examples, we seek conditions under which
$i : \Lambda \to \Pi (\Lambda )$
is injective.
3.2 Embedding singly connected higher-rank graphs
Definition 3.4. A k-graph
$\Lambda $
is singly connected if there is at most one path between any two vertices; that is, for all
$u,v \in \Lambda ^0$
, we have
$| u {\Lambda }v | \le 1$
.
Singly connected k-graphs need not be connected. The vertex set of a singly connected k-graph is partially ordered by the relation
$\le $
given by
$u \le v$
if and only if
$u \Lambda v \neq \emptyset $
.
Example 3.5. Write
$\{ t_i : i=1 , \ldots , n\}$
for the generators of the free group
$\mathbb {F}_n$
. Let
$c :B_n \to \mathbb {F}_n$
be the
$1$
-cocycle such that
$c(f_i) = t_i$
for all i. Then
$\mathbb {F}_n \times _c B_n$
is singly connected.
There is a relationship between singly connected k-graphs and the simply connected k-graphs of Section 2.3, though neither condition implies the other.
Proposition 3.6. Let
$\Lambda $
be a connected k-graph and suppose that
$i : \Lambda \to \Pi (\Lambda )$
is injective. If
$\Lambda $
is simply connected, then it is singly connected.
Proof. Suppose that
$\Lambda $
is not singly connected. Then there exist distinct elements
$\lambda , \mu \in \Lambda $
such that
$s(\lambda ) = s(\mu )$
and
$r(\lambda ) = r(\mu )$
. Since
$i : \Lambda \to \Pi (\Lambda )$
is injective,
${i(\lambda ) \ne i(\mu )}$
and thus
$i(\lambda )^{-1}i(\mu ) \in \pi _1 (\Lambda , s(\lambda )) \backslash \{ s(\lambda )\}$
. Hence,
$\Lambda $
is not simply connected.
The reverse implication fails, as the following example illustrates.
Example 3.7. Let E be the directed graph with
$E^0=\{u, v, w, x\}$
and
$E^1= \{ e, f, g, h \}$
such that
$s(e) = u = s(f)$
,
$s(g) = w = s(h)$
,
$r(e) = v = r(h)$
and
$r(f) = x = r(g)$
. Then the
$1$
-graph
$E^*$
is a singly connected
$1$
-graph that is not simply connected since
$\pi _1(E^*, u) \cong \mathbb {Z}$
. Adding tails at both u and w as in [Reference Bates, Pask, Raeburn and Szymański4, Lemma 1.2] yields a source-free
$1$
-graph with the same property.
We use the next theorem, which exploits the universal property of the fundamental groupoid from Section 2.2, to show that singly connected k-graphs embed in their fundamental groupoids.
Theorem 3.8. Let
$\Lambda $
be a k-graph and let
$\mathcal {G}$
be a groupoid. If there is an injective functor
$F : \Lambda \to \mathcal {G}$
, then
$i : \Lambda \to \Pi (\Lambda )$
is injective.
Proof. The universal property of the fundamental groupoid yields a homomorphism
$\tilde {F} : \Pi (\Lambda ) \to \mathcal {G}$
such that
$F = \tilde {F} \circ i$
. Hence, if F is injective, then i is injective.
Proposition 3.9. Let
$\Lambda $
be a connected k-graph. Then:
-
(i) the canonical map
$\iota : \Lambda \to T(\Lambda ^0)$
is injective if and only if
$\Lambda $
is singly connected; -
(ii) if
$\Lambda $
is singly connected, then
$i : \Lambda \to \Pi (\Lambda )$
is injective.
Proof. The first assertion follows by definition and the second follows from Theorem 3.8.
Theorem 3.8 also allows us to deduce embeddability from the existence of a suitable
$1$
-cocycle.
Definition 3.10. Let
$\Lambda $
be a k-graph, G a countable group and
$c : \Lambda \to G$
a 1-cocycle. We say that c is essential if the restriction of c to
$u{\Lambda }v$
is injective for all
$u, v \in \Lambda ^0$
.
Example 3.11. The
$1$
-cocycle
$c : B_n \to \mathbb {F}_n$
described in Example 3.5 is essential.
Proposition 3.12. Let
$\Lambda $
be a connected k-graph and let
$v \in \Lambda ^0$
. Then the following are equivalent:
-
(i) the
$1$
-cocycle
$\kappa : \Lambda \to \pi _1(\Lambda , v)$
given in Remark 2.13 is essential; -
(ii)
$\Lambda $
admits an essential cocycle
$c : \Lambda \to G$
to a group G; and -
(iii)
$i : \Lambda \to \Pi (\Lambda )$
is injective.
For any essential cocycle
$c : \Lambda \to G$
as in part (ii),
$G \times _c \Lambda $
is singly connected.
Proof. The implication (i)
$\implies $
(ii) is obvious.
For (ii)
$\implies $
(iii), suppose that
$c : \Lambda \to G$
is an essential cocycle into a group. Note that
$G \times T(\Lambda ^0)$
is a groupoid. Define
$j : \Lambda \to G \times T(\Lambda ^0)$
by
$j(\lambda ) := (c(\lambda ), (r(\lambda ), s(\lambda )))$
; then j is a functor. Since c is essential, j is injective, so
$i : \Lambda \to \Pi (\Lambda )$
is injective by Theorem 3.8.
For (iii)
$\implies $
(i), suppose that
$i : \Lambda \to \Pi (\Lambda )$
is injective. Recall that by Remark 2.13,
$(\kappa (\lambda ), (r(\lambda ), s(\lambda ))) = (\phi _\gamma \circ {i})(\lambda )$
for all
$\lambda \in \Lambda $
. Hence,
$\kappa $
is essential, since
$\phi _\gamma \circ i$
is injective.
For the final statement, suppose that
$c : \Lambda \to G$
is essential, and that
${r(g, \lambda ) = r(h,\mu )}$
and
$s(g,\lambda ) = s(h,\mu )$
in
$G \times _c \Lambda $
. Then
$$ \begin{align*} (g, r(\lambda)) &= r(g,\lambda) = r(h, \mu) = (h, r(\mu)) \quad \text{and} \\ (c(\lambda) g, s(\lambda)) &= s(g,\lambda) = s(h,\mu) = (c(\mu) h, s(\mu)). \end{align*} $$
So
$r(\lambda ) = r(\mu )$
,
$s(\lambda ) = s(\mu )$
,
$g = h$
and
$c(\lambda )g = c(\mu )h$
. These last two equalities give
$c(\lambda ) = c(\mu )$
. Thus,
$j(\lambda ) = j(\mu )$
and hence
$\lambda = \mu $
. Therefore,
$(g, \lambda ) = (h,\mu )$
and so
$G \times _c \Lambda $
is singly connected.
3.3 More general embedding results
In this section, we investigate embeddability of k-graphs that are not singly connected. We start with one of the most useful results in our toolkit, which relies on the universal property of the fundamental groupoid given in Definition 2.9.
Proposition 3.13 (Lifting embeddability).
Let
$\Lambda , \Sigma $
be connected k-graphs and let
$p : \Sigma \to \Lambda $
be a covering. Then
$i_\Lambda : \Lambda \to \Pi ( \Lambda )$
is injective if and only if
$i_\Sigma : \Sigma \to \Pi (\Sigma )$
is injective.
Proof. Suppose that
$i_\Lambda : \Lambda \to \Pi (\Lambda )$
is injective and that
$\sigma , \sigma ' \in \Sigma $
satisfy
$ {i_\Sigma (\sigma ) = i_\Sigma ( \sigma ' )}$
. In particular,
$s(\sigma ) = s(\sigma ')$
; let
$u := s(\sigma )$
. By universality of
$\Pi (\Sigma )$
, there is a unique groupoid morphism
$\tilde {p} : \Pi (\Sigma ) \to \Pi (\Lambda )$
such that
$\tilde {p} \circ i_\Sigma = i_\Lambda \circ p$
. Hence,
$$ \begin{align*} i_\Lambda(p(\sigma)) = \tilde{p} ( i_\Sigma( \sigma)) = \tilde{p} ( i_\Sigma( \sigma')) = i_\Lambda(p(\sigma')). \end{align*} $$
Injectivity of
$i_\Lambda $
forces
$p(\sigma ) = p(\sigma ')$
. Since p is a covering, it is injective on
$s^{-1}(u)$
. So
$\sigma = \sigma '$
, and hence
$i_\Sigma : \Sigma \to \Pi (\Sigma )$
is injective.
For the reverse implication, suppose that
$i_\Lambda : \Lambda \to \Pi ( \Lambda )$
is not injective. Then there are distinct
$\lambda , \lambda ' \in \Lambda $
such that
$i_\Lambda ( \lambda ) = i_\Lambda ( \lambda ' )$
. We may assume without loss of generality that
$\Sigma $
is the universal covering of
$\Lambda $
so that
$\Sigma $
is simply connected. Since
$\Sigma $
is connected,
$r \times s : \Pi ( \Sigma ) \to \Sigma ^0 \times \Sigma ^0$
is an isomorphism, so
$\Pi ( \Sigma ) \cong T( \Sigma ^0 )$
. By Theorem 2.18(ii), given
$u \in \Lambda ^0$
, there is a cocycle
$\eta : \Lambda \to \pi _1(\Lambda , u)$
such that
$\Sigma \cong \pi _1(\Lambda , u) \times _\eta \Lambda $
and p is given by projection onto the second factor. It follows that
$s( \lambda ) = s( \lambda ' )$
,
${r( \lambda ) = r( \lambda ' )}$
and
$\eta ( \lambda ) = \eta ( \lambda ' )$
(since
$\eta $
factors through
$i_\Lambda $
and
$i_\Lambda ( \lambda ) = i_\Lambda ( \lambda ' )$
). Identifying
$\Sigma $
with the skew-product as above, set
$\sigma = (1, \lambda ), \sigma ' = (1, \lambda ') \in \Sigma $
, so
$\sigma , \sigma '$
are distinct. We have
$$ \begin{align*} s(\sigma) = s(1, \lambda) = (\eta(\lambda), s(\lambda)) = (\eta(\lambda'), s(\lambda')) = s(1, \lambda') = s(\sigma') \end{align*} $$
and similarly,
$r(\sigma ) = r(\sigma ')$
. So
$(r \times s)(i_\Sigma (\sigma )) = (r \times s)(i_\Sigma (\sigma '))$
. Since
$r \times s$
is injective on
$i_\Sigma $
, we deduce that
$i_\Sigma : \Sigma \to \Pi (\Sigma )$
is not injective.
Our later results say that embeddability is preserved by various constructions of new k-graphs from old ones. So we need to know that some basic classes of k-graphs, like
$1$
-graphs, embed.
Theorem 3.14. Let
$\Lambda $
be a
$1$
-graph. Then
$i_\Lambda : \Lambda \to \Pi ( \Lambda )$
is injective.
Proof. Write
$\Lambda = \bigsqcup _{i=1}^n \Lambda _i $
as a disjoint union of connected graphs. For
$i=1 , \ldots , n$
, let
$\Sigma _i$
be the universal cover of
$\Lambda _i$
. Since
$\Sigma = \bigsqcup _{i=1}^n \Sigma _i$
is (the path category of) a disjoint union of trees, there is at most one undirected path connecting any two distinct vertices. It follows that
$\Sigma $
is singly connected and therefore embeddable by Proposition 3.9(ii). Hence,
$i_\Lambda : \Lambda \to \Pi ( \Lambda )$
is injective.
Corollary 3.15. Let
$\Lambda $
be a k-graph and suppose that
$i_\Lambda : \Lambda \to \Pi (\Lambda )$
is injective.
-
(i) Let
$f : \mathbb {N}^\ell \to \mathbb {N}^k$
be an affine map. Then
$i : f^* ( \Lambda ) \to \Pi (f^* ( \Lambda ) )$
is injective. -
(ii) If
$\Gamma $
is an
$\ell $
-graph and
$i_\Gamma : \Gamma \to \Pi (\Gamma )$
is injective, then
$i_{\Lambda \times \Gamma } : \Lambda \times \Gamma \to \Pi (\Lambda \times \Gamma )$
is injective. -
(iii) If
$c : \Lambda \to G$
is a
$1$
-cocycle into a group, then
$i_{G\times \Lambda } : G \times _c \Lambda \to \Pi ( G \times _c \Lambda )$
is injective. -
(iv) If
$\alpha : \mathbb {N}^\ell \to \operatorname {Aut} (\Lambda )$
is an action, then there is an action
$\tilde {\alpha } : \mathbb {Z}^\ell \to \operatorname {Aut}(\Pi (\Lambda ))$
such that
$\tilde {\alpha _n} \circ i_\Lambda = i_{\Lambda } \circ \alpha _n$
for
$n \in \mathbb {N}^\ell $
. Both are injective. Moreover,
$$ \begin{align*} i_\Lambda \times i_{\mathbb{N}^\ell} : \Lambda \times_\alpha \mathbb{N}^\ell \to \Pi ( \Lambda) \times_\alpha \mathbb{Z}^\ell \quad\text{and}\quad i_{\Lambda \times_\alpha \mathbb{N}^\ell} : \Lambda \times_\alpha \mathbb{N}^\ell \to \Pi(\Lambda \times_\alpha \mathbb{N}^\ell) \end{align*} $$
$i_\Lambda \times i_{\mathbb {N}^\ell }$
induces an isomorphism
$\Pi ( \Lambda \times _\alpha \mathbb {N}^\ell ) \cong \Pi ( \Lambda ) \times _\alpha \mathbb {Z}^\ell $
.
Proof. (i) Define
$i \times \operatorname {id} : f^*( \Lambda ) \to \Pi (\Lambda ) \times \mathbb {Z}^\ell $
by
$(i \times \operatorname {id})(\lambda , n) = (i(\lambda ), n)$
. Then
$i \times \operatorname {id}$
is an injective functor into a groupoid, so the result follows from Theorem 3.8.
(ii) The map
$i_\Lambda \times i_\Gamma : \Lambda \times \Gamma \to \Pi (\Lambda ) \times \Pi (\Gamma )$
is an injective functor into a groupoid, so the result follows from Theorem 3.8.
(iii) By universality of
$\Pi (\Lambda )$
, there is a cocycle
$\tilde {c} : \Pi (\Lambda ) \to G$
such that
$\tilde {c} \circ i_\Lambda = c$
. The skew-product groupoid
$G \times _{\tilde {c}} \Pi (\Lambda )$
is equal as a set to
$G \times \Pi (\Lambda )$
, and
$\operatorname {id}_G \times i_\Lambda : G \times _c \Lambda \to G \times _{\tilde {c}} \Pi (\Lambda )$
is a functor. Since
$i_\Lambda $
is injective, so is
$\operatorname {id}_G \times i_\Lambda $
, so the result follows from Theorem 3.8.
(iv) Since the action
$\alpha $
of
$\mathbb {N}^\ell $
on
$\Lambda $
is determined by
$\ell $
commuting automorphisms, it extends to an action (also called
$\alpha $
) of
$\mathbb {Z}^\ell $
on
$\Lambda $
. By functoriality, this extends to an action
$\tilde {\alpha } : \mathbb {Z}^\ell \to \operatorname {Aut}(\Pi (\Lambda ))$
such that
$\tilde {\alpha _n} \circ i_\Lambda = i_{\Lambda } \circ \alpha _n$
for
$n \in \mathbb {N}^\ell $
. It is routine to check that
$i_\Lambda \times i_{\mathbb {N}^\ell }$
is a functor; it is injective because
$i_\Lambda $
and
$i_{\mathbb {N}^\ell }$
are injective. So Theorem 3.8 implies that
$i_{\Lambda \times _\alpha \mathbb {N}^\ell }$
is injective.
To see that
$i_\Lambda \times i_{\mathbb {N}^\ell }$
induces an isomorphism
$\Pi ( \Lambda \times _\alpha \mathbb {N}^\ell ) \cong \Pi ( \Lambda ) \times _\alpha \mathbb {Z}^\ell $
, note that the universal property of
$\Pi ( \Lambda \times _\alpha \mathbb {N}^\ell )$
implies that
$i_\Lambda \times i_{\mathbb {N}^\ell }$
induces a homomorphism
$\tilde {i} : \Pi ( \Lambda \times _\alpha \mathbb {N}^\ell ) \to \Pi ( \Lambda ) \times _\alpha \mathbb {Z}^\ell $
such that
$\tilde {i} \circ i_{\Lambda \times _\alpha \mathbb {N}^\ell } = i_\Lambda \times i_{\mathbb {N}^\ell }$
. We construct an inverse. The restriction
$c_1 := i_{\Lambda \times _\alpha \mathbb {N}^\ell }|_{\Lambda \times \{0\}} : \Lambda \to \Pi (\Lambda \times _\alpha \mathbb {N}^\ell )$
is a functor, as is
${c_2 := i_{\Lambda \times _\alpha \mathbb {N}^\ell }|_{\Lambda ^0 \times \mathbb {N}^\ell }}$
. The universal property of
$\Pi (\Lambda )$
implies that
$i_{\Lambda \times _\alpha \mathbb {N}^\ell }|_{\Lambda \times \{0\}}$
induces a homomorphism
$\tilde {c}_1 : \Pi (\Lambda ) \to \Pi ( \Lambda \times _\alpha \mathbb {N}^\ell )$
; and
$i_{\Lambda \times _\alpha \mathbb {N}^\ell }|_{\Lambda ^0 \times \mathbb {N}^\ell }$
extends to a homomorphism
$\tilde {c}_2 : \Lambda ^0 \times \mathbb {Z}^\ell \to \Pi ( \Lambda \times _\alpha \mathbb {N}^\ell )$
. Routine calculations show that
$\tilde {c}_1 \times \tilde {c}_2 : \Pi (\Lambda ) \times _{\tilde {\alpha }} \mathbb {Z}^\ell \to \Pi (\Lambda \times _\alpha \mathbb {N}^\ell )$
is a homomorphism inverse to
$\tilde {i}$
.
Remark 3.16. Combining Theorem 3.14 and Corollary 3.15(iv), we see that crossed-product graphs of
$1$
-graphs always embed in their fundamental groupoids.
Examples 3.17. We present two examples of Corollary 3.15(i).
-
(i) Define
$f : \mathbb {N}^2 \to \mathbb {N}$
by
$f(a,b)=a+b$
. Let
$\Lambda = f^* (B_n)$
. Corollary 3.15 implies that
$f^* (\Lambda )$
embeds in its fundamental group since Theorem 3.14 impies that the 1-graph
$B_n$
does. Indeed, for
$\theta : [n] \times [n] \to [n] \times [n]$
given by
$\theta (i,j)=(i, j)$
, we have
$\Lambda \cong \mathbb {F}_\theta ^+$
. -
(ii) Let
$\Lambda $
be a
$2$
-graph and define
$f : \mathbb {N}^2 \to \mathbb {N}^2$
by
$f(a,b)=(a,b)+\mathbf {1}$
. Then
$f^*(\Lambda )$
is the dual graph
$\mathbf {1} \Lambda $
described in [Reference Allen, Pask and Sims1, Definition 3.1]. So for the
$2$
-graph
$\Lambda = \mathbb {F}_\theta ^+$
from example (i) above,
$f^* ( \mathbb {F}_\theta ^+ )$
embeds in its fundamental group by Corollary 3.15(i).
Corollary 3.18 (Action graphs).
Let
$\Lambda $
be a k-graph. Let
$B_n$
be the
$1$
-graph described in Example 2.3(i). Let
$\mu \mapsto \alpha _\mu $
be a functor from
$B_n$
to
$\operatorname {Aut}(\Lambda )$
. Let
${\Gamma = B_n \times \Lambda }$
; define
$d : \Gamma \to \mathbb {N}^{k+1}$
by
$d(\mu , \lambda ) = (|\mu |, d(\lambda ))$
; define
$r, s : \Gamma \to \Gamma ^0$
by
${r(\mu , \lambda ) = (u, \alpha _\mu (r(\lambda )))}$
and
$s(\mu , \lambda ) = (u, s(\lambda ))$
; and define composition in
$\Gamma $
by
$$ \begin{align} (\mu,\alpha_\nu(\lambda))(\nu,\xi) = (\mu\nu, \lambda\xi). \end{align} $$
Then
$(\Gamma ,d)$
is a
$(k+1)$
-graph. If
$i_\Lambda : \Lambda \to \Pi ( \Lambda )$
is injective, then
$i_\Gamma : \Gamma \to \Pi ( \Gamma )$
is injective.
Proof. It is routine to check that (3-3) determines an associative composition. The map d is clearly a functor and if
$d(\mu ,\lambda ) = (a+b, m+n)$
, then factorising
$\mu = \mu _a \mu _b$
and
$\lambda = \lambda _m \lambda _n$
with the appropriate degrees, the factorisation
$(\mu ,\lambda ) = (\mu _a, \alpha _{\mu _b}(\lambda _m))(\mu _b,\lambda _n)$
is the unique factorisation of
$(\mu ,\lambda )$
into morphisms of degrees
$(a,m)$
and
$(b,n)$
. So
$\Gamma $
is a
$(k+1)$
-graph.
Universality of
$\Pi ( \Lambda )$
implies that each
$\alpha _\mu $
extends to an automorphism of
$\Pi ( \Lambda )$
. So
$\alpha $
extends to an action of
$\Pi (B_n) \cong \mathbb {F}_n$
on
$\Pi ( \Lambda )$
, with semidirect product groupoid
$\Pi (B_n) \ltimes _{\tilde {\alpha }} \Pi ( \Lambda )$
. Then
$i_{B_n} \times i_\Lambda $
is an embedding of
$\Gamma $
in
$\Pi (B_n) \ltimes _{\tilde {\alpha }} \Pi ( \Lambda )$
. The result now follows by Theorem 3.8.
Examples 3.19.
-
(i) Fix
$m,n \ge 2$
. Let
$\Lambda = B_m$
be the
$1$
-graph described in Example 2.3(i). For each
$f \in B_n^1$
, let
$\alpha _f$
be a permutation of
$B_m^1$
and extend this to a
$1$
-cocycle
$B_n \to \operatorname {Aut}(\Lambda )$
in the only possible way. By Theorem 3.18, these data give rise to a
$2$
-graph
$\Gamma $
that embeds in its fundamental group. Define
$\theta : [n] \times [m] \to [m] \times [n]$
by
$\theta (i,j) = (j' , i)$
if and only if
$\alpha _{f_i}(f_j) = f_{j'}$
. Then
$\Gamma $
is isomorphic to the
$2$
-graph
$\mathbb {F}_\theta ^+$
of Example 2.5. In particular,
$\mathbb {F}_\theta ^+$
embeds in its fundamental group.
Example 3.20. Fix
$n \ge 2$
and a permutation
$\sigma \in \operatorname {Bij} ( [n] )$
, the group of all bijections of the set
$[n]$
. Define
$\theta : [n] \times [n] \to [n] \times [n]$
by
$\theta (i,j) = (\sigma (i) ,j )$
. This fits into the situation of Example 3.19, so
$\mathbb {F}_\theta ^+$
embeds in its fundamental group.
Definition 3.21. Let X be a nonempty set. A map
$R : X^2 \to X^2$
is a (set-theoretic) Yang–Baxter solution if
$$ \begin{align*} (R \times \text{id}_X)(\text{id}_X \times R)(R \times \text{id}_X) = (\text{id}_X \times R)(R \times \text{id}_X) (\text{id}_X \times R) \end{align*} $$
as maps on
$X^3$
. For every permutation
$\sigma $
of X, there is a Yang–Baxter solution R given by
$R(e, f) = (\sigma (f), e)$
; such solutions are called permutation-type Yang–Baxter solutions.
For the interplay between the Yang–Baxter equation and k-graphs, see [Reference Yang47].
Lemma 3.22. Fix a finite set X and a Yang–Baxter solution
$R : X^2 \to X^2$
on X. Fix
$k \ge 2$
. Let
$\Lambda _{k, R}^0 = \{v\}$
. For
$i \le k$
, let
$\Lambda _{k, R}^{\varepsilon _i}=\{i\} \times X$
. For
$(i, e) \in \Lambda _{k, R}^{\varepsilon _i}$
and
$(j, f) \in \Lambda _{k, R}^{\varepsilon _j}$
with
$i < j$
, set
$$ \begin{align*} (i,e)(j,f) = (j, f')(i, e') \quad\text{if } R(e,f) = ( f',e'). \end{align*} $$
There is a unique k-graph
$\Lambda _{k, R}$
with these edges and factorisation rules. If R is a permutation-type Yang–Baxter solution, then
$i : \Lambda _{k, R} \to \Pi ( \Lambda _{k, R} )$
is injective.
Proof. The first statement follows from [Reference Yang47, Section 4.1]. For the second statement, we proceed by induction. For
$k=2$
, this follows from [Reference Yang47, Section 4.1]. Now suppose inductively that
$\Lambda _{k-1, R}$
embeds in its fundamental groupoid. There is an automorphism
$\alpha $
of
$\Lambda _{k-1, R}$
such that
$\alpha (i, e) = (i, \sigma (e))$
for all
$i \le k-1$
and
$e \in X$
. For
$e \in B^1_{|X|}$
, let
${\alpha _{e} := \alpha \in \operatorname {Aut}(\Lambda _{k-1, R})}$
. Corollary 3.18 yields a k-graph
$\Gamma = B_{|X|} \times _\alpha \Lambda _{k-1, R}$
. Choose a bijection
$\phi : B_{|X|}^1 \to \Lambda _{k, R}^{\varepsilon _1}$
. Then there is an isomorphism
$\Gamma \to \Lambda _{k, R}$
that agrees with
$\phi $
on
${B^1_{|X|} \subseteq \Gamma }$
and takes each
$(i, e) \in \Lambda ^{\varepsilon _i}_{k-1, R} \subseteq \Gamma $
to
$(i+1, e) \in \Lambda ^{\varepsilon _{i+1}}_{k, R}$
. Corollary 3.18 implies that
$\Gamma $
embeds in its fundamental groupoid, so
$\Lambda _{k, R}$
does too.
Remark 3.23. For a long time, the literature on k-graphs lacked concrete examples with
$k \ge 3$
not obtained from lower-rank graphs via the constructions of Corollary 3.15. Yang’s important insight [Reference Yang47] remedied this situation: every Yang–Baxter solution yields k-graphs for arbitrary k, typically not of the forms from Corollary 3.15. In particular, Lemma 3.22 uses Yang’s construction to see that every finite permutation
$\sigma $
yields a k-graph that embeds in its fundamental groupoid for each
$k \ge 1$
. Taking
$\sigma = \operatorname {id}$
yields Cartesian-product k-graphs, but most other choices of
$\sigma $
yield k-graphs that do not arise from the constructions of Corollary 3.15.
Remark 3.24. Work of Lawson and Vdovina also yields many embeddable k-graphs. A monoidal k-graph is rigid [Reference Lawson and Vdovina29, page 37] if whenever e and f are edges of different degrees, there are unique edges
$e', e", f', f"$
such that
$e'f = f'e$
and
$ef" = fe"$
. Theorem 3.8 and [Reference Lawson and Vdovina29, Theorem 11.14] combined imply that every rigid monoidal k-graph
$\Lambda $
embeds in
$\Pi (\Lambda )$
.
We finish the section by showing that a strongly connected k-graph
$\Lambda $
embeds in
$\Pi (\Lambda )$
whenever the submonoid of endomorphisms at any vertex embeds in a group.
Proposition 3.25. Let
$\Lambda $
be a strongly connected k-graph and H a group. Fix
$v \in \Lambda ^0$
. If there exists an injective monoid homomorphism
$c : v\Lambda v \to H$
, then
$i : \Lambda \to \Pi (\Lambda )$
is injective.
Proof. The universal property of
$\Pi (\Lambda )$
given in Definition 2.9 implies that there is a homomorphism
$\tilde {c} : i(v)\Pi (\Lambda )i(v) \to H$
such that
$\tilde {c} \circ i = c$
. Since
$\Lambda $
is strongly connected and since
$\Pi (\Lambda )$
is a discrete groupoid,
$\Pi (\Lambda )$
is isomorphic to
$T(\Lambda ^0) \times i(v)\Pi (\Lambda )i(v)$
. Post-composing this isomorphism with
$\operatorname {id}_{T(\Lambda ^0)} \times \tilde {c}$
yields a groupoid homomorphism
$q : \Pi (\Lambda ) \to T(\Lambda ^0) \times H$
. Suppose that
$q(i(\mu )) = q(i(\nu ))$
. Fix
$\lambda \in v\Lambda r(\mu )$
and
$\tau \in s(\mu )\Lambda v$
. We have
$$ \begin{align*} ((v,v), c(\lambda\mu\tau)) &= ((v,v), \tilde{c}(i(\lambda\mu\tau))) = q(i(\lambda\mu\tau))\\ &= q(i(\lambda\nu\tau)) = ((v,v), \tilde{c}(i(\lambda\nu\tau))) ((v,v), c(\lambda\nu\tau)). \end{align*} $$
Since c is injective,
$\lambda \mu \tau = \lambda \nu \tau $
and so
$\mu = \nu $
. Thus,
$q \circ i$
, and therefore i, is injective.
Example 3.26. Consider the
$2$
-graph
$\Lambda $
below with relations
$a_0e=fa_1, \ a_1e=fa_0 , bf=eb$
.
Then
$\Lambda $
is strongly connected. None of our results before Proposition 3.25 apply to show that
$\Lambda $
embeds in
$\Pi (\Lambda )$
. Since
$e b a_i = b f a_i = ba_{1-i} e$
for each i, the monoid
${u\Lambda u \subseteq \Lambda }$
has presentation
$$ \begin{align*} u \Lambda u = \langle e, b a_0 , b a_1 : eba_i = ba_{1-i} e, i=0,1 \rangle, \end{align*} $$
so is isomorphic to the semidirect product
$\mathbb {F}^+_2 \times _\alpha \mathbb {N}$
for the action
$\alpha $
that interchanges
$\{ b a_0 , b a_1 \}$
, the generators of
$\mathbb {F}^+_2$
. The action
$\alpha $
extends uniquely to an action
$\tilde {\alpha }$
of
$\mathbb {Z}$
on
$\mathbb {F}_2$
, and
$u \Lambda u \cong \mathbb {F}^+_2 \times _\alpha \mathbb {N}$
embeds in
$\mathbb {F}_2 \times _{\tilde {\alpha }} \mathbb {Z}$
. So Proposition 3.25 implies that
${i : \Lambda \to \Pi (\Lambda )}$
is injective.
4
$C^*$
-algebraic results
Here, we generalise [Reference Kumjian and Pask25, Corollary 4.14], which says that the
$C^*$
-algebra of a connected row-finite
$1$
-graph is Rieffel–Morita equivalent to a crossed product of a commutative
$C^*$
-algebra by the fundamental group of the graph. The situation is much more complicated in higher dimensions.
Let
$\Lambda $
be a connected row-finite source-free k-graph. Fix
$v\in \Lambda ^0$
. By Theorem 2.18 (see [Reference Pask, Raeburn and Quigg33, Corollary 6.5]), there is a cocycle
$\eta : \Lambda \to \pi _1 (\Lambda , v)$
such that the skew-product
$ \pi _1 ( \Lambda , v ) \times _\eta \Lambda $
is isomorphic to the universal cover
$\Sigma $
of
$\Lambda $
. It then follows from [Reference Kumjian and Pask26, Theorem 5.7] that
$C^*(\Lambda )$
is Rieffel–Morita equivalent to
$C^*(\Sigma )\rtimes \pi _1(\Lambda ,v)$
. Our main theorem describes the coefficient algebra
$C^*(\Sigma )$
of this crossed product.
Theorem 4.1. Let
$\Lambda $
be a connected row-finite source-free k-graph and let
${\Sigma = \pi _1(\Lambda , v) \times _\eta \Lambda }$
be as above so that
$C^*(\Lambda )$
is Rieffel–Morita equivalent to
$C^*(\Sigma ) \rtimes \pi _1(\Lambda , v)$
.
-
(i) The
$C^*$
-algebra
$C^*(\Sigma )$
is AF. -
(ii) If
$\Lambda $
embeds in its fundamental groupoid
$\Pi (\Lambda )$
, then
$C^*(\Sigma )$
is type I
$_0$
and its spectrum has a cover by zero-dimensional compact open Hausdorff subsets. -
(iii) If
$\Lambda $
embeds in its fundamental groupoid
$\Pi (\Lambda )$
and
$\Sigma ^{\mathbb {N}\mathbf {1}}$
is simply connected, then
$C^*(\Sigma )$
is Rieffel–Morita equivalent to a commutative
$C^*$
-algebra.
We use the next two results to prove parts (i) and (ii) of Theorem 4.1.
Proposition 4.2. Let
$\Gamma $
be a row-finite source-free k-graph. If
$\Gamma $
is simply connected, then there is a map
$f: \Gamma ^0 \to \mathbb {Z}^k$
such that
$d(\lambda ) = f(s(\lambda )) - f(r(\lambda ))$
for all
$\lambda \in \Gamma $
. Moreover,
$C^*(\Gamma )$
is AF.
Proof. Since
$d : \Gamma \to \mathbb {Z}^k$
is a cocycle, Lemma 2.19 ensures the existence of f. Now, [Reference Kumjian and Pask26, Lemma 5.4] implies that
$C^*(\Gamma )$
is AF.
Example 4.3. Let
$\Lambda $
be the
$2$
-graph of Example 3.1 (see [Reference Pask, Raeburn and Quigg32, Example 7.1]). Recall that
$\Lambda $
does not embed in its fundamental groupoid and the universal cover
${\Sigma = \mathbb {Z}^2 \times _d \Lambda }$
is simply connected. We claim that
$C^*(\Sigma )$
is Rieffel–Morita equivalent to the UHF algebra
$M_{6^{\infty }}$
(in fact,
$C^*(\Sigma ) \cong M_{6^{\infty }} \otimes \mathcal {K}$
). For each
$n \in \mathbb {N}$
, set
$v_n := n\mathbf {1}$
and observe that as in the proof of [Reference Kumjian and Pask26, Lemma 5.4],
$A_n := C^*(\{ s_\lambda : s(\lambda ) = v_n \}) \cong \mathcal {K}(\ell ^2(s^{-1}(v_n)))$
. Moreover, for all n,
$A_n \subset A_{n+1}$
and the multiplicity of the embedding is 6 (since
$|v_n\Lambda v_{n+1}| = 6$
). Since the sequence
$(v_n)^\infty_{n=1}$
is cofinal in
$\mathbb {Z}^2$
,
$$ \begin{align*} C^*(\Sigma) \cong \varinjlim A_n \cong \varinjlim \mathcal{K}(\ell^2(s^{-1}(v_n))). \end{align*} $$
Hence,
$C^*(\Sigma )$
is Rieffel–Morita equivalent to the UHF algebra
$M_{6^{\infty }}$
as claimed.
Proposition 4.4. Let
$\Gamma $
be a row-finite source-free k-graph. If
$\Gamma $
is singly connected, then, for each
$v \in \Gamma ^0$
, the corner
$s_vC^*(\Gamma )s_v$
is an abelian
$C^*$
-algebra isomorphic to
$C(Z(v))$
. Moreover,
$C^*(\Gamma )$
is type I
$_0$
and
$\operatorname {Prim} C^*(\Gamma )$
admits a cover by zero-dimensional compact open Hausdorff sets.
Proof. Fix
$x, y \in \Gamma ^\infty $
such that
$x(0) = y(0)$
and
$p,q\in \mathbb {N}^k$
. We claim that if
$\sigma ^p(x) = \sigma ^q(y)$
, then
$p =q$
and
$x = y$
. To see this, suppose that
$\sigma ^p(x) = \sigma ^q(y)$
. Then
$\sigma ^p(x)(0) = \sigma ^q(y)(0)$
. Let
$u := x(0) = y(0)$
and
$v := \sigma ^p(x)(0) = \sigma ^q(y)(0)$
. Then
$x(0, p), y(0, q) \in u{\Gamma }v$
. Since
$\Gamma $
is singly connected,
$x(0, p) = y(0, q)$
. Hence,
${x = x(0, p)\sigma ^p(x) = y(0, q)\sigma ^q(y)(0) = y}$
, and the claim holds.
Now, recall from [Reference Kumjian and Pask26] that
$\Gamma ^\infty = \mathcal {G}_\Gamma ^0$
and that for
$v \in \Gamma ^0$
,
$$ \begin{align*} \mathcal{G}_\Gamma|_{Z(v)} := \{ \gamma \in \mathcal{G}_\Gamma : s(\gamma), r(\gamma) \in Z(v) \}, \end{align*} $$
$s_v = \chi _{Z(v)} \subset C_0(\mathcal {G}_\Gamma ^0)$
and
$s_vC^*(\Gamma )s_v \cong C^*( \mathcal {G}_\Gamma |_{Z(v)})$
. By the first paragraph,
${\mathcal {G}_\Gamma |_{Z(v)}^0 \cong Z(v)}$
. Hence,
$C^*( \mathcal {G}_\Gamma |_{Z(v)}) \cong C(Z(v))$
. So for each
$v \in \Gamma ^0$
, the ideal
$I_v$
generated by
$s_v$
is Rieffel–Morita equivalent to the abelian
$C^*$
-algebra
$C(Z(v))$
. Since
$C^*(\Gamma )$
is generated by the ideals
$I_v$
,
$C^*(\Gamma )$
is type I
$_0$
.
By definition of the hull-kernel topology, the ideals
$I_v$
yield a cover of
$\operatorname {Prim}(C^*(\Gamma ))$
by open sets
$\widehat {I}_v \cong \operatorname {Prim}(I_v)$
. Since each
$I_v$
is Rieffel–Morita equivalent to
$C(Z(v))$
, each
$\operatorname {Prim}(I_v) \cong Z(v)$
is a zero-dimensional compact open Hausdorff subspace of
$\operatorname {Prim}(C^*(\Gamma ))$
.
Proof of Theorem 4.1(i) and (ii).
Proposition 4.2 for
$\Gamma =\Sigma $
gives part (i). If
$\Lambda \to \Pi (\Lambda )$
is injective, then so is
$\Sigma \to \Pi (\Sigma )$
by Proposition 3.13. Since
$\Sigma $
is simply connected, Proposition 3.6 implies that
$\Sigma $
is singly connected; so Proposition 4.4 for
$\Gamma =\Sigma $
gives part (ii).
To prove Theorem 4.1(iii), we argue that the spectrum
$C^*(\Sigma )^{\wedge }$
is Hausdorff: then Theorem 4.1(ii) shows that
$C^*(\Sigma )$
is Rieffel–Morita equivalent to a continuous-trace
$C^*$
-algebra with totally disconnected spectrum, and the Dixmier–Douady theorem shows that
$C^*(\Sigma )$
is Rieffel–Morita equivalent to
$C_0(C^*(\Sigma )^{\wedge })$
. We argue that
$C_0(C^*(\Sigma )^{\wedge }) \cong \Sigma ^{\mathbb {N}\mathbf {1}}/\mathcal {G}_{\Sigma ^{\mathbb {N}\mathbf {1}}}$
and use the additional hypothesis that
$\Sigma ^{\mathbb {N}\mathbf {1}}$
is simply connected to prove Theorem 4.1(iii). We do not know whether this additional hypothesis is automatic; certainly, even for
$1$
-graphs, being singly connected does not guarantee that the associated
$C^*$
-algebra has Hausdorff spectrum.
Example 4.5. Let E be the directed graph (pictured below) such that:
-
•
$E^0 = \{u_n, v_n : n \in \mathbb {Z}\} \cup \{w_{n, i} : n \in \mathbb {Z} \text { and }i \ge 0\}$
; and -
•
$E^1 = \{e_n, f_n, g_n, h_n : n \in \mathbb {Z}\} \cup \{k_{n, i} : n \in \mathbb {Z}\text { and }i \ge 0\}$
,
and such that for
$n \in \mathbb {Z}$
and
$i \ge 0$
,
$$ \begin{align*} r(e_n) &= s(e_{n-1}) = r(g_n) = u_n,\quad r(f_n) = s(f_{n-1}) = r(h_n) = v_n,\\ s(g_n) &= s(h_n) = r(k_{n,0}) = w_{n,0}, \quad s(k_{n,i}) = r(k_{n, i+1}) = w_{n, i+1}. \end{align*} $$
This graph E is singly connected. Define
$x,y \in E^\infty $
by
$x = e_0 e_1 e_2 \cdots $
and
${y = f_0 f_1 f_2\cdots }$
. Then
$[x] \not = [y]$
in
$\mathcal {G}_E^{(0)}/\mathcal {G}_E$
. We claim that they cannot be separated by disjoint open sets. To see this, for
$n \in \mathbb {Z}$
, let
$z_n = k_{n,0} k_{n, 1} k_{n, 2} \cdots $
. We show that
$[z_n] \to [x]$
and
$z_n \to [y]$
as
$n \to \infty $
. By symmetry, we just have to show that
$[z_n] \to [x]$
. For this, just note that
$[z_n] = [e_0 e_1 \cdots e_{n-1} g_n z_n]$
and we have
$\lim _{n \to \infty } e_0 e_1 \cdots e_{n-1} g_n z_n = e_0 e_1 e_2 \cdots = x$
.
We have
$C^*(E) \cong C^*(\mathcal {G}_E)$
by [Reference Kumjian, Pask, Raeburn and Renault27, Proposition 4.1]. Since
$C^*(E)$
is type I
$_0$
, its spectrum is homeomorphic, by [Reference Clark11, Corollary 4.2], to the orbit space
$\mathcal {G}_E^{(0)}/\mathcal {G}_E$
of
$\mathcal {G}_E$
, which we just saw is not Hausdorff. Note that E is not simply connected (for example,
$e_0g_1h_1^{-1}f_0^{-1}h_0g_0^{-1} \in \Pi (E^*)^{u_0}_{u_0} \setminus \{u_0\}$
).
Example 4.5 suggests a Hausdorffness criterion (Lemma 4.7). As this criterion is not easy to check, in Theorem 4.8, we specialise to singly connected k-graphs and recast it in terms of the following relation on vertices, which permeates analyses of ideals of k-graph
$C^*$
-algebras [Reference Raeburn, Sims and Yeend36].
Notation 4.6. For
$\Gamma $
a k-graph, we define a relation
$\le $
on
$\Gamma ^0$
by
$v \le w$
if and only if
$v \Gamma w \not = \emptyset $
.
Lemma 4.7. Let
$\Gamma $
be a row-finite source-free k-graph and let
$\mathcal {G}_\Gamma $
be its k-graph groupoid. The orbit space
$\Gamma ^\infty /\mathcal {G}_\Gamma $
is Hausdorff if and only if for every pair of infinite paths
$x,y \in \Gamma ^\infty $
such that
$[x]\not =[y]$
, there exists
$N \in \mathbb {N}^k$
such that the vertices
$x(N)$
on x and
$y(N)$
on y have no common upper bound with respect to
$\le $
, in the sense that
$s(\mu ) \not = s(\nu )$
for all
$\mu \in x(N)\Gamma $
and
$\nu \in y(N)\Gamma $
.
Proof. We have
$[x] = [y]$
if and only if
$\sigma ^m(x) = \sigma ^n(y)$
for some
$m,n$
. So it suffices to fix
$x,y$
such that
$\sigma ^m(x) \not = \sigma ^n(y)$
for all
$m,n$
, and show that
$[x]$
and
$[y]$
can be separated if and only if there exists N as in the statement. Suppose that there is no such N. For each
$N \in \mathbb {N}^k$
, choose
$\mu _N \in x(N)\Gamma $
and
$\nu _N \in y(N)\Gamma $
with
$s(\mu _N) = s(\nu _N)$
, and
${z_N \in s(\mu _N)\Gamma ^\infty }$
. Then
$x(0,N)\mu _Nz_N \to x$
and
$y(0, N)\nu _N z_N \to y$
. Since each
$[x(0,N)\mu _Nz_N] = [z_N] = [y(0,N)\nu _Nz_N]$
, this forces
$[z_N] \to [x]$
and
$[z_N] \to [y]$
. Now suppose that there exists N as in the statement. Then
$q(Z(x(0, N)))$
and
$q(Z(y(0,N)))$
are disjoint open neighbourhoods of
$[x]$
and
$[y]$
in
$\Gamma ^\infty /\mathcal {G}_\Gamma $
.
Recall that a filter for a partially ordered set
$(X, \preceq )$
is a nonempty subset
$F \subseteq X$
such that:
-
(a) for all
$u,v \in F$
, there exists
$w \in F$
such that
$u,v \preceq w$
; -
(b) if
$v \in F$
and
$u \preceq v$
, then
$u \in F$
.
A filter F for
$\preceq $
is called an ultrafilter if:
-
(c) F is not properly contained in any other filter
$F'$
for
$(X, \preceq )$
.
If
$\Gamma $
is singly connected, then
$\le $
is a partial order on
$\Gamma ^0$
. We show that elements of
$\Gamma ^\infty /\mathcal {G}_\Gamma $
correspond with ultrafilters for
$(\Gamma ^0, \le )$
and use this to characterise Hausdorffness of
$\Gamma ^\infty /\mathcal {G}_\Gamma $
.
Theorem 4.8. Let
$\Gamma $
be a singly connected row-finite source-free k-graph. Then the ultrafilters for
$(\Gamma ^0, \le )$
are exactly the sets
$[x]^0 := \{r(y) : y \in [x]\}$
indexed by elements
$x \in \Gamma ^\infty $
. Moreover,
$\Gamma ^\infty /\mathcal {G}_\Gamma $
is Hausdorff if and only if for every pair
$U, V$
of distinct ultrafilters of
$(\Gamma ^0, \le )$
, there is a pair
$u \in U$
and
$v \in V$
with no common upper bound with respect to
$\le $
.
Proof. For the first statement, first fix
$x \in \Gamma ^\infty $
. If
$v_1, v_2 \in [x]^0$
, then
$v_1 = r(\alpha \sigma ^m(x))$
and
$v_2 = r(\beta \sigma ^n(x))$
for some
$\alpha ,\beta , m,n$
, and then
$w = r(\sigma ^{m+n}(x)) \in [x]^0$
satisfies
${v_i \Gamma w \not = \emptyset }$
by definition; so
$[x]^0$
satisfies part (a). If
$w \in [x]^0$
and
$v \in \Gamma ^0$
satisfy
$v \le w$
, say
${\alpha \in v \Gamma w}$
, then since
$w \in [x]^0$
, we have
$w = r(\beta \sigma ^{n}(x))$
for some
$\beta , n$
and so
$v = r(\alpha \beta \sigma ^n(x)) \in [x]^0$
; so
$[x]^0$
satisfies part (b). Suppose that F is a filter for
$(\Gamma ^0, \le )$
containing
$[x]^0$
. Fix
$v \in F$
; we must show that
$v \in [x]^0$
. Since
$v, x(0) \in F$
, there exists
$w \in F$
with
$v \le w$
and
$x(0) \le w$
, and by part (b), if
$w \in [x]^0$
, then
$v \in [x]^0$
; so we just have to show that
$w \in [x]^0$
. Fix
$\alpha \in x(0)\Gamma w$
. Then
$r(\sigma ^{d(\alpha )}(x)) \in [x]^0 \subseteq F$
. So there exists
$w' \in F$
such that
$r(\sigma ^{d(\alpha )}(x)), w \le w'$
; say
$\rho \in r(\sigma ^{d(\alpha )}(x)) \Gamma w'$
and
$\tau \in w \Gamma w'$
. So
$\alpha \tau $
and
$x(0, d(\alpha ))\rho $
both belong to
$x(0) \Gamma w'$
. Since
$\Gamma $
is singly connected, this forces
$\alpha \tau = x(0, d(\alpha ))\rho $
, so the factorisation property forces
$x(0, d(\alpha )) = \alpha $
; hence,
$w = s(\alpha ) = s(x(0,d(\alpha ))) \in [x]^0$
.
Now, fix an ultrafilter F for
$(\Gamma ^0, \le )$
. Enumerate
$F = (v_1, v_2, \ldots )$
, put
$w_1 = v_1$
and inductively use part (a) to choose
$w_{i+1} \in F$
such that
$v_{i+1}, w_i \le w_{i+1}$
. So
$(w_i)_i$
is an increasing sequence such that every
$v \in F$
satisfies
$v \le w_i$
for some i. For each i, use that
$w_i \le w_{i+1}$
to fix
$\alpha _i \in w_i \Gamma w_{i+1}$
, let
$\rho _i := \alpha _1\cdots \alpha _i$
and choose
$y_i \in Z(\rho _i) \subseteq \Gamma ^\infty $
. Since
$(y_i)_i$
belongs to the compact set
$Z(v_1)$
, it has a convergent subsequence
$y_{i_l} \to y \in Z(v_1)$
. We claim that
$F = [y]^0$
. By part (c), it suffices to show that
$F \subseteq [y]^0$
. So fix
$v \in F$
. Then
$v = v_m \le w_m$
for some
$m \in \mathbb {N}$
. Choose l so that
$i_l \ge m$
. For
$l' \ge l$
, we have
${y_{i_{l'}} \in Z(\rho _{i_{l'}}) \subseteq Z(\rho _{i_l})}$
. Hence,
$y \in Z(\rho _{i_l})$
. So
$w_{i_l} = s(\rho _{i_l}) = r(\sigma ^{d(\rho _{i_l})}(y)) \in [y]^0$
. By choice of
$(w_j)_j$
and l, we have
$v \le w_m \le w_{i_l}$
. So part (b) gives
$v \in [y]^0$
. This proves the first statement.
For the second statement, by Lemma 4.7, it suffices to show that for all
$x,y \in \Gamma ^\infty $
with
$[x] \not = [y]$
, there exists N such that
$s(x(N)\Gamma ) \cap s(y(N)\Gamma ) = \emptyset $
if and only if, for all pairs
$U \not = V$
of ultrafilters of
$(\Gamma ^0, \le )$
, there exist
$u \in U$
and
$v \in V$
with no common upper bound with respect to
$\le $
.
First, suppose that for every pair
$x,y \in \Gamma ^\infty $
, there exists N such that
$s(\rho ) \not = s(\tau )$
for every
$\rho \in x(N)\Gamma $
and
$\tau \in y(N)\Gamma $
. Fix ultrafilters
$U \not = V$
, and fix
$x,y \in \Gamma ^\infty $
with
$U = [x]^0$
and
$V = [y]^0$
. Fix N such that
$s(\rho ) \not = s(\tau )$
for every
$\rho \in x(N)\Gamma $
and
$\tau \in y(N)\Gamma $
. Then
$u = x(N) \in U$
and
$v = y(N) \in V$
have no common upper bound. Now, suppose that for every pair
$U \not = V$
of ultrafilters, there exist
$u \in U$
and
$v \in V$
with no common upper bound. Fix
$x,y \in \Gamma ^\infty $
with
$U = [x]^0$
and
$V = [y]^0$
. Fix
$u \in U$
and
$v \in V$
with no common upper bound. Fix
$x' \in [x]$
and
$y' \in [y]$
with
$r(x') = u$
and
$r(y') = v$
, and
$m, m'$
and
$n, n'$
such that
$\sigma ^m(x) = \sigma ^{m'}(x')$
and
$\sigma ^n(y) = \sigma ^{n'}(y')$
. Fix
$N \ge m,n$
. Then
$u \Gamma x(N) \not = \emptyset $
and
$v \Gamma y(N) \not = \emptyset $
. Since
$u,v$
have no common upper bound, nor do
$x(N)$
and
$y(N)$
; so
${s(\rho ) \not = s(\tau )}$
for all
$\rho \in x(N)\Gamma $
and
$\tau \in y(N)\Gamma $
.
Remark 4.9. Lemma 4.7 gels with [Reference Kumjian and Pask25, Proposition 4.3]: if E is a simply connected row-finite source-free directed graph, then
$E^\infty /\mathcal {G}_E$
is Hausdorff. We prove the contrapositive. Suppose that
$E^\infty /\mathcal {G}_E$
is not Hausdorff. Since E is a
$1$
-graph,
$i : E^* \to \Pi (E^*)$
is injective. Corollary 4.7 gives
$x,y \in E^\infty $
such that
$\sigma ^m(x) \not = \sigma ^n(y)$
for all
$m,n$
, and, for all
$N \ge 0$
,
$\rho _N \in x(N)E^*$
and
$\tau _N \in y(N) E^*$
such that
$s(\rho _N) = s(\tau _N) =: w_N$
. We first claim that there exists
$N_0$
such that
$x(n) \not = y(m)$
for all
$m,n \ge N_0$
. To see this, suppose that there are increasing sequences
$(n_i), (m_i)$
such that
$x(n_i) = y(m_i)$
for all i. Since E is singly connected,
$x(n_i, n_{i+1}) = y(m_i, m_{i+1})$
for all i; so
$\sigma ^{n_0}(x) = \sigma ^{n_0}(y)$
, which is a contradiction. So by replacing
$x, y$
with
$\sigma ^{N_0}(x)$
and
$\sigma ^{N_0}(y)$
, we may assume that
$x(m) \not = y(n)$
for all
$m,n$
.
Hence, each
$w_N$
is on exactly one of
$x, y$
; without loss of generality,
$w_0$
is not on x. Let
$\alpha = x(0, |\rho _0|)$
and
$\beta = y(0, |\tau _0|)$
. Then
$\rho _0^{-1}\alpha \rho _{|\rho _0|}\tau _{|\tau _0|}^{-1}\beta ^{-1}\tau _0 \in (\Pi (E))^{w_0}_{w_0}$
. We show that
$\rho _0^{-1}\alpha \rho _{|\rho _0|}\tau _{|\tau _0|}^{-1}\beta ^{-1}\tau _0 \not = w_0$
. Since
$r(\rho _{|\rho _0|}) = x(|\rho _0|) \not = y(|\tau _0|) = r(\tau _{|\tau _0|})$
in reduced form [Reference Higgins20, Proposition 4.9],
$\rho _{|\rho _0|} \tau _{|\tau _0|}^{-1} = e \gamma \lambda ^{-1}$
, where
$e \in E^1$
is the first edge of
$\rho _{|\rho _0|}$
. Similarly, since
$w_0 = s(\tau _0) \not = x(|\rho _0|) = s(\alpha )$
in reduced form,
$\rho _0^{-1}\alpha = \zeta ^{-1}\eta f$
, where f is the last edge of
$\alpha $
and
$\zeta ,\eta \in E^*$
. So in reduced form,
$\rho _0^{-1}\alpha \rho _{|\rho _0|}\tau _{|\tau _0|}^{-1} = \zeta ^{-1} \eta f e \gamma \lambda ^{-1}$
. In particular, the word
$fe$
appears in the reduced form of
$\rho _0^{-1}\alpha \rho _{|\rho _0|}\tau _{|\tau _0|}^{-1}\beta ^{-1}\tau _0$
, so this is a nontrivial element of
$(\Pi (E))^{w_0}_{w_0}$
. Hence, E is not simply connected.
Remark 4.10. The argument of the preceding remark does not go through for k-graphs because there is no canonical reduced form for elements of the fundamental groupoid of a k-graph.
For
$1$
-graphs
$E^*$
, we can use Remark 4.9 to check Hausdorffness of
$E^\infty /\mathcal {G}_E \cong C^*(E^*)^{\wedge }$
. So it helps to relate Hausdorffness of the orbit space of a k-graph to that of a natural sub-
$1$
-graph
Proposition 4.11. Let
$\Gamma $
be a row-finite source-free k-graph. Suppose that
$\Gamma $
is simply connected. Then
$\Gamma ^\infty /\mathcal {G}_\Gamma $
is Hausdorff if and only if
$(\Gamma ^{\mathbb {N}\mathbf {1}})^\infty /\mathcal {G}_{\Gamma ^{\mathbb {N}\mathbf {1}}}$
is Hausdorff.
To prove this, we show that
$\Gamma ^\infty /\mathcal {G}_\Gamma $
is homeomorphic to a clopen subset of
$(\Gamma ^{\mathbb {N}\mathbf {1}})^\infty /\mathcal {G}_{\Gamma ^{\mathbb {N}\mathbf {1}}}$
.
Lemma 4.12. Let
$\Gamma $
be a row-finite source-free k-graph. Suppose that
$\Gamma $
is simply connected. Let
$f : \Gamma ^0 \to \mathbb {Z}^k$
be a function such that
$d(\lambda ) = f(s(\lambda )) - f(r(\lambda ))$
for all
$\lambda \in \Gamma $
as in Proposition 4.2. Let E be the directed graph such that
$E^0 = f^{-1}(\mathbb {Z}\mathbf {1})$
and
$E^1 = E^0 \Gamma ^{\mathbf {1}}$
. Let
$j : E^\infty \to \Gamma ^\infty $
be the map such that
$j(x)$
is the unique infinite path such that
$j(x)(0, n\cdot \mathbf {1}) = x_1 x_2\cdots x_n$
for all
$n \in \mathbb {N}$
(see [Reference Kumjian and Pask26, Remark 2.2]). Then j descends to a homeomorphism
$\widetilde {j} : E^\infty /\mathcal {G}_E \to \Gamma ^\infty /\mathcal {G}_\Gamma $
.
Proof. The map j restricts to a homeomorphism
$vE^\infty \to v\Gamma ^\infty $
for each
$v \in E^0$
, so is continuous.
We claim that if
$x,y \in E^\infty $
, then
$j(x) \sim _{\mathcal {G}_\Gamma } j(y)$
if and only if
$x \sim _{\mathcal {G}_E} y$
. To see this, fix
$x,y \in E^\infty $
. Then
$j(x) \sim _{\mathcal {G}_\Gamma } j(y)$
if and only if there exist
$m,n \in \mathbb {N}^k$
such that
$\sigma ^m(j(x)) = \sigma ^n(j(y))$
. Since
$f(r(\sigma ^m(j(x)))) = f(r(x)) + m$
for all
$m \in \mathbb {N}^k$
and similarly for y, and since
$f(r(x)), f(r(y)) \in \mathbb {Z}\mathbf {1}$
, we deduce that
$j(x) \sim _{\mathcal {G}_\Gamma } j(y)$
if and only if there exist
$m,n \in \mathbb {N}^k$
such that
$\sigma ^m(j(x)) = \sigma ^n(j(y))$
and
$m - n \in \mathbb {Z}\mathbf {1}$
. Since
$m - n \in \mathbb {Z}\mathbf {1}$
if and only if there exists
$p \in \mathbb {N}^k$
such that
$m+p, n+p \in \mathbb {N}\mathbf {1}$
, we deduce that
$j(x) \sim _{\mathcal {G}_\Gamma } j(y)$
if and only if
$\sigma ^{a\mathbf {1}}(j(x)) = \sigma ^{b\mathbf {1}}(j(y))$
for some
$a,b \in \mathbb {N}$
; that is, if and only if
$\sigma ^a(x) = \sigma ^b(y)$
for some
$a,b \in \mathbb {N}$
. Hence,
$j(x) \sim _{\mathcal {G}_\Gamma } j(y)$
if and only if
$x \sim _{\mathcal {G}_E} y$
. It follows that j descends to a continuous function
$\widetilde {j} : E^\infty /\mathcal {G}_E \to \Gamma ^\infty /\mathcal {G}_\Gamma $
.
Fix
$p : \Gamma ^0 \to \mathbb {N}^k$
satisfying
$f(v) + p(v) \in \mathbb {Z}\mathbf {1}$
for all v. For
$x \in \Gamma ^\infty $
and
$j \in \mathbb {N}$
, let
$\tilde {x}_j := \sigma ^{p(r(x))}((j-1)\mathbf {1}, j\mathbf {1}) \in E^1$
and define
$h(x) := \tilde {x}_1 \tilde {x}_2 \cdots \tilde {x}_n \cdots \in E^\infty $
. As
$x \mapsto p(r(x))$
is locally constant, h is continuous. Since
$\sigma ^{p(x)}(x) \sim _{\mathcal {G}_\Gamma } x$
for all x, the claim above shows that
$x \sim _{\mathcal {G}_\Gamma } y$
if and only if
$h(x) \sim _{\mathcal {G}_E} h(y)$
, so h descends to a continuous function
$\tilde {h} : \Gamma ^\infty /\mathcal {G}_\Gamma \to E^\infty /\mathcal {G}_E$
. It is routine to check that
$\tilde {h}$
and
$\tilde {j}$
are mutually inverse:
$h \circ j = \operatorname {id}_{E^\infty }$
and
$[j \circ h(x)] = [\sigma ^{p(x)}(x)] = [x]$
for all
$x \in \Gamma ^\infty $
. In particular, j descends to a homeomorphism as claimed.
Proof of Proposition 4.11.
Resume the notation of Lemma 4.12. It suffices for us to show that
$(\Gamma ^{\mathbb {N}\mathbf {1}})^\infty /\mathcal {G}_{\Gamma ^{\mathbb {N}\mathbf {1}}}$
is Hausdorff if and only if
$E^\infty /\mathcal {G}_E$
is Hausdorff.
For
$p \in \mathbb {Z}^k$
, let
$V_p := f^{-1}(p + \mathbb {Z}\mathbf {1}) \subseteq \Gamma ^0$
(so
$V_0$
is V in Lemma 4.12). If
$p-q \not \in \mathbb {Z}\mathbf {1}$
, then
$V_p\Gamma ^{\mathbb {N}\mathbf {1}} V_q = \emptyset $
. So if
$x \in V_p \Gamma ^{\mathbb {N}\mathbf {1}}$
and
$y \in V_q \Gamma ^{\mathbb {N}\mathbf {1}}$
, then
$\sigma ^{a\mathbf {1}}(x) \not = \sigma ^{b\mathbf {1}}(y)$
for all
$a,b \in \mathbb {N}$
and hence,
$[x]_{\mathcal {G}_{\Gamma ^{\mathbb {N}\mathbf {1}}}} \not = [y]_{\mathcal {G}_{\Gamma ^{\mathbb {N}\mathbf {1}}}}$
. Hence, the sets
$\big \{V_p (\Gamma ^{\mathbb {N}\mathbf {1}})^{\infty } : p \in \mathbb {Z}^{k-1} \times \{0\}\big \}$
have mutually disjoint open images in
$(\Gamma ^{\mathbb {N}\mathbf {1}})^\infty /\mathcal {G}_{\Gamma ^{\mathbb {N}\mathbf {1}}}$
. So it suffices to show that each of these images is Hausdorff.
Let
$q : (\Gamma ^{\mathbb {N}\mathbf {1}})^\infty \to (\Gamma ^{\mathbb {N}\mathbf {1}})^\infty /\mathcal {G}_{\Gamma ^{\mathbb {N}\mathbf {1}}}$
be the quotient map. By assumption,
$E^\infty /\mathcal {G}_E = q(V_0 (\Gamma ^{\mathbb {N}\mathbf {1}})^{\infty })$
is Hausdorff, so it suffices to fix
$p \in \mathbb {Z}^{k-1} \setminus \{0\}$
and show that
$(q(V_p\Gamma ^{\mathbb {N}\mathbf {1}})^\infty ) \cong (q(V_0\Gamma ^{\mathbb {N}\mathbf {1}})^\infty )$
.
Since
$V_p = V_{p + a\mathbf {1}}$
for all
$a \in \mathbb {N}$
, we may assume that
$p \ge 0$
. Fix
$n \in \mathbb {N}^k$
such that
$p + n \in \mathbb {Z}\mathbf {1}$
. Then
$\sigma ^p : V_0 \Gamma ^\infty \to V_p\Gamma ^\infty $
and
$\sigma ^n : V_p \Gamma ^\infty \to V_{p+n}\Gamma ^\infty = V_0\Gamma ^\infty $
are continuous. Using [Reference Kumjian and Pask26, Remark 2.2], we can identify
$V_p \Gamma ^\infty $
with
$V_p(\Gamma ^{\mathbb {N}\mathbf {1}})^\infty $
and
$V_0 \Gamma ^\infty $
with
$V_0 (\Gamma ^{\mathbb {N}\mathbf {1}})^\infty $
, and these identifications are compatible with the shift maps.
If
$x \sim _{\mathcal {G}_{\Gamma ^{\mathbb {N}\mathbf {1}}}} y$
, then
$\sigma ^p(x) \sim _{\mathcal {G}_{\Gamma ^{\mathbb {N}\mathbf {1}}}} \sigma ^p(y)$
and similarly for n, so
$\sigma ^p$
and
$\sigma ^n$
descend to continuous maps
$\tilde \sigma ^p : q(V_0\Gamma ^{\mathbb {N}\mathbf {1}})^\infty \to q(V_p\Gamma ^{\mathbb {N}\mathbf {1}})^\infty $
and
$\tilde \sigma ^n : q(V_p\Gamma ^{\mathbb {N}\mathbf {1}})^\infty \to q(V_0\Gamma ^{\mathbb {N}\mathbf {1}})^\infty $
. Since
$x \sim _{\mathcal {G}_{\Gamma ^{\mathbb {N}\mathbf {1}}}} \sigma ^{p+n}(x) = \sigma ^p(\sigma ^n(x))$
, we see that
$\tilde \sigma ^p \circ \tilde \sigma ^n$
is the identity map on
$q(V_p\Gamma ^{\mathbb {N}\mathbf {1}})^\infty $
and, similarly,
$\tilde \sigma ^p \circ \tilde \sigma ^n$
is the identity map on
$q(V_0\Gamma ^{\mathbb {N}\mathbf {1}})^\infty $
. So
$\tilde \sigma ^p$
and
$\tilde \sigma ^n$
are mutually inverse and hence homeomorphisms.
Corollary 4.13. Let
$\Gamma $
be a row-finite source-free k-graph. Suppose that both
$\Gamma $
and the sub-
$1$
-graph
$\Gamma ^{\mathbb {N}\mathbf {1}}$
are simply connected. Then
$\Gamma ^\infty /\mathcal {G}_\Gamma $
is Hausdorff.
Proof. Proposition 4.2 gives
$f: \Gamma ^0 \to \mathbb {Z}^k$
such that
$d(\lambda ) = f(s(\lambda )) - f(r(\lambda ))$
for all
$\lambda \in \Gamma $
. Let E be the directed graph such that
$E^0 = f^{-1}(\mathbb {Z}\mathbf {1})$
and
$E^1 = E^0 \Gamma ^{\mathbf {1}}$
. Lemma 4.12 gives
$\Gamma ^\infty /\mathcal {G}_\Gamma \cong E^\infty /\mathcal {G}_E$
. Since
$E^*$
is a sub-1-graph of the simply connected graph
$\Gamma ^{\mathbb {N} \mathbf {1}}$
, it is simply connected. Hence,
$E^\infty /\mathcal {G}_E$
is Hausdorff by [Reference Kumjian and Pask25, Lemma 4.2] (see Remark 4.9) and thus,
$\Gamma ^\infty /\mathcal {G}_\Gamma $
is Hausdorff.
Example 4.14. Surprisingly, simple connectedness of
$\Gamma $
and of
$\Gamma ^{\mathbb {N}\mathbf {1}}$
are independent conditions. For the monoidal
$2$
-graph
$\Lambda $
of [Reference Pask, Raeburn and Quigg32, Example 7.1] (Example 3.1), we have an isomorphism
$\Pi (\Lambda ) \cong \mathbb {Z}^2$
that intertwines
$i : \Lambda \to \Pi (\Lambda )$
with
${d : \Lambda \to \mathbb {N}^2 \subseteq \mathbb {Z}^2}$
. So
$\Gamma := \mathbb {Z}^2 \times _d \Lambda \cong \pi ( \Lambda ) \times _i \Lambda $
is simply connected. However,
$\Gamma ^{\mathbb {N}\mathbf {1}}$
is the graph with vertices
$\{v_m : m \in \mathbb {Z}^2\}$
and six parallel edges from
$v_{m + \mathbf {1}}$
to
$v_m$
for each
${m \in \mathbb {Z}^2}$
, so is not simply connected. In the other direction, let
$\Delta _1$
be the
$1$
-graph with vertices
$\mathbb {Z}$
and edges
$e_n$
with
$s(e_n) = n+1$
and
$r(e_n) = n$
, and define
$l : \mathbb {N}^2 \to \mathbb {N}$
by
$l(m,n) = m+n$
. Then the
$2$
-graph
$\Gamma := l^*(\Omega _1)$
has fundamental group
$\mathbb {Z}$
generated by
$(e_0, (1,0))(e_0, (0,1))^{-1}$
, so is not simply connected, but
$\Gamma ^{\mathbb {N}\mathbf {1}}$
is a disjoint union of copies of
$\Omega _1$
, so is simply connected.
Remark 4.15. In the context of Corollary 4.13, simple connectedness of
$\Gamma ^{\mathbb {N}\mathbf {1}}$
is equivalent to that of
$E^*$
as in Lemma 4.12. Also, as in the proof of Proposition 4.11, the orbit space
$(\Gamma ^{\mathbb {N}\mathbf {1}})^\infty / \mathcal {G}_{\Gamma ^{\mathbb {N}\mathbf {1}}}$
is a topological disjoint union of copies of
$E^\infty /\mathcal {G}_E$
indexed
$\mathbb {Z}^k/\mathbb {Z}\mathbf {1}$
.
Proof of Theorem 4.1(iii).
As in the proof of part (ii), since
$\Lambda \to \Pi (\Lambda )$
is injective,
$\Sigma $
is singly connected, and
$C^*(\Sigma )$
is type I
$_0$
. The proof of Proposition 4.4 shows that
$\mathcal {G}_\Sigma $
has trivial isotropy. Hence, the spectrum of
$C^*(\Sigma )$
is homeomorphic to the orbit space
$\Sigma ^\infty /\mathcal {G}_\Sigma $
[Reference Clark11, Corollary 4.2]. Now, since
$\Sigma ^{\mathbb {N}\mathbf {1}}$
is simply connected, Corollary 4.13 implies that
$\Sigma ^\infty /\mathcal {G}_\Sigma $
is Hausdorff. So
$C^*(\Sigma )$
is a continuous-trace
$C^*$
-algebra. Since
$X := \Sigma ^\infty /\mathcal {G}_\Sigma $
is zero-dimensional,
$\check {H}_3(X, \mathbb {Z}) = \{0\}$
, and hence the Dixmier–Douady invariant
$\delta (C^*(\Sigma )) \in \check {H}_3(X, \mathbb {Z})$
is trivial. So by the Dixmier–Douady theorem [Reference Raeburn and Williams37, Corollary 5.58],
$C^*(\Sigma )$
is Rieffel–Morita equivalent to
$C(\Sigma ^\infty /\mathcal {G}_\Sigma )$
.
Remark 4.16. A related realisation of
$C^*$
-algebras of k-graphs (and more general categories) as crossed products of abelian algebras by partial actions of their fundamental groups appears in [Reference Brix, Bruce and Dor On7, Theorem 4.17]. Interestingly, embeddability also crops up there for different reasons.
Remark 4.17. It seems hard to nail down the relationships between the key hypotheses in this section: simple connectedness of
$\Gamma $
and of
$\Gamma ^{\mathbb {N}\mathbf {1}}$
, and embedding of
$\Gamma $
in
$\Pi (\Gamma )$
.
For example, the following two assertions both seem reasonable: that if
$\Gamma $
is simply connected, then the
$\mathbf {1}$
-dual
$\mathbf {1}\Gamma $
obtained from Proposition 2.7 for
$f : n \mapsto n + \mathbf {1}$
is also simply connected; and that
$\mathbf {1}\Gamma $
always embeds in
$\Pi (\mathbf {1}\Gamma )$
(after all,
$\mathbf {1}\Gamma \owns \lambda \mapsto (r(\lambda ), d(\lambda ), s(\lambda ))$
is injective on
$\bigcup _{n \le \mathbf {1}} \Gamma ^n$
, and this map descends to
$\Pi (\Gamma )$
, so the skeleton and factorisation rules are preserved in
$\Pi (\Gamma )$
). However, at most one of these assertions is true in general: consider the skew-product
$\Gamma := \mathbb {Z}^2 \times _d \Lambda $
of Example 3.1; we show that if
$\mathbf {1}\Gamma $
is simply connected, then it does not embed in
$\Pi (\mathbf {1}\Gamma )$
.
Since
$\mathbf {1}\Gamma $
is canonically isomorphic to the skew-product
$\mathbb {Z}^2 \times _d (\mathbf {1}\Lambda )$
, if
$\mathbf {1}\Gamma $
is simply connected, then
$\mathbb {Z}^2 \times _d (\mathbf {1}\Lambda )$
is simply connected, forcing
$\pi _1(\Lambda , v) \cong \mathbb {Z}^2$
. However, inspection of the skeleton of
$\mathbf {1}\Lambda $
shows that
$eeec$
and
$eedec$
are distinct blue cycles based at the vertex
$ec \in \mathbf {1}\Lambda $
, so generate a sub-semigroup of
$\mathbf {1}\Lambda $
isomorphic to
$\mathbb {F}^+_2$
, which cannot embed in
$\mathbb {Z}^2$
.
Remark 4.18. The preceding remark is exemplary of a number of seemingly elementary questions that we have been unable to resolve.
-
(i) If
$\Gamma $
is simply connected and embeds in
$\Pi (\Gamma )$
, is
$\Gamma ^\infty /\mathcal {G}_\Gamma $
Hausdorff? -
(ii) If both
$\Gamma $
and
$\Gamma ^{\mathbb {N}\mathbf {1}}$
are simply connected, does
$\Gamma $
necessarily embed in
$\Pi (\Gamma )$
? -
(iii) Which, if either, of the two assertions mentioned in Remark 4.17 is correct?
-
(iv) Does
$\mathbf {1}\Gamma $
always embed in
$\Pi (\mathbf {1}\Gamma )$
?
5
$\tilde{A_2}$
-groups
In this section, we construct coverings
$\Sigma _{\mathcal {T}} \to \Lambda _{\mathcal {T}}$
of
$2$
-graphs corresponding to
$\tilde {A_2}$
-groups
$\Gamma _{\mathcal {T}}$
. These groups arise from free, vertex-transitive actions on buildings. We show that
$\Sigma _{\mathcal {T}}$
and
$\Lambda _{\mathcal {T}}$
both embed in their fundamental groupoids, and that
$\Sigma _{\mathcal {T}}$
is always singly connected so that its
$C^*$
-algebra is of Type I
$_0$
.
The
$\tilde {A_2}$
-groups are built from finite projective planes. A finite projective plane
$(P,L)$
of order q consists of finite sets P of points and L of lines with
$|P| = |L| = q^2 + q + 1$
, and a relation
$\in $
from P to L—if
$p \in l$
, we say p lies on l and that l contains p—such that any two points lie on exactly one common line, any two lines contain exactly one common point and there exist four distinct points of which no single line contains more than two. Each line necessarily contains exactly q points and each point necessarily lies on exactly q lines.
We begin with a brief introduction of the groups we wish to study and by collecting some structural results that we need for our construction.
5.1
$\tilde{A_2}$
-group basics
Following [Reference Cartwright, Mantero, Steger and Zappa8, Section 2], given a finite projective plane
$(P,L)$
and a bijection
$\lambda : P \to L$
, we define a triella compatible with
$\lambda $
to be a set
$\mathcal {T} \subset P \times P \times P$
such that:
-
(T1) given
$x,y \in P$
, there exists
$z \in P$
such that
$(x,y,z) \in \mathcal {T}$
if and only if
$y \in \lambda (x)$
; -
(T2)
$(x,y,z) \in \mathcal {T} \Rightarrow (y,z,x) \in \mathcal {T}$
; -
(T3) for any
$x,y \in P$
, there is at most one
$z \in P$
such that
$(x,y,z) \in \mathcal {T}$
.
Definition 5.1. Given a finite projective plane
$(P,L)$
, a bijection
$\lambda : P \to L$
and a triella
$\mathcal {T}$
compatible with
$\lambda $
as above, we define the associated
$\tilde {A_2}$
-group by
$$ \begin{align*} \Gamma = \Gamma_{\mathcal{T}} := \langle a_x, x \in P \mid a_x a_y a_z = 1 \text{ for each } (x,y,z) \in \mathcal{T} \rangle. \end{align*} $$
Remarks 5.2.
-
(i) The associated
$\tilde {A_2}$
-building is an oriented simplicial
$2$
-complex constructed from the Cayley graph of
$\Gamma _{\mathcal {T}}$
: the vertices or 0-simplices are identified with
$\Gamma _{\mathcal {T}}$
, the
$1$
-simplices are identified with pairs
$(w, wa_x)$
where
$w \in \Gamma _{\mathcal {T}}$
and
$x \in P$
. The
$2$
-simplices are identified with triples
$(w, wa_x, wa_xa_y)$
where
$w \in \Gamma _{\mathcal {T}}$
,
$x \in P$
and
$y \in \lambda (x)$
. The free and transitive action of
$\Gamma _{\mathcal {T}}$
on 0-simplices by left multiplication extends to a free action on the building. -
(ii) In [Reference Konter and Vdovina24, Reference Vdovina45], Vdovina et al. start with similar data to produce an object they call a polyhedron satisfying rules that have the flavour of a triella. We discovered this point of view late in our investigation and plan to look into it more deeply in future work.
Example 5.3. Many examples are considered in [Reference Cartwright, Mantero, Steger and Zappa9]. The following illustrative example with
$q = 2$
was first described in [Reference Cartwright, Mantero, Steger and Zappa9, Section 4]:
$$ \begin{align*} \Gamma_{A.1} = \langle a_0 , \ldots , a_6: a_{[i]_7} a_{[i+1]_7} a_{[i+3]_7} =1 \rangle \quad\text{where }[i]_7 = (i \text{ mod }7). \end{align*} $$
We describe elements of
$\Gamma _{\mathcal {T}}$
as products of generators and their inverses. The following standard terminology for finitely generated groups helps us discuss such expressions.
Definition 5.4. Let
$\Gamma _{\mathcal {T}}$
be an
$\tilde {A_2}$
-group with generators
$\{ a_x : x \in P\}$
. By a word in
$\Gamma _{\mathcal {T}}$
, we mean a string of the form
$g_1 g_2 \cdots g_k$
such that each
$g_i \in \{a_x, a_x^{-1} : x \in P\}$
. The word
$g_1 \cdots g_k$
represents the element
$w \in \Gamma _{\mathcal {T}}$
if the product
$\prod ^k_{i=1} g_i$
in
$\Gamma _{\mathcal {T}}$
is equal to w. We typically indicate the group law by juxtaposition, so we write
$w = g_1 \cdots g_k$
when the word
$g_1 \cdots g_k$
represents w. Context dictates whether a string
$g_1 \cdots g_k$
is being regarded as a word or as a product.
It is helpful to express elements of
$\Gamma _{\mathcal {T}}$
in a standard form.
Proposition 5.5. Let
$\Gamma _{\mathcal {T}}$
be an
$\tilde {A_2}$
-group with generators
$\{ a_x : x \in P \}$
. Let
$w \in \Gamma _{\mathcal {T}}$
. Then there are unique integers
$m, n \ge 0$
and unique elements
$x_1 , \ldots , x_m, y_1 , \ldots , y_n \in P$
such that
$$ \begin{align} w = a_{x_1} \cdots a_{x_{m}} a_{y_1}^{-1} \cdots a_{y_{n}}^{-1}, \text{ and } \end{align} $$
$$ \begin{align*} &\text{(a) } x_{i+1} \not\in \lambda (x_{i} ) \text{ for }1 \le i < m;\quad \text{(b) } y_{j} \not\in \lambda ( y_{j+1} ) \text{ for }1 \le j < n;\text{ and}\quad\\ &\text{(c) } x_m \neq y_1 \text{ if }m,n \geq 1. \end{align*} $$
For the same
$m,n$
, there are also unique
$t_1 , \ldots , t_n, s_1 , \ldots , s_m \in P$
such that
$$ \begin{align} w = a_{t_1}^{-1} \cdots a_{t_{n}}^{-1} a_{s_1} \cdots a_{s_{m}}, \text{ and } \end{align} $$
$$ \begin{align*} &(\text{a})^\prime s_{i+1} \not\in \lambda (s_{i}) \text{ for }1 \le i < m;\quad (\text{b})^\prime t_{j} \not\in \lambda ( t_{j+1} ) \text{ for }1 \le j < n;\text{ and}\quad\\ &(\text{c})^{\prime} t_{\ell} \neq s_1 \text{ if }m, n \geq 1. \end{align*} $$
We call the expressions above the right normal form and left normal form of w, respectively. Both have minimal length amongst words in the generators and their inverses that represent w. Moreover, every minimal-length word in the generators and their inverses that represents w contains m generators and n generator-inverses.
Proof. See [Reference Cartwright, Mantero, Steger and Zappa8, Proposition 3.2] and [Reference Cartwright and Młotkowski10, Lemma 6.2].
Corollary 5.6. Let
$\Gamma _{\mathcal {T}}$
be an
$\tilde {A_2}$
-group with generators
$\{ a_x : x \in P\}$
. For all
$x , y \in P$
such that
$x \neq y$
, there exist unique
$s, t \in P$
with
$s \ne t$
such that
$a_x^{-1} a_y= a_s a_t^{-1}$
.
Proposition 5.5 allows us to define a degree functor for a
$2$
-graph structure on
$\Gamma _{\mathcal {T}}$
in terms of the number of generators and their inverses in a minimal representative of an element.
Definition 5.7. Let
$\Gamma _{\mathcal {T}}$
be an
$\tilde {A_2}$
-group. Define
$\delta : \Gamma _{\mathcal {T}} \to \mathbb {N}^2$
by
$\delta (w) = (m,n)$
if its right normal form is as in (5-1) (equivalently, its left normal form is as in (5-2)). We define the length of w to be
$|\delta (w)| = m + n$
. We call
$\delta $
the shape function.
Remark 5.8. The shape function
$\delta $
is not additive. For example, in the
$\widetilde{A}_2$
-group
$\Gamma _{A.1} = \langle a_0 , \ldots , a_6: a_{[i]_7} a_{[i+1]_7} a_{[i+3]_7} =1 \rangle $
,
$$ \begin{align*} \delta (a_1 a_2) = \delta ( a_4^{-1} ) = (0,1) \neq (2,0) = \delta ( a_1 ) + \delta ( a_2 ). \end{align*} $$
The shape function
$\delta $
gives rise to a natural notion of a reduced word.
Definition 5.9. A word
$g_1 \cdots g_k$
in
$\Gamma _{\mathcal {T}}$
is said to be reduced if it has minimal length among words that represent the same element of
$\Gamma _{\mathcal {T}}$
. That is,
$g_1 \cdots g_k$
is reduced if
$|\delta (g_1 \cdots g_k)| = k$
.
Remarks 5.10.
-
(i) The final statement of Proposition 5.5 shows that words in right normal form or left normal form are reduced words.
-
(ii) Not all words that have no ‘obvious cancellations’ are reduced: the word
${w_1 = a_0{a_4}^{-1}a_6}$
in
$\Gamma _{A.1} = \langle a_0 , \ldots , a_6: a_{[i]_7} a_{[i+1]_7} a_{[i+3]_7} =1 \rangle $
is not reduced since
$$ \begin{align*} a_0 {a_4}^{-1} a_6 = a_0 a_1a _2 a_6 = a_0 a_1{a_0}^{-1} = {a_3}^{-1}{a_0}^{-1}. \end{align*} $$
-
(iii) Every subword of a reduced word is reduced.
-
(iv) If
$w = g_1 \cdots g_k$
is reduced and for some i,
$g_i = a_x$
and
$g_{i+1}= a_y^{-1}$
for some
${x, y \in P}$
with
$x \ne y$
, then by Corollary 5.6, there exist unique
$s, t \in P$
with
$s \ne t$
such that
$a_x{a_y}^{-1} = {a_s}^{-1}a_t$
. The word obtained from w by replacing
$g_ig_{i+1}= a_x a_y^{-1}$
with
${a_s}^{-1}a_t$
is also reduced.
Example 5.11. Consider
$\Gamma _{\mathcal {T}} := \Gamma _{A.1} = \langle a_0 , \ldots , a_6: a_{[i]_7} a_{[i+1]_7} a_{[i+3]_7} =1 \rangle $
from Example 5.3. For
$w = a_0 a_2^{-1} a_5^{-1} \in \Gamma _{\mathcal {T}}$
, we have
$\delta (w) = (1,2)$
; the reduced expressions for w and the corresponding segment of the reversed Cayley graph of
$\Gamma _{\mathcal {T}}$
(the Cayley graph of
$\Gamma _{\mathcal {T}}^{\mathrm{op}}$
) are illustrated below.
To obtain
$2$
-graphs from
$\tilde {A_2}$
groups, we relate the shape function
$\delta $
to the group law.
Lemma 5.12 (Unique factorisation).
Let
$\Gamma _{\mathcal {T}}$
be an
$\tilde {A_2}$
-group and suppose that
${m,n \in \mathbb {N}^2}$
and
$w \in \Gamma _{\mathcal {T}}$
satisfy
$\delta (w) = m+n$
. Then there exist unique
$h,k \in \Gamma _{\mathcal {T}}$
such that
$\delta (h)= m$
,
$\delta (k)=n$
and
$w=hk$
. More generally, if
$n_i \in \mathbb {N}^2$
satisfy
$\delta (w) = n_1 + \cdots + n_k$
, then there exist unique
$h_i \in \Gamma _{\mathcal {T}}$
such that each
$\delta (h_i)= n_i$
and
$w= h_1 \cdots h_k$
.
Given
$w, h, k \in \Gamma _{\mathcal {T}}$
such that
$\delta (whk) = \delta (w) + \delta (h) + \delta (k)$
, we have
$\delta (wh) = \delta (w) + \delta (h)$
and
$\delta (hk) = \delta (h) + \delta (k)$
.
Proof. This follows from repeated applications of Corollary 5.6.
Notation 5.13. If
$\delta (w) = (m,n) \ge \mathbf {1}$
, then Lemma 5.12 yields unique
$a,b,c,d\in \Gamma _{\mathcal {T}}$
such that
$$ \begin{align*} w = bd = c a, \quad \delta ( a ) = \delta( b ) = \mathbf{1} \quad\text{and} \quad \delta ( d ) = \delta (c) = \delta (w) - \mathbf{1}. \end{align*} $$
We adopt the notation
$s(w)=a$
,
$r(w)=b$
,
$c(w)=c$
,
$d(w)=d$
. Note that if
$\delta (w) = \mathbf {1}$
, then
$r (w)= s (w)$
and
$b(w) = c(w)= 1$
.
We provide a criterion for determining when a concatenation of three reduced words is reduced.
Proposition 5.14. Let
$\Gamma _{\mathcal {T}}$
be an
$\tilde {A_2}$
-group and fix
$w_0, w_1, w_2 \in \Gamma _{\mathcal {T}}$
. Suppose that
$\delta (w_0w_1) = \delta (w_0) + \delta (w_1)$
,
$\delta (w_1w_2) = \delta (w_1) + \delta (w_2)$
and
$\delta (w_1) \ge \mathbf {1}$
. Then
$$ \begin{align*} \delta(w_0w_1w_2) = \delta(w_0) + \delta(w_1) + \delta(w_2). \end{align*} $$
Proof. We induct on
$|\delta (w_2)|$
. Suppose that
$|\delta (w_2)|=1$
, so
$\delta (w_2) \in \{(1,0), (0,1)\}$
. If
$\delta (w_2) = (1, 0)$
, then
$w_2 = a_x$
for some
$x \in P$
, so
$w_0w_1w_2 = w_0w_1a_x$
. By Proposition 5.5, if
$\delta ( w_0w_1 ) = (m, n)$
, then in left normal form,
$w_0w_1 = a_{s_1}^{-1} \cdots a_{s_{n}}^{-1} a_{t_1} \cdots a_{t_{m}}$
and
${w_1 = a_{p_1}^{-1} \cdots a_{p_{k}}^{-1} a_{q_1} \cdots a_{q_{\ell }}}$
. Lemma 5.12 gives
$q_{\ell } = t_m$
(as
$\delta (w_0w_1) = \delta (w_0) + \delta (w_1)$
) and
$x \notin \lambda (q_{\ell })$
(as
$\delta (w_1w_2) = \delta (w_1) + \delta (w_2)$
). Hence,
$$ \begin{align*} w_0w_1w_2= a_{s_1}^{-1} \cdots a_{s_{n}}^{-1} a_{t_1} \cdots a_{t_{m}}a_x \end{align*} $$
is the left normal form of
$w_0w_1w_2$
and so
$$ \begin{align*} \delta(w_0w_1w_2) = (m+1, n)= (m, n) + (1, 0) = \delta(w_0w_1) + \delta(w_2)= \delta(w_0) + \delta(w_1) + \delta(w_2). \end{align*} $$
If
$\delta (w_2) = (0,1)$
, arguing similarly with right normal forms gives
$\delta (w_0w_1w_2) = \delta (w_0) + \delta (w_1) + \delta (w_2)$
.
Now suppose that the result holds for
$|\delta (w_2)| = n \ge 1$
, suppose that
$|\delta (w_2)| = n + 1$
. Then there exist unique
$h, k \in \Gamma _{\mathcal {T}}$
such that
$w_2 = hk$
,
$\delta (w_2) = \delta (h) + \delta (k)$
and
$|\delta (k)|= 1$
. Since
$$ \begin{align*} \delta(w_1hk) = \delta(w_1w_2) = \delta(w_1) + \delta(w_2) = \delta(w_1) + \delta(h) + \delta(k), \end{align*} $$
Lemma 5.12 gives
$\delta (w_1h) = \delta (w_1) + \delta (h)$
. Since
$|\delta (h)| = n$
, the induction hypothesis gives
$\delta (w_0(w_1h)) = \delta (w_0) + \delta (w_1) + \delta (h) = \delta (w_0) + \delta (w_1h)$
. Moreover,
$$ \begin{align*} \delta((w_1h)k) &= \delta(w_1w_2) = \delta(w_1) + \delta(w_2) = \delta(w_1) + \delta(h) + \delta(k) = \delta(w_1h) + \delta(k). \end{align*} $$
Therefore, since
$\delta (w_1h) \ge \mathbf {1}$
and
$|\delta (k)| = 1$
, it follows that
$$ \begin{align*} \delta(w_0w_1w_2) &= \delta(w_0(w_1h)k) = \delta(w_0) + \delta(w_1h) + \delta(k) = \delta(w_0) + \delta(w_1) + \delta(h)+ \delta(k) \\ &= \delta(w_0) + \delta(w_1) + \delta(w_2).\\[-35pt] \end{align*} $$
The following extends the above criterion to an arbitrary concatenation of reduced words.
Corollary 5.15. Fix
$w_0, w_1, \ldots , w_n \in \Gamma _{\mathcal {T}}$
. Suppose that
$\delta (w_iw_{i+1}) = \delta (w_i) + \delta (w_{i+1})$
for all
$0 \le i < n$
and
$\delta (w_i) \ge \mathbf {1}$
for all
$0 < i < n$
. Then
$$ \begin{align*} \delta(w_0w_1 \cdots w_{n-1} w_n) = \delta(w_0) + \delta(w_1) + \cdots + \delta(w_{n-1}) + \delta(w_{n}). \end{align*} $$
Proof. We induct on n. This is trivial for
$n = 1$
. Fix
$n \ge 1$
, suppose the result holds for all
$k \le n$
, and fix
$w_0, w_1, \ldots , w_n, w_{n+1} \in \Gamma _{\mathcal {T}}$
with
$\delta (w_iw_{i+1}) = \delta (w_i) + \delta (w_{i+1})$
for all
$i = 0, 1, \ldots , n$
and
$\delta (w_i) \ge \mathbf {1}$
for all
$i = 1, \ldots , n$
. Then
$$ \begin{align*} \delta(w_0w_1 \cdots w_n) = \delta(w_0) + \delta(w_1) + \cdots + \delta(w_n) = \delta(w_0w_1 \cdots w_{n-1}) + \delta(w_{n}), \end{align*} $$
and since
$\delta (w_n w_{n+1}) = \delta (w_n)+\delta (w_{n+1}) $
,
$$ \begin{align*} \delta(w_0w_1 \cdots w_nw_{n+1}) &= \delta(w_0w_1 \cdots w_{n-1}) + \delta(w_{n}) + \delta ( w_{n+1})\\ &= \delta(w_0) + \delta(w_1) + \cdots + \delta(w_n) + \delta ( w_{n+1}) \end{align*} $$
by Proposition 5.14. Thus, the result holds by induction.
5.2 The
$2$
-graph associated to an
$\tilde {A_2}$
-group
Given an
$\tilde {A_2}$
-group
$\Gamma _{\mathcal {T}}$
, we now construct a
$2$
-graph
$\Lambda _{\mathcal {T}}$
using the relation between the multiplicative structure of its reduced words and the shape function discussed in the previous section.
Definition 5.16. Fix an
$\tilde {A_2}$
-group
$\Gamma _{\mathcal {T}}$
. We define
$$ \begin{align*} \Lambda_{\mathcal{T}} = \{ w \in \Gamma_{\mathcal{T}} : \delta (w) \ge \mathbf{1} \} \quad\text{and} \quad \Lambda_{\mathcal{T}}^0 = \{ u \in \Gamma_{\mathcal{T}} : \delta (u) = \mathbf{1} \}. \end{align*} $$
We define
$r, s : \Lambda _{\mathcal {T}} \to \Lambda _{\mathcal {T}}^0$
as in Notation 5.13 and
$d : \Lambda _{\mathcal {T}} \to \mathbb {N}^2$
by
$d(\lambda ) = \delta (\lambda ) -\mathbf {1}$
. For
$\lambda ,\mu \in \Lambda _{\mathcal {T}}$
such that
$s(\lambda ) = r(\mu )$
, we define
$\lambda \circ \mu $
as follows: write
$\lambda = c(\lambda ) s(\lambda )$
and
$\mu = r (\mu ) b(\mu )$
as in Notation 5.13; we define
$$ \begin{align} \lambda \circ \mu := c(\lambda)s (\lambda) b(\mu). \end{align} $$
Our definition of
$\lambda \circ \mu $
in (5-3) emphasises the overlap of
$\lambda = c(\lambda )s(\lambda )$
and
${\mu = r(\mu )c(\mu )}$
in the element
$s(\lambda ) = r(\mu )$
of
$\delta ^{-1}(\mathbf {1}) \subseteq \Gamma _{\mathcal {T}}$
. We can also express it to emphasise its compatibility with the maps b and c: for
$\lambda , \mu \in \Lambda _{\mathcal {T}}$
as above with
$s(\lambda ) = r(\mu )$
,
$$ \begin{align} \lambda \circ \mu = c(\lambda)s(\lambda)b(\mu) = r(\lambda) b(\lambda) b(\mu) \quad\text{and}\quad \lambda \circ \mu = c(\lambda) r ( \mu ) b(\mu) = c(\lambda) c(\mu) s(\mu). \end{align} $$
Our main result in this subsection is that Definition 5.16 defines a
$2$
-graph.
Theorem 5.17. With definitions and notation as above,
$(\Lambda _{\mathcal {T}}, d )$
is a
$2$
-graph and the maps
$b, c : \Lambda _{\mathcal {T}} \to \Gamma _{\mathcal {T}}$
of Notation 5.13 are
$1$
-cocycles.
Proof. Associativity of multiplication in
$\Gamma _{\mathcal {T}}$
ensures that
$\Lambda _{\mathcal {T}}$
is a category under
$\circ $
. To see that
$d : \Lambda _{\mathcal {T}} \to \mathbb {N}^2$
is a functor, fix
$\lambda ,\mu \in \Lambda _{\mathcal {T}}$
with
$s(\lambda )=r(\mu )$
. We have
${\lambda \circ \mu = c(\lambda )s(\lambda )b(\mu )}$
, where
$\delta (s(\lambda ))) = \mathbf {1}$
. So the first part of Lemma 5.12 gives
$$ \begin{align*} \delta(c(\lambda)s(\lambda)) &= d(\lambda) + \mathbf{1} = \delta(c(\lambda)) + \delta(s(\lambda)) \\ \delta(s(\lambda)b(\mu)) &= d(\mu) + \mathbf{1} = \delta(s(\lambda)) + \delta(b(\mu)). \end{align*} $$
Hence, by Proposition 5.14 and since
$\delta (s(\lambda )) = \mathbf {1}$
by definition,
$$ \begin{align*} d(\lambda \circ \mu) = \delta(c(\lambda)s(\lambda) b(\mu)) - \mathbf{1} = \delta(c(\lambda)) + \delta(s(\lambda)) + \delta( b(\mu)) - \mathbf{1} = d(\lambda) + d(\mu). \end{align*} $$
It remains to show that
$( \Lambda _{\mathcal {T}},d)$
satisfies the factorisation property. Suppose that
$d(\lambda ) = (m_1+m_2,n_1+n_2)$
. Then
$\delta ( \lambda ) = (m_1+m_2,n_1+n_2)+\mathbf {1}$
. Hence, by Lemma 5.12, there exist unique
$g, h, k$
such that
$ \lambda = ghk$
,
$\delta (g) = (m_1, n_1)$
,
$\delta (h) = \mathbf {1}$
and
${\delta (k) = (m_2, n_2)}$
. Thus,
$\lambda = \mu \circ \nu $
, where
$\mu = gh$
,
$\nu = hk$
,
$d(\mu ) = (m_1, n_1)$
and
$d(\nu ) = (m_2, n_2)$
, and this is the unique such factorisation.
Fix
$\lambda , \mu \in \Lambda _{\mathcal {T}}$
with
$s(\lambda ) = r(\mu )$
. Equation (5-4) and the definition of c give
$c(\lambda \circ \mu )s(\lambda \circ \mu ) = \lambda \circ \mu = c(\lambda )c(\mu )s(\mu )$
and
$r(\lambda \circ \mu )b(\lambda \circ \mu ) = \lambda \circ \mu = r(\lambda ) b(\lambda )b(\mu )$
. We already saw that
$s(\lambda \circ \mu ) = s(\mu )$
and
$r(\lambda \circ \mu ) = r(\lambda )$
, so cancellativity in
$\Gamma _{\mathcal {T}}$
gives
$c(\lambda )c(\mu ) = c(\lambda \circ \mu )$
and
$b(\lambda )b(\mu ) = b(\lambda \circ \mu )$
.
Remark 5.18. Resume the notation of [Reference Robertson and Steger38, Section 7]. Let
$M_1, M_2$
be the matrices [Reference Robertson and Steger38, page 135] obtained from the Cayley graph
$\mathscr {B}_{\mathcal {T}}$
of
$\Gamma _{\mathcal {T}}$
regarded as a building as in [Reference Cartwright, Mantero, Steger and Zappa8]. Then
$\Lambda _{\mathcal {T}}$
is isomorphic to the
$2$
-graph
$\Lambda _{M_1, M_2}$
obtained from the
$M_i$
as in [Reference Kumjian and Pask26, Example 1.7(iv)]. Indeed, as
$\Gamma _{\mathcal {T}}$
acts transitively on vertices of
$\mathscr {B}_{\mathcal {T}}$
, we can identify the alphabet
$A = \Gamma /\mathcal {I}$
[Reference Robertson and Steger38, page 135] with type-rotating isometries
${i : t \to \mathscr {B}_{\mathcal {T}}}$
such that
$i((0,0)) = e_{\Gamma _{\mathcal {T}}}$
. By Proposition 5.5,
$i \mapsto i((1,1))$
is a bijection between such isometries and
$\delta ^{-1}(\mathbf {1}) = \Lambda _{\mathcal {T}}^0$
. Likewise, for
$w_1, w_2 \in \Lambda _{\mathcal {T}}^0$
, the set
$w_1\Lambda _{\mathcal {T}}^{\varepsilon _i}w_2$
is in bijection with type-rotating isometries
$i : \mathfrak {p}_{\varepsilon _i} \to \mathscr {B}_{\mathcal {T}}$
such that
$i((0,0)) = e_{\Gamma _{\mathcal {T}}}$
,
$t(\mathbf {1}) = w_1$
and
$t(\mathbf {1}+\varepsilon _i)t(\varepsilon _i)^{-1} = w_2$
; that is, diagrams as in [Reference Robertson and Steger38, Figure 9]. So the adjacency matrices of
$\Lambda _{\mathcal {T}}$
are the
$M_i$
. Since
$M_1, M_2$
satisfy (H0)–(H3) [Reference Robertson and Steger38, Proposition 7.9 and Theorem 7.10],
$M_1M_2$
is a
$0,1$
-matrix, so [Reference Hazlewood, Raeburn, Sims and Webster19, Theorems 4.4 and 4.5] gives
$\Lambda _{\mathcal {T}} \cong \Lambda _{M_1, M_2}$
.
Corollary 5.19. With notation as above, the cocycle
$c: \Lambda _{\mathcal {T}} \to \Gamma _{\mathcal {T}}$
of Theorem 5.17 is essential and the canonical map
$i : \Lambda _{\mathcal {T}} \to \Pi (\Lambda _{\mathcal {T}})$
is injective.
5.3 The covering
$2$
-graph
$\Sigma _{\mathcal {T}}$
In this section, we construct a covering
$2$
-graph
$\Sigma _{\mathcal {T}}$
for
$\Lambda _{\mathcal {T}}$
.
We define
$\Sigma _{\mathcal {T}} \subseteq \Gamma _{\mathcal {T}} \times \Gamma _{\mathcal {T}}$
as follows. Let
$$ \begin{align*} \Sigma_{\mathcal{T}} := \{ (x, y) \in \Gamma_{\mathcal{T}} \times \Gamma_{\mathcal{T}} : \mathbf{1} \le \delta(x^{-1}y) \} \quad\text{and} \quad \Sigma^0_{\mathcal{T}} := \{ (x, y) \in \Gamma_{\mathcal{T}} \times \Gamma_{\mathcal{T}} : \mathbf{1} = \delta(x^{-1}y) \} \end{align*} $$
with
$d(x, y) := \delta (x^{-1}y) - \mathbf {1}$
for all
$(x, y) \in \Sigma _{\mathcal {T}}$
. By Lemma 5.12, for each
$(x, y) \in \Sigma _{\mathcal {T}}$
, there exist unique
$z_{x,y}, w_{x,y} \in \Gamma _{\mathcal {T}}$
such that
$$ \begin{align} \delta(x^{-1}z_{x,y}) = \delta(w_{x,y}^{-1}y) = \mathbf{1} \quad\text{and} \quad \delta(x^{-1}y) = \delta(x^{-1}z_{x,y}) + \delta(z_{x,y}^{-1}y) = \delta(x^{-1}w_{x,y}) + \delta(w_{x,y}^{-1}y). \end{align} $$
We define
$r(x, y) := (x, z_{x,y})$
and
$s(x, y) := (w_{x,y}, y)$
. If
$(u, v) \in \Sigma _{\mathcal {T}}$
satisfies
${s(x, y) = r(u, v)}$
, we define
$(x, y)(u, v) := (x, v)$
. We show that
$\Sigma _{\mathcal {T}} \cong \Gamma _{\mathcal {T}} \times _{c} \Lambda _{\mathcal {T}}$
(see Definition 2.4).
Proposition 5.20. With the above structure,
$\Sigma _{\mathcal {T}}$
is a
$2$
-graph. Let
$c : \Lambda _{\mathcal {T}} \to \Gamma _{\mathcal {T}}$
be the cocycle of Theorem 5.17. There is an isomorphism
$\phi : \Sigma _{\mathcal {T}} \to \Gamma _{\mathcal {T}} \times _{c} \Lambda _{\mathcal {T}}$
such that
$\phi (x, y) = (x, x^{-1}y)$
for all
$(x, y) \in \Sigma _{\mathcal {T}}$
. The inverse satisfies
$\phi ^{-1}(x, \lambda ) = (x, x\lambda )$
. There is a free action of
$\Gamma _{\mathcal {T}}$
on
$\Sigma _{\mathcal {T}}$
given by
$g \cdot (x, y) := (gx, gy)$
, and
$\phi $
is equivariant for this action and the left action of
$\Gamma _{\mathcal {T}}$
on
$\Gamma _{\mathcal {T}} \times _{c} \Lambda _{\mathcal {T}}$
by translation in the first coordinate. In particular,
$\phi $
descends to an isomorphism
$\widetilde {\phi } : \Gamma _{\mathcal {T}} \backslash \Sigma _{\mathcal {T}} \to \Lambda _{\mathcal {T}}$
such that
$\widetilde {\phi }([x,y]) = x^{-1}y$
.
Proof. We prove that
$\psi : \Gamma _{\mathcal {T}} \times _{c} \Lambda _{\mathcal {T}} \to \Sigma _{\mathcal {T}}$
defined by
$\psi (x, \lambda ) = (x, x\lambda )$
for
${(x, \lambda ) \in \Gamma _{\mathcal {T}} \times _{c} \Lambda _{\mathcal {T}}}$
is an isomorphism, and deduce that
$\phi = \psi ^{-1}$
has the desired properties. To see that
$\psi $
is a functor, fix
$(x, \lambda ) \in \Gamma _{\mathcal {T}} \times _{c} \Lambda _{\mathcal {T}}$
. Then
$d(x, \lambda ) = d(\lambda ) = d(x, x\lambda )= d(\psi (x, \lambda ) )$
. Recall that
$c(\lambda )s(\lambda ) = \lambda = r(\lambda )b(\lambda )$
,
$s(x, \lambda ) = (xc(\lambda ), s(\lambda ))$
and
$s(x, x\lambda ) = (xc(\lambda ), x\lambda )$
since
$w = xc(\lambda )$
satisfies
$\delta (w^{-1}x\lambda ) = \delta (s(\lambda )) = \mathbf {1}$
and
$$ \begin{align*} \delta(x^{-1}x\lambda) = \delta(\lambda) =\delta(c(\lambda)) + \delta(s(\lambda)) = \delta(x^{-1}w) + \delta(w^{-1}x\lambda). \end{align*} $$
Hence,
$$ \begin{align*} \psi(s(x, \lambda)) = \psi(xc(\lambda), s(\lambda)) = (xc(\lambda), xc(\lambda)s(\lambda)) = (xc(\lambda), x\lambda) = s(x, x\lambda) = s(\psi(x, \lambda)). \end{align*} $$
A similar computation shows that
$\psi (r(x, \lambda )) = (x, xr(\lambda )) = r(\psi (x, \lambda ))$
.
Given composable elements
$(x, \lambda ), (xc(\lambda ), \mu ) \in \Gamma _{\mathcal {T}} \times _{c} \Lambda _{\mathcal {T}}$
, the above argument shows that
$\psi (x, \lambda ), \psi (xc(\lambda ), \mu )$
are composable in
$\Sigma _{\mathcal {T}}$
. We have
$$ \begin{align*} \psi((x, \lambda) (xc(\lambda), \mu)) &= \psi(x, \lambda\circ\mu) = (x, x(\lambda\circ\mu))\\ &= (x, xc(\lambda)\mu) = (x, x\lambda)(xc(\lambda), xc(\lambda)\mu) = \psi(x, \lambda) \psi(xc(\lambda), \mu). \end{align*} $$
Hence,
$\psi $
is an isomorphism and thus
$\Sigma _{\mathcal {T}}$
is a
$2$
-graph. That
$\phi $
is equivariant follows from its definition and the last assertion follows from [Reference Kumjian and Pask26, Remark 5.6].
Proposition 5.21. Let
$(x, z), (w, y) \in \Sigma _{\mathcal {T}}^0$
. Then
$(x, z)\Sigma _{\mathcal {T}}(w, y) \not = \emptyset $
if and only if
$$ \begin{align*} \delta(x^{-1}w) + \mathbf{1} = \delta(x^{-1}y) = \delta(z^{-1}y) + \mathbf{1}, \end{align*} $$
and then
$(x, z)\Sigma _{\mathcal {T}}(w, y) = \{(x, y)\}$
. In particular,
$\Sigma _{\mathcal {T}}$
is singly connected and
$C^*(\Sigma _{\mathcal {T}})$
is type I
$_0$
.
Proof. If
$\sigma \in (x, z)\Sigma _{\mathcal {T}}(w, y)$
, then
$r(\sigma ) = (x, z)$
and
$s(\sigma ) = (w, y)$
, so
$\sigma = (x, y)$
,
$w = w_{x, y}$
and
$z = z_{x, y}$
by (5-5). In particular,
$(x, z)\Sigma _{\mathcal {T}}(w, y)$
is either empty or equal to
$\{(x, y)\}$
.
If
$\delta (x^{-1}y) \ge \mathbf {1}$
, then
$(x,y) \in \Sigma _{\mathcal {T}}$
if and only if
$s(x, y) = (w, y)$
and
$r(x, y) = (x, z)$
. Moreover,
$s(x, y) = (w, y)$
if and only if
$w = w_{x, y}$
, that is (see (5-5)),
$$ \begin{align*} \delta(x^{-1}y) &= \delta(x^{-1}w) + \delta(w^{-1}y) = \delta(x^{-1}w) + \mathbf{1} \end{align*} $$
and
$r(x, y) = (x, z)$
if and only if
$z = z_{x, y}$
, that is,
$$ \begin{align*} \delta(x^{-1}y) &= \delta(x^{-1}z) + \delta(z^{-1}y) = \delta(z^{-1}y) + \mathbf{1}. \end{align*} $$
The final assertion follows from the first paragraph of the proof and Proposition 4.4.
Remark 5.22. That
$\Sigma _{\mathcal {T}}$
is singly connected also follows from the facts that
${\Sigma _{\mathcal {T}} \cong \Gamma _{\mathcal {T}} \times _{c} \Lambda _{\mathcal {T}}}$
(by Proposition 5.20), c is essential and
$\Gamma _{\mathcal {T}} \times _{c} \Lambda _{\mathcal {T}}$
is singly connected (by Corollary 5.19).
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