Download - Coverings of k-graphs - COnnecting REpositories · In Section 6, we construct universal coverings using skew products. We also show that every connected covering is a relative skew

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Journal of Algebra 289 (2005) 161–191

www.elsevier.com/locate/jalgebr

Coverings ofk-graphs

David Paska, John Quiggb, Iain Raeburna,∗

a School of Mathematical and Physical Sciences, University of Newcastle, NSW 2308, Australiab Department of Mathematics and Statistics, Arizona State University, Tempe, AZ 85287, USA

Received 30 July 2004

Available online 9 April 2005

Communicated by Michel Broué

Abstract

k-graphs are higher-rank analogues of directed graphs which were first developed to pcombinatorial models for operator algebras of Cuntz–Krieger type. Here we develop the thecovering spaces fork-graphs, obtaining a satisfactory version of the usual topological classtion in terms of subgroups of a fundamental group. We then use this classification to descrC∗-algebras of coveringk-graphs as crossed products by coactions of homogeneous spaces,alizing recent results on theC∗-algebras of graphs. 2005 Elsevier Inc. All rights reserved.

Keywords:k-Graph; Small category; Covering; Fundamental group;C∗-algebra; Coaction

1. Introduction

k-graphsare combinatorial structures which arek-dimensional analogues of (directegraphs. They were introduced by Kumjian and the first author [18] to help understandof Robertson and Steger on higher-rank analogues of the Cuntz–Krieger algebras [The theory ofk-graphs and theirC∗-algebras parallels in many respects that of graphs

This research was supported by grants from the Australian Research Council and the University of Ne* Corresponding author.

E-mail addresses:[email protected] (D. Pask), [email protected] (J. Quigg),
[email protected], [email protected] (I. Raeburn).
0021-8693/$ – see front matter 2005 Elsevier Inc. All rights reserved.doi:10.1016/j.jalgebra.2005.01.051

162 D. Pask et al. / Journal of Algebra 289 (2005) 161–191

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their Cuntz–Krieger algebras [18,23,24]. Here we investigate to what extent thereanalogue fork-graphs of the theory of coverings of graphs, and the implications oftheory for theC∗-algebras ofk-graphs.

A coveringof a graphF is by definition a surjective graph morphismp :E → F whichis a local isomorphism. As for coverings of topological spaces, the coverings ofF are clas-sified by the conjugacy classes of subgroups of the fundamental groupπ1(F ), and everyconnected covering arises as a quotient of a universal covering (see [4,12], for exaThis last theorem has interesting ramifications for the Cuntz–Krieger algebrasC∗(E) ofcovering graphs: ifp :E → F is a covering, then there is a coactionδ of the fundamen-tal groupπ1(F ) on C∗(F ) and a subgroupH of π1(F ) such thatC∗(E) is isomorphicto the crossed productC∗(F ) ×δ (π1(F )/H) [4, Theorem 3.2]. This theorem has in tubeen of considerable interest in nonabelian duality forC∗-algebras: it provided a family ocrossed products by homogeneous spaces which we could analyse using our unding of graph algebras, and this analysis inspired substantial improvements in MansImprimitivity Theorem [14].

We seek, therefore, an analogue of this theory of covering graphs fork-graphs, anda generalisation of [4, Theorem 3.2] which describes theC∗-algebras of the coveringk-graphs. Any theory of coverings must involve the fundamental group, and a majorityauthors prefer to use the whole fundamental groupoid. We showed in [19] that themental groupoids ofk-graphs do not behave as well as one might hope, and in partithat the path category need not embed faithfully in the fundamental groupoid whenk > 1.So it is something of a relief that our final results on coverings mirror in every respeclassical topological theory.

Our approach is to exploit an equivalence between the coverings of ak-graph and actions of its fundamental groupoid, under which the connected coverings correspotransitive actions. Thus we deduce many of our main theorems from a classificationtransitive actions of an arbitrary groupoid.

Because every small category is isomorphic to a quotient of a path category, it wclear from the proofs that all our results carry over to arbitrary small categories; howwe eschew such a generalization since we have no useful applications.

After we completed this paper, we learned of the existence of [1,3,13,16], whichtain results similar to some of ours. In [1, Appendix], Bridson and Haefliger develoelementary theory of the fundamental group and coverings of a small category andresults similar to some of ours. Bridson and Haefliger concentrate on the fundamgroup—indeed, they stop just short of defining the fundamental groupoid. In [3,13], Band Higgins investigate coverings of groupoids, and prove the equivalence with groactions. Our work was done completely independently of these other sources, andlieve our methods are of interest, especially our use of skew products. In [16], Kudevelops, in the specific context ofk-graphs, the fundamental groupoid and the existeof the universal covering, and proves that, under reasonable hypotheses, theC∗-algebra ofthe universal coveringk-graph is Rieffel–Morita equivalent to a commutative algebra.thank Kumjian for bringing [1] to our attention.

We begin in Section 2 by introducing our notion of covering, and stating our mainsification theorems. Analogues of these theorems for coverings of groupoids were
in [3, Chapter 9]. In Section 3, we briefly discuss actions of groupoids on sets, and prove

D. Pask et al. / Journal of Algebra 289 (2005) 161–191 163

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the equivalence between the category of coverings of ak-graphΛ and the category of actions of its fundamental groupoidG(Λ) (Theorem 3.5). The main theorem of this sectis a technical result (Theorem 3.7) which implies that the connected coverings ofΛ corre-spond to transitive actions ofG(Λ), and that the fundamental groups of coverings ofΛ canbe identified with the stability groups of the corresponding actions ofG(Λ). In Section 4,we state and prove analogues for groupoid actions of most of our main theorems, ain 5 we prove the main theorems themselves. Many of them follow from the general rin the previous section, but when it seemed easier to prove a result about coverings dwe did so.

In Section 6, we construct universal coverings using skew products. We also shoevery connected covering is arelative skew product(Corollary 6.10), and prove a versioof the Gross–Tucker Theorem which identifies thek-graphs which admit free actions ofgroup as skew products.

It seems to us thatk-graphs are likely to be of interest in their own right, so we hbeen careful to limit our discussion ofC∗-algebras to a final section on the applicatioof our theory. Our sought-after generalisation of [4, Theorem 3.2] is Corollary 7.2main idea in the proof of Corollary 7.2 comes from [15]: every group-valued cocycleη ona k-graphΛ induces a normal and maximal coactionδη of the group onC∗(Λ), and everyk-graph carries a suitable cocycle with values in the fundamental group. We also pdecomposition theorem which generalises [4, Corollary 3.6], prove that theC∗-algebra ofeveryk-graph is nuclear, and prove that theC∗-algebra of the skew product by the degrmap is always AF.

2. Main results

For k-graphs and groupoids we adopt the conventions of [18,19,23], except thatnot require them to be countable. Briefly, ak-graph is a small categoryΛ equipped with afunctord :Λ → N

k satisfying thefactorization property: for all α ∈ Λ andn, l ∈ Nk such

thatd(α) = n + l there exist uniqueβ,γ ∈ Λ such thatd(β) = n, d(γ ) = l, andα = βγ .Whend(α) = n we sayα hasdegreen. A groupoid is a small category in which evermorphism has an inverse. All groupoids and groups in this paper are discrete, in thethat they carry no topology.

If C is either ak-graph or a groupoid, theverticesare the objects, andC0 denotes the seof vertices. Forα ∈ C, thesources(α) is the domain, and theranger(α) is the codomainFor u,v ∈ C0 we writeuC = r−1(u), Cv = s−1(v), anduCv = uC ∩ Cv. C is connectedifthe equivalence relation onC0 generated by(u, v) | uCv = ∅ is C0 × C0; for a groupoidthis just meansuCv = ∅ for all u,v ∈ C0. If Λ is ak-graph,u,v ∈ Λ0, andn ∈ N

k , we writeΛn = d−1(n), uΛn = uΛ ∩ Λn, andΛnv = Λv ∩ Λn. A morphismbetweenk-graphs is adegree-preserving functor.

In general we often write composition of maps as juxtaposition, especially when wchasing around commutative diagrams.

Definition 2.1. A coveringof a k-graphΛ is a surjectivek-graph morphismp :Ω → Λ

such that for allv ∈ Ω0, p mapsΩv 1–1 ontoΛp(v) andvΩ 1–1 ontop(v)Λ. If (Ω,p)

and(Σ,q) are coverings ofΛ, amorphismfrom (Ω,p) to (Σ,q) is ak-graph morphism
φ :Ω → Σ making the diagram


o,s de-

uncto-

ry,

Ωφ

p

Σ

q

Λ

commute; we writeφ : (Ω,p) → (Σ,q). A coveringp :Ω → Λ is connectedif Ω (hencek-graphΛ) is connected.

Remark. If Λ is connected then surjectivity ofp is implied by the other properties. Alsany functorφ :Ω → Σ making the above diagram commute automatically preservegrees, hence is a morphism of coverings.

Every k-graphΛ has afundamental groupoid, which is a groupoidG(Λ) such thatG(Λ)0 = Λ0 together with a canonical functori :Λ → G(Λ) which is the identity onΛ0

and has the following universal property: for every functorT from Λ into a groupoidHthere exists a unique groupoid morphismT ′ making the diagram

Λi

T

G(Λ)

T ′

H

commute. The assignmentΛ → G(Λ) is functorial fromk-graphs to groupoids. Thefun-damental groupof Λ at a vertexx ∈ Λ0 is the isotropy group

π(Λ,x) := xG(Λ)x.

(The subscript 1 in the standard notation seems redundant in this context.) By friality of Λ → G(Λ), a coveringp :Ω → Λ induces homomorphismsp∗ :π(Ω,v) →π(Λ,p(v)).

Theorem 2.2. Let (Ω,p) and (Σ,q) be connected coverings of ak-graph Λ. For allx ∈ Λ0, the familyp∗π(Ω,v) | p(v) = x is a conjugacy class of subgroups ofπ(Λ,x).For all x ∈ Λ0, v ∈ p−1(x), andu ∈ q−1(x), there is a morphism(Ω,p) → (Σ,q) takingv to u if and only ifp∗π(Ω,v) ⊂ q∗π(Σ,u). Consequently,(Ω,p) ∼= (Σ,q) if and only ifthe subgroupsp∗π(Ω,v) andq∗π(Σ,u) of π(Λ,x) are conjugate for some, hence evex ∈ Λ0, v ∈ p−1(x), andu ∈ q−1(x).

For our next result we will need to enlarge our supply of morphisms.

Definition 2.3. If p :Ω → Λ andq :Ω → Γ are coverings, amorphismfrom (Ω,p) to(Ω,q) is ak-graph morphismφ :Λ → Γ making the diagram

Ωp q

Λφ

Γ

commute; we writeφ : (Ω,p) → (Ω,q).


hereced,e

,

g

-

ing

From the context it is always clear which type of morphism of coverings we mean. Tis an obvious notion of morphism which would unify the two kinds we have introdubut since we have no use for it we omit it. In the above definition, observe that sincp issurjective, there is at most one morphismφ : (Ω,p) → (Ω,q).

We will also need to know about quotients by group actions: let Aut(Ω,p) denote theautomorphism group of a connected coveringp :Ω → Λ. As we shall show in Section 5the quotient mapΩ → Ω/Aut(Ω,p) gives rise to a commuting diagram

Ω

p

Ω/Aut(Ω,p)

Λ

of connected coverings.

Corollary 2.4. Letp :Ω → Λ be a connected covering,x ∈ Λ0, andv ∈ p−1(x). Then thefollowing are equivalent:

(i) the subgroupp∗π(Ω,v) of π(Λ,x) is normal;(ii) Aut(Ω,p) acts transitively onp−1(x);

(iii) the coveringΩ/Aut(Ω,p) → Λ is an isomorphism;(iv) (Ω,p) is isomorphic to the coveringΩ → Ω/Aut(Ω,p).

Theorem 2.5. Letp :Ω → Λ be a connected covering,x ∈ Λ0, andv ∈ p−1(x). Then thenormalizerN(p∗π(Ω,v)) of p∗π(Ω,v) in π(Λ,x) acts on(Ω,p), and in fact

Aut(Ω,p) ∼= N(p∗π(Ω,v)

)/p∗π(Ω,v).

Theorem 2.2 shows how isomorphism classes of connected coverings of ak-graphΛ areinversely related to conjugacy classes of subgroups of the fundamental groupπ(Λ,x) (forany choice of vertexx). The identity map onΛ gives a minimal covering, correspondinto the improper subgroupπ(Λ,x). At the opposite extreme:

Definition 2.6. A coveringp :Ω → Λ is universalif it is connected and for every connected coveringq :Σ → Λ there exists a morphism(Ω,p) → (Σ,q).

For coverings of small categories, the following result is [1, Proposition A.19].

Theorem 2.7. Every connectedk-graphΛ has a universal covering. A connected coverp :Ω → Λ is universal if and only ifp∗π(Ω,v) = x for some, hence every,x ∈ Λ0 andv ∈ p−1(x).

The following result shows that every subgroup ofπ(Λ,x) occurs in the form
p∗π(Ω,v) for some connected covering(Ω,p).


d

gs

a

en

Theorem 2.8. Let p :Ω → Λ be a universal covering,x ∈ Λ0, v ∈ p−1(x), and H asubgroup ofπ(Λ,x). LetH act on(Ω,p) according to Theorem2.5. Then the associatecoveringq :Ω/H → Λ is connected, and

H = q∗π(Ω/H,vH).

Moreover, every connected covering ofΛ is isomorphic to one of these coverinΩ/H → Λ.

3. Coverings and actions

Definition 3.1 (cf. [3,13]). An actionof a groupoidG on a setV is a functorT from G tothe category of sets such thatV is the disjoint union of the setsT (x) for x ∈ G0. Put:

• Vx = T (x) for x ∈ G0;• G ∗ V = (a, v) | a ∈ G, v ∈ Vs(a);• av = T (a)(v) for (a, v) ∈ G ∗ V .

Thetransformation groupoidis the setG ∗ V with operations

s(a, v) = (s(a), v

), r(a, v) = (

r(a), av), (a, bv)(b, v) = (ab, v).

Thestability groupat v ∈ V is

Sv := a ∈ G | av = v.

The action(V ,G) is

• transitiveif V = Gv for some (hence every)v ∈ V ;• free if Sv = x for all x ∈ G0 andv ∈ Vx .

Thus, for each objectx ∈ G0 we have a setVx , and for eacha ∈ xGy we have a bijectionv → av from Vy to Vx . Since we require the setsVx to be pairwise disjoint, we havebundleV → G0, andG acts as bijections among the fibers of this bundle.

Definition 3.2. If G acts on bothV andU , a morphismfrom (V ,G) to (U,G) is a mapφ :V → U which isG-equivariantin the sense that

φ(av) = aφ(v) for all (a, v) ∈ G ∗ V.

Remark. Thus, a morphism between actions ofG is just a natural transformation betwe
the functors.


ch will

the

We are ready to begin forging the connection between coverings and actions, whitake the form of an equivalence between the categories of coverings of ak-graphΛ (andtheir morphisms) and actions ofG(Λ) (and their morphisms).

Proposition 3.3. Let Λ be a k-graph, and let its fundamental groupoidG(Λ) act on asetV . Put

Λ ∗ V = (α, v) ∈ Λ × V | v ∈ Vs(α)

and αv = i(α)v for (α, v) ∈ Λ ∗ V.

ThenΛ ∗ V becomes ak-graph with operations

s(α, v) = (s(α), v

), r(α, v) = (

r(α),αv),

(α,βv)(β, v) = (αβ, v), d(α, v) = d(α),

and the coordinate projectionpΛ :Λ ∗ V → Λ is a covering.Moreover, the assignments(

V,G(Λ)) → (Λ ∗ V,pΛ) and φ → idΛ ∗ φ (1)

give a functor from actions ofG(Λ) to coverings ofΛ.

Proof. Routine computations (similar to those showingG(Λ) ∗ V is a groupoid) verifythatΛ ∗ V is a category. Also,pΛ is clearly a surjective morphism. To see that it hascovering property, just note that for(x, v) ∈ Λ0 ∗ V = (Λ ∗ V )0 we have

(Λ ∗ V )(x, v) = (α, v) | α ∈ Λx

,

(x, v)(Λ ∗ V ) = (α,u) | α ∈ xΛ, αu = v

.

The mapd :Λ ∗ V → Nk is the composition of the functorsd :Λ → N

k andpΛ :Λ ∗V → Λ, and hence is a functor. We verify the factorization property: let(λ, v) ∈ Λ ∗ V

andn, l ∈ Nk with d(λ, v) = n + l. Thend(λ) = n + l, so there existµ ∈ Λn, ν ∈ Λl such

thatλ = µν. Then(µ, νv), (ν, v) ∈ Λ ∗ V , and since

s(µ, νv) = (s(µ), νv

) = (r(ν), νv

) = r(ν, v),

we can multiply:

(µ, νv)(ν, v) = (µν, v) = (λ, v).

Since

d(µ, νv) = d(µ) = n and d(ν, v) = d(ν) = l,


s

ue

ts

ed

a

For

gory

the factorization has the right degrees. For uniqueness, if also

(λ, v) = (α,βv)(β, v) with d(α,βv) = n andd(β, v) = l,

thenλ = αβ with d(α) = n andd(β) = l, so we must haveα = µ andβ = ν, hence

(α,βv) = (µ, νv) and (β, v) = (ν, v).

ThusΛ ∗ V is ak-graph.pΛ preserves degrees by construction, hence is a covering.For the other part, routine computations show that ifφ :V → U is a morphism of action

of G(Λ), then idΛ ∗ φ is a morphism of the corresponding coverings ofΛ, and that (1) isfunctorial.

In the opposite direction:

Proposition 3.4. Let p :Ω → Λ be a covering ofk-graphs. Then there exists a uniqaction ofG(Λ) onΩ0 such that

i(α)v = r(λ) if α = p(λ) andv = s(λ),

wherei :Λ → G(Λ) is the canonical functor.Moreover, the assignments

(Ω,p) → (Ω0,G(Λ)

)and φ → φ|Ω0 (2)

give a functor from coverings ofΛ to actions ofG(Λ).

Proof. For α ∈ xΛy, the range and source maps ofΩ takep−1(α) 1–1 ontop−1(x) andp−1(y), respectively, thus affording a bijectionT (α) :p−1(y) → p−1(x). Routine compu-tations show that the resulting mapT from Λ to the groupoid of bijections among the sein the familyp−1(x) | x ∈ Λ0 is functorial. SinceT is a functor fromΛ into a groupoid,it factors uniquely through a morphism fromG(Λ) to the same groupoid, giving the desiraction ofG(Λ).

For the other part, a routine computation shows that ifφ : (Ω,p) → (Σ,q) is a mor-phism of coverings ofΛ, thenφ(αv) = αφ(v) for (α, v) ∈ Λ ∗ Ω0. Thus if T andS arethe functors giving the actions ofG(Λ) on Ω0 andΣ0, respectively, thenφ gives a nat-ural transformation fromT i to Si (wherei :Λ → G(Λ) is the canonical functor), hencenatural transformation fromT to S by universality ofi. Thereforeφ|Ω0 is a morphism ofgroupoid actions. Routine computations show that (2) is functorial.

For coverings of groupoids, the following result is [3, Section 9.4, Exercise 3].coverings of small categories, it is similar to [1, Proposition A.23].

Theorem 3.5. The functors described in the preceding two propositions give a cate
equivalence between coverings ofΛ and actions ofG(Λ).


p

a-

natural

ps.

ults,sition

ng

Proof. First, given a covering(Ω,p), it follows from the definitions that the ma(p, s) :Ω → Λ ∗ Ω0 is bijective, and clearlyp = pΛ(p, s). A computation using theidentity r(λ) = p(λ)s(λ) for λ ∈ Ω verifies that(p, s) is functorial. Thus(Ω,p) ∼=(Λ ∗ Ω0,pΛ).

Next, given an action(V ,G(Λ)), it is obvious that the map fromV to Λ0∗V = (Λ∗V )0

takingv ∈ Vx to (x, v) is bijective, and it follows straight from the definitions that it isΛ-equivariant, henceG(Λ)-equivariant. Therefore(V ,G(Λ)) ∼= (Λ0 ∗ V,G(Λ)).

If φ : (Ω,p) → (Σ,q) andψ : (V ,G(Λ)) → (U,G(Λ)) are morphisms, then the digrams

(Ω,p)∼=

φ

(Λ ∗ Ω0,pΛ)

idΛ∗φ|Ω0

(Σ,q) ∼= (Λ ∗ Σ0,pΛ)

and

(V ,G(Λ))∼=

ψ

(Λ0 ∗ V,G(Λ))

idΛ0∗ψ

(U,G(Λ)) ∼= (Λ0 ∗ U,G(Λ))

commute. Thus the isomorphisms of the preceding two paragraphs implement aequivalence between the functors of Propositions 3.3 and 3.4.

The next result says that above equivalence matches up the automorphism grou

Corollary 3.6. If p :Ω → Λ is a covering then the mapφ → φ|Ω0 gives an isomorphismAut(Ω,p) ∼= Aut(Ω0,G(Λ)).

Theorem 3.7. Letp :Ω → Λ be a covering, and letG(Λ) act onΩ0 as in Proposition3.4.Then the map

(p∗, s) :G(Ω) → G(Λ) ∗ Ω0

is a groupoid isomorphism.

Before giving the proof of this theorem, let us use it to deduce the following two reswhich are similar to [3, 9.4.2] in the case of coverings of groupoids, and to [1, PropoA.22] in the case of coverings of small categories.

Corollary 3.8. A coveringp :Ω → Λ is connected if and only if the correspondigroupoid action(Ω0,G(Λ)) is transitive.

Proof. Ω is connected if and only ifG(Ω), equivalentlyG(Λ) ∗ Ω0, is. For any(a, v) ∈G(Λ) ∗ Ω0, we haves(a, v) = (s(a), v) andr(a, v) = (r(a), av). It follows thatG(Ω) isconnected if and only if for allu,v ∈ Ω0 there existsa ∈ G(Λ) such thatu = av, i.e., if

0
and only ifG(Λ) acts transitively onΩ .


-

-

feem

mutingram

n de-the,

e ofec-

Corollary 3.9. Let p :Ω → Λ be a covering,x ∈ Λ0, andv ∈ p−1(x). Thenp∗ maps thefundamental groupπ(Ω,v) isomorphically onto the stability groupSv of the corresponding groupoid action(Ω0,G(Λ)).

Proof. The isomorphism of Theorem 3.7 takes the morphismp∗ to the coordinate projection G(Λ) ∗ Ω0 → G(Λ). It follows thatp∗ mapsG(Ω)v 1–1 ontoG(Λ)x. Thusp∗ mapsπ(Ω,v) isomorphically ontosomesubgroup ofπ(Λ,x). For c ∈ G(Ω)v anda = p∗(c)we haveav = r(c), soc ∈ π(Ω,v) if and only if a ∈ Sv . The result follows.

In [1, Proposition A.17] (for coverings of small categories), the above injectivity op∗is asserted to follow “directly from the definition of a covering,” but to us it does not sso immediate.

Proof of Theorem 3.7. Our strategy is to present the fundamental groupoids ofΩ andΛ as path categories of augmented graphs modulo cancellation relations and comsquares, and match up the kernels. More precisely, we will build a commutative diag

P(F+)R

onto

(q∗,s) ∼=

G(Ω)

(p∗,s)

P(E+) ∗ Ω0Q∗id

ontoG(Λ) ∗ Ω0

of functors, where the left-hand vertical isomorphism takes the equivalence relatiotermined by the top horizontal functorR onto the equivalence relation determined bybottom horizontal functorQ ∗ id. This will suffice to show that(p∗, s) is an isomorphismsince the horizontal functors are surjective.

Let E be the1-skeletonof Λ, that is, the graph whose vertices coincide with thosΛ and whose edgesE1 comprise all elements ofΛ whose degree is a standard basis vtor in N

k . We need to recall a few things from [19]. Adiagram of typeE in a categoryC is a mapD :E → C which is a morphism fromE to the underlying graph ofC. Thereis a (small)path categoryP(E) and acanonical diagram∆ :E → P(E) with the uni-versal property that for every diagramD :E → C there is a unique functorT making thediagram

E∆

D

P(E)

T

C

commute. The assignmentE → P(E) is functorial from graphs to small categories. Are-lation for E is a pair(α,β) of paths inP(E) with s(α) = s(β) and r(α) = r(β). If K

is a set of relations forE, a diagramD of type E satisfiesK if D(α) = D(β) for all
(α,β) ∈ K . Let SΛ denote the set of allcommuting squaresfor Λ, i.e., relations forE of


d

be

l

ri-

ism

-by

the form(ef, gh), wheree andf are composable edges inE with orthogonal degrees ang andh are the unique edges such thatd(g) = d(f ), d(h) = d(e) andef = gh.

The augmented graphE+ = E ∪ E−1, whereE−1 denotes the inverse edges, canused to give a presentation of the fundamental groupoid; more precisely, lettingCE bethe set(e−1e, s(e)) | e ∈ E1 ∪ E−1 of cancellation relationsfor E, there is a surjectivefunctor, which we denote for this proof byQ, making the diagram

E+

∆

E

i|E

P(E+)onto

QG(Λ)

commute, such that the associated equivalence relation onP(E+) is generated byCE ∪SΛ, where∆ :E+ → P(E+) is the canonical diagram andi :Λ → G(Λ) is the canonicafunctor. In particular, the diagramQ∆ :E+ → G(Λ) satisfiesCE , i.e.,

Q∆(e−1) = Q∆(e)−1 for all e ∈ E1.

Let R :P(F+) → G(Ω) be the corresponding surjective functor for the 1-skeletonF ofthe coveringk-graphΩ . Consider the diagram

P(F+)R

q∗

G(Ω)

p∗

F+∆

q

F

p|FΩ

i

p

E+∆

E Λi

P(E+)Q

G(Λ)

The∆’s are canonical diagrams and thei ’s are canonical functors. The right-hand quadlateral commutes by functoriality ofΛ → G(Λ). The restrictionp|F is a graph mor-phism, and takesF ontoE by definition of covering and 1-skeleton. The graph morphq :F+ → E+ is the extension ofp|F defined by

q(f −1) = p(f )−1.

The inside squares commute by definition ofp|F andq. The left-hand quadrilateral commutes by functoriality ofE → P(E). The top and bottom 5-sided diagrams commuteconstruction ofR andQ. Thus

p∗R∆|F = Qq∗∆|F.


icatedap

onby

SinceQ∆ satisfiesCE , Q∆q = Qq∗∆ satisfiesCF ; sinceR∆ also satisfiesCF andp∗ isa groupoid morphism, it follows thatp∗R∆ = Qq∗∆, hencep∗R = Qq∗ by universalityof ∆.

Thus the diagram

P(F+)R

q∗

G(Ω)

p∗

P(E+)Q

G(Λ)

commutes. It is easy to see that the slightly enlarged diagram

P(F+)R

(q∗,s)

G(Ω)

(p∗,s)

P(E+) ∗ Ω0Q∗id

G(Λ) ∗ Ω0

also commutes—the sources just come along for the ride. This is the diagram indat the beginning of the proof.q∗ :P(F+) → P(E+) is a 1-graph covering, and the m(q∗, s) is the associated isomorphism from the proof of Theorem 3.5.

The horizontal functorsR andQ ∗ id are surjective, so it remains to prove that(q∗, s)takes the equivalence relation determined byR, which is generated by the cancellatirelationsCF and the commuting squaresSΩ , onto the equivalence relation determinedQ∗ id, which is generated by the pairs of the form((α, v), (β, v)), where(α,β) ∈ CE ∪SΛ.

Let (f −1f, s(f )) ∈ CF , and pute = q∗(f ) ∈ E+. We have

q∗(f −1f

) = q∗(f )−1q∗(f ) = e−1e,

ands(f −1f ) = s(f ), so

(q∗, s)(f −1f

) = (e−1e, s(f )

).

On the other hand,q∗(s(f )) = s(e), so

(q∗, s)(s(f )

) = (s(e), s(f )

).

Thus

(q∗, s)(f −1f, s(f )

) = ((e−1e, s(f )

),(s(e), s(f )

)); (3)

note that(e−1e, s(e)) is a typical element ofCE .


ed thectionce of

Now let (ab, cd) ∈ SΩ , so thata, b, c, d ∈ F 1, the common degree ofa andd is orthog-onal to the common degree ofb andc, andab = cd in thek-graphΩ . Put

e = q(a), f = q(b), g = q(c), and h = q(d).

Then

q∗(ab) = ef and q∗(cd) = gh,

andef = gh in thek-graphΛ because the diagram

F

p|FΩ

p

E Λ

commutes. On the other hand,s(ab) = s(cd), so

(q∗, s)(ab, cd) = ((ef, s(ab)

),(gh, s(ab)

)); (4)

sinceq :F → E is a covering,(ef, gh) is a typical element ofSΛ.Together, Eqs. (3) and (4) show that(q∗, s) takesCF ∪SΩ onto the set((α, v), (β, v)) |

(α,β) ∈ CE ∪ SΛ, which suffices. It follows from Theorem 3.7 that ifp :Ω → Λ is ak-graph covering, thenp∗ :G(Ω) →

G(Λ) is agroupoid coveringin the sense of [3,13].

4. Classification of transitive groupoid actions

By the results of the preceding section, to classify connected coverings we only newell-known classification of transitive groupoid actions. The results we state in this seare elementary and we claim no originality. We supply the proofs for the convenienthe reader.

Proposition 4.1. Let (V ,G) be a transitive groupoid action andx ∈ G0. Then the familySv | v ∈ Vx is a conjugacy class of subgroups ofxGx.

Proof. Just note thatSav = aSva−1 for (a, v) ∈ G ∗ V .

Proposition 4.2. Let a groupoidG act transitively on bothV andU , and letx ∈ G0, v ∈ Vx ,andu ∈ Ux . Then there is a morphism(V ,G) → (U,G) takingv to u if and only ifSv ⊂ Su.

Proof. If φ :V → U is equivariant andφ(v) = u, then

φ(av) = aφ(v) = au for all a ∈ Gx,


,

sm of

r thanmorest, we

soSv ⊂ Su. Conversely, assumeSv ⊂ Su, and defineφ :V → U by

φ(av) = au for a ∈ Gx.

This is well defined because ifa, b ∈ Gx andav = bv, then

b−1a ∈ Sv ⊂ Su,

soau = bu. Clearlyφ is equivariant. Proposition 4.3. Let (V ,G) be a transitive groupoid action,x ∈ G0, andv ∈ Vx . Then thenormalizerN(Sv) of Sv in xGx acts on the right of the action(V ,G) by automorphismsand in fact

Aut(V ,G) ∼= N(Sv)/Sv.

Proof. The computations in the proof of Proposition 4.2 show that every automorphi(V ,G) is of the formav → acv, wherec ∈ xGx satisfies

Sv = Scv = cSvc−1,

i.e., c ∈ N(Sv), and conversely every suchc gives rise to an automorphism of(V ,G) inthis manner. Define(av)c = acv. ThenN(Sv) acts on the right, since forc, d ∈ N(Sv) wehave (

(av)c)d = (acv)d = acdv = (av)cd.

Clearly(av)c = av if and only if c ∈ Sv . ThusN(Sv)/Sv acts freely on the right of(V ,G).The result follows.

Our next result is the groupoid-action analogue of Theorem 2.7. However, rathemerely asserting the existence of a certain kind of action of a groupoid, we givedetail, because this will be useful when we apply it to the analogue for coverings. Firneed the following.

Definition 4.4. A cocycleon a groupoidG is a functorη :G → G whereG is a group. Thecocycle actionof G on the Cartesian productG0 × G is given by

a(s(a), g

) = (r(a), η(a)g

).

We writeG0 ×η G to indicateG0 × G equipped with the cocycle action.

Proposition 4.5. Let G be a connected groupoid andx ∈ G0. There is a cocycleη :G →
xGx such that the associated cocycle action(G0 ×η xGx,G) is free and transitive.


sen

roup

d

ion

r-

Proof. For eachy ∈ G0 pick ty ∈ yGx, with tx = x. Then η(a) = t−1r(a)ats(a) defines a

surjective cocycleG → xGx which is the identity map onxGx.Fora ∈ yGz andg ∈ xGx we havea(z, g) = (y, η(a)g). The action is transitive, becau

if y ∈ G0 andg ∈ xGx we havetyg(x, x) = (y, g). By transitivity, to show that the actiois free it suffices to check the stability group at(x, x): for a ∈ yGx we havea(x, x) =(y, η(a)), so if a ∈ S(x,x) theny = x andη(a) = x, so a ∈ xGx, henceη(a) = a, thusa = x.

We next give a groupoid-action analogue of Theorem 2.8. First note that if a gG acts on (the right of) a groupoid action(V ,G) by automorphisms, thenG acts on thequotient setV/G by a(vG) = (av)G.

Proposition 4.6. Let (V ,G) be a free transitive groupoid action,x ∈ G0, v ∈ Vx , andH asubgroup ofxGx. Let H act on(V ,G) according to Proposition4.3. Then the associateaction(V/H,G) is transitive, andH = SvH .

Proof. The action ofG on V/H is transitive since it is a quotient of the transitive acton V . We haveV = Gv, H acts onV by (av)h = ahv, andG acts onV/H by a(vH) =avH . Thusa(vH) = vH if and only if av ∈ Hv, equivalentlya ∈ H by freeness.

5. Proofs of main results

Proof of Theorem 2.2. The corresponding groupoid actions(Ω0,G(Λ)) and(Σ0,G(Λ))

are transitive, by Corollary 3.8. We haveπ(Λ,x) = xG(Λ)x, and forv ∈ p−1(x) we havep∗π(Ω,v) = Sv , so the first statement follows from Proposition 4.1.

By Theorem 3.5 a morphismφ : (Ω,p) → (Σ,q) with φ(v) = u corresponds to a mophismψ : (Ω0,G(Λ)) → (Σ0,G(Λ)) with ψ(v) = u, and we havep∗π(Ω,v) = Sv andq∗π(Σ,u) = Su, so the second statement follows from Proposition 4.2.

The last statement now follows quickly from the above.Proposition 5.1. Let G be a group acting freely by automorphisms on the right of ak-graphΩ . Then the quotient setΩ/G becomes ak-graph with operations

s(λG) = s(λ)G, r(λG) = r(λ)G,

(λG)(µG) = (λµ)G, d(λG) = d(λ),

and the quotient mapΩ → Ω/G is a covering.

Proof. More precisely, the composition is defined as follows: ifs(λ)G = r(µ)G, then itfollows from freeness of the action that the set

αβ | α ∈ λG,β ∈ µG,s(α) = r(β)


position-of

r

ve

nt

ight

aster

. Itctors,

comprises a single orbit, which coincides with(λµ)G if we adjustλ,µ within their respec-tive orbits so that they are composable (and freeness is needed to show that the comof orbits is well defined). Routine computations show thatΩ/G is a category, and the quotient map is then a surjective functor. BecauseG acts by automorphisms, all elementsany orbitαG have the same degree, which we define to bed(αG). This gives a functod :Ω/G → N

k , and by construction the quotient map intertwines the twod ’s.We verify the factorization property: letλG ∈ Ω/G andn, l ∈ N

k with d(λG) = n + l.Thend(λ) = n + l, so there exist uniqueµ ∈ Ωn andν ∈ Ωl such thatλ = µν, and then

d(µG) = n, d(νG) = l, and λG = µGνG.

For the uniqueness ofλG andµG, suppose

d(αG) = n, d(βG) = l, and λG = αGβG.

Thend(α) = n andd(β) = l, and we can adjustβ in theG-orbit so thats(α) = r(β). ThenαGβG = (αβ)G, so there existsg ∈ G such that

(αg)(βg) = (αβ)g = λ.

But d(αg) = n andd(βg) = l, so we must haveαg = µ andβg = ν, henceαG = µG andβG = νG.

ThusΩ/G is ak-graph, and the quotient map is ak-graph morphism. Forv ∈ Ω0, thequotient map takesΩv onto(Ω/G)(vG) by construction; we must show that it is injection this set. Letλ,µ ∈ Ωv such thatλG = µG. Then there existsg ∈ G such thatλ = µg.Thus

v = s(λ) = s(µg) = s(µ)g = vg.

SinceG acts freely, we must haveg = e, henceλ = µ. Similar arguments show the quotiemap is injective on(vG)(Ω/G). Proposition 5.2. Let G be a group acting freely by covering automorphisms on the rof a k-graph coveringp :Ω → Λ. Then the mapλG → p(λ) :Ω/G → Λ is a covering.

Proof. We could deduce this from a corresponding groupoid-action result, but it is fto prove this one directly. We certainly have a commuting diagram

Ω

p

Ω/G

Λ

of surjective functors, whereΩ → Ω/G is the covering from the preceding propositionis easy to verify that, whenever we have such a commuting diagram of surjective fun
if two of the maps are coverings, then so is the third.


hism

sitive,ction

gnly

d let

ver-

For the proof of our next main result, we wish to apply the above to the automorpgroup of a connected covering. For this we need to know that this group acts freely.

Proposition 5.3. Every automorphism of a connectedk-graph covering acts freely.

Proof. Since the covering is connected, the corresponding groupoid action is tranand it is straightforward to verify that every automorphism of a transitive groupoid aacts freely. Proof of Corollary 2.4. The equivalence (i)⇔ (ii) follows quickly from Theorem 2.2.

The quotientk-graph Ω/Aut(Ω,p) is connected sinceΩ is, hence the coverinq :Ω/Aut(Ω,p) → Λ is connected. Thus this covering is an isomorphism if and oif q−1(x) = vG. Since the setq−1(x) coincides with the set of Aut(Ω,p)-orbits of ele-ments ofp−1(x), we have (ii)⇔ (iii).

Finally, for (iii) ⇔ (iv), just note that the coveringΩ/Aut(Ω,p) → Λ is the uniquemorphism from the coveringΩ → Ω/Aut(Ω,p) to the given coveringΩ → Λ. Proof of Theorem 2.5. Passing to the corresponding groupoid action(Ω0,G(Λ)), theresult follows from Proposition 4.3.

Later we will need a precise description of the action ofN(p∗π(Ω,v)) on (Ω,p) cor-responding to the action ofN(Sv) on (Ω0,G(Λ)), and we record this here: letλ ∈ Ω andc ∈ N(p∗π(Ω,v)). Since the covering(Ω,p) is connected, by Corollary 3.8G(Λ) actstransitively onΩ0, so there existsa ∈ G(Λ) such thats(λ) = av. Thenλc is the uniqueelement ofΩ such that

p(λc) = p(λ) and s(λc) = acv.

This is well defined becausec normalizesp∗π(Ω,v).

Proof of Theorem 2.7. By [19, Proposition 5.9], the fundamental groupoidG(Λ) is con-nected sinceΛ is. Proposition 4.5 gives a certain free and transitive action(V ,G(Λ)). Letp :Ω → Λ be the corresponding covering, which is connected by Corollary 3.8, anx ∈ Λ0 andv ∈ p−1(x). Then

p∗π(Ω,v) = Sv = x,

so the covering(Ω,p) is universal by Theorem 2.2.Moreover, again by Theorem 2.2, if(Σ,q) is any universal covering ofΛ, then be-

cause there is a morphism(Σ,q) → (Ω,p), we must haveq∗π(Σ,u) = x for allu ∈ q−1(x).

It will be useful to record the following alternative characterization of universal co
ings. But first:


t

) of are is atnerally

witherticesnceampleunate,hether

2 wet isid

llows

gs, andn 4.5

esired

Definition 5.4. A k-tree is a connectedk-graphΩ with π(Ω,v) = v for some, henceevery, vertexv ∈ Ω0.

Corollary 5.5. If p :Ω → Λ is a connected covering, then the following are equivalen:

(i) the covering(Ω,p) is universal;(ii) the corresponding groupoid action(Ω0,G(Λ)) is free;

(iii) Ω is a k-tree.

Proof. This follows from Theorem 2.7 sincep∗|π(Ω,v) is injective and the stabilizerSv

is p∗π(Ω,v). Remark. A 1-tree is the path category (modulo conventions regarding compositiongraph which is a tree in the usual sense. In a 1-tree, between any 2 vertices themost 1 undirected path, hence certainly at most 1 directed path; this does not gehold in k-trees, as illustrated by one of our basic examples [19, Example 7.2] of ak-graphΛ which does not embed faithfully in its fundamental groupoid. This was a 2-graph4 vertices, 4 horizontal edges, and 6 vertical edges. All multiple edges between vcollapsed under the canonical functori, making the fundamental groupoid an equivalerelation on 4 objects, thus the fundamental groups were all trivial. Hence this is an exof a 2-tree with multiple morphisms with the same source and range. This is unfortbecause it means that in practice we have no effective algorithm for determining wa givenk-graph is ak-tree, short of computing the fundamental group.

Proof of Theorem 2.8. First of all, since the covering is universal,p∗π(Ω,v) = x. Thusthe action ofH on (Ω,p) guaranteed by Theorem 2.5 is free, hence by Proposition 5.really do have a coveringq :Ω/H → Λ. Moreover, this covering is connected since ia quotient of the connected covering(Ω,p). By Corollary 5.5 the corresponding groupoaction(Ω0,G(Λ)) is free. We havexG(Λ)x = π(Λ,x), soH acts on(Ω0,G(Λ)) accord-ing to Proposition 4.3. By Proposition 4.6 we haveH = SvH . Sinceq∗π(Ω/H,vH) = SvH

by Corollary 3.9, we have shown the first part of the theorem. The other part now foimmediately from Theorem 2.2.

6. Skew products

Our statement of Theorem 2.7 merely asserted the existence of universal coverinthe proof merely showed how this existence followed from the analogous Propositiofor groupoid actions. However, Proposition 4.5 gave a specific construction of the dgroupoid action using cocycles. The analogue fork-graph coverings is a skew product.

Definition 6.1. A cocycleon ak-graphΛ is a functor fromΛ to a group.

Observation 6.2. Since a cocycleη :Λ → G is a functor into a group(oid), there is a
unique cocycleκ making the diagram


ocy-is

actions for,

r-at

Λη

i

G

G(Λ)

κ

commute. In fact, this gives a1–1correspondence between cocycles onΛ and onG(Λ).

Proposition 6.3. Let η :Λ → G be ak-graph cocycle. Then the Cartesian productΛ × G

becomes ak-graph with operations

s(α, g) = (s(α), g

), r(α, g) = (

r(α), η(α)g),(

α,η(β)g)(β, g) = (αβ,g), d(α,g) = d(α),

and the coordinate projectionΛ × G → Λ is a covering.

Proof. Let η′ :G(Λ) → G be the corresponding groupoid cocycle. The associated ccle action ofG(Λ) is on the setΛ0 × G, and the coveringk-graph corresponding to thgroupoid action isΛ ∗ (Λ0 × G). The map(

α, (x, g)) → (α, g) :Λ ∗ (

Λ0 × G) → Λ × G

is bijective, transforms thek-graph operations onΛ∗ (Λ0 ×G) into the operations onΛ×G indicated in the proposition, and transforms the corresponding coveringΛ ∗ (Λ0 × G)

→ Λ into the coordinate projectionΛ × G → Λ. The following definition is a variation of [18, Definition 5.1].

Definition 6.4. The skew productk-graph associated to a cocycleη :Λ → G is Λ × G

with the operations from Proposition 6.3. We writeΛ ×η G to indicate thisk-graph. Theskew-product coveringis the coordinate projectionpΛ :Λ ×η G → Λ.

Note that in the proof of the above Proposition 6.3, the skew-product coveringΛ ×η

G → Λ was not exactly the same as the covering corresponding to the cocycle(G(Λ)0 ×η G,G(Λ)), rather these coverings were merely isomorphic—neverthelesconvenience we regard the skew-product covering ascorrespondingto the cocycle actionthus committing a mild abuse.

We can now apply this to construct universal coverings.

Corollary 6.5. LetΛ be a connectedk-graph andx ∈ Λ0. Then there is a cocycleη :Λ →π(Λ,x) such that the skew-product coveringΛ ×η π(Λ,x) → Λ is universal.

Proof. In the proof of Proposition 4.5 we constructed a groupoid cocycleG(Λ) →π(Λ,x); let η :Λ → π(Λ,x) be the associatedk-graph cocycle. The skew-product coveing corresponds to the cocycle action ofG(Λ), and the proof of Theorem 2.7 showed th
this corresponding covering is universal.


-

sal

d by

g

g

Proposition 6.6. Let η :Λ → G be ak-graph cocycle. ThenG acts freely on the skewproduct coveringΛ ×η G → Λ via

(λ, g)h = (λ, gh) for λ ∈ Λ,g,h ∈ G.

Proof. This can be checked directly without pain, but it is even easier to note thatG actson the corresponding groupoid action(Λ0 × G,G(Λ)) by

(x, g)h = (x, gh) for x ∈ Λ0, g,h ∈ G,

and then apply Corollary 3.6. It is obvious that the action is free.For a connectedk-graphΛ, Corollary 6.5 gives a specific construction of a univer

coveringΛ×η π(Λ,x) → Λ, so Theorem 2.5 gives an action ofπ(Λ,x) on this covering.In the following result we verify that this action coincides with the one guaranteeProposition 6.6.

Proposition 6.7. LetΛ be a connectedk-graph, and letΛ×η π(Λ,x) → Λ be the univer-sal covering as in Corollary6.5. Then the action ofπ(Λ,x) on the skew-product coverinΛ×η π(Λ,x) → Λ given by Theorem2.5agrees with the action given by Proposition6.6.

Proof. The corresponding action ofG(Λ) is on (Λ ×η π(Λ,x))0 = Λ0 × π(Λ,x). Let’ssee how the proof of Theorem 2.5 tells usπ(Λ,x) acts onΛ ×η π(Λ,x) → Λ. Denotingthe skew-product covering mapΛ ×η π(Λ,x) → Λ by pΛ, we must start by choosina vertexv ∈ p−1

Λ (x). Then the action of an elementh ∈ π(Λ,x) on an element(λ, g) ∈Λ ×η π(Λ,x) is computed as follows: findb ∈ G(Λ)x such thatbv = s(λ, g), and then(λ, g)h is the unique element ofΛ ×η π(Λ,x) such that both

pΛ

((λ, g)h

) = pΛ(λ,g) = λ and s((λ, g)h

) = bhv.

For v we choose(x, x). By definition of the cocycle action, for anyb ∈ G(Λ)x we haveb(x, x) = (r(b), η(b)). Given(λ, g), puty = s(λ). Thens(λ, g) = (y, g), so we wantb ∈G(Λ)x such that (

r(b), η(b)) = (y, g).

The cocycleη constructed in the proof of Proposition 4.5 takesyG(Λ)x ontoπ(Λ,x), sosuch an elementb exists. Then for such ab we have

bh(x, x) = (r(bh), η(bh)

) = (r(b), η(b)η(h)

) = (y, gh).

Therefore(λ, g)h = (λ, gh), as desired, since

pΛ(λ,gh) = λ and s(λ, gh) = (y, gh).


vering

ts forts for

ents of

ng

show

ver-

Theorem 2.8 concerns an action of a subgroupH of the fundamental groupπ(Λ,x)

on a universal covering ofΛ; we want to see how this looks whenΛ is a skew prod-uct Λ ×η π(Λ,x) as in Corollary 6.5. It is cleaner to do it in the abstract: letη :Λ → G

be a cocycle andH a subgroup ofG, and letH act on the skew productΛ ×η G ac-cording to Proposition 6.6. Since this action is free, we can form the associated co(Λ ×η G)/H → Λ. The map

(λ, g)H → (λ, gH): (Λ ×η G)/H → Λ × (G/H)

is bijective, transforms thek-graph operations on the quotient(Λ ×η G)/H into

s(λ, gH) = (s(λ), gH

), r(λ, gH) = (

r(λ), η(λ)gH),(

λ,η(µ)gH)(µ,gH) = (λµ,gH), d(λ,gH) = d(λ),

and transforms the covering(Λ ×η G)/H → Λ into the coordinate projectionΛ ×G/H → Λ.

Definition 6.8. If η :Λ → G is ak-graph cocycle andH is a subgroup ofG, therelativeskew productk-graph, denotedΛ ×η G/H , is the Cartesian productΛ × G/H with theabove operations, and therelative skew-product coveringis the coordinate projectionΛ×η

G/H → Λ.

We should point out that this concept is not new: a version of relative skew producgraphs appears in, for example, [4,12]. While we did not need relative skew producthe general theory of coverings—for us they arose as just a particular case of quotiskew products—they will be important for us in our application toC∗-coactions.

Let’s formalize the above discussion.

Proposition 6.9. If η :Λ → G is a k-graph cocycle andH a subgroup ofG, then theassociated covering(Λ×η G)/H → Λ is isomorphic to the relative skew-product coveriΛ ×η G/H → Λ via the map

(a, g)H → (a, gH).

The value of the above definition is that it captures all connected coverings, as wein the following result, a graph version of which appeared in [4, Proposition 2.2].

Corollary 6.10. Every connected covering is isomorphic to a relative skew-product coing.

Proof. Let p :Ω → Λ be a connected covering,x ∈ Λ0, andv ∈ p−1(x). It follows fromTheorem 2.8, Corollary 6.5, and Propositions 6.7 and 6.9 that(Ω,p) is isomorphic to a
relative skew-product coveringΛ ×η π(Λ,x)/p∗π(Ω,v).


-n.4,

gs, ithee [12,proof.ience

uct

here

ion

ep

Gross–Tucker Theorem

If the subgroupH of G is normal, then a relative skew productΛ ×η G/H may beregarded as an ordinary skew product associated to the cocycleΛ → G/H obtained fromη by composing with the quotient homomorphismG → G/H . In particular, with the notation from the above proof, if the subgroupp∗π(Ω,v) of π(Λ,x) is normal, then the giveconnected covering(Ω,p) is isomorphic to a skew-product covering. From Corollary 2we know that this will happen if and only if Aut(Ω,p) acts transitively onp−1(x). Onesituation where this is obviously true is for a coveringΩ → Ω/G, whereG is a groupacting freely on a connectedk-graphΩ .

While all this is a nice application of the general theory of connected coverincheats us out of the full truth: connectedness ofΩ is unnecessary, as we will show in tfollowing result, a version of the Gross–Tucker Theorem (for the graph version, seTheorem 2.2.2]). In the disconnected case it is more efficient to give a “bare-hands”Actually, this result appears in [18, Remark 5.6], but we prove it here for the convenof the reader, since we will need this more general result forC∗-coactions.

Theorem 6.11 (Gross–Tucker Theorem). LetG be a group acting freely on ak-graphΣ .Then the coveringΣ → Σ/G given by the quotient map is isomorphic to a skew-prodcovering(Σ/G) ×η G → Σ/G.

Proof. The corresponding groupoid-action result, Lemma 6.12 below, is easier, sowe merely indicate how the Gross–Tucker Theorem will follow. PutΛ = Σ/G, and lett :Σ → Λ be the quotient map. ThenG is a subgroup of Aut(Σ, t) acting freely andtransitively on each sett−1(x) for x ∈ Λ0. By Lemma 6.12, the associated groupoid act(Σ0,G(Λ)) is isomorphic to a cocycle action(Λ0 ×η G,G(Λ)), so the coveringt :Σ → Λ

is isomorphic to a skew-product coveringΛ ×η G → Λ. We must pay the debt we incurred in the above proof.

Lemma 6.12. LetG be a group acting on the right of a groupoid action(V ,G), freely andtransitively on each setVx for x ∈ G0. Then(V ,G) is isomorphic to a cocycle action.

Proof. We begin by choosing a cross-section of the mapV → G0: for eachx ∈ G0 pickvx ∈ Vx . Letx, y ∈ G0 anda ∈ xGy. Then bothavy andvx are inVx , so by hypothesis therexists a unique elementη(a) of G such thatvxη(a) = avy . We verify that the resulting maη :G → G is a cocycle: ifx, y, z ∈ G0, a ∈ xGy, andb ∈ yGz, then

vxη(a)η(b) = avyη(b) = abvz = vxη(ab),

soη(a)η(b) = η(ab) sinceG acts freely.Defineφ :G0 ×η G → V by φ(x,g) = vxg. To see thatφ is injective, let(x, g), (y,h) ∈

G0 × G, and assumeφ(x,g) = φ(y,h). Then

vyhg−1 = vx ∈ Vx.


usualenessrality,

theto the

of

Sincevy ∈ Vy , so isvyhg−1. Thus we must havex = y, henceg = h sinceG acts freely.To see thatφ is surjective, letx ∈ G0 andv ∈ Vx . SinceG acts transitively onVx , we canchooseg ∈ G such thatv = vxg, and thenv = φ(x,g). Forx, y ∈ G0, a ∈ xGy, andg ∈ G

we have

φ(a(y, g)

) = φ(x,η(a)g

) = vxη(a)g = avyg = aφ(y, g),

so φ intertwines the cocycle action and the given action. Thereforeφ : (G0 ×η G,G) →(V ,G) is an isomorphism.

7. Coactions

In this section we discuss the implications of our results for theC∗-algebras ofk-graphs.There are by now several different classes ofk-graphsΛ whoseC∗-algebrasC∗(Λ) admita satisfactory structure theory, and our results apply to all of them. Indeed, of thetheory we need to know only that the core is AF and that the Gauge-Invariant UniquTheorem holds. Thus the results of this section apply to, in increasing order of genethe row-finitek-graphs without sources of [18], the locally convex row-finitek-graphsof [23], and the finitely alignedk-graphs of [24].

The main point of [15] is that a labelling of a graph gives rise to a coaction ongraphC∗-algebra, and that, moreover, the coaction crossed product is isomorphicC∗-algebra of the skew-product graph. Here we adapt this tok-graphs.

For C∗-coactions we adopt the conventions of [6,7,21,22]. Acoactionof a groupG

on aC∗-algebraA is an injective nondegenerate homomorphismδ of A into the spatialtensor productA ⊗ C∗(G) satisfying thecoaction identity(id ⊗ δG)δ = (δ ⊗ id)δ, whereδG is the comultiplication onC∗(G). For g ∈ G the associatedspectral subspaceof A isAg := a ∈ A | δ(a) = a ⊗ g, and thefixed point algebrais Aδ := Ae. The disjoint unionA := ⊔

g∈G Ag is a Fell bundlein the sense thatAgAh ⊂ Agh andA∗g = Ag−1, and the

linear spanΓc(A) is a dense∗-subalgebra ofA. Defineρ :A → G by ρ(a) = g if a ∈Ag .The coactionδ is calledmaximalif the norm ofA is the largestC∗-norm on the∗-algebraΓc(A) [6], andnormalif (id ⊗ λ)δ is injective, whereλ is the left regular representationG [21]. For a subgroupH of G, the Cartesian productA× G/H is a Fell bundle over thetransformation groupoidG × G/H , with operations(

a,ρ(b)gH)(b, gH) = (ab, gH) and (a, gH)∗ = (

a∗, ρ(a)gH),

and the linear spanΓc(A × G/H) is a ∗-algebra, whose completionA ×δ| G/H in thelargestC∗-norm is therestricted crossed productof A by δ. WhenH = e thedual actionof G on the crossed productA ×δ G is given byδh(a, g) = (a, gh−1).

Let η :Λ → G be ak-graph cocycle. The right action ofG on Λ ×η G discussed inProposition 6.6 induces an actionγ of G onC∗(Λ ×η G) such that( )

γh s(λ,g) = s(λ,gh−1).


and

or-tity,

d

er

Theorem 7.1. Letη :Λ → G be ak-graph cocycle andH a subgroup ofG. Then:

(i) there exists a unique coactionδ = δη of G onC∗(Λ) such that

δ(sλ) = sλ ⊗ η(λ) for λ ∈ Λ;

(ii) C∗(Λ ×η G/H) ∼= C∗(Λ) ×δ| G/H ;(iv) if H = e the above isomorphism is equivariant for the actionγ of G onC∗(Λ×η G)

and the dual actionδ onC∗(Λ) ×δ G;(v) the coactionδ is both maximal and normal.

Our desired extension of [4, Theorem 3.2] follows immediately from Theorem 7.1Corollary 6.10.

Corollary 7.2. Let p :Ω → Λ be a connected covering,x ∈ Λ0, and v ∈ p−1(x). Thenthere exists a coactionδ of π(Λ,x) onC∗(Λ) such that

C∗(Ω) ∼= C∗(Λ) ×δ| π(Λ,x)/p∗π(Ω,v).

Proof of Theorem 7.1. (i) It is routine to verify that the assignmentλ → sλ ⊗ η(λ) givesa Cuntz–KriegerΛ-family in C∗(Λ) ⊗ C∗(G), and hence determines a unique homomphismδ :C∗(Λ) → C∗(Λ)⊗C∗(G); δ is nondegenerate and satisfies the coaction idenand the Gauge-Invariant Uniqueness Theorem shows thatδ is injective.

(ii) Define θ :Λ ×η G/H → C∗(Λ) ×δ| G/H by

θ(λ, gH) = (sλ, gH).

It is routine to verify that this gives a Cuntz–Krieger(Λ×η G/H)-family in theC∗-algebraC∗(Λ) ×δ| G/H , hence determines a homomorphism

θ :C∗(Λ ×η G/H) → C∗(Λ) ×δ| G/H.

The Gauge-Invariant Uniqueness Theorem shows thatθ is injective, and it is obviouslysurjective.

(iii) For the equivariance,

θγh

(s(λ,g)

) = θ(s(λ,gh−1)

) = (sλ, gh−1) = δh(sλ, g) = δhθ

(s(λ,g)

).

(iv) We first show that the coactionδ is maximal. LetA be the Fell bundle associateto the coactionδ. Since the spectral subspaces are linearly independent inC∗(Λ), Γc(A)

sits insideC∗(Λ) as a∗-subalgebra, giving an obvious representation ofA in C∗(Λ),which, in turn, extends uniquely to a homomorphismπ :C∗(A) → C∗(Λ). For maximalityit suffices, by [6, Proposition 4.2], to show thatπ is injective. The inclusionΛ →A gives amapρ0 :Λ → C∗(A); the image is a Cuntz–KriegerΛ-family, because the Cuntz–Krieg
relations can be expressed within the Fell bundleA. Thus there is a unique homomorphism


-ss

ar

o-uired.

d

aveion

ρ :C∗(Λ) → C∗(A) such thatρ(sλ) = ρ0(λ) for λ ∈ Λ. Forλ ∈ Λ we haveπρ0(λ) = λ. Itfollows thatπ is injective sinceρ is a right inverse.

The homomorphismπ := (id ⊗ λ)δ intertwines the gauge actionα and the tensorproduct actionα⊗ id, andπ(pv) = 0 for every vertexv, so the Gauge-Invariant UniqueneTheorem implies thatπ is faithful. Thusδ is normal. Corollary 7.3. For the Fell bundleA of the coactionδ :C∗(Λ) → C∗(Λ) ⊗ C∗(G), wehaveC∗(A) = C∗

r (A) (so thatA is amenable in the sense of Exel[10]).

Proof. The maximality ofδ says thatC∗(A) = C∗(Λ), and the normality that the regulrepresentation(id ⊗ λ) δ is an isomorphism ofC∗(Λ) onto

C∗r (A) := range

((id ⊗ λ) δ

).

Decomposition

We apply Theorem 7.1 to give an analogue fork-graphs of Green’s decomposition therem [11, Proposition 1] in which the subgroup need not be normal and no twist is req

Corollary 7.4. Letη :Λ → G be ak-graph cocycle,δ = δη the associated coaction ofG onC∗(Λ), andH a subgroup ofG. Then there is a coactionε of H on the restricted crosseproductC∗(Λ) ×δ| G/H such that

C∗(Λ) ×δ| G/H ×ε H ∼= C∗(Λ) ×δ G,

equivariantly for the dual actionε and the restricted dual actionδ|H .

Proof. Since our aim is to apply Theorem 7.1, we need a cocycle.H acts freely onΛ×η G,and we have

(Λ ×η G)/H ∼= Λ ×η G/H.

Thus by Gross–Tucker Theorem 6.11 (twice!) there is a cocycleκ :Λ ×η G/H → H suchthat

Λ ×η G/H ×κ H ∼= Λ ×η G.

By Theorem 7.1, lettingδκ denote the corresponding coaction ofH on C∗(Λ ×η G/H),we have

C∗(Λ ×η G/H) ×δκ H ∼= C∗(Λ) ×δ G,

equivariantly forδκ andδ|H . Appealing to the Gross–Tucker Theorem once more we hC∗(Λ ×η G/H) ∼= C∗(Λ)×δ| G/H ; this isomorphism is equivariant for a unique coact

∗
ε of H onC (Λ) ×δ| G/H , and the result follows.


s thel resultrossedthat

t.

ty, and

tion ofhe

rons

a

Remark. For 1-graphs, Corollary 7.4 reduces to [4, Corollary 3.6], except that it usefull rather than reduced crossed product. Since [4, Corollary 3.6] motivated a generafor decompositions of crossed products by normal coactions using the reduced cproduct [4, Theorem 4.2], it is tempting to conjecture on the basis of Corollary 7.4there is a similar decomposition for maximal coactions using the full crossed produc

The next corollary extends [18, Theorem 5.7].

Corollary 7.5. With the above hypotheses, andγ the action ofG onC∗(Λ×η G) describedbefore Theorem7.1, we have

C∗(Λ ×η G) ×γ | H ∼= C∗(Λ ×η G/H) ⊗K(l2(H)

).

Proof. We have:

C∗(Λ ×η G) ×γ | H ∼= C∗(Λ) ×δ G ×δH

∼= (C∗(Λ) ×δ| G/H

) ×ε H ×ε H

∼= (C∗(Λ) ×δ| G/H

) ⊗K(l2(H)

)∼= C∗(Λ ×η G/H) ⊗K

(l2(H)

),

where we successively applied: Theorem 7.1, Corollary 7.4, crossed-product dualiTheorem 7.1 again. Cohomology

The theories of both graphs and groupoids (see, e.g., [12,20,25]), contain a nocohomology of cocycles. This is easily adapted tok-graphs, and has ramifications for tassociated coverings and coactions: we call cocyclesη, κ :Λ → G cohomologousif thereexists a mapx → τx :Λ0 → G such that

τxη(a) = κ(a)τy for all a ∈ xΛy.

If we regardη andκ as functors then the mapτ is just a natural isomorphism fromη to κ .It is routine to verify that the map(x, g) → (x, τxg) gives ak-graph isomorphismΛ ×η

G ∼= Λ×κ G which is equivariant for the associated actions ofG, and the unitary multiplie∑x∈Λ0(x ⊗ τx) implements an exterior equivalence between the associated coactiδη

andδκ .

The gauge coaction

We can view the degree functor as a cocycled :Λ → Zk . By Theorem 7.1 there is

unique coactionδ = δd of Zk onC∗(Λ) such that

δ(sλ) = sλ ⊗ d(λ) for λ ∈ Λ.


The-

e7]

a

the

)

hen,

ell

e

al

causeAF isns doult

We call δ the gauge coactionbecause the corresponding action ofTk = Zk is the usual

gauge action.WhenΛ is row-finite and has no sources, the following result is contained in [18,

orem 5.5].

Theorem 7.6. Suppose thatΛ is a countable finitely alignedk-graph. ThenC∗(Λ) isnuclear, andC∗(Λ ×d Z

k) is AF.

Proof of nuclearity. The fixed-point algebraC∗(Λ)δ is the core, which is AF (see thproof of [24, Theorem 3.1]). ThusC∗(Λ)δ is in particular nuclear, and [22, Corollary 2.1implies thatC∗(Λ) is also nuclear.

We will prove that is AF by proving thatC∗(Λ) ×δ Zk , the isomorphic algebr

C∗(Λ ×d Zk) is AF. The proof would not be hard if we hadsaturation(see below for

the definition), for then the crossed product would be Morita–Rieffel equivalent tofixed-point algebra. However, in the general case we require a digression.

Recall from [22] that anideal propertyis a propertyP of C∗-algebras such that (1every C∗-algebra has a largest ideal withP , (2) P is inherited by ideals, and (3)P isinvariant under Morita–Rieffel equivalence. The motivation for this definition was tand remains for us here, that ifδ is a coaction of a discrete groupG on aC∗-algebraA,then for any ideal propertyP , the crossed productA ×δ G hasP if and only if the fixed-point algebraAδ does. It is shown in [22] that nuclearity is an ideal property, and it is wknown that liminality and postliminality are ideal properties.

Proposition 7.7. Among separableC∗-algebras, AF is an ideal property.

Proof. For invariance under Morita–Rieffel equivalence, letA ∼ B with A being AF.SinceA and B are separable, we haveA ⊗ K ∼= B ⊗ K. SinceA is AF, so isA ⊗ K,henceB ⊗K. Thus the hereditary subalgebraB is also AF, by [9, Theorem 3.1]. The samresult of Elliott shows that AF is inherited by ideals.

We finish by showing that everyC∗-algebraA has a largest AF ideal, i.e., an AF idewhich contains every AF ideal. Claim: ifI andJ are AF ideals ofA, then the idealI + J

is AF. Since

(I + J )/I ∼= I/(I ∩ J ),

the quotient(I + J )/I is AF. Thus the extensionI + J of (I + J )/I by I is AF, byresults of Brown [2] and Elliott [9]. We pause to make this reference more precise, bethe required result must be pieced together. Elliott proved in [9, Corollary 3.3] thatclosed under extensions provided projections lift, and Brown proved that projectioindeed lift (from an AF quotient by an AF ideal)—actually, the full proof of Brown’s resis in [8, Section 9].

Now let I be the closed span of all AF ideals ofA. ThenI is certainly an ideal ofA. Bythe above,I is the closure of an upward-directed union of AF ideals. ThereforeI is AF.
By construction, every AF ideal ofA is contained inI .


.

tral

We are now ready to prove that the crossed product by the gauge coaction is AF.

Back to the proof of Theorem 7.6. SinceC∗(Λ)δ is AF, C∗(Λ) ×δ Zk ∼= C∗(Λ ×d Z

k),and AF is an ideal property, the result follows from [22, Corollary 2.17].

When k = 1, the skew-product graph has no cycles, so itsC∗-algebra is AF by, forexample, [5, Corollary 2.13]. If we had a corresponding result concerning cycles fork > 1,this would give an alternate proof of the second part of Theorem 7.6—but we do not

Saturation

Let δ = δd be the gauge coaction ofZk on C∗(Λ). Recall from [22] thatδ is calledsaturatedif

C∗(Λ)n+m = spanC∗(Λ)nC∗(Λ)m for everyn,m ∈ Z

k,

or equivalently if

C∗(Λ)δ = spanC∗(Λ)nC∗(Λ)∗n for everyn ∈ Z

k.

If δ is saturated thenC∗(Λ)δ is Morita–Rieffel equivalent toC∗(Λ) ×δ Zk [22].

Recall that a vertexv of Λ is called asourceif vΛn = ∅ for somen ∈ Nk , and similarly

asink if someΛnv is empty. The following result generalizes [17, Proposition 2.8].

Proposition 7.8. Let δ be the gauge coaction ofZk onC∗(Λ).

(i) If δ is surjective, in particular ifΛ has either no sources or no sinks, every specsubspaceC∗(Λ)n for n ∈ Z

k is nontrivial.(ii) If Λ is row-finite and has neither sources nor sinks, thenδ is saturated.

Proof. (i) Let n ∈ Zk . Choosel ∈ N

k with n+ l 0, thenj ∈ Nk with j n+ l andj l,

and thenλ ∈ Λj . We can factor

λ = µν = αβ with d(ν) = n + l andd(β) = l.

Thus

0 = sλs∗λ = sµsνs

∗βs∗

α,

so thatsνs∗β is a nonzero element of

C∗(Λ)n+lC∗(Λ)∗l ⊂ C∗(Λ)n.

(ii) Now assume thatΛ is row-finite and has neither sources nor sinks. Letl ∈ Zk . To

see thatδ is saturated, we must show that

C∗(Λ)δ ⊂ spanC∗(Λ)lC∗(Λ)∗l .


es for

d

By Lemma 7.9 below, we have

C∗(Λ)δ = C∗(Λ)0 = spansλs

∗µ | d(λ) = d(µ), s(λ) = s(µ)

,

so it suffices to show that if

d(λ) = d(µ) = n and s(λ) = s(µ) = v,

thensλs∗µ ∈ spanC∗(Λ)lC

∗(Λ)∗l . Choosem ∈ Nk with m l andm n. SinceΛ is row-

finite and has no sources,

pv =∑

α∈vΛm−n

sαs∗α.

SinceΛ has no sinks, for eachα ∈ vΛm−n we can chooseνα ∈ Λm−ls(α). Thenps(α) =s∗να

sνα , so

sλs∗µ = sλpvs

∗µ

=∑

α∈vΛm−n

sλsαs∗να

sνα s∗αs∗

µ

∈ spanC∗(Λ)nC∗(Λ)m−nC

∗(Λ)∗m−lC∗(Λ)m−lC

∗(Λ)∗m−nC∗(Λ)∗n

⊂ spanC∗(Λ)lC∗(Λ)∗l .

In the above proof, we applied the following characterization of spectral subspacthe gauge coaction.

Lemma 7.9. Letη :Λ → G be a cocycle on thek-graphΛ, and letδ = δη be the associatecoaction ofG onC∗(Λ). Then for allg ∈ G,

C∗(Λ)g = spansλs

∗µ | η(λ)η(µ)−1 = g

.

Proof. Obviously a productsλs∗µ is in C∗(Λ)g if and only if η(λ)η(µ)−1 = g, so the left-

hand side contains the right.Recall from [22] that there is a bounded linear projection

Eg = (id ⊗ χg

)δ :C∗(Λ) → C∗(Λ)g,

where here the characteristic functionχg is regarded as a linear functional onC∗(G).Any a ∈ C∗(Λ)g can be approximated by a linear combination

∑n1 cisλi

s∗µi

, and then

a = Eg(a) ≈ Eg

(n∑1

cisλis∗µi

)=

n∑1

ciEg

(sλi

s∗µi

) =∑

cisλis∗µi

| η(λi)η(µi)−1 = g

,

which is in the right-hand side. The result follows.


to

ity ofr their

r Al-ce, RI,

-

135.

9)

ovi-

)

2, Van

Math.

tor The-

Example 7.10. Row-finiteness is necessary in Proposition 7.8(ii). To see this, letΛ be the1-graph

•· · · • • •v

•∞u

• • · · ·

in which there are infinitely many edges fromu to v (and the graph extends indefinitelythe right and left). The projectionpv is in the fixed-point algebraC∗(Λ)δ , and cannot beapproximated in norm by a linear combination of productsses

∗f for edgese, f ∈ vΛu. It

follows that

C∗(Λ)δ = spanC∗(Λ)1C∗(Λ)∗1,

soδ is not saturated.

Acknowledgments

Most of this research was done while the second author visited the UniversNewcastle, and he thanks his hosts, particularly Iain Raeburn and David Pask, fohospitality.

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