Almost finiteness and homology of certain non-free actions
aa r X i v : . [ m a t h . OA ] J u l ALMOST FINITENESS AND HOMOLOGY OF CERTAINNON-FREE ACTIONS
EDUARD ORTEGA AND EDUARDO SCARPARO
Abstract.
We show that Cantor minimal
Z ⋊ Z -systems and essentially freeamenable odometers are almost finite. We also compute the homology groupsof Cantor minimal Z⋊Z -systems and show that the associated transformationgroupoids satisfy the HK conjecture if and only if the action is free. Introduction
The property of almost finiteness for ample groupoids was introduced by Matuiand applied to questions in groupoid homology in [13]. Furthermore, in [2], [10],[11] and [19], this property was applied to problems in classification of C ∗ -algebras.Kerr and Szab´o in [11] and Suzuki in [19] observed that it is a consequence of workof Downarowicz and Zhang [5] that any free action of a group with subexponentialgrowth on the Cantor set is almost finite. Moreover, free odometers arising fromsequences of finite index normal subgroups of an amenable group were shown to bealmost finite by Kerr in [10].On the other hand, there exist interesting examples of non-free odometers. Forexample, in [18], certain non-free Z ⋊ Z -odometers were shown to be counterex-amples to the HK conjecture , which is a conjecture posted by Matui in [14] thatrelates the homology groups of an ample groupoid G with the K-theory of its re-duced C ∗ -algebra.In Section 2 of this work, we give two different proofs that these Z⋊Z -odometersare almost finite. First, we prove that an amenable odometer is essentially free ifand only if it is almost finite. The forward implication uses the fact proven byKar et al ([8]) that for such odometers the acting group admits a Følner sequenceconsisting of complete sets of coset representatives.In the case of the Z ⋊ Z -odometers mentioned above, this can be done usinga specific Følner sequence for Z ⋊ Z (Example 2.7), which we employ next forshowing that any Cantor minimal Z ⋊ Z -system is almost finite.In Section 3, we compute the homology groups of Cantor minimal Z⋊Z -systemsand conclude that these systems satisfy the HK conjecture if and only if the actionis free. It should come as no surprise that, in determining the homology groups, weare able to mainly follow the ideas introduced by Bratteli et al in [3] and Thomsenin [21] in their computation of the K-theory of the associated crossed productssince, although the HK conjecture is now known to be false, it has been verifiedin many cases (see the work of Proietti and Yamashita [16, Remark 4.7] and thereferences therein). This work was carried out during the tenure of an ERCIM ‘Alain Bensoussan’ FellowshipProgramme. Almost finiteness
In this section, we review some terminology about ´etale groupoids and verifyalmost finiteness for certain classes of non-principal groupoids.2.1.
Almost finite groupoids.
Let G be a Hausdorff ´etale groupoid with rangeand source maps denoted by r and s . A bisection is a subset S ⊂ G such that r | S and s | S are injective maps.Let G ′ := { g ∈ G : r ( g ) = s ( g ) } and G (0) be the unit space of G . Then G issaid to be principal if G ′ = G (0) and effective if the interior of G ′ equals G (0) . The orbit of a point x ∈ G (0) is the set G ( x ) := r ( s − ( x )). We say that G minimal ifthe orbit of each x ∈ G (0) is dense in G (0) . Also G is said to be ample if its unitspace is totally disconnected. Example 2.1.
Let X be a locally compact Hausdorff space and let α : Γ y X be an action of a discrete group Γ on X . Given g ∈ Γ and x ∈ X we will denoteby gx := α ( g )( x ). As a space, the transformation groupoid G associated with α is G := Γ × X equipped with the product topology. The product of two elements( h, y ) , ( g, x ) ∈ G is defined if and only if y = gx , in which case ( h, gx )( g, x ) :=( hg, x ). Inversion is given by ( g, x ) − := ( g − , gx ). The unit space G (0) is naturallyidentified with X . Note that G is principal if and only if α is free ( gx = x ⇒ g = e ).If X is totally disconnected, then G is ample.Let G be an ample groupoid with compact unit space. A subgroupoid K ⊂ G is said to be elementary if K is compact-open, principal and K (0) = G (0) . Thegroupoid G is said to be almost finite if for any compact subset C ⊂ G and ǫ > K ⊂ G elementary subgroupoid such that, for any x ∈ G (0) , we have that | CKx \ Kx || K ( x ) | < ǫ. For compact groupoids, the following holds:
Proposition 2.2.
Let G be an ample almost finite groupoid. If G is compact, then G is principal.Proof. Compactness of G implies that there is a finite partition of G into compact-open bisections. Hence, there is M > x ∈ G (0) , we have that | G ( x ) | < M .If G is not principal, there is g ∈ G \ G (0) such that r ( g ) = s ( g ). Then, for anyelementary subgroupoid K ⊂ G , we have that g / ∈ K and | gKs ( g ) \ Ks ( g ) || K ( s ( g )) | > M .
Therefore, G is not almost finite. (cid:3) Given an ´etale groupoid G with compact unit space, denote by M ( G ) the set of G -invariant probability measures on G (0) . Remark 2.3.
It was observed in [13, Remark 6.6] that if G is an almost finitegroupoid, then M ( G ) = ∅ and, given µ ∈ M ( G ), we have that(1) µ ( r ( U ∩ ( G ′ \ G (0) ))) = 0 , for any U ⊂ G compact-open bisection. LMOST FINITENESS AND HOMOLOGY OF CERTAIN NON-FREE ACTIONS 3
Almost finite actions.
Let us recall the characterization of almost finitenessfor transformation groupoids presented in [11] and [19]. We will restrict ourselvesto actions on totally disconnected spaces.Let Γ be a group acting on a compact Hausdorff totally disconnected space X .A clopen tower is a pair ( V, S ) consisting of a clopen subset V of X and a finitesubset S of Γ such that the sets sV for s ∈ S are pairwise disjoint. The set S issaid to be the shape of the tower. A clopen castle is a finite collection of clopentowers { ( V i , S i ) } i ∈ I such that the sets S i V i = { gV i : g ∈ S i } for i ∈ I are pairwisedisjoint.Given ǫ > K ⊂ Γ finite, we say that a finite set F ⊂ Γ is (
K, ǫ ) -invariant if | KF △ F | < ǫ | F | .Let G be the transformation groupoid associated to the action of Γ on X . Then G is almost finite if and only if, given K ⊂ Γ finite and ǫ >
0, there is a clopencastle which partitions X , and whose shapes are ( K, ǫ )-invariant (see [19, Lemma5.2] for a proof of this fact). In this case, we say that the action is almost finite .Recall that an action of a group Γ on a locally compact Hausdorff space X is saidto be minimal if the orbit of any x ∈ X is dense in X . Clearly, this is equivalentto the associated transformation groupoid being minimal. It is also equivalentto every open (or closed) Γ-invariant set being trivial. If the action is minimal,then any f ∈ C ( X, Z ) which is Γ-invariant must be constant. We will say that ahomeomorphism ϕ on X is minimal if the Z -action induced by ϕ is minimal. By a Cantor minimal Γ -system , we mean a minimal action of Γ on the Cantor set.Given Γ a group acting on a compact Hausdorff space X , we denote by M Γ ( X )the set of Γ-invariant probability measures on X . For g ∈ Γ, let Fix g ⊂ X be theset of points fixed by g .If µ ∈ M Γ ( X ), we say that the action of Γ on ( X, µ ) is essentially free if, for any g ∈ Γ \ { e } , we have µ (Fix g ) = 0. The action is said to be topologically free if theinterior of Fix g is empty for each g ∈ Γ \ { e } . Notice that topological freeness isequivalent to effectiveness of the associated transformation groupoid.Also observe that if Γ y X is a minimal action, then any µ ∈ M Γ ( X ) has fullsupport. Therefore, for minimal actions, essential freeness of Γ y ( X, µ ) impliestopological freeness of Γ y X . Remark 2.4. If G is a transformation groupoid associated to an action of a groupΓ on a compact Hausdorff space X and µ ∈ M ( G ) ≈ M Γ ( X ), then the condition in(1) is equivalent to the action of Γ on ( X, µ ) being essentially free. It follows fromRemark 2.3 that if the action is almost finite, then Γ y ( X, µ ) is essentially free.2.3.
Odometers.
Let Γ be a group and (Γ i ) i ∈ N a sequence of finite index subgroupsof Γ such that, for every i ∈ N , Γ i (cid:13) Γ i +1 .For each i ∈ N , let p i : Γ / Γ i +1 → Γ / Γ i be the surjection given by(2) p i ( γ Γ i +1 ) := γ Γ i , for γ ∈ Γ . Let X := lim ←− (Γ / Γ i , p i ) = { ( x i ) ∈ Q Γ / Γ i : p i ( x i +1 ) = x i , ∀ i ∈ N } . Then X ishomeomorphic to the Cantor set and Γ acts in a minimal way on X by γ ( x i ) :=( γx i ), for γ ∈ Γ and ( x i ) ∈ X . This action is called an odometer .Given j ≥ g Γ j ∈ Γ / Γ j , let U ( j, g Γ j ) := { ( x i ) ∈ X : x j = g Γ j } . Then { U ( j, g Γ j ) : j ∈ N , g Γ j ∈ Γ / Γ j } is a basis for X consisting of compact-open sets.Notice that X admits a unique Γ-invariant probability measure. Hence, thereis no ambiguity in calling an odometer action essentially free. It is known (see, EDUARD ORTEGA AND EDUARDO SCARPARO for example, [7, (1.6)]) that the odometer is essentially free if and only if, for each g ∈ Γ \ { e } , we have lim i |{ x ∈ Γ / Γ i : gx = x }|| Γ / Γ i | = 0 . Let us now give a characterization of almost finite odometers. First recall thata Følner sequence ( F n ) for Γ is a sequence of finite subsets of Γ such that for everyfinite g ∈ Γ we have that | gF n ∆ F n || F n | → Theorem 2.5 ([8],[9]) . Let Γ be a countable amenable group and Γ y X :=lim ←− Γ / Γ i an odometer. The following conditions are equivalent: (i) The action is essentially free; (ii)
There is a Følner sequence ( F n ) for Γ such that each F n is a complete setof representatives for Γ / Γ n ; (iii) The action is almost finite; (iv)
There is a unique tracial state on C ( X ) ⋊ Γ .Proof. The equivalence of (i) and (iv) is a consequence of [9, Corollary 2.8]. That(iii) implies (i) is a consequence of Remark 2.4. The implication from (i) to (ii) isthe content of [8, Theorem 7]. Let us show then that (ii) implies (iii).Take C ⊂ Γ × X compact and ǫ >
0. We are going to find K ⊂ Γ × X elementarysubgroupoid such that, for each x ∈ X , | CKx \ Kx | K ( x ) < ǫ. By enlarging C , we may assume that C = D × X for some D ⊂ Γ finite.Take n ∈ N such that | DF n \ F n || F n | < ǫ. Let K ⊂ Γ × X be K := [ σ ∈ S Fn [ γ ∈ F n { σ ( γ ) γ − } × U ( k, γ Γ k ) . It is straightforward to check that K is an elementary subgroupoid of Γ × X .Furthermore, given γ ∈ F n and x ∈ U ( k, γ Γ k ), we have that | CKx \ Kx | K ( x ) = | DF n γ − \ F n γ − || F n | < ǫ. Therefore, Γ × X is almost finite. (cid:3) A few remarks are in order about the result above:
Remark 2.6. (i) It follows from Theorem 2.5 and the results in [2] that crossedproducts associated to essentially free odometers in which the acting group is count-able and amenable are classifiable by their Elliott-invariant.(ii) It is a consequence of [12, Proposition 4.7] that if Γ y lim ←− Γ / Γ i is an almostfinite odometer, then Γ is amenable.(iii) In [18, Proposition 2.1], it was shown that an odometer Γ y X := lim ←− Γ / Γ i is topologically free if and only if, for every γ ∈ T Γ i \ { e } and j ≥
1, there exists b ∈ Γ j such that b − γb / ∈ T Γ i . LMOST FINITENESS AND HOMOLOGY OF CERTAIN NON-FREE ACTIONS 5
We do not know whether, for a countable amenable group Γ, topological freenessof a Γ-odometer implies essential freeness. If one drops the amenability assumption,there are several counterexamples in the literature (see, for example, [1, Theorem1]).
Example 2.7.
Recall that the infinite dihedral group is the semidirect product
Z ⋊ Z associated to the action of Z on Z by multiplication by − n i ) be a strictly increasing sequence of natural numbers such that n i | n i +1 ,for every i ∈ N . Define Γ := Z ⋊ Z and, for i ≥
1, Γ i := n i Z ⋊ Z .By [18, Lemma 3.2] and the fact that any element of the form ( n, ∈ Z ⋊ Z is conjugate to either (0 ,
1) or (1 , g ∈ Γ \ { e } fixes at mostfinitely many points in lim ←− Γ / Γ i . Hence, this odometer is essentially free.Let us describe a Følner sequence for Z ⋊ Z satisfying condition (ii) in Theorem2.5. Given m ∈ N , let F m ⊂ Z ⋊ Z be defined by F m = ( ([ − m , − × { } ) ∪ ([0 , m ) × { } ) , if m is even([ − m − , − × { } ) ∪ ([0 , m − ] × { } ) , if m is odd.(3)Then ( F m ) m ∈ N is a Følner sequence for Z ⋊ Z such that each F m is a set ofrepresentatives for Z⋊Z ( m Z ) ⋊Z .2.4. Almost finiteness of Cantor minimal
Z ⋊ Z -systems. Notice that anyaction α of Z ⋊ Z on a set X is given by a pair of bijections on X ( ϕ, σ ) such that σ = Id X and σϕσ = ϕ − , so that α n,i = ϕ n σ i , for ( n, i ) ∈ Z ⋊ Z .There is an isomorphism Z ⋊ Z ≃ Z ∗ Z which takes the canonical generators a, b ∈ Z ∗ Z to (1 , , (0 , ∈ Z ⋊ Z . Proposition 2.8.
Let α := ( ϕ, σ ) be a minimal action of Z ⋊ Z on the Cantorset X . The following holds:(i) The Z -action induced by ϕ is free and α is topologically free. If α is not free,then either σ or ϕσ has at least one fixed point.(ii) If the Z -action induced by ϕ is not minimal, then there exists a clopen set Y such that Y ∩ σ ( Y ) = ∅ , Y ∪ σ ( Y ) = X , ϕ ( Y ) = Y and ϕ | Y is minimal. Inparticular, α is free.Proof. (i) Suppose that the Z -action induced by ϕ is not free. Then there is n ∈ Z \ { } and x ∈ X such that ϕ n ( x ) = x , then ϕ n σ ( x ) = σϕ − n ( x ) = σ ( x ). Hence,the α -orbit of x is finite, which contradicts the fact that α is minimal. Therefore,the Z -action induced by ϕ is free.Suppose that α is not topologically free. Then there is a non-empty open set U ⊂ X and n ∈ Z such that ϕ n σ fixes U pointwise.Fix x ∈ U . By minimality of α , there is ( m, i ) ∈ Z ⋊ Z such that ϕ m σ i ( x ) ∈ U and ϕ m σ i ( x ) = x . Furthermore, by multiplying ( m, i ) on the left by ( n, i = 0.Since ϕ m ( x ) ∈ U , we get that x = ϕ n σϕ m ( x ) = ϕ − m ϕ n σ ( x ) = ϕ − m ( x ) , whichis a contradiction with the fact that ϕ m ( x ) = x . Therefore, α is topologically free.If α is not free, then there is n ∈ Z and x ∈ X such that ϕ n σ ( x ) = x . Since anyelement of the form ( n, ∈ Z ⋊ Z is conjugate to (0 ,
1) or (1 , σ or ϕσ has at least one fixed point.(ii) This is the content of [15, Proposition 2.10] (see also [21, Lemma 4.28]). (cid:3) EDUARD ORTEGA AND EDUARDO SCARPARO
The following lemma is a slight modification of [3, Lemma 1.4], and we includethe proof for the sake of completeness.
Lemma 2.9.
Let ( ϕ, σ ) be a minimal action of Z ⋊ Z on the Cantor set X and Y ⊂ X a non-empty clopen set such that σ ( Y ) = Y .Then there is a partition of Y into clopen sets Y , . . . , Y K and positive integers J < J · · · < J K such that { ϕ k ( Y i ) : i = 1 , . . . , K, k = 0 , . . . , J i − } is a partitionof X , and σϕ J i ( Y i ) = Y i for each i = 1 , . . . , K .Proof. Define λ ( y ) := min { n > ϕ n ( y ) ∈ Y } , for y ∈ Y . From Proposition 2.8,it follows that the map λ is well-defined, in the sense that for each y ∈ Y there is n > ϕ n ( y ) ∈ Y .It is easy to check that λ is continuous. Hence, it has a finite range { J , . . . , J K } ,where J < J < · · · < J K . Define Y i = λ − ( J i ) for each i . Then the sets { ϕ k ( Y i ) : i = 1 , . . . , K, k = 0 , . . . , J i − } are pairwise disjoint. From Proposition2.8, it follows that the union of these sets is ϕ and σ -invariant, hence minimalityof the action implies that this is a partition of X .Let us show that σϕ J i ( Y i ) = Y i for each i . Observe first that σϕ J i ( Y i ) ⊂ Y ,since ϕ J i ( Y i ) ⊂ Y and Y is σ -invariant.Take y ∈ σϕ J i ( Y i ). Then there is x ∈ Y i such that y = σϕ J i ( x ). In particular, ϕ J i ( y ) = σ ( x ) ∈ Y . Therefore, λ ( y ) ≤ J i .If i = 1, we must then have λ ( y ) = J . Therefore, σϕ J ( Y ) ⊂ Y . Since σϕ J isinvolutive, we conclude that σϕ J ( Y ) = Y . Arguing the same way for i = 2 , . . . , K ,the result follows. (cid:3) Theorem 2.10.
Any minimal action of
Z ⋊ Z on the Cantor set is almost finite.Proof. Let α = ( ϕ, σ ) be a minimal action of Z ⋊ Z on the Cantor set X . Fromthe fact that α is topologically free (Proposition 2.8) and [12, Lemma 2.4], weobtain that there is y ∈ X such that, if ( n, i ) , ( m, j ) ∈ Z ⋊ Z are distinct, then ϕ n σ i ( x ) = ϕ m σ j ( x ).Let Z be a clopen neighborhood of y and Y := Z ∪ σ ( Z ). Then σ ( Y ) = Y and, given N ∈ N , if we take Z sufficiently small, we can assume that the sets { Y, ϕ ( Y ) , . . . , ϕ N − ( Y ) } are disjoint. Then, by Lemma 2.9, we can partition Y intoclopen sets Y , . . . , Y K such that X = ⊔ Ki =1 ⊔ J i − k =0 ϕ k ( Y i )and σϕ J i ( Y i ) = Y i for each i = 1 , . . . , K . In particular, ϕ − l σ ( Y i ) = ϕ J i − l ( Y i ) forany l .Consider the Følner sequence ( F m ) introduced in (3), and notice that(4) X = K G i =1 G g ∈ F Ji α g ( Y i ) . Furthermore, each J i ≥ N . By taking N sufficiently big, we can make the shapesof the castle (4) arbitrarily invariant. (cid:3) LMOST FINITENESS AND HOMOLOGY OF CERTAIN NON-FREE ACTIONS 7 Homology of Cantor minimal
Z ⋊ Z -systems In this section, we compute the homology groups of Cantor minimal
Z ⋊ Z -systems.Given a group Γ and a Γ-module M , we denote by M Γ the quotient of M by thesubgroup generated by elements of the form m − mg , for m ∈ M and g ∈ Γ. Recallthat M Γ is canonically isomorphic to H (Γ , M ). If i : Λ → Γ is an embedding, wedenote the canonical map i ∗ : H ∗ (Λ , M ) → H ∗ (Γ , M ) by cor.We will use the following result about the homology of free products, whoseproof can be found in [20, Theorem 2.3]. Theorem 3.1.
Let Γ and Γ be groups and M a (Γ ∗ Γ ) -module. Then, for n ≥ , H n (Γ , M ) ⊕ H n (Γ , M ) ≃ H n (Γ ∗ Γ , M ) and there is an exact sequence −→ H (Γ , M ) ⊕ H (Γ , M ) (cor , cor) −→ H (Γ ∗ Γ , M ) −→ M (cor , − cor) −→ M Γ ⊕ M Γ (cor , cor) −→ M Γ ∗ Γ −→ X , then C ( X, Z ) has astructure of Γ-module given by f a ( x ) := f ( ax ) for every f ∈ C ( X, Z ) and a ∈ Γ. Lemma 3.2.
Let a and b be involutive homeomorphisms on the Cantor set X andsuppose that the Z -action induced by ab is minimal. Denote by A and B the abeliangroup C ( X, Z ) endowed with the Z -action given by a and b , respectively.Then (cor , cor) : H ( Z , A ) ⊕ H ( Z , B ) → H ( Z ∗ Z , C ( X, Z )) is an isomor-phism and the following sequence is exact: −→ C ( X, Z ) (cor , − cor) −→ A Z ⊕ B Z (cor , cor) −→ C ( X, Z ) Z ∗ Z −→ Proof.
By Theorem 3.1, we only need to show that(cor , − cor) : C ( X, Z ) → A Z ⊕ B Z is injective.Take f ∈ C ( X, Z ) such that (cor , − cor)( f ) = 0. This implies that there exist g, h ∈ C ( X, Z ) such that f = g − ga = h − hb . In particular, f a = f b = − f .Therefore, f ab = f . Since the Z -action induced by ab is minimal, we conclude that f is constant. Finally, as f a = − f , we must have f = 0. (cid:3) In order to apply Theorem 3.1 and Lemma 3.2 to Cantor minimal Z ∗ Z -systems,we need to compute homology groups of the form H ∗ ( Z , C ( X, Z )). Lemma 3.3.
Let a be an involutive homeomorphism on a compact, Hausdorff,totally disconnected space X . Then, for k ≥ , we have H k +1 ( Z , C ( X, Z )) ≃ C (Fix a , Z ) .Proof. By [22, Theorem 6.2.2], we have that H k +1 ( Z , C ( X, Z )) = { f ∈ C ( X, Z ) : f = f a }{ f + f a : f ∈ C ( X, Z ) } . EDUARD ORTEGA AND EDUARDO SCARPARO
Let E : { f ∈ C ( X, Z ): f = fa }{ f + fa : f ∈ C ( X, Z ) } → C (Fix a , Z ) be the map given by restriction to Fix a .Clearly, this is a well-defined homomorphism, and we will show that it is bijective.Given F ⊂ Fix a clopen, take for each x ∈ F a clopen set U x ⊂ X such that x ∈ U x , U x ∩ Fix a ⊂ F and a ( U x ) = U x . Then there exist x , . . . , x n ∈ F such that F = S ( U x i ∩ Fix a ). Let U := S U x i . Notice that a ( U ) = U . Hence, E ([1 U ]) = 1 F .Therefore, E is surjective.Let us show now injectivity of E . Take f ∈ C ( X, Z ) such that f = f a and E ([ f ]) = 0, and we will show that [ f ] = 0. We claim that we can assume that f | Fix a = 0. Indeed, let U , . . . , U n be a -invariant clopen subsets of X whose unioncover Fix a , and such that f is constant on each U i . By taking differences, we canassume that these sets are disjoint. By summing f with functions of the form2 m i U i , we get our claim.Assume then that f | Fix a = 0. We have f = P q ∈ Z \{ } q f − ( q ) . Since the supportof f does not intersect Fix a , we can, for each q ∈ Z \ { } , partition f − ( q ) as f − ( q ) = A q ⊔ a ( A q ), for some A q clopen. Hence, [ f ] = 0. (cid:3) Lemma 3.4.
Let a be an involutive homeomorphism on a compact, Hausdorff,totally disconnected space X and G a := { f ∈ C ( X, Z ) : f a = f and f (Fix a ) ⊂ Z } .Then ψ : C ( X, Z ) Z → G a [ f ] f + f a is an isomorphism and, for k ≥ , we have H k ( Z , C ( X, Z )) = 0 .Proof. Clearly, ψ is a well-defined homomorphism, and we will show that it isbijective.Given f ∈ C ( X, Z ), suppose f + f a = 0. Then f | Fix a = 0. Take A clopenneighborhood of Fix a such that a ( A ) = A and f vanishes on A .Then we can partition A c as A c = B ⊔ a ( B ) for some clopen set B . Let g := f B .Then f = g − ga . Hence, ψ is injective.Let us now show surjectivity of ψ . Take f ∈ C ( X, Z ) such that f a = f and f (Fix a ) ⊂ Z , and let us show that f ∈ Im ψ .Let U , . . . , U n be a -invariant clopen subsets of X whose union cover Fix a , andsuch that f is constant on each U i . By taking differences, we can assume that thesesets are disjoint. By summing f with functions of the form 2 m i U i , we may assumethat f | Fix a = 0. We have f = P q ∈ Z \{ } q f − ( q ) . Since the support of f does notintersect Fix a , we can, for each q ∈ Z \{ } , partition f − ( q ) as f − ( q ) = A q ⊔ a ( A q ),for some A q clopen. Let g := P q ∈ Z \{ } qA q . Then ψ ([ g ]) = f .Finally, notice that by [22, Theorem 6.2.2], we have that, for k ≥ H k ( Z , C ( X, Z )) = { f ∈ C ( X, Z ) : f + f σ = 0 }{ f − f σ : f ∈ C ( X, Z ) } . Since the right-hand side of (5) is equal to ker ψ , the result follows. (cid:3) Given a finite index subgroup Λ of a group Γ, and M a Γ-module, there is ahomomorphism tr : M Γ → M Λ given by tr([ m ]) = [ P g i m ], where { g , . . . , g [Γ:Λ] } isa complete set of right coset representatives (this is the so called transfer map , andit does not depened on the choice of representatives). LMOST FINITENESS AND HOMOLOGY OF CERTAIN NON-FREE ACTIONS 9
Theorem 3.5.
Let α := ( ϕ, σ ) be an action of Z ⋊ Z on the Cantor set X suchthat the restricted Z -action is minimal. Let tr : H ( Z ⋊ Z , C ( X, Z )) → (1 + σ ∗ ) H ( Z , C ( X, Z ))[ f ] [ f + f σ ] . If α is not free, then tr is an isomorphism. If α is free, then ker tr ≃ Z and it isgenerated by [1 K ] − [1 L ] where K and L are clopen sets such that X = K ⊔ σ ( K ) = L ⊔ ϕσ ( L ) .Proof. Let f ∈ ker tr. In this case, there is h ∈ C ( X, Z ) such that(6) f + f σ = h − hϕ. This implies that h − hϕ = ( h − hϕ ) σ = hσ − hϕσ .Therefore, h + hϕσ = hσ + hϕ . Composing the right-hand side of this equationwith ϕ − , we obtain that h + hϕσ is ϕ -invariant. Since, ϕ is minimal, we concludethat there is an integer z such that h + hϕσ = − z (7)Furthermore, from (6), we get that f + f σ − h = − hϕ = − hϕ − hσ + hσ = z + hσ. Therefore, f + f σ − h − hσ = z. (8)Let G σ and G ϕσ be as in Lemma 3.4. By Lemmas 3.2 and 3.4, there is anisomorphism ψ : C ( X, Z ) Z⋊Z → G σ ⊕ G ϕσ { ( g + gσ, − g − gϕσ ) : g ∈ C ( X, Z ) } such that, for f , f ∈ C ( X, Z ), we have ψ ([ f + f ]) = [( f + f σ, f + f ϕσ )].From (7) and (8) we get that ψ ([ f ]) = ψ ([ f + 0]) = [( f + f σ, z, − z )].If z is even, then clearly [( z, − z )] = 0. Besides, if α is not free, then it followsfrom (7), (8) and Proposition 2.8 that z is even.Now suppose α is free and take K and L clopen sets as in the statement. Clearly,[1 K − L ] ∈ ker tr and ψ ([1 K − L ]) = [(1 , − , − = 0.Indeed, if there is g ∈ C ( X, Z ) such that 1 = g + gσ = g + gϕσ , then gσ = gϕσ ,which implies that g is ϕ -invariant, hence constant. But this contradicts the factthat 1 = g + gσ . (cid:3) The next result is essentially a summary of what we have obtained so far.
Theorem 3.6.
Let α := ( ϕ, σ ) be a minimal action of Z ⋊ Z on the Cantor set X . (i) If ϕ is not minimal, then there exists Y ⊂ X clopen and ϕ -invariant suchthat X = Y ⊔ σ ( Y ) and ϕ | Y is minimal. Furthermore, H ( Z ⋊ Z , C ( X, Z )) ≃ H ( Z , C ( Y, Z )) ,H ( Z ⋊ Z , C ( X, Z )) ≃ Z ,H n ( Z ⋊ Z , C ( X, Z )) = 0 , for n ≥ (ii) If ϕ is minimal and α is free, then H ( Z ⋊ Z , C ( X, Z )) ≃ Z ⊕ (1 + σ ∗ ) H ( Z , C ( X, Z )) ,H n ( Z ⋊ Z , C ( X, Z )) = 0 , for n ≥ If α is not free, then H ( Z ⋊ Z , C ( X, Z )) ≃ (1 + σ ∗ ) H ( Z , C ( X, Z )) ,H n − ( Z ⋊ Z , C ( X, Z )) ≃ C (Fix σ ⊔ Fix ϕσ , Z ) , for n ≥ ,H n ( Z ⋊ Z , C ( X, Z )) = 0 , for n ≥ . Proof. (i) The existence of Y is the content of Proposition 2.8. Notice that C ( X, Z ) ≃ Ind
Z⋊Z Z C ( Y, Z ) . Shapiro’s Lemma [4, Proposition 6.2] implies then that H ∗ ( Z ⋊ Z , C ( X, Z )) ≃ H ∗ ( Z , C ( Y, Z )) . Finally, the homology groups of a Cantor minimal Z -system are easy to compute.(ii) Since H ( Z , C ( X, Z )) is torsion-free, it follows from [6, Theorem 24.1] thatthe map tr in Theorem 3.5 admits right inverse.The remaining computations of the homology groups in cases (ii) and (iii) are aconsequence of Theorem 3.1 and Lemmas 3.2, 3.3 and 3.4. In case (iii), since the Z -action induced by ϕ is free, notice that Fix σ is disjoint from Fix ϕσ . (cid:3) Let us now apply Theorem 3.6 to an example which had its K-theory computedin [3, Corollary 4.4].
Example 3.7.
Fix θ ∈ (0 ,
1) an irrational number. Let ˜ X be the set obtainedfrom R by replacing each t ∈ Z + θ Z by two elements { t − , t + } , and endow ˜ X with the order topology. Notice that there is an action Z α y ˜ X by translations( α n ( x ) = n + x ). We let X := ˜ X Z . Then X is homeomorphic to the Cantor set andthere is a minimal homeomorphism R θ : X → X given by R θ ( x ) = x + θ .Furthermore, there is an involutive homeomorphism σ on X given by σ ( x ) = − x and σ ( t ± ) = t ∓ . Notice that the only fixed point of σ is , and the only fixed pointsof R θ ◦ σ are θ and θ .It was shown in [17, Theorem 2.1] that H ( Z , C ( X, Z )) ≃ Z , and the generatorsare [1 [0 + ,θ + )] ] and [1 [ θ + , + ) ]. Observe that σ ∗ acts trivially on these two elements.From these observations and Theorem 3.6, it follows that, for k ≥ H ( Z ⋊ Z , C ( X, Z )) ≃ Z H k − ( Z ⋊ Z , C ( X, Z )) ≃ Z × Z × Z H k ( Z ⋊ Z , C ( X, Z )) = 0 . Remark 3.8.
In [14], Matui conjectured that if G is an ample, effective, secondcountable, minimal groupoid with unit space homemorphic to the Cantor set, then K ∗ ( C ∗ r ( G )) ≃ L ∞ n =0 H n + ∗ ( G ) for ∗ = 0 , HK conjecture ). In [18], the (transfor-mation groupoids of the)
Z ⋊ Z -odometers from Example 2.7 were shown to becounterexamples to the HK conjecture.Let α be a minimal action of Z ⋊ Z on the Cantor set X . If α is not free,then the associated crossed product C ( X ) ⋊ Z ⋊ Z is AF [3, Theorem 3.5], hence LMOST FINITENESS AND HOMOLOGY OF CERTAIN NON-FREE ACTIONS 11 K ( C ( X ) ⋊ Z ⋊ Z ) = 0. On the other hand, it follows from Proposition 2.8 andTheorem 3.6 that H n +1 ( Z ⋊ Z , C ( X, Z )) = 0. Therefore, α is a counterexampleto the HK conjecture.If α is free, it follows from [21, Theorems 4.30 and 4.42] and Theorem 3.6 that α satisfies the HK conjecture. Alternatively, this also follows from Theorem 3.6 and[16, Remark 4.7]. Remark 3.9.
Given an ample groupoid G with compact unit space, the topologicalfull group of G , denoted by [[ G ]] is the the group of compact-open bisections U suchthat r ( U ) = s ( U ) = G (0) .Given an effective, minimal, second countable groupoid G with compact unitspace homeomorphic to the Cantor set, Matui conjectured in [14] that the indexmap I : [[ G ]] ab → H ( G ) is surjective and that its kernel is a quotient of H ( G ) ⊗ Z under a certain canonical map ( AH conjecture ). Assuming that G is also minimaland almost finite, Matui proved that the index map is surjective and if G is alsoprincipal, then it satisfies AH conjecture.We have not been able to verify whether the AH conjecture holds for non-freeCantor minimal Z ⋊ Z -systems in general, but note that in [18] it was shown thatthe Z ⋊ Z -odometers from Example 2.7 satisfy it. References [1]
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