arXiv:1704.00699v2 [math.DS] 14 Jan 2018

FØLNER TILINGS FOR ACTIONS OF AMENABLE GROUPS
CLINTON T. CONLEY, STEVE C. JACKSON, DAVID KERR, ANDREW S. MARKS,
BRANDON SEWARD, AND ROBIN D. TUCKER-DROB
Abstract. We show that every probability-measure-preserving action of a countable
amenable group G can be tiled, modulo a null set, using finitely many finite subsets
of G (“shapes”) with prescribed approximate invariance so that the collection of tiling
centers for each shape is Borel. This is a dynamical version of the Downarowicz–Huczek–
Zhang tiling theorem for countable amenable groups and strengthens the Ornstein–Weiss
Rokhlin lemma. As an application we prove that, for every countably infinite amenable
group G, the crossed product of a generic free minimal action of G on the Cantor set is
Z-stable.

1. Introduction
A discrete group G is said to be amenable if it admits a finitely additive probability
measure which is invariant under the action of G on itself by left translation, or equivalently
if there exists a unital positive linear functional ℓ∞ (G) → C which is invariant under
the action of G on ℓ∞ (G) induced by left translation (such a functional is called a left
invariant mean). This definition was introduced by von Neumann in connection with the
Banach–Tarski paradox and shown by Tarski to be equivalent to the absence of paradoxical
decompositions of the group. Amenability has come to be most usefully leveraged through
its combinatorial expression as the Følner property, which asks that for every finite set
K ⊆ G and δ > 0 there exists a nonempty finite set F ⊆ G which is (K, δ)-invariant in
the sense that |KF ∆F | < δ|F |.
The concept of amenability appears as a common thread throughout much of ergodic
theory as well as the related subject of operator algebras, where it is known via a number
of avatars like injectivity, hyperfiniteness, and nuclearity. It forms the cornerstone of
the theory of orbit equivalence, and also underpins both Kolmogorov–Sinai entropy and
the classical ergodic theorems, whether explicitly in their most general formulations or
implicitly in the original setting of single transformations (see Chapters 4 and 9 of [11]).
A key tool in applying amenability to dynamics is the Rokhlin lemma of Ornstein and
Weiss, which in one of its simpler forms says that for every free probability-measurepreserving action G y (X, µ) of a countably infinite amenable group and every finite set
K ⊆ G and δ > 0 there exist (K, δ)-invariant finite sets T1 , . . . , Tn ⊆ G and measurable
sets A1 , . . . , An ⊆ X such that the sets sAi for i = 1, . . . , n and s ∈ Ti are pairwise disjoint
and have union of measure at least 1 − δ [17].
The proportionality in terms of which approximate invariance is expressed in the Følner
condition makes it clear that amenability is a measure-theoretic property, and it is not
Date: November 9, 2017.
1

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CONLEY, JACKSON, KERR, MARKS, SEWARD, AND TUCKER-DROB

surprising that the most influential and definitive applications of these ideas in dynamics (e.g., the Connes–Feldman–Weiss theorem) occur in the presence of an invariant or
quasi-invariant measure. Nevertheless, amenability also has significant ramifications for
topological dynamics, for instance in guaranteeing the existence of invariant probability
measures when the space is compact and in providing the basis for the theory of topological
entropy. In the realm of operator algebras, similar comments can be made concerning the
relative significance of amenability for von Neumann algebras (measure) and C∗ -algebras
(topology).
While the subjects of von Neumann algebras and C∗ -algebras have long enjoyed a symbiotic relationship sustained in large part through the lens of analogy, and a similar relationship has historically bound together ergodic theory and topological dynamics, the last
few years have witnessed the emergence of a new and structurally more direct kind of rapport between topology and measure in these domains, beginning on the operator algebra
side with the groundbreaking work of Matui and Sato on strict comparison, Z-stability,
and decomposition rank [14, 15]. On the side of groups and dynamics, Downarowicz,
Huczek, and Zhang recently showed that if G is a countable amenable group then for
every finite set K ⊆ G and δ > 0 one can partition (or “tile”) G by left translates of
finitely many (K, δ)-invariant finite sets [3]. The consequences that they derive from this
tileability are topological and include the existence, for every such G, of a free minimal
action with zero entropy. One of the aims of the present paper is to provide some insight
into how these advances in operator algebras and dynamics, while seemingly unrelated
at first glance, actually fit together as part of a common circle of ideas that we expect,
among other things, to lead to further progress in the structure and classification theory
of crossed product C∗ -algebras.
Our main theorem is a version of the Downarowicz–Huczek–Zhang tiling result for
free p.m.p. (probability-measure-preserving) actions of countable amenable groups which
strengthens the Ornstein–Weiss Rokhlin lemma in the form recalled above by shrinking
the leftover piece down to a null set (Theorem 3.6). As in the case of groups, one does
not expect the utility of this dynamical tileability to be found in the measure setting,
where the Ornstein–Weiss machinery generally suffices, but rather in the derivation of
topological consequences. Indeed we will apply our tiling result to show that, for every
countably infinite amenable group G, the crossed product C(X) ⋊ G of a generic free
minimal action G y X on the Cantor set possesses the regularity property of Z-stability
(Theorem 5.4). The strategy is to first prove that such an action admits clopen tower
decompositions with arbitrarily good Følner shapes (Theorem 4.2), and then to demonstrate that the existence of such tower decompositions implies that the crossed product is
Z-stable (Theorem 5.3). The significance of Z-stability within the classification program
for simple separable nuclear C∗ -algebras is explained at the beginning of Section 5.
It is a curious irony in the theory of amenability that the Hall–Rado matching theorem
can be used not only to show that the failure of the Følner property for a discrete group
implies the formally stronger Tarski characterization of nonamenability in terms of the
existence of paradoxical decompositions [2] but also to show, in the opposite direction,
that the Følner property itself implies the formally stronger Downarowicz–Huczek–Zhang
characterization of amenability which guarantees the existence of tilings of the group
by translates of finitely many Følner sets [3]. This Janus-like scenario will be reprised

FØLNER TILINGS FOR ACTIONS OF AMENABLE GROUPS

3

here in the dynamical context through the use of a measurable matching argument of
Lyons and Nazarov that was originally developed to prove that for every simple bipartite
nonamenable Cayley graph of a discrete group G there is a factor of a Bernoulli action of
G which is an a.e. perfect matching of the graph [13]. Accordingly the basic scheme for
proving Theorem 3.6 will be the same as that of Downarowicz, Huczek, and Zhang and
divides into two parts:
(i) using an Ornstein–Weiss-type argument to show that a subset of the space of
lower Banach density close to one can be tiled by dynamical translates of Følner
sets, and
(ii) using a Lyons–Nazarov-type measurable matching to distribute almost all remaining points to existing tiles with only a small proportional increase in the size of
the Følner sets, so that the approximate invariance is preserved.
We begin in Section 2 with the measurable matching result (Lemma 2.6), which is a
variation on the Lyons–Nazarov theorem from [13] and is established along similar lines.
In Section 3 we establish the appropriate variant of the Ornstein–Weiss Rokhlin lemma
(Lemma 3.4) and put everything together in Theorem 3.6. Section 4 contains the genericity
result for free minimal actions on the Cantor set, while Section 5 is devoted to the material
on Z-stability.
Acknowledgements. C.C. was partially supported by NSF grant DMS-1500906. D.K. was
partially supported by NSF grant DMS-1500593. Part of this work was carried out while
he was visiting the Erwin Schrödinger Institute (January–February 2016) and the Mittag–
Leffler Institute (February–March 2016). A.M. was partially supported by NSF grant
DMS-1500974. B.S. was partially supported by ERC grant 306494. R.T.D. was partially
supported by NSF grant DMS-1600904. Part of this work was carried out during the AIM
SQuaRE: Measurable Graph Theory.
2. Measurable matchings
Given sets X and Y and a subset R ⊆ X × Y , with each x ∈ X we associate its vertical
section Rx = {y ∈ Y : (x, y) ∈ R} and with each y ∈ Y we associate S
its horizontal section
Ry = {x ∈ X : (x, y) ∈ R}. Analogously, for A ⊆ X we put RA = x∈A Rx = {y ∈ Y :
∃x ∈ A (x, y) ∈ R}. We say that R is locally finite if for all x ∈ X and y ∈ Y the sets Rx
and Ry are finite.
If now X and Y are standard Borel spaces equipped with respective Borel measures
µ and ν, we say that R ⊆ X × Y is (µ, ν)-preserving if whenever f : A → B is a Borel
bijection between subsets A ⊆ X and B ⊆ Y with graph(f ) ⊆ R we have µ(A) = ν(B).
We say that R is expansive if there is some c > 1 such that for all Borel A ⊆ X we have
ν(RA ) ≥ cµ(A).
We use the notation f : X ⇀ Y to denote a partial function from X to Y . We say that
such a partial function f is compatible with R ⊆ X × Y if graph(f ) ⊆ R.
Proposition 2.1 (ess. Lyons–Nazarov [13, Theorem 1.1]). Suppose that X and Y are
standard Borel spaces, that µ is a Borel probability measure on X, and that ν is a Borel
measure on Y . Suppose that R ⊆ X × Y is Borel, locally finite, (µ, ν)-preserving, and

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CONLEY, JACKSON, KERR, MARKS, SEWARD, AND TUCKER-DROB

expansive. Then there is a µ-conull X ′ ⊆ X and a Borel injection f : X ′ → Y compatible
with R.
Proof. Fix a constant of expansivity c > 1 for R.
We construct a sequence (fn )n∈N of Borel partial injections from X to Y which are
compatible with R. Moreover, we will guarantee that the set X ′ = {x ∈ X : ∃m ∈ N ∀n ≥
m x ∈ dom(fn ) and fn (x) = fm (x)} is µ-conull, establishing that the limiting function
satisfies the conclusion of the lemma.
Given a Borel partial injection g : X ⇀ Y we say that a sequence (x0 , y0 , . . . , xn , yn ) ∈
X × Y × · · · × X × Y is a g-augmenting path if
• x0 ∈ X is not in the domain of g,
• for all distinct i, j < n, yi 6= yj ,
• for all i < n, (xi , yi ) ∈ R,
• for all i < n, yi = g(xi+1 ),
• yn ∈ Y is not in the image of g.
We call n the length of such a g-augmenting path and x0 the origin of the path. Note
that the sequence (x0 , y0 , y1 , . . . , yn ) in fact determines the entire g-augmenting path, and
moreover that xi 6= xj for distinct i, j < n.
In order to proceed we require a few lemmas.
Lemma 2.2. Suppose that n ∈ N and g : X ⇀ Y is a Borel partial injection compatible
with R admitting no augmenting paths of length less than n. Then µ(X \ dom(g)) ≤ c−n .
Proof. Put A0 = X \ dom(g). Define recursively for i < n sets Bi = RAi and Ai+1 =
Ai ∪ g −1 (Bi ). Note that the assumption that there are no augmenting paths of length
less than n implies that each Bi is contained in the image of g. Expansivity of R yields
ν(Bi ) ≥ cµ(Ai ) and (µ, ν)-preservation of R then implies that µ(Ai+1 ) ≥ ν(Bi ) ≥ cµ(Ai ).
Consequently, 1 ≥ µ(An ) ≥ cn µ(A0 ), and hence µ(A0 ) ≤ c−n .

We say that a graph G on a standard Borel space X has a Borel N-coloring if there is
a Borel function c : X → N such that if x and y are G-adjacent then c(x) 6= c(y).
Lemma 2.3 (Kechris–Solecki–Todorcevic [12, Proposition 4.5]). Every locally finite Borel
graph on a standard Borel space has a Borel N-coloring.
Proof. Fix a countable algebra {Bn : n ∈ N} of Borel sets which separates points (for
example, the algebra generated by the basic open sets of a compatible Polish topology),
and color each vertex x by the least n ∈ N such that Bn contains x and none of its
neighbors.

Analogously, for k ∈ N, we say that a graph on a standard Borel X has a Borel kcoloring if there is a Borel function c : X → {1, . . . , k} giving adjacent points distinct
colors.
Lemma 2.4 (Kechris–Solecki–Todorcevic [12, Proposition 4.6]). If a Borel graph on a
standard Borel X has degree bounded by d ∈ N, then it has a Borel (d + 1)-coloring.
Proof. By Lemma 2.3, the graph has a Borel N-coloring c : X → N. We recursively
build sets An for n ∈ N by A0 = {x ∈ X : c(x) = 0} and An+1 = An ∪ {x ∈ X :

FØLNER TILINGS FOR ACTIONS OF AMENABLE GROUPS

5

S
c(x) = n + 1 and no neighbor of x is in An }. Then A = n An is a Borel set which is
G-independent, and moreover is maximal with this property. So the restriction of G to
X \ A has degree less than d, and the result follows by induction.

Lemma 2.5 (ess. Elek–Lippner [5, Proposition 1.1]). Suppose that g : X ⇀ Y is a Borel
partial injection compatible with R, and let n ≥ 1. Then there is a Borel partial injection
g′ : X ⇀ Y compatible with R such that
• dom(g ′ ) ⊇ dom(g),
• g′ admits no augmenting paths of length less than n,
• µ({x ∈ X : g ′ (x) 6= g(x)} ≤ nµ(X \ dom(g)).
Proof. Consider the set Z of injective sequences (x0 , y0 , x1 , y1 , . . . , xm , ym ), where m < n,
such that for all i ≤ m we have (xi , yi ) ∈ R and for all i < m we have (xi+1 , yi ) ∈ R. Equip
Z with the standard Borel structure it inherits as a Borel subset of (X × Y )≤n . Consider
also the locally finite Borel graph G on Z rendering adjacent two distinct
sequences in
F
Z if they share any entries. By Lemma 2.3 there is a partition Z = k∈N Zk of Z into
Borel sets such that for all k, no two elements of Zk are G-adjacent. In other words, we
partition potential augmenting paths into countably many colors, where no two paths of
the same color intersect. Thus we may flip paths of the same color simultaneously without
risk of causing conflicts. Towards that end, fix a bookkeeping function s : N → N such
that s−1 (k) is infinite for all k ∈ N in order to consider each color class infinitely often.
Given a g-augmenting path z = (x0 , y0 , . . . , xm , ym ), define the flip along z to be the
Borel partial function gz : X ⇀ Y given by
(
yi
if ∃i ≤ m x = xi ,
gz (x) =
g(x) otherwise.
The fact that z is g-augmenting ensures that gz is injective. More generally, for any Borel
G-independent set Z aug ⊆ Z of g-augmenting paths, we may simultaneously flip g along
all paths in Z aug to obtain another Borel partial injection (g)Z aug .
We iterate this construction. Put g0 = g. Recursively assuming that gk : X ⇀ Y has
been defined, let Zkaug be the set of gk -augmenting paths in Zs(k) , and let gk+1 = (gk )Z aug
k
be the result of flipping gk along all paths in Zkaug . As each x ∈ X is contained in only
finitely many elements of Z, and since each path in Z can be flipped at most once (after the
first flip its origin is always in the domain of the subsequent partial injections), it follows
that the sequence (gk (x))k∈N is eventually constant. Defining g′ (x) to be the limiting
value, it is routine to check that there are no g′ -augmenting paths of length less than n.
Finally, to verify the third item of the lemma, put A = {x ∈ X : g′ (x) 6= g(x)}.
With each x ∈ A associate the origin of the first augmenting path along which it was
flipped. This is an at most n-to-1 Borel function from A to X \ dom(g), and since R is
(µ, ν)-preserving the bound follows.

We are now in position to follow the strategy outlined at the beginning of the proof.
Let f0 : X ⇀ Y be the empty function. Recursively assuming the Borel partial injection
fn : X ⇀ Y has been defined to have no augmenting paths of length less than n, let fn+1
be the Borel partial injection (fn )′ granted by applying Lemma 2.5 to fn . Thus fn+1 has
no augmenting paths of length less than n + 1 and the recursive construction continues.

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CONLEY, JACKSON, KERR, MARKS, SEWARD, AND TUCKER-DROB

Lemma 2.2 ensures that µ(X \ dom(fn )) ≤ c−n , and thus the third item of Lemma 2.5
ensures that µ({x ∈ X : fn+1 (x) 6= fn (x)}) ≤ (n + 1)c−n . As the sequence (n + 1)c−n is
summable, the Borel–Cantelli lemma implies that X ′ = {x ∈ X : ∃m ∈ N ∀n ≥ m x ∈
dom(fn ) and fn (x) = fm (x)} is µ-conull. Finally, f = limn→∞ fn ↾ X ′ is as desired.

Lemma 2.6. Suppose X and Y are standard Borel spaces, that µ is a Borel measure on
X, and that ν is a Borel measure on Y . Suppose R ⊆ X × Y is Borel, locally finite,
(µ, ν)-preserving graph. Assume that there exist numbers a, b > 0 such that |Rx | ≥ a for
µ-a.e. x ∈ X and |Ry | ≤ b for ν-a.e. y ∈ Y . Then ν(RA ) ≥ ab µ(A) for all Borel subsets
A ⊆ X.
R
R
Proof. Since R is (µ, ν)-preserving we have A |Rx | dµ = RA |Ry ∩ A| dν. Hence
Z
Z
Z
Z
y

b dν = bν(RA ).
|R ∩ A| dν ≤
|Rx | dµ =
a dµ ≤
aµ(A) =
A

A

RA

RA

3. Følner tilings
Fix a countable group G. For finite sets K, F ⊆ G and δ > 0, we say that F is (K, δ)invariant if |KF △F | < δ|F |. Note this condition implies |KF | < (1 + δ)|F |. Recall that
G is amenable if for every finite K ⊆ G and δ > 0 there exists a (K, δ)-invariant set F . A
Følner sequence is a sequence of finite sets Fn ⊆ G with the property that for every finite
K ⊆ G and δ > 0 the set Fn is (K, δ)-invariant for all but finitely many n. Below, we
always assume that G is amenable.
Fix a free action G y X. For A ⊆ X we define the lower and upper Banach densities
of A to be
|A ∩ F x|
|A ∩ F x|
D (A) = sup inf
and
D̄(A) = inf sup
.
F ⊆G x∈X
¯
|F |
|F |
F ⊆G x∈X
F finite

F finite

Equivalently [3, Lemma 2.9], if (Fn )n∈N is a Følner sequence then
D (A) = lim inf
n→∞ x∈X
¯

|A ∩ Fn x|
|Fn |

and

D̄(A) = lim sup

n→∞ x∈X

|A ∩ Fn x|
.
|Fn |

We now define an analogue of ‘(K, δ)-invariant’ for infinite subsets of X. A set A ⊆ X
(possibly infinite) is (K, δ)∗ -invariant if there is a finite set F ⊆ G such that |(KA△A) ∩
F x| < δ|A ∩ F x| for all x ∈ X. Equivalently, A is (K, δ)∗ -invariant if and only if for every
Følner sequence (Fn )n∈N we have limn supx |(KA△A) ∩ Fn x|/|A ∩ Fn x| < δ.
A collection {Fi : i ∈ I} of finite subsets of X is called ǫ-disjoint if for each i there is
an Fi′ ⊆ Fi such that |Fi′ | > (1 − ǫ)|Fi | and such that the sets {Fi′ : i ∈ I} are pairwise
disjoint.
Lemma 3.1. Let K, W ⊆ G be finite, let ǫ, δ > 0, let C ⊆ X, and for c ∈ CS let Fc ⊆ W
be (K, δ(1 − ǫ))-invariant. If the collection
{Fc c : c ∈ C} is ǫ-disjoint and c∈C Fc c has
S
positive lower Banach density, then c∈C Fc c is (K, δ)∗ -invariant.
S
Proof. Set A = c∈C Fc c and set T = W W −1 ({1G } ∪ K)−1 . Since W is finite and each
Fc ⊆ W , there is 0 < δ0 < δ such that each Fc is (K, δ0 (1 − ǫ))-invariant. Fix a finite set
D (A)
(A)
U ⊆ G which is (T, D¯2|T
| (δ − δ0 ))-invariant and satisfies inf x∈X |A ∩ U x| > ¯ 2 |U |. Now

FØLNER TILINGS FOR ACTIONS OF AMENABLE GROUPS

7

fix x ∈ X. Let B be the set of b ∈ U x such that T b 6⊆ U x. Note that B ⊆ T −1 (T U x△U x)
and thus
|U |
|T U △U |
·
· |A ∩ U x| < (δ − δ0 )|A ∩ U x|.
|B| ≤ |T | ·
|U |
|A ∩ U x|
P
Set C ′ = {c ∈ C : Fc c ⊆ U x}. Note that the ǫ-disjoint assumption gives (1−ǫ) c∈C ′ |Fc | ≤
|A∩U x|. Also, our definitions of C ′ , T , and B imply that if c ∈ C \C ′ and
S ({1G }∪K)Fc c∩
U x 6= ∅ then (({1G }∪K)Fc c)∩U x ⊆ B. Therefore (KA△A)∩U x ⊆ B∪ c∈C ′ (KFc c△Fc c).
Combining this with the fact that each set Fc is (K, δ0 (1 − ǫ))-invariant, we obtain
X
|(KA△A) ∩ U x| ≤ |B| +
|KFc c△Fc c|
c∈C ′

< (δ − δ0 )|A ∩ U x| +

X

δ0 (1 − ǫ)|Fc |

c∈C ′

≤ (δ − δ0 )|A ∩ U x| + δ0 |A ∩ U x|
= δ|A ∩ U x|.
Since x was arbitrary, we conclude that A is (K, δ)∗ -invariant.



Lemma 3.2. Let T ⊆ G be finite and let ǫ, δ > 0 with ǫ(1 + δ) < 1. Suppose that A ⊆ X
is (T −1 , δ)∗ -invariant. If B ⊇ A and |B ∩ T x| ≥ ǫ|T | for all x ∈ X, then
D (B) ≥ (1 − ǫ(1 + δ)) · D (A) + ǫ.
¯
¯
Proof. This is implicitly demonstrated in [3, Proof of Lemma 4.1]. As a convenience to
the reader, we include a proof here. Fix θ > 0. Since A is (T −1 , δ)∗ -invariant, we can pick
a finite set U ⊆ G which is (T, θ)-invariant and satisfies
inf

x∈X

|A ∩ U x|
> D(A) − θ
¯
|U |

and

sup
x∈X

|T −1 A ∩ U x|
< 1 + δ.
|A ∩ U x|

|A∩U x|
|U |

> D (A) − θ, and set U ′ = {u ∈ U : A ∩ T ux = ∅}. Notice that
¯
|U | − |T −1 A ∩ U x|
|T −1 A ∩ U x| |A ∩ U x|
|U ′ |
=
=1−
·
> 1 − (1 + δ)α.
|U |
|U |
|A ∩ U x|
|U |

Fix x ∈ X, set α =

Since A ∩ T U ′ x = ∅ and |B ∩ T y| ≥ ǫ|T | for all y ∈ X, it follows that |(B \ A) ∩ T ux| ≥ ǫ|T |
for all u ∈ U ′ . Thus there are ǫ|T ||U ′ | many pairs (t, u) ∈ T × U ′ with tux ∈ B \ A. It
follows there is t∗ ∈ T with |(B \ A) ∩ t∗ U ′ x| ≥ ǫ · |U ′ |. Therefore


|A ∩ U x| |(B \ A) ∩ t∗ U ′ x| |U ′ |
|U |
|B ∩ T U x|
≥
+
·
·
′
|T U |
|U |
|U |
|U |
|T U |


−1
> α + ǫ(1 − (1 + δ)α) · (1 + θ)


= (1 − ǫ(1 + δ))α + ǫ · (1 + θ)−1


> (1 − ǫ(1 + δ))(D (A) − θ) + ǫ · (1 + θ)−1 .
¯
Letting θ tend to 0 completes the proof.


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CONLEY, JACKSON, KERR, MARKS, SEWARD, AND TUCKER-DROB

Lemma 3.3. Let X be a standard Borel space and let G y X be a free Borel action. Let
Y ⊆ X be Borel, let T ⊆ G be finite, and let ǫ ∈ (0, 1/2). Then there is a Borel set C ⊆ X
and a Borel function c ∈ C 7→ Tc ⊆ T suchSthat |Tc | > (1− ǫ)|T |, the sets {Tc c : c ∈ C} are
pairwise disjoint and disjoint with Y , Y ∪ c∈C Tc c = Y ∪ T C, and |(Y ∪ T C) ∩ T x| ≥ ǫ|T |
for all x ∈ X.

Proof. Using Lemma 2.4, fix a Borel partition P = {P1 , . . . , Pm } of X such that T x∩T x′ =
∅ for S
all x 6= x′ ∈ Pi and all 1 ≤ i ≤ m. We will pick Borel sets Ci ⊆ Pi and set
C = 1≤i≤m Ci . Set Y0 = Y . Let 1 ≤ i ≤ m and inductively assume that Yi−1 has
been defined. Define Ci = {c ∈ Pi : |Yi−1 ∩ T c| < ǫ|T |}, defineSYi = Yi−1 ∪ T Ci , and for
c ∈ Ci set Tc = {t ∈ T : tc 6∈ Yi−1 }. It is easily seen that C = 1≤i≤m Ci has the desired
properties.


The following lemma is mainly due to Ornstein–Weiss [17], who proved it with an
invariant probability measure taking the place of Banach density. Ornstein and Weiss also
established a purely group-theoretic counterpart of this result which was later adapted to
the Banach density setting by Downarowicz–Huczek–Zhang in [3] and will be heavily used
in Section 5, where it is recorded as Theorem 5.2. The only difference between this lemma
and prior versions is that we simultaneously work in the Borel setting and use Banach
density.

Lemma 3.4. [17, II.§2. Theorem 5] [3, Lemma 4.1] Let X be a standard Borel space and
let G y X be a free Borel action. Let K ⊆ G be finite, let ǫ ∈ (0, 1/2), and let n satisfy
(1 − ǫ)n < ǫ. Then there exist (K, ǫ)-invariant sets F1 , . . . , Fn , a Borel set C ⊆ X, and a
Borel function c ∈ C 7→ Fc ⊆ G such that:
(i) for every c ∈ C there is 1 ≤ i ≤ n with Fc ⊆ Fi and |Fc | > (1 − ǫ)|Fi |;
(ii) theSsets Fc c, c ∈ C, are pairwise disjoint; and
(iii) D ( c∈C Fc c) > 1 − ǫ.
¯
Proof. Fix δ > 0 satisfying (1 + δ)−1 (1 − (1 + δ)ǫ)n < ǫ − 1 + (1 + δ)−1 . Fix a sequence
of (K, ǫ)-invariant sets F1 , . . . , Fn such that Fi is (Fj−1 , δ(1 − ǫ))-invariant for all 1 ≤ j <
i ≤ n.
The set C will be the disjoint union of sets Ci , 1 ≤ i ≤ n. The construction
S
S will be
such that Fc ⊆ Fi and |Fc | > (1 − ǫ)|Fi | for c ∈ Ci . We will define Ai = i≤k≤n c∈Ck Fc c
S
and arrange that Ai+1 ∪ Fi Ci = Ai+1 ∪ c∈Ci Fc c and

D (Ai ) ≥ (1 + δ)−1 − (1 + δ)−1 (1 − ǫ(1 + δ))n+1−i .
¯
S
In particular, we will have Ai = i≤k≤n Fk Ck .
To begin, apply Lemma 3.3 with Y = ∅ and T = Fn to get a Borel set Cn and a Borel
map c ∈ S
Cn 7→ Fc ⊆ Fn such that |Fc | > (1 − ǫ)|Fn |, the sets {Fc c : c ∈ Cn } are pairwise
disjoint, c∈Cn Fc c = Fn Cn , and |Fn Cn ∩ Fn x| ≥ ǫ|Fn | for all x ∈ X. Applying Lemma
3.2 with A = ∅ and B = Fn Cn we find that the set An = Fn Cn satisfies D (An ) ≥ ǫ.
¯
Inductively assume that Cn through Ci+1 have been defined and An through Ai+1 are
defined as above and satisfy (3.1). Using Y = Ai+1 and T = Fi , apply Lemma 3.3 to get
a Borel set Ci and a Borel map c ∈ Ci 7→ Fc ⊆ Fi such that |FcS| > (1 − ǫ)|Fi |, the sets
{Fc c : c ∈ Ci } are pairwise disjoint and disjoint with Ai+1 , Ai+1 ∪ c∈Ci Fc c = Ai+1 ∪ Fi Ci ,
and |(Ai+1 ∪ Fi Ci ) ∩ Fi x| ≥ ǫ|Fi | for all x ∈ X. The set Ai+1 is the union of an ǫ-disjoint
(3.1)

FØLNER TILINGS FOR ACTIONS OF AMENABLE GROUPS

9

collection of (Fi−1 , δ(1 − ǫ))-invariant sets and has positive lower Banach density. So by
Lemma 3.1 Ai+1 is (Fi−1 , δ)∗ -invariant. Applying Lemma 3.2 with A = Ai+1 , we find that
Ai = Ai+1 ∪ Fi Ci satisfies
D (Ai ) ≥ (1 − ǫ(1 + δ)) · D (Ai+1 ) + ǫ
¯
¯
ǫ(1 + δ)
(1 − ǫ(1 + δ))
− (1 + δ)−1 (1 − ǫ(1 + δ))n+1−i +
≥
(1 + δ)
1+δ
= (1 + δ)−1 − (1 + δ)−1 (1 − ǫ(1 + δ))n+1−i .
This completes the inductive step and completes the definition of C. It is immediate from
the construction that (i) and (ii) are satisfied. Clause (iii) also follows by noting that (3.1)
is greater than 1 − ǫ when i = 1.

We recall the following simple fact.
Lemma 3.5. [3, Lemma 2.3] If F ⊆ G is (K, δ)-invariant and F ′ satisfies |F ′ △F | < ǫ|F |
then F ′ is (K, (|K|+1)ǫ+δ
)-invariant.
1−ǫ
Now we present the main theorem.
Theorem 3.6. Let G be a countable amenable group, let (X, µ) be a standard probability
space, and let G y (X, µ) be a free p.m.p. action. For every finite K ⊆ G and every δ > 0
there exist a µ-conull G-invariant Borel set X ′ ⊆ X, a collection {Ci : 0 ≤ i ≤ m} of
Borel subsets of X ′ , and a collection {Fi : 0 ≤ i ≤ m} of (K, δ)-invariant sets such that
{Fi c : 0 ≤ i ≤ m, c ∈ Ci } partitions X ′ .
< δ. Apply Lemma 3.4 to get (K, ǫ)-invariant
Proof. Fix ǫ ∈ (0, 1/2) satisfying (|K|+1)6ǫ+ǫ
1−6ǫ
′
′
sets F1 , . . . , Fn , a Borel set C ⊆ X, and a Borel function c ∈ C 7→ Fc ⊆ G satisfying
(i) for every c ∈ C there is 1 ≤ i ≤ n with Fc ⊆ Fi′ and |Fc | > (1 − ǫ)|Fi′ |;
(ii) theSsets Fc c, c ∈ C, are pairwise disjoint; and
(iii) D ( c∈C Fc c) > 1 − ǫ.
¯
S
Set Y = X \ c∈C Fc c. If µ(Y ) = 0 then we are done. So we assume µ(Y ) > 0 and
we let ν denote the restriction of µ to Y . Fix a Borel map c ∈ C 7→ Zc ⊆ Fc satisfying
4ǫ|Fc | < |Zc | < 5ǫ|Fc | for all c ∈ C (it’s clear from the proof of Lemma
S 3.4 that we may
choose the sets Fi′ so that ǫ|Fc | > mini ǫ(1 − ǫ)|Fi′ | > 1). Set Z = c∈C Zc c and let ζ
denote the restriction
of µ to Z (note that µ(Z) > 0).
S
Set W = ni=1 Fi′ and W ′ = W W −1 . Fix a finite set U ⊆ G which is (W ′ , (1/2−ǫ)/|W ′ |)invariant and satisfies inf x∈X |(X \ Y ) ∩ U x| > (1 − ǫ)|U |. Since every amenable group
admits a Følner sequence consisting of symmetric sets, we may assume that U = U −1 [16,
Corollary 5.3]. Define R ⊆ Y ×Z by declaring (y, z) ∈ R if and only if y ∈ U z (equivalently
z ∈ U y). Then R is Borel, locally finite, and (ν, ζ)-preserving. We now check that R is
expansive. We automatically have |Rz | = |Y ∩ U z| < ǫ|U | for all z ∈ Z. By Lemma 2.6 it
suffices to show that |Ry | = |Z ∩ U y| ≥ 2ǫ|U | for all y ∈ Y . Fix y ∈ Y . Let B be the set
of b ∈ U y such that W ′ b 6⊆ U y. Then B ⊆ W ′ (W ′ U y△U y) and thus
|W ′ U △U |
|B|
≤ |W ′ | ·
< 1/2 − ǫ.
|U |
|U |

10

CONLEY, JACKSON, KERR, MARKS, SEWARD, AND TUCKER-DROB

Let A be the union of those sets Fc c, c ∈ C, which are contained in U y. Notice that
(X \ Y ) ∩ U y ⊆ B ∪ A. Therefore
|A|
|(B ∪ A) ∩ U y|
|(X \ Y ) ∩ U y|
1
−ǫ+
>
≥
> 1 − ǫ,
2
|U |
|U |
|U |
hence |A| > |U |/2. By construction |Z ∩ A| > 4ǫ|A|. So |Z ∩ U y| ≥ |Z ∩ A| > 2ǫ|U |. We
conclude that R is expansive.
Apply Proposition 2.1 to obtain a G-invariant µ-conull set X ′ ⊆ X and a Borel injection
ρ : Y ∩X ′ → Z with graph(ρ) ⊆ R. Consider the sets Fc ∪{g ∈ U : gc ∈ Y and ρ(gc) ∈ Fc c}
as c ∈ C varies. These are subsets of W ∪ U and thus there are only finitely many such sets
which we can enumerate as F1 , . . . , Fm . We partition C ∩ X ′ into Borel sets C1 , . . . , Cm
with c ∈ Ci if and only if c ∈ X ′ and Fc ∪ {g ∈ U : gc ∈ Y and ρ(gc) ∈ Fc c} = Fi . Since
ρ is defined on all of Y ∩ X ′ , we see that the sets {Fi c : 1 ≤ i ≤ m, c ∈ Ci } partition X ′ .
Finally, for c ∈ Ci ∩ X ′ , if we let Fj′ be such that |Fc △Fj′ | < ǫ|Fj′ |, then
|Fi △Fj′ | ≤ |Fi △Fc | + |Fc △Fj′ | ≤ |ρ−1 (Fc c)| + ǫ|Fj′ |
≤ |Zc | + ǫ|Fj′ | < 5ǫ|Fc | + ǫ|Fj′ | ≤ 6ǫ|Fj′ |.
Using Lemma 3.5 and our choice of ǫ, this implies that each set Fi is (K, δ)-invariant.



4. Clopen tower decompositions with Følner shapes
Let G y X be an action of a group on a compact space. By a clopen tower we mean
a pair (B, S) where B is a clopen subset of X (the base of the tower) and S is a finite
subset of G (the shape of the tower) such that the sets sB for s ∈ S are pairwise disjoint.
By a clopen tower decomposition of X we mean a finite collection {(Bi , Si )}ni=1 of clopen
towers such that the sets S1 B1 , . . . , Sn Bn form a partition of X. We also similarly speak
of measurable towers and measurable tower decompositions for an action G y (X, µ) on a
measure space, with the bases now being measurable sets instead of clopen sets. In this
terminology, Theorem 3.6 says that if G y (X, µ) is a free p.m.p. action of a countable
amenable group on a standard probability space then for every finite set K ⊆ G and
δ > 0 there exists, modulo a null set, a measurable tower decomposition of X with (K, δ)invariant shapes.
Lemma 4.1. Let G be a countably infinite amenable group and G y X a free minimal
action on the Cantor set. Then this action has a free minimal extension G y Y on
the Cantor set such that for every finite set F ⊆ G and δ > 0 there is a clopen tower
decomposition of Y with (F, δ)-invariant shapes.
Proof. Let F1 ⊆ F2 ⊆ . . . be an increasing sequence of finite subsets of G whose union is
equal to G. Fix a G-invariant Borel probability measure µ on X (such a measure exists
by amenability). The freeness of the action G y X means that for each n ∈ N we can
apply Theorem 3.6 to produce, modulo a null set, a measurable tower decomposition Un
for the p.m.p. action G y (X, µ) such that each shape is (Fn , 1/n)-invariant. Let A be the
unital G-invariant C∗ -algebra of L∞ (X, µ) generated by C(X) and the indicator functions
of the levels of each of the tower decompositions Un . Since there are countably many
such indicator functions and the group G is countable, the C∗ -algebra A is separable.
Therefore by the Gelfand–Naimark theorem we have A = C(Z) for some zero-dimensional

FØLNER TILINGS FOR ACTIONS OF AMENABLE GROUPS

11

metrizable space Z and a G-factor map ϕ : Z → X. By a standard fact which can be
established using Zorn’s lemma, there exists a nonempty closed G-invariant set Y ⊆ Z
such that the restriction action G y Y is minimal. Note that Y is necessarily a Cantor
set, since G is infinite. Also, the action G y Y is free, since it is an extension of a free
action. Since the action on X is minimal, the restriction ϕ|Y : Y → X is surjective and
hence a G-factor map. For each n we get from Un a clopen tower decomposition Vn of
Y with (Fn , 1/n)-invariant shapes, and by intersecting the levels of the towers in Vn with
Y we obtain a clopen tower decomposition of Y with (Fn , 1/n)-invariant shapes, showing
that the extension G y Y has the desired property.

Let X be the Cantor set and let G be a countable infinite amenable group. The set
Act(G, X) is a Polish space under the topology which has as a basis the sets
Uα,P,F = {β ∈ Act(G, X) : αs A = βs A for all A ∈ P and s ∈ F }
where α ∈ Act(G, X), P is a clopen partition of X, and F is a finite subset of G.
Write FrMin(G, X) for the set of actions in Act(G, X) which are free and minimal. Then
FrMin(G, X) is a Gδ set. To see this, fix an enumeration s1 , s2 , s3 , . . . of G \ {e} (where
e denotes the identity element of the group) and for every n ∈ N and
S nonempty clopen
set A ⊆ X define the set Wn,A of all α ∈ Act(G, X) such that (i) s∈F αs A = X for
some finite set F ⊆ G, and (ii) there exists a clopen partition {A1 , . . . , Ak } of A such that
αsn Ai ∩ Ai = ∅ for all i = 1, . . . , k. Then each Wn,A is open, which means, with A ranging
T over
T the countable collection of nonempty clopen subsets of X, that the intersection
n∈N A Wn,A , which is equal to FrMin(G, X), is a Gδ set. It follows that FrMin(G, X)
is a Polish space.
Theorem 4.2. Let G be a countably infinite amenable group. Let C be the collection of
actions in FrMin(G, X) with the property that for every finite set F ⊆ G and δ > 0 there
is a clopen tower decomposition of X with (F, δ)-invariant shapes. Then C is a dense Gδ
subset of FrMin(G, X).
α

Proof. That C is a Gδ set is a simple exercise. Let G y X be any action in FrMin(G, X).
β

By Lemma 4.1 this action has a free minimal extension G y Y with the property in the
theorem statement, where Y is the Cantor set. Let P be a clopen partition of X and F
a nonempty finite subset of G. Write A1 , . . . , An for the members of the clopen partition
W
−1
−1
s∈F s P. Then for each i = 1, . . . , n the set Ai and its inverse image ϕ (Ai ) under
the extension map ϕ : Y → X are Cantor sets, and so we can find a homeomorphism
ψi : Ai → ϕ−1 (Ai ). Let ψ : X → Y be the homeomorphism which is equal to ψi on Ai for
γ
each i. Then the action G y X defined by γs = ψ −1 ◦ βs ◦ ψ for s ∈ G belongs to C as
well as to the basic open neighborhood Uα,P,F of α, establishing the density of C.

5. Applications to Z-stability
A C∗ -algebra A is said to be Z-stable if A ⊗ Z ∼
= A where Z is the Jiang–Su algebra [10],
with the C∗ -tensor product being unique in this case because Z is nuclear. Z-stability has
become an important regularity property in the classification program for simple separable
nuclear C∗ -algebras, which has recently witnessed some spectacular advances. Thanks to
recent work of Gong–Lin–Niu [6], Elliott–Gong–Lin–Niu [4], and Tikuisis–White–Winter

12

CONLEY, JACKSON, KERR, MARKS, SEWARD, AND TUCKER-DROB

[22], it is now known that simple separable unital C∗ -algebras satisfying the universal
coefficient theorem and having finite nuclear dimension are classified by ordered K-theory
paired with tracial states. Although Z-stability does not appear in the hypotheses of this
classification theorem, it does play an important technical role in the proof. Moreover, it
is a conjecture of Toms and Winter that for simple separable infinite-dimensional unital
nuclear C∗ -algebras the following properties are equivalent:
(i) Z-stability,
(ii) finite nuclear dimension,
(iii) strict comparison.
Implications between (i), (ii), and (iii) are known to hold in various degrees of generality.
In particular, the implication (ii)⇒(i) was established in [23] while the converse is known
to hold when the extreme boundary of the convex set of tracial states is compact [1]. It
remains a problem to determine whether any of the crossed products of the actions in Theorem 5.4 falls within the purview of these positive results on the Toms–Winter conjecture,
and in particular whether any of them has finite nuclear dimension (see Question 5.5).
By now there exist highly effectively methods for establishing finite nuclear dimension
for crossed products of free actions on compact metrizable spaces of finite covering dimension [20, 21, 7], but their utility is structurally restricted to groups with finite asymptotic
dimension and hence excludes many amenable examples like the Grigorchuk group. One
can show using the technology from [7] that, for a countably infinite amenable group with
finite asymptotic dimension, the crossed product of a generic free minimal action on the
Cantor set has finite nuclear dimension. Our intention here has been to remove the restriction of finite asymptotic dimension by means of a different approach that establishes
instead the conjecturally equivalent property of Z-stability but for arbitrary countably
infinite amenable groups.
To verify Z-stability in the proof of Theorem 5.3 we will use the following result of
Hirshberg and Orovitz [8]. Recall that a linear map ϕ : A → B between C∗ -algebras is
said to be complete positive if its tensor product id ⊗ ϕ : Mn ⊗ A → Mn ⊗ B with the
identity map on the n × n matrix algebra Mn is positive for every n ∈ N. It is of order zero
if ϕ(a)ϕ(b) = 0 for all a, b ∈ A satisfying ab = 0. One can show that ϕ is an order-zero
completely positive map if and only if there is an embedding B ⊆ D of B into a larger
C∗ -algebra, a ∗ -homomorphism π : A → D, and a positive element h ∈ D commuting with
the image of π such that ϕ(a) = hπ(a) for all a ∈ A [24]. Below - denotes the relation
of Cuntz subequivalence, so that a - b for positive elements a, b in a C∗ -algebra A means
that there is a sequence (vn ) in A such that limn→∞ ka − vn bvn∗ k = 0.
Theorem 5.1. Let A be a simple separable unital nuclear C∗ -algebra not isomorphic to
C. Suppose that for every n ∈ N, finite set Ω ⊆ A, ε > 0, and nonzero positive element
a ∈ A there exists an order-zero complete positive contractive linear map ϕ : Mn → A
such that
(i) 1 − ϕ(1) - a,
(ii) k[b, ϕ(z)]k < ε for all b ∈ Ω and norm-one z ∈ Mn .
Then A is Z-stable.

FØLNER TILINGS FOR ACTIONS OF AMENABLE GROUPS

13

The following is the Ornstein–Weiss quasitiling theorem [17] as formulated in Theorem 3.36 of [11]. For finite sets A, F ⊆ G we write
∂F A = {s ∈ A : F s ∩ A 6= ∅ and F s ∩ (G \ A) 6= ∅}.
For λS≤ 1, a collection C of finite subsets of G is said to λ-cover a finite subset A of G if
|A ∩ C| ≥ λ|A|. For β ≥ 0, a collection C of finite subsets of G is said to be β-disjoint if
for each C ∈ C there is a set C ′ ⊆ C with |C ′ | ≥ (1 − β)|C| so that the sets C ′ for C ∈ C
are pairwise disjoint.
Theorem 5.2. Let 0 < β < 12 and let n be a positive integer such that (1 − β/2)n <
β. Then whenever e ∈ T1 ⊆ T2 ⊆ · · · ⊆ Tn are finite subsets of a group G such that
|∂Ti−1 Ti | ≤ (η/8)|Ti | for i = 2, . . . , n, for every (Tn , β/4)-invariant nonempty finite set
E ⊆ G there exist C1 , . . . , Cn ⊆ G such that
S
(i) ni=1 Ti Ci ⊆ E, and
S
(ii) the collection of right translates ni=1 {Ti c : c ∈ Ci } is β-disjoint and (1−β)-covers
E.
Theorem 5.3. Let G be a countably infinite amenable group and let G y X be a free
minimal action on the Cantor set such that for every finite set F ⊆ G and δ > 0 there is a
clopen tower decomposition of X with (F, δ)-invariant shapes. Then C(X) ⋊ G is Z-stable.
Proof. Let n ∈ N. Let Υ be a finite subset of the unit ball of C(X), F a symmetric finite
subset of G containing the identity element e, and ε > 0. Let a be a nonzero positive
element of C(X) ⋊ G. We will show the existence of an order-zero completely positive
contractive linear map ϕ : Mn → C(X) ⋊ G satisfying (i) and (ii) in Theorem 5.1 where
the finite set Ω there is taken to be Υ ∪ {us : s ∈ F }. Since C(X) ⋊ G is generated as a
C∗ -algebra by the unit ball of C(X) and the unitaries us for s ∈ G, we will thereafter be
able to conclude by Theorem 5.1 that C(X) ⋊ G is Z-stable.
By Lemma 7.9 in [18] we may assume that a is a function in C(X). Taking a clopen set
A ⊆ X on which a is nonzero, we may furthermore assume that a is equal to the indicator
function 1A . Minimality implies that the clopen sets sA for s ∈ G cover X, and so by
compactness there is a finite set D ⊆ G such that D −1 A = X.
Equip X with a compatible metric d. Choose an integer Q > n2 /ε.
Let γ > 0, to be determined. Take a 0 < β < 1/n which is small enough so that if T is
a nonempty finite subset of G which is sufficiently invariant
under left translation by F Q
T
and T ′ is a subset of T with |T ′ | ≥ (1 − nβ)|T | then | s∈F Q s−1 T ′ | ≥ (1 − γ)|T |.
Choose an L ∈ N large enough so that (1 − β/2)L < β. By amenability there exist
finite subsets e ∈ T1 ⊆ T2 ⊆ · · · ⊆ TL of G such that |∂Tl−1 Tl | ≤ (β/8)|Tl | for l = 2, . . . , L.
By the previous paragraph, we may also assume that for each l the set Tl is sufficiently
invariant under left translation by F Q so that for all T ⊆ Tl satisfying |T | ≥ (1 − nβ)|Tl |
one has
\
s−1 T ≥ (1 − γ)|Tl |.
(5.1)
s∈F Q

By uniform continuity there is a η > 0 such that |f (x)−f (y)| < ε/(3n2 ) for all f ∈ Υ∪Υ2
and all x, y ∈ X satisfying d(x, y) < η. Again by uniform continuity there is an η ′ > 0

14

CONLEY, JACKSON, KERR, MARKS, SEWARD, AND TUCKER-DROB

S
such that d(tx, ty) < η for all x, y ∈ X satisfying d(x, y) < η ′ and all t ∈ L
l=1 Tl . Fix a
clopen partition {A1 , . . . , AM } of X whose members all have diameter less that η ′ .
Let E be a finite subset of G containing TL and let δ > 0 be such that δ ≤ β/4. We will
further specify E and δ below. By hypothesis there is a collection {(Vk , Sk )}K
k=1 of clopen
towers such that the shapes S1 , . . . , SK are (E, δ)-invariant and the sets S1 V1 , . . . , SK VK
partition X. We may assume that for each k = 1, . . . , K the set Sk is large enough so that
X

L
Mn
(5.2)
|Tl | ≤ β|Sk |.
l=1

By a simple procedure we can construct, for each k, a clopen partition Pk of Vk such that
each level of every one of the towers (V, Sk ) for V ∈ Pk is contained in one of the sets
A1 , . . . , AM as well as in one of the sets A and X \ A. By replacing (Vk , Sk ) with these
thinner towers for each k, we may therefore assume that each level in every one of the
towers (V1 , S1 ), . . . , (VK , SK ) is contained in one of the sets A1 , . . . , AM and in one of the
sets A and X \ A.
Let 1 ≤ k ≤ K. Since Sk is (TL , β/4)-invariant, by Theorem 5.2 and our choice of
the sets T1 , . . . , TL we can find Ck,1 , . . . , Ck,L ⊆ Sk such that the collection {Tl c : l =
1, . . . , L, c ∈ Ck,l } is β-disjoint, has union contained in Sk , and (1 − β)-covers Sk . By
β-disjointness, for every l = 1, . . . , L and c ∈ Ck,l we can find a Tk,l,c ⊆ Tl satisfying
|Tk,l,c | ≥ (1 − β)|Tl | so that the collection of sets Tk,l,cc for l = 1, . . . , L and c ∈ Ck,l is
disjoint and has the same union as the sets Tl c for l = 1, . . . , L and c ∈ Ck,l , so that it
(1 − β)-covers Sk .
For each l = 1, . . . , L and m = 1, . . . , M write Ck,l,m for the set of all c ∈ Ck,l such that
(n)
(1)
cVk ⊆ Am , and choose pairwise disjoint subsets Ck,l,m, . . . , Ck,l,m of Ck,l,m such that each
has cardinality ⌊|Ck,l,m |/n⌋. For each i = 2, . . . , n choose a bijection
G (1)
G (i)
Λk,i :
Ck,l,m →
Ck,l,m
l,m

(1)

l,m

(i)

which sends Ck,l,m to Ck,l,m for all l, m. Also, define Λk,1 to be the identity map from
F
(1)
l,m Ck,l,m to itself.
T
F
(1)
′
= ni=1 Tk,l,Λk,i (c) , which satisfies
Let 1 ≤ l ≤ L and c ∈ m Ck,l,m. Define the set Tk,l,c

(5.3)

′
| ≥ (1 − nβ)|Tl | ≥ (1 − nβ)|Tk,l,c |
|Tk,l,c

since each Tk,l,Λk,i (c) is a subset of Tl of cardinality at least (1 − β)|Tl |. Set
\
′
′
Bk,l,c,Q =
sTk,l,c
, Bk,l,c,0 = Tk,l,c
\ F Q−1 Bk,l,c,Q,
s∈F Q

and, for q = 1, . . . , Q − 1, using the convention F 0 = {e},

Bk,l,c,q = F Q−q Bk,l,c,Q \ F Q−q−1 Bk,l,c,Q.
′
. For s ∈ F we have
Then the sets Bk,l,c,0, . . . , Bk,l,c,Q partition Tk,l,c

(5.4)

sBk,l,c,Q ⊆ Bk,l,c,Q−1 ∪ Bk,l,c,Q,

FØLNER TILINGS FOR ACTIONS OF AMENABLE GROUPS

15

while for q = 1, . . . , Q − 1 we have
sBk,l,c,q ⊆ Bk,l,c,q−1 ∪ Bk,l,c,q ∪ Bk,l,c,q+1,

(5.5)

for if we are given a t ∈ Bk,l,c,q then st ∈ F Q−q+1 Bk,l,c,Q, while if st ∈ F Q−q−2 Bk,l,c,Q then
t ∈ F Q−q−1 Bk,l,c,Q since F is symmetric, contradicting the membership of t in Bk,l,c,q .
Also, from (5.1) and (5.3) we get
|Bk,l,c,Q| ≥ (1 − γ)|Tl |.

(5.6)
For i = 2, . . . , n, c ∈

F

(i)
m Ck,l,m ,

and q = 0, . . . , Q we set Bk,l,c,q = Bk,l,λ−1(c),q .
k,i

Write Λk,i,j for the composition Λk,i ◦ Λ−1
k,j . Define a linear map ψ : Mn → C(X) ⋊ G
by declaring it on the standard matrix units {eij }ni,j=1 of Mn to be given by
X
X X
utΛk,i,j (c)c−1 t−1 1tcVk
ψ(eij ) =
k,l,m c∈C (j)

k,l,m

′
t∈Tk,l,c

and extending linearly. Then ψ(eij )∗ = ψ(eji ) for all i, j and the product ψ(eij )ψ(ei′ j ′ ) is
1 or 0 depending on whether i = i′ , so that ψ is a ∗ -homomorphism.
F
(j)
For all k and l, all 1 ≤ i, j ≤ n, and all c ∈ m Ck,l,m we set
hk,l,c,i,j =

Q
X

X

q=1 t∈Bk,l,c,q

q
u
−1 −1 1tcV
k
Q tΛk,i,j (c)c t

and put
h=

n
X
X X

hk,l,c,i,i.

k,l,m i=1 c∈C (i)
k,l,m

Then h is a norm-one function which commutes with the image of ψ, and so we can define
an order-zero completely positive contractive linear map ϕ : Mn → C(X) ⋊ G by setting
ϕ(z) = hψ(z).
Note that ϕ(eij ) =

P

k,l,m

P

(j)

c∈Ck,l,m

hk,l,c,i,j .

We now verify condition (ii) in Theorem 5.1 for the elements of the set {us : s ∈ F }.
F
(j)
Let 1 ≤ i, j ≤ n. For all k and l, all c ∈ m Ck,l,m, and all s ∈ F we have
us hk,l,c,i,j u−1
s − hk,l,c,i,j =

Q
X

X

q=1 t∈Bk,l,c,q

q
u
−1
−1 1stcV
k
Q stΛk,i,j (c)c (st)

−

Q
X

X

q=1 t∈Bk,l,c,q

q
u
−1 −1 1tcV ,
k
Q tΛk,i,j (c)c t

and so in view of (5.4) and (5.5) we obtain
kus hk,l,c,i,j u−1
s − hk,l,c,i,j k ≤

1
ε
< 2.
Q
n

16

CONLEY, JACKSON, KERR, MARKS, SEWARD, AND TUCKER-DROB

∗
∗
Since each of the elements b = us hk,l,c,i,j u−1
s −hk,l,c,i,j is such that b b and bb are dominated
′
′
by twice the indicator functions of Tk,l,Λ−1 (c) cVk and Tk,l,Λ−1 (c) Λk,i,j (c)Vk , respectively, and
j
j
F
(i′ )
′
′
the sets Tk,l,Λ−1(c) cVk over all k, l, all i = 1, . . . , n, and all c ∈ m Ck,l,m are pairwise
i′

disjoint, this yields

−1
kus ϕ(eij )u−1
s − ϕ(eij )k = max max kus hk,l,c,i,j us − hk,l,c,i,j k <
k,l,m c∈C (j)
k,l,m

ε
n2

and hence, for every norm-one element z = (zij ) ∈ Mn ,
k[us , ϕ(z)]k = kus ϕ(z)u−1
s − ϕ(z)k ≤

n
X

|zij |kus ϕ(eij )u−1
s − ϕ(eij )k

i,j=1

ε
= ε.
n2
Next we verify condition (ii) in Theorem 5.1 for the functions in Υ. Let 1 ≤ i, j ≤ n.
(j)
Let g ∈ Υ ∪ Υ2 . Let 1 ≤ k ≤ K, 1 ≤ l ≤ L, 1 ≤ m ≤ M , and c ∈ Ck,l,m. Then
< n2 ·

(5.7)

h∗k,l,c,i,j ghk,l,c,i,j =

Q
X

X

q=1 t∈Bk,l,c,q

q2
(tcΛk,i,j (c)−1 t−1 g)1tcVk
Q2

and
(5.8)

gh∗k,l,c,i,j hk,l,c,i,j

=

Q
X

X

q=1 t∈Bk,l,c,q

q2
g1tcVk .
Q2

Now let x ∈ Vk . Since Λk,i,j (c)x and cx both belong to Am , we have d(Λk,i,j (c)x, cx) < η ′ .
It follows that for every t ∈ Tl we have d(tΛk,i,j (c)x, tcx) < η by our choice of η ′ , so that
|g(tΛk,i,j (c)x) − g(tcx)| < ε/(3n2 ) by our choice of η, in which case
k(tcΛk,i,j (c)−1 t−1 g − g)1tcVk k = kc−1 t−1 ((tcΛk,i,j (c)−1 t−1 g − g)1tcVk )k
= k(Λk,i,j (c)−1 t−1 g − c−1 t−1 g)1Vk k
= sup |g(tΛk,i,j (c)x) − g(tcx)|
x∈Vk

<

ε
.
3n2

Using (5.7) and (5.8) this gives us
(5.9)

kh∗k,l,c,i,j ghk,l,c,i,j − gh∗k,l,c,i,j hk,l,c,i,j k
= max

max

q=1,...,Q t∈Bk,l,c,q

q2
ε
k(tcΛk,i,j (c)−1 t−1 g − g)1tcVk k < 2 .
Q2
3n

Set w = ϕ(eij ) for brevity. Let f ∈ Υ. For g ∈ {f, f 2 } the functions h∗k,l,c,i,j ghk,l,c,i,j −
(j)

gh∗k,l,c,i,j hk,l,c,i,j over all k, l, and m and all c ∈ Ck,l,m have pairwise disjoint supports, so
that (5.9) yields
ε
kw∗ gw − gw∗ wk < 2 .
3n

FØLNER TILINGS FOR ACTIONS OF AMENABLE GROUPS

17

It follows that
kw∗ f 2 w − f w∗ f wk ≤ kw∗ f 2 w − f 2 w∗ wk + kf (f w∗ w − w∗ f w)k <

2ε
3n2

and so
kf w − wf k2 = k(f w − wf )∗ (f w − wf )k
= kw∗ f 2 w − f w∗ f w + f w∗ wf − w∗ f wf k
≤ kw∗ f 2 w − f w∗ f wk + k(f w∗ w − w∗ f w)f k
2ε
ε
ε
< 2 + 2 = 2.
3n
3n
n
Therefore for every norm-one element z = (zij ) ∈ Mn we have
k[f, ϕ(z)]k ≤

n
X

|zij |k[f, ϕ(eij )]k < n2 ·

i,j=1

ε
= ε.
n2

Finally, we verify that the parameters in the construction of ϕ can be chosen so that
1 − ϕ(1) - 1A . By taking the sets S1 , . . . , SK to be sufficiently left invariant (by enlarging
E and shrinking δ if necessary) we may assume that for every k = 1, . . . , K there is an
Sk′ ⊆ Sk such that the set {s ∈ Sk′ : Ds ⊆ Sk } has cardinality at least |Sk |/2. Let
1 ≤ k ≤ K. Take a maximal set Sk′′ ⊆ Sk′ such that the sets Ds for s ∈ Sk′′ are pairwise
disjoint, and note that |Sk′′ | ≥ |Sk′ |/|D −1 D| ≥ |Sk |/(2|D|2 ). Since D −1 A = X, each of the
sets DsVk for s ∈ Sk′′ intersects A, and so the set Sk♯ of all s ∈ Sk such that sVk ⊆ A has
F F F
cardinality at least |Sk |/(2|D|2 ). Define Sk,1 = l,m ni=1 c∈C (i) Bk,l,c,Qc, which is the
k,l,m

set of all s ∈ Sk such that the function ϕ(1) takes the value 1 on sVk . Set Sk,0 = Sk \ Sk,1 .
P
(i)
Since ni=1 |Ck,l,m | ≥ |Ck,l,m | − n for every l and m, by (5.2) we have
n
XX

(i)

|Tl ||Ck,l,m | ≥

l,m i=1

X

|Tl ||Ck,l,m | − M n

l,m

≥

[

X

|Tl |

l

Tl Ck,l − β|Sk |

l

≥ (1 − 2β)|Sk |.
(i)

Since for all l and i and all c ∈ Ck,l,m we have |Bk,l,c,Q| ≥ (1 − γ)|Tl | by (5.6), it follows,
putting λ = (1 − γ)(1 − 2β), that
|Sk,1 | ≥ (1 − γ)

n
XX

(i)

|Tl ||Ck,l,m | ≥ λ|Sk |.

l,m i=1

By taking γ and β small enough we can guarantee that 1 − λ ≤ 1/(2|D|2 ) and hence
|Sk,0 | = |Sk | − |Sk,1 | ≤ (1 − λ)|Sk | ≤ |Sk♯ |,

18

CONLEY, JACKSON, KERR, MARKS, SEWARD, AND TUCKER-DROB

so that there exists an injection θk : Sk,0 → Sk♯ . Define
z=

K X
X

uθk (s)s−1 1sVk .

k=1 s∈Sk,0

A simple computation shows that z ∗ 1A z is the indicator function of
is the support of 1 − ϕ(1), and so putting v = (1 − ϕ(1))1/2 z ∗ we get

FK

k=1 Sk,0 Vk ,

which

v1A v ∗ = (1 − ϕ(1))1/2 z ∗ 1A z(1 − ϕ(1))1/2 = 1 − ϕ(1).
This demonstrates that 1 − ϕ(1) - 1A , as desired.



Combining Theorems 5.1 and 4.2 yields the following.
Theorem 5.4. Let G be a countably infinite amenable group and X the Cantor set. Then
the set of all actions in FrMin(G, X) whose crossed product is Z-stable is comeager, and
in particular nonempty.
Question 5.5. Do any of the crossed products in Theorem 5.4 have tracial state space
with compact extreme boundary (from which we would be able to conclude finite nuclear
dimension by [1] and hence classifiability)? For G = Z a generic action in FrMin(G, X)
is uniquely ergodic, so that the crossed product has a unique tracial state [9]. However,
already for Z2 nothing of this nature seems to be known. On the other hand, it is known
that the crossed products of free minimal actions of finitely generated nilpotent groups on
compact metrizable spaces of finite covering dimension have finite nuclear dimension, and
in particular are Z-stable [21].
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Clinton T. Conley, Department of Mathematical Sciences, Carnegie Mellon University,
Pittsburgh, PA 15213, U.S.A.
E-mail address: clintonc@andrew.cmu.edu
Steve Jackson, Department of Mathematics, University of North Texas, Denton, TX 762035017, U.S.A.
E-mail address: jackson@unt.edu
David Kerr, Department of Mathematics, Texas A&M University, College Station, TX
77843-3368, U.S.A.
E-mail address: kerr@math.tamu.edu
Andrew Marks, UCLA Department of Mathematics, Los Angeles, CA 90095-1555, U.S.A.
E-mail address: marks@math.ucla.edu
Brandon Seward, Courant Institute of Mathematical Sciences, New York, NY 10012, U.S.A.
E-mail address: bseward@cims.nyu.edu
Robin Tucker-Drob, Department of Mathematics, Texas A&M University, College Station,
TX 77843-3368, U.S.A.
E-mail address: rtuckerd@math.tamu.edu

