Submitted to
Journal of the Mathematical Society of Japan

Essential holonomy of Cantor actions
By Steven Hurder and Olga Lukina(Received Jan. 20, 2023)(Revised Oct. 02, 2023)

Abstract. A minimal group action has essential holonomy if the set ofpoints with non-trivial holonomy has positive measure. If such an action is
topologically free, then having essential holonomy is equivalent to the action
not being essentially free, which means that the set of points with non-trivial
stabilizer has positive measure. In this paper, we investigate the relation between the property of having essential holonomy and structure of the acting
group for minimal equicontinuous actions on Cantor sets. We show that if
such a group action is locally quasi-analytic and has essential holonomy, thenevery commutator subgroup in the group lower central series has elementswith positive measure set of points with non-trivial holonomy. In particular,
we prove that a minimal equicontinuous Cantor action by a nilpotent group
has no essential holonomy. We also show that the property of having essential
holonomy is preserved under return equivalence and continuous orbit equivalence of minimal equicontinuous Cantor actions. Finally, we give examples
to show that the assumption on the action that it is locally quasi-analytic isnecessary.

1.

Introduction

We say that (X, Γ, Φ) is a Cantor action if Γ is a countable group, X is a Cantor
space, and Φ : Γ × X → X is an action by homeomorphisms. In this paper we consider
Cantor actions that are minimal and equicontinuous. Under these assumptions, (X, Γ, Φ)
has a unique ergodic invariant probability measure µ. We recall further basic definitions
and constructions for Cantor actions in Section 2, as used in the formulation of our results
below.
The dynamical properties of Cantor actions, even under assumptions of minimality
and equicontinuity, can have surprisingly subtle aspects, especially for the case whereΓ is non-abelian, as revealed by the many examples in the literature. One approach toclassifying minimal equicontinuous Cantor actions is via their dynamical properties. The
main result of this work makes a new contribution to this classification scheme, as it
relates the algebraic properties of Γ and the dynamics of the action, through the studyof the property that an action has essential holonomy; see Definition 1.3 below.
First recall some standard notions concerning the fixed point sets for a Cantor action(X, Γ, Φ). We use the notation g · x = Φ(g)(x), for g ∈ Γ and x ∈ X. The set Xg = {x ∈X | g · x = x} consists of fixed points for g, and the stabilizer of a point x ∈ X is the
subgroup Γx = {g ∈ Γ | g · x = x}. Let2020 Mathematics Subject Classification. Primary: 20E18, 22F10, 37A15, 37B05; Secondary: 57S10.
Key Words and Phrases. Equicontinuous group actions, topologically free, essentially free, holonomy, locally quasi-analytic actions, nilpotent actions, lower central series, commutator subgroup.

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S. Hurder and O. Lukina

XΓ =

[

Xg = {x ∈ X | Γx 6= {e}}(1)e6=g∈Γ

be the set of all points fixed by some non-identity element g ∈ Γ.
DEFINITION 1.1. A minimal Cantor action (X, Γ, Φ) with invariant probability
measure µ is:
1. free if XΓ is empty.
2. topologically free if XΓ is a meager set in X.
3. essentially free if µ(XΓ ) = 0.
Recall that a Cantor action (X, Γ, Φ) is effective if the only element of Γ that acts
as the identity on X is the identity of Γ. It is elementary to show that if Γ is abelian,then every effective minimal Cantor action of Γ must be free. The topologically free
Cantor actions have an important role in the study of the C∗ -algebras associated withthe actions, as studied for example in [4, 8, 24, 27, 30].
Kambites, Silva and Steinberg showed in [25, Theorem 4.3] that the action of agroup generated by finite automata on a rooted tree is topologically free if and only if
it is essentially free. Joseph proved in [22, Corollary 2.4] that if Γ has countably many
subgroups, then a minimal Cantor action of Γ is topologically free if and only if it isessentially free.
Bergeron and Gaboriau [7] showed that if Γ is a non-amenable group which is a freeproduct of two residually finite groups, then Γ admits a minimal equicontinuous Cantor
action which is topologically free and not essentially free. Abért and Elek [1] proved asimilar result for finitely generated non-abelian free groups Γ. Joseph [22] proved thatany non-amenable surface group admits a continuum of pairwise non-conjugate and measurably non-isomorphic minimal equicontinuous Cantor actions which are topologically
free and not essentially free.
Examples of effective minimal equicontinuous Cantor actions which are not topologically free include some actions of branch and weakly branch groups on the boundaries ofrooted trees [15, 29, 31], some actions of nilpotent groups [21], and actions of topologicalfull groups [10, 15].
The work by Gröger and Lukina [16] introduced a refinement of the notion of essentially free actions, called the essential holonomy property. The idea is to consider
the dynamics of the action in small neighborhoods of fixed points. In place of the setof points with trivial (resp. non-trivial) stabilizers, one considers the set of points withtrivial (resp. non-trivial) holonomy. The analog of an essentially free Cantor action isan action which has no essential holonomy.
DEFINITION 1.2. Let (X, Γ, Φ) be a Cantor action. Say that x ∈ X is a point of
non-trivial holonomy for g ∈ Γ if Φ(g)(x) = x, and for each open set U ⊂ X with x ∈ U ,there exists y ∈ U such that Φ(g)(y) 6= y.
We say that a fixed point x ∈ X is a point of trivial holonomy for g ∈ Γ, if x is fixedby Φ(g), and x has an open neighborhood Ux,g where every point is fixed by Φ(g). We

Essential holonomy of Cantor actions

3

say that x ∈ X is a point of trivial holonomy, if x is a point of trivial holonomy for allg ∈ Γ with g · x = x. We say that x ∈ X is a point of non-trivial holonomy, if x is a pointof non-trivial holonomy for some g ∈ Γ with g · x = x.
Let Xholdenote the (possibly empty) subset of points of non-trivial holonomy in Xg ,gand set
[
[
Xhol
Xhol
⊂
Xg ⊂ XΓ .
(2)
Γ =
g
g∈Γ

g∈Γ

The set Xhol
is invariant under the action of Γ, thus if (X, Γ, Φ) admits an ergodicΓ
invariant probability measure, then Xhol
has either µ-measure 0 or 1. The followingΓconcept was formulated in [16]:DEFINITION 1.3. A measure-preserving Cantor action (X, Γ, Φ) with ergodic invariant probability measure µ has essential holonomy if the set Xhol
Γ of points with nontrivial holonomy has positive µ-measure. Otherwise, it has no essential holonomy.
is always ‘small’, namely, it is a meager
We note that topologically the set XholΓset. Recall that a set is meager if it is contained in a countable union of nowhere dense
need not be ‘small’ with respect to the measure µ, and this paper
sets. However, XholΓis measurably
investigates minimal equicontinuous Cantor actions where the set XholΓ‘large’, namely, has full µ-measure.
Vorobets [32] showed that the standard action of the Grigorchuk group on theboundary of a binary rooted tree has only a countable set of points with non-trivial
holonomy, which implies that it has no essential holonomy. Gröger and Lukina provedin [16] that the action of a group generated by finite automata on a rooted tree has no
essential holonomy; their proof does not require the Cantor action to be topologically
free. They also gave a criterion for when a group action on a rooted tree has no essential
holonomy for its boundary Cantor action.
The decomposition (2) of XholΓ as a countable union of fixed point sets implies thatan action has essential holonomy if and only if, for at least one g ∈ Γ, the set Xholghas positive µ-measure. In particular, this implies that whether or not a Cantor action
has essential holonomy is a local property. This remark is the basis for the following
invariance result, with details and proof given in Section 3.
PROPOSITION 1.4.
1. Suppose that minimal equicontinuous Cantor actions(Xi , Γi , Φi ), for i = 1, 2, are continuous orbit equivalent. Then either both haveessential holonomy, or both have no essential holonomy.
2. Suppose that minimal equicontinuous Cantor actions (Xi , Γi , Φi ), for i = 1, 2, arereturn equivalent. Then either both have essential holonomy, or both have no essential holonomy.
For a Cantor action (X, Γ, Φ), the assumption that µ is a probability measure, andthat the action is minimal, implies that µ is a continuous measure; that is, the measureof any point x ∈ X is zero. Thus, if a Cantor action has essential holonomy, then theintersection of Xhol
with the support of µ must be an uncountable set. This remark isg

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S. Hurder and O. Lukina

the basis for the result by Joseph in [22, Corollary 2.4] that if Γ has only countably
many subgroups, then a topologically free minimal Cantor action by Γ is essentially free,and hence has no essential holonomy. The argument in [22] holds for minimal actions
by homeomorphisms on any compact Hausdorff space with an invariant finite ergodic
measure.
Our main result below relates the dynamical properties of a minimal equicontinuous Cantor action of a finitely generated group Γ, with the lower central seriesΓ = γ1 (Γ) ⊃ γ2 (Γ) ⊃ · · · ⊃ γn (Γ) ⊃ · · · . Intuitively, a locally quasi-analytic minimal Cantor action (X, Γ, Φ) is an action such that every homeomorphism Φ(g) has aunique extension from small open sets to open sets of uniform diameter over X, see Section 2.5 for a precise definition and discussion of this property. In particular, the locallyquasi-analytic property is a localized form of the topologically free property.
THEOREM 1.5. Let (X, Γ, Φ) be a locally quasi-analytic minimal equicontinuous
Cantor action with invariant ergodic probability measure µ. If the action has essential
holonomy, then for every n ≥ 1 there exists φn ∈ γn (Γ) such that the action of φn has apositive measure set of points with non-trivial holonomy.
The proof of Theorem 1.5 is given in Section 4. We observe the following corollary
for n = 2:
COROLLARY 1.6. Let (X, Γ, Φ) be a locally quasi-analytic minimal equicontinuous
Cantor action with ergodic invariant probability measure µ. Let [Γ, Γ] ⊂ Γ be the commutator subgroup of Γ. If the action has essential holonomy, then there exists φ ∈ [Γ, Γ]such that the action of φ has a positive measure set of points with non-trivial holonomy.
It is tempting to apply Corollary 1.6 inductively, so that the conclusion of Theorem 1.5 applies to the derived series of Γ. This argument does not go through, though,because while the action of the commutator subgroup [Γ, Γ] on X is again equicontinuousand locally quasi-analytic, it need not be minimal, and the minimality assumption on
the action is critical for the proof of Theorem 1.5.
The family of examples constructed in the proof of Theorem 5.1 show that theassumption that a minimal equicontinuous Cantor action is locally quasi-analytic is essential for the conclusions of Theorem 1.5 and Corollary 1.6. The actions of groups Γand the commutator subgroups [Γ, Γ] in Theorem 5.1 are minimal and not locally quasianalytic, and the actions of the group Γ have essential holonomy, while the actions of[Γ, Γ] have no essential holonomy.
If Γ is a Noetherian group, then every Cantor action by Γ must be locally quasianalytic by [19, Theorem 1.6]. For a nilpotent group Γ, the lower central series terminates, and Γ is Noetherian, so as a consequence we obtain the following.
COROLLARY 1.7. Let (X, Γ, Φ) be a minimal equicontinuous Cantor action. If Γ
is a finitely generated nilpotent group, then (X, Γ, Φ) has no essential holonomy.
In the case when (X, Γ, Φ) is topologically free, Corollary 1.7 recovers the result ofJoseph [22], who proved that a minimal Cantor action of a finitely generated nilpotent

Essential holonomy of Cantor actions

5group (and more generally, of a group with countable number of subgroups acting on acompact Hausdorff topological space) is topologically free if and only if it is essentially
free. The paper [22] uses the method of invariant random subgroups, which is very
different from our proof. While the result in [22] applies to topologically free minimal
actions which are not necessarily equicontinuous, the result of Corollary 1.7 applies tominimal actions that are equicontinuous but need not be topologically free.
Up until recently, it was an open problem to show that if Γ is a finitely generated
amenable group, then every minimal equicontinuous Cantor action of Γ has no essential
holonomy. In the paper [23], Joseph constructs a family of wreath products of two finitely
generated amenable groups which are amenable groups, and which admit topologically
free and not essentially free minimal equicontinuous actions. Actions in Theorem 5.1
in our paper are not locally quasi-analytic (and therefore not topologically free) actionsof infinitely generated amenable groups which have essential holonomy. It would beinteresting to obtain a criterion for when an amenable group admits an action withessential holonomy.
Theorem 1.5 suggests that the property of having essential holonomy is an interestinginvariant of Cantor actions, intrinsically related to the structure of the acting group, tobe further explored.
2.

Cantor actions

We recall some basic properties of Cantor actions. More details can be found in[5, 9, 10, 19, 20, 26].
2.1. Basic notions
Let (X, Γ, Φ) denote a topological action Φ : Γ × X → X. We write g · x for Φ(g)(x)when appropriate. The orbit of x ∈ X is the subset O(x) = {g · x | g ∈ Γ}. The action isminimal if for all x ∈ X, its orbit O(x) is dense in X. The action is said to be effective,
or faithful, if the homomorphism Φ : Γ → Homeo(X) is injective.
An action (X, Γ, Φ) is equicontinuous with respect to a metric dX on X, if for all
ε > 0 there exists δ > 0, such that for all x, y ∈ X and g ∈ Γ we have that dX (x, y) < δ
implies dX (g · x, g · y) < ε. The property of being equicontinuous is independent of the
choice of the metric on X which is compatible with the topology of X.
For the rest of the section, we assume that a Cantor action (X, Γ, Φ) is minimal andequicontinuous. We say that U ⊂ X is adapted to the action if U is a non-empty clopen
subset, and for any g ∈ Γ, if Φ(g)(U ) ∩ U 6= ∅ then Φ(g)(U ) = U . Recall a basic propertyof minimal equicontinuous Cantor actions (see [19, Section 3]).
PROPOSITION 2.1. Let (X, Γ, Φ) be a minimal equicontinuous Cantor action, andlet dX be an invariant metric on X compatible with the topology on X. Given x ∈ X andε > 0, there exists an adapted clopen set U ⊂ X with x ∈ U and diam(U ) < ε.
COROLLARY 2.2. For a minimal equicontinuous Cantor action (X, Γ, Φ), theadapted clopen sets form a subbasis for the topology of X.
For an adapted set U , the set of “return times” to U ,6

S. Hurder and O. Lukina

ΓU = {g ∈ Γ | g · U ∩ U 6= ∅}(3)

is a subgroup of Γ, called the stabilizer of U . Indeed, for g, g 0 ∈ Γ with g · U ∩ g 0 · U 6= ∅we have g −1 g 0 · U = U , hence g −1 g 0 ∈ ΓU . Moreover, the translates {g · U | g ∈ Γ} forma finite clopen partition of X, and are in 1-1 correspondence with the quotient space
XU = Γ/ΓU . Then Γ acts by permutations of the finite set XU and so the stabilizer
group ΓU ⊂ Γ has finite index. Note that this implies that if V ⊂ U is a proper inclusionof adapted sets, then the inclusion ΓV ⊂ ΓU is also proper.
Let U be an adapted set for the action (X, Γ, Φ), then the action of Γ restricts on U
to an action of ΓU , so we have a homomorphism ΦU : ΓU → Homeo(U ). Let HU denotethe image of this action. Note that the map ΦU : ΓU → HU is injective if the action is
topologically free.
2.2. Group chains
Given a basepoint x, by iterating the process in Proposition 2.1 one can always
construct the following:
DEFINITION 2.3. Let (X, Γ, Φ) be a minimal equicontinuous Cantor action. A
properly descending chain of clopen sets U = {U` ⊂ X | ` ≥ 0}, with U0 = X, is said tobe an adapted neighborhood basis at x ∈ X for the action Φ if each U` is adapted to theaction Φ, U`+1 ⊂ U` for all ` ≥ 0, and ∩ U` = {x}.
Let Γ` = ΓU` denote the stabilizer group of U` given by (3). Then we obtain adescending chain of finite index subgroups GUx : Γ = Γ0 ⊃ Γ1 ⊃ Γ2 ⊃ · · ·. Note that eachΓ` has finite index in Γ, and is not assumed to be a normal subgroup. Also note thatwhile the intersection of the chain U is a single point {x}, the intersection of the stabilizer
groups in GUx need not be the trivial group.
Next, set X` = Γ/Γ` and note that Γ acts transitively on the left on X` . The
inclusion Γ`+1 ⊂ Γ` induces a natural Γ-invariant quotient map p`+1 : X`+1 → X` .
Introduce the inverse limit
X = lim {p`+1 : X`+1 → X` | ` ≥ 0} = {(x` ) = (x0 , x1 , . . .) | p`+1 (x`+1 ) = x` }←−(4)which is a Cantor space with the Tychonoff topology. Thus elements of X are infinite
sequences with entries in X` , ` ≥ 0. The actions of Γ on the factors X` induce a minimal
equicontinuous action, denoted by Φx : Γ × X → X, which reads(g, (x` )) 7→ g · (x` ) = (g · x` ) = (g · x0 , g · x1 , . . .).

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For each ` ≥ 0, we have the “partition coding map” Θ` : X → X` which is Γ-equivariant.
The maps {Θ` } are compatible with the map on quotients in (4), and so define a limit map
Θx : X → X. The fact that the diameters of the clopen sets {U` } tend to zero, implies
that Θx is a homeomorphism. This is proved in detail in [12, Appendix A]. Moreover,Θx (x) = e∞ = (eΓ` ) ∈ X, the basepoint of the inverse limit (4), where eΓ` = Γ` is thecoset of the identity e ∈ Γ. Let X have an ultrametric metric such that Γ acts on X by
isometries, for instance, letEssential holonomy of Cantor actionsdX ((x` ), (y` )) = 2−m ,where m = max{` | x` = y` , ` ≥ 0}.

7(6)

Then let dX be the ultrametric metric on X induced from dX by the homeomorphism Θx .
The minimal equicontinuous action (X, Γ, Φx ) is called the odometer model centered at
x for (X, Γ, Φ).
The group chain GUx depends on x and U, and one can introduce an equivalence
relation which, for a given group Γ, identifies the class of group chains with topologicallyconjugate associated odometer models. We refer the interested reader to [12].
2.3. Unique ergodic invariant measure
Given a minimal equicontinuous Cantor action (X, Γ, Φ), choose an adapted neighborhood basis U and consider the corresponding group chain GUx and the odometer model(4). The group Γ acts transitively on the coset space X` = Γ/Γ` and we define a Γinvariant probability measure µ` on X` by giving equal weight to each point (coset) inX` . Thus one hasµ` (hΓ` ) =1, for all hΓ` ∈ X` and all ` ≥ 0,
|Γ : Γ` |(7)

where |Γ : Γ` | denotes the index of Γ` in Γ. The unique Γ-invariant measure on the
inverse limit X is defined as the limit of the pull-backs of these measures under the
projection maps X → X` . Then the invariant measure µ on X is the pull-back via thehomeomorphism Θx : X → X.
Alternately, consider the closure E = Φ(Γ) ⊂ Homeo(X) in the uniform topology. It
is a profinite compact group, called the Ellis or enveloping group [5, 13]. (If the action(X, Γ, Φ) is not assumed to be equicontinuous, then Φ(Γ) is only a semi-group.) The
group E acts on X with isotropy group Ex = {g ∈ E | g · x = x} for x ∈ X. Then Ex is a
closed subgroup of E, and we have X = E/Ex . The profinite group E has a unique Haar
measure µb, which is invariant with respect to the action of E on itself. The measure µbon E pushes down to the measure µ on X.
2.4. Lebesgue density theorem
Let (X, Γ, Φ) be a minimal equicontinuous Cantor action with probability measureµ and ultrametric dX induced from (6). Let B(x, ) = {y ∈ X | dX (x, y) < } denote theopen ball with center x of radius ε > 0. The proof of the Lebesgue Density Theorem inthe formulation below can be found, for instance, in [28, Proposition 2.10].
THEOREM 2.4. Let X be a Polish space, and suppose X has an ultrametric dX
compatible with its topology. Let µ be a probability measure on X, and let A be a Borel
set of positive measure. Then for µ-almost every x ∈ A, the Lebesgue density of x in A,
given bylim

→0µ(A ∩ B(x, ))µ(B(x, ))exists and is equal to 1.
We give an important consequence of the Lebesgue Density Theorem.

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LEMMA 2.5. Let (X, Γ, Φ) be a minimal equicontinuous Cantor action, with invariant probability measure µ. Assume there exists an element g ∈ Γ for which Xhol
has
g
hol
positive µ-measure. Then there exists x ∈ Xg such that, for all 0 < ε < 1, there exists
an adapted set Uε with x ∈ Uε and µ(Uε ∩ Xholg ) ≥ (1 − ε) · µ(Uε ).
Proof. Since Xhol
has positive µ-measure, by the Lebesgue Density Theorem 2.4
gthere exists a point x ∈ Xholof full Lebesgue density. For this point, choose an adaptedgneighborhood basis U = {U` ⊂ X | ` ≥ 0} at x. By the convergence of the limit in
Theorem 2.4 there exists `ε so that µ(U` ∩ Xholg ) ≥ (1 − ε) · µ(U` ) for ` ≥ `ε . Then set
Uε = U`ε .
2.5. Locally quasi-analytic
The quasi-analytic property for Cantor actions was introduced by Álvarez López
and Candel in [3, Definition 9.4] as a generalization of the notion of a quasi-analytic
action studied by Haefliger for actions of pseudogroups of real-analytic diffeomorphisms
in [17]. The authors introduced a local form of the quasi-analytic property in [18, 19]:
DEFINITION 2.6. [19, Definition 2.1] A topological action (X, Γ, Φ) on a metric
Cantor space X, is locally quasi-analytic if there exists ε > 0 such that for any nonempty open set U ⊂ X with diam(U ) < ε, and for any non-empty open subset V ⊂ U ,and elements g1 , g2 ∈ Γ
if the restrictions Φ(g1 )|V = Φ(g2 )|V, then Φ(g1 )|U = Φ(g2 )|U.

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The action is said to be quasi-analytic if (9) holds for U = X.
In other words, (X, Γ, Φ) is locally quasi-analytic if for every g ∈ Γ, the homeomorphism Φ(g) has unique extensions on the sets of diameter ε > 0 in X, with ε uniform
over X. We note that an effective minimal equicontinuous Cantor action (X, Γ, Φ) is
topologically free if and only if it is quasi-analytic [19, Proposition 2.2].
Recall that a group Γ is Noetherian [6] if every increasing chain of subgroups has amaximal element. Equivalently, a group is Noetherian if every subgroup of Γ is finitely
generated.
THEOREM 2.7. [19, Theorem 1.6] Let Γ be a Noetherian group. Then a minimal
equicontinuous Cantor action (X, Γ, Φ) is locally quasi-analytic.
A finitely generated nilpotent group is Noetherian, so as a corollary we obtain that allminimal equicontinuous Cantor actions by finitely generated nilpotent groups are locallyquasi-analytic. Examples of locally quasi-analytic actions which are not quasi-analytic
are easy to construct; see for instance [19, Example A.4].
3.

Invariance

We recall notions of equivalence for Cantor actions, considered as topological dynamical systems. For each notion considered, we show that the measurable dynamicalsystems property that a minimal equicontinuous Cantor action has essential holonomy isEssential holonomy of Cantor actions

9preserved for equivalent actions. The work [14] gives a comparison of the various notionsof equivalence for the case of Cantor actions by Γ = Zn .
First, we recall the most basic equivalence of actions.
DEFINITION 3.1. Cantor actions (X1 , Γ1 , Φ1 ) and (X2 , Γ2 , Φ2 ) are said to be isomorphic, or conjugate, if there is a homeomorphism h : X1 → X2 and a group isomorphism Θ : Γ1 → Γ2 so that
Φ1 (γ) = h−1 ◦ Φ2 (Θ(γ)) ◦ h for all γ ∈ Γ1 .

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The statement below holds for measure-preserving actions which are not necessarily
equicontinuous.
PROPOSITION 3.2. Let (X1 , Γ1 , Φ1 ) and (X2 , Γ2 , Φ2 ) be isomorphic minimal Cantor actions, with respective unique ergodic invariant probability measures µ1 and µ2 . Then
(X1 , Γ1 , Φ1 ) has essential holonomy if and only if (X2 , Γ2 , Φ2 ) has essential holonomy.
Proof. Let a homeomorphism h : X1 → X2 and an isomorphism Θ : Γ1 → Γ2 be
as in Definition 3.1. Then h∗ (µ2 ) is an invariant probability measure for (X1 , Γ1 , Φ1 ),hence by uniqueness µ1 = h∗ (µ2 ). Moreover, for g ∈ Γ1 the action Φ1 (g) has essentialholholonomy if and only if Φ2 (Θ(g)) has essential holonomy, so h(XholΓ1 ) = XΓ2 , and theclaim follows.

The notion of return equivalence is the analog for Cantor actions of Morita equivalence for C∗ -algebras. This equivalence is weaker than the notion of isomorphism, andis natural when considering the Cantor actions defined by the holonomy actions formatchbox manifolds, as in [18, 19]. For convenience in the definition below we restrict
to minimal equicontinuous Cantor actions, but the notion of return equivalence can be
defined for more general minimal actions. In a more general case there are no adapted
neighborhoods, and the closure of the action in Homeo(Ui ), i = 1, 2, need not be a group,and so more care is required in the definition.
DEFINITION 3.3. Minimal equicontinuous Cantor actions (X1 , Γ1 , Φ1 ) and(X2 , Γ2 , Φ2 ) are said to be return equivalent if there exist non-empty clopen subsets
Ui ⊂ Xi , for i = 1, 2, such that Ui is adapted to the action Φi , and there is a homeomorphism h : U1 → U2 whose induced homomorphism h∗ : Homeo(U1 ) → Homeo(U2 )restricts to an isomorphism Θ : HU1 → HU2 .
Note that when Ui = Xi and both actions are effective, then this definition reduces
to the usual notion of isomorphism of the actions, with induced group isomorphismΘ : Γ1 ∼= HX1 → HX2 ∼= Γ2 .
PROPOSITION 3.4. Let (X1 , Γ1 , Φ1 ) and (X2 , Γ2 , Φ2 ) be return equivalent minimal equicontinuous Cantor actions, with respective unique ergodic invariant probability
measures µ1 and µ2 . Then (X1 , Γ1 , Φ1 ) has essential holonomy if and only if (X2 , Γ2 , Φ2 )has essential holonomy.

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S. Hurder and O. Lukina

Proof. Assume that (X1 , Γ1 , Φ1 ) has essential holonomy, and let U1 ⊂ X1 andU2 ⊂ X2 be clopen sets such that the restricted actions are isomorphic by Θ : HU1 → HU2 .
Let g ∈ Γ1 be such that Xhol
has positive µ1 -measure, and thus there exists x ∈ X1 suchgthat x is fixed by Φ1 (g) and Xhol
1,g has Lebesgue density 1 at x. The action Φ1 is minimal
on X1 so there exists k ∈ Γ1 such that k · x ∈ U1 . Then g 0 = kgk −1 ∈ Γ1 has a fixed point00x0 = kx which is a point of Lebesgue density 1 in the set Xhol
1,g 0 . As x ∈ U1 ∩ Φ1 (g )(U1 )and U1 is adapted, we have U1 = Φ1 (g 0 )(U1 ) and so g 0 ∈ Γ1,U .
The action of Γ1,U1 on U1 is minimal, so the renormalized measure µ01 =µ(U1 )−1 (µ1 |U1 ) is the unique invariant probability measure for the restricted action ofΓ1,U1 on U1 .
00hol
The set Xhol
1,g 0 ∩U1 has Lebesgue density 1 at x , hence its image h(x ) ∈ h(X1,g 0 ∩U1 ) ⊂0
U2 is also a point of Lebesgue density 1 for the action of Θ(g ) on U2 , with corresponding
renormalized measure µ02 on U2 . Thus, (X2 , Γ2 , Φ2 ) has essential holonomy. The converse
follows similarly.
The notion of continuous orbit equivalence for Cantor actions was introduced in[8]. It is the analogue for topological dynamics of measurable orbit equivalence formeasurable actions, as first introduced by Dye [11]. Continuous orbit equivalence plays
a fundamental role in the classification of group actions on Cantor sets (see for example[30]).
DEFINITION 3.5. Minimal Cantor actions (X1 , Γ1 , Φ1 ) and (X2 , Γ2 , Φ2 ) are said
to be continuously orbit equivalent if there exists a homeomorphism h : X1 → X2 andcontinuous functions
1. α : G1 × X1 → G2 , h(Φ1 (g1 )(x1 )) = Φ2 (α(g1 , x1 ))(h(x1 )) for all g1 ∈ G1 andx1 ∈ X 1 ;2. β : G2 × X2 → G1 , h−1 (Φ2 (g2 )(x2 )) = Φ1 (β(g2 , x2 ))(h−1 (x2 )) for all g2 ∈ G2 andx2 ∈ X2 .
The homeomorphism h is called a continuous orbit equivalence between the two
actions. Note that the functions α and β are not assumed to satisfy the cocycle property.
We have the following result of Cortez and Medynets:
THEOREM 3.6. [10] Let (X1 , Γ1 , Φ1 ) and (X2 , Γ2 , Φ2 ) be topologically free minimal equicontinuous Cantor actions. If the actions Φ1 and Φ2 are continuously orbit
equivalent, then they are return equivalent.
This result was generalized by the authors:
THEOREM 3.7. [19] Let (X1 , Γ1 , Φ1 ) and (X2 , Γ2 , Φ2 ) be locally quasi-analytic
minimal equicontinuous Cantor actions. If the actions Φ1 and Φ2 are continuously orbit
equivalent, then they are return equivalent.
In fact, equicontinuity of minimal Cantor actions is preserved under continuous orbit
equivalence, as shown in the authors’ work [21]:Essential holonomy of Cantor actions

11

PROPOSITION 3.8. [21, Proposition 3.1] Suppose that minimal Cantor actions(X1 , Γ1 , Φ1 ) and (X2 , Γ2 , Φ2 ) are continuously orbit equivalent. If both Γ1 and Γ2 are
finitely generated groups, and (X1 , Γ1 , Φ1 ) is equicontinuous, then (X2 , Γ2 , Φ2 ) is equicontinuous.
For the general case (not necessarily locally quasi-analytic) of minimal Cantor actions with unique invariant probability measures, we have the following:
PROPOSITION 3.9. Let (X1 , Γ1 , Φ1 ) and (X2 , Γ2 , Φ2 ) be continuously orbit equivalent minimal Cantor actions, with respective unique ergodic invariant probability measures µ1 and µ2 . Then (X1 , Γ1 , Φ1 ) has essential holonomy if and only if (X2 , Γ2 , Φ2 ) has
essential holonomy.
Proof. Let h : X1 → X2 be the homeomorphism given by Definition 3.5. Thenthe pull-back µ
e2 = h∗ (µ2 ) is a probability measure on X1 . Condition (1) of Definition
3.5 implies that µ
e2 is invariant under the action (X1 , Γ1 , Φ1 ), hence by uniqueness ofmeasure µ1 = µe2 .
Assume that (X1 , Γ1 , Φ1 ) has essential holonomy, and let g ∈ Γ1 be such that Xholghas
has positive µ1 -measure, so there exists x ∈ X1 such that Φ1 (g)(x) = x and Xholgµ1 -Lebesgue density 1 at x.
Let α : G1 × X1 → G2 be the map given by condition (1) so that h(Φ1 (g1 )(x1 )) =
Φ2 (α(g1 , x1 ))(h(x1 )) for all g1 ∈ G1 and x1 ∈ X1 . Then for g1 = g there exists a clopen
set U1 ⊂ X1 so that x ∈ U1 and α(g, y) ∈ Γ2 is continuous for y ∈ U1 . Since Γ2 is adiscrete space, we can assume that U1 is sufficiently small so that g2 = α(g, y) ∈ Γ2 isconstant for y ∈ U .
We then have h(Φ1 (g)(y)) = Φ2 (g2 )(h(y)) for all y ∈ U1 . Set U2 = h(U1 ) thenthis states that hU1 : U1 → U2 conjugates the action of Φ1 (g1 ) on U1 with the action ofΦ2 (g2 ) on U2 . Thus, Φ2 (g2 ) has non-trivial holonomy at the fixed point z = h(x) andthis is a point of Lebesgue density 1 for the action of Φ2 (g2 ). In particular, the action(X2 , Γ2 , Φ2 ) has essential holonomy.
The converse implication is proved similarly, using condition (2) of Definition 3.5. 
4.

Dynamics and the lower central series

Theorem 1.5 relates the non-trivial essential holonomy property for a minimal
equicontinuous Cantor action (X, Γ, Φ), with the lower central series of Γ. In this section,
we show that if a minimal equicontinuous Cantor action of Γ is locally quasi-analytic andhas essential holonomy, then every commutator subgroup in the lower central series of Γ
has elements with positive measure sets of points with non-trivial holonomy.
First, recall the construction of the lower central series for Γ.
Set
γ1 (Γ) = Γ,andfori ≥ 1,
let
γi+1 (Γ) = [Γ, γi (Γ)] be the commutator subgroup, which is a normal subgroup of
Γ. Then for a ∈ γi (Γ) and b ∈ γj (Γ) the commutator [a, b] ∈ γi+j (Γ). Moreover, thesesubgroups form a descending chainΓ = γ1 (Γ) ⊃ γ2 (Γ) ⊃ · · · ⊃ γn (Γ) ⊃ · · · .

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12

S. Hurder and O. Lukina

Note that each quotient group γi (Γ)/γi+1 (Γ) is abelian.
The group Γ is nilpotent of length nΓ if there exists an nΓ > 0 such that γn (Γ) is
the trivial group for n > nΓ , and γnΓ (Γ) is non-trivial. It follows that every element in
γnΓ (Γ) commutes with every element of Γ; that is, γnΓ (Γ) is contained in the center ofΓ.
Denote by Φn : γn (Γ) × X → X the restriction of the action Φ to the subgroup γn (Γ).
DEFINITION 4.1. A minimal equicontinuous Cantor action (X, Γ, Φ), with uniqueergodic invariant probability measure µ, is said to have essential holonomy at depth nt if
the restricted action (X, γnt (Γ), Φnt ) has essential holonomy, but (X, γn (Γ), Φn ) has no
essential holonomy for n > nt . The action has essential holonomy at infinite depth iffor all n ≥ 1, the restricted action (X, γn (Γ), Φn ) has essential holonomy.
Here is our main technical result.
PROPOSITION 4.2. Let (X, Γ, Φ) be a minimal equicontinuous Cantor action
which is locally quasi-analytic. If the action has essential holonomy, then it has essential
holonomy of infinite depth.
Proof. Suppose that (X, Γ, Φ) has essential holonomy at depth nt . We show that
this leads to a contradiction by localizing the action to a sufficiently small adapted set.
Recall that µ denotes the unique ergodic invariant probability measure for the action.
hol
The set Xhol
γn (Γ) has positive µ-measure, while the set Xγ(n +1) (Γ) has µ-measure zero.
ttThus, there exists g ∈ γnt (Γ) such that µ(Xholg ) > 0. Moreover, for all τ ∈ γ(nt +1) (Γ) we
holhave µ(Xτ ) = 0.
Let x ∈ X be such that g · x = x, and Xhol
has Lebesgue density 1 at x. Letg
U ⊂ X be an adapted clopen set with x ∈ U , and so g ∈ ΓU , and sufficiently small
diameter such that the restricted action ΦU : ΓU × U → U is quasi-analytic, and we haveµ(Xholg ∩ U ) ≥ 3/4 · µ(U ), as in Lemma 2.5.
By the assumption that x ∈ Xhol
g , there exists y ∈ U so that z = g · y 6= y. Notethat z 6= x as g · x = x. Let V ⊂ U with y ∈ V be a sufficiently small clopen set suchthat (g · V ) ∩ V = ∅. Then choose ky ∈ ΓU such that ky · x ∈ V . Then replace y withky · x and we have g · y 6= y.
Let τ = [g, ky ] be the commutator. Then g ∈ γnt (Γ) implies that τ ∈ γ(nt +1) (Γ)∩ΓU .
By definition we have g · ky = τ · ky · g. Then for w ∈ Xholg ∩ U calculateg · (ky · w) = τ · ky · g · w = τ · (ky · w) .

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We use the identity (12) to prove the key observation:
hol
LEMMA 4.3. The sets ky · (Xholg ∩ U ) and (Xg ∩ U ) are disjoint. In particular,holµ ky · (Xholg ∩ U ) ∩ (Xg ∩ U ) = 0.
hol
Proof. Suppose that w ∈ Xholg ∩ U satisfies ky · w ∈ Xg ∩ U . Then ky · w is a
fixed point for the action of g, and so by (12) we have ky · w is a fixed point for the actionof τ = [g, ky ].

Essential holonomy of Cantor actions

13

As g ∈ γnt (Γ), we have that τ ∈ γ(nt +1) (Γ). Then by the definition of the index nt ,for µ-almost all w ∈ Xholg ∩ U , we have that ΦU (τ ) has trivial holonomy at ky · w, as theaction of Φ(ky ) is a measure preserving homeomorphism.
Thus, there exists a clopen set Ww with ky · w ∈ Ww ⊂ U such that ΦU (τ ) acts as
the identity on Ww . As we chose U so that the action ΦU is quasi-analytic on U , this
implies that ΦU (τ ) acts as the identity on U . However, we also have g · y 6= y, so using
the identity (12) again, we have τ · y = τ · (ky · x) 6= ky · x, and thus ΦU (τ ) does not act ashol
the identity on U , which is a contradiction. Therefore, ky ·(Xholg ∩U )∩(Xg ∩U ) = ∅. 
We now complete the proofof Proposition 4.2. As µ is invariant under the action
holof ΦU we have µ ky · (Xholg ∩ U ) = µ Xg ∩ U ≥ 3/4 · µ(U ). But then by Lemma 4.3
we obtain the contradictionholµ(U ) ≥ µ ky · (Xhol(13)
g ∩ U ) + µ Xg ∩ U ≥ (3/4 + 3/4)µ(U ) > µ(U ) .
Thus, the action (X, Γ, Φ) cannot have essential holonomy at finite depth.



COROLLARY 4.4. Let (X, Γ, Φ) be a minimal equicontinuous Cantor action, andsuppose that Γ is a finitely generated nilpotent group. Then the set of points with nontrivial holonomy has measure 0; that is, (X, Γ, Φ) does not have essential holonomy.
Proof. By Theorem 2.7, the action (X, Γ, Φ) is locally quasi-analytic. As Γ isnilpotent, the commutator group γnΓ (Γ) is in the center of Γ, hence the set of points
with non-trivial holonomy for the action of γnΓ (Γ) on X has measure 0; that is, it has
no essential holonomy. Thus by Proposition 4.2 the action of Γ on X has no essential
holonomy.

5.

Commutators and not locally quasi-analytic actions

In this section, we exhibit a family of examples to show that the assumption that aminimal equicontinuous Cantor action (X, Γ, Φ) is locally quasi-analytic is essential forthe conclusions of Theorem 1.5 and Corollary 1.6.
THEOREM 5.1. Let {n` }`≥1 , n` ≥ 5 be a sequence of positive numbers, such thatQn` > 2n`−1 . Let S` be a set with n` elements, and let X = `≥1 S` . Let Sym(S` )be the symmetric group on n` symbols. There exists a countably generated subgroupQΓ ⊂ `≥1 Sym(S` ) with the following properties.
1. The lower central series of Γ stabilizes, i.e. γn (Γ) = [Γ, Γ] for n ≥ 2.
2. The minimal equicontinuous action (X, Γ, Φ) is not locally quasi-analytic and hasessential holonomy, i.e. the set of points with non-trivial holonomy has full measure.
3. The induced action (X, [Γ, Γ], Φ) of the commutator subgroup is minimal, not locallyquasi-analytic and it has no essential holonomy.
At the moment we are not aware of an action of a finitely generated group which
exhibits similar properties for the commutator action.

14

S. Hurder and O. Lukina

The family of examples in Theorem 5.1 is an amalgam of the family considered in[2, Theorem 1.5] and [16, Theorem 6.1]. The idea to use actions of alternating groups
comes from [2, Theorem 1.5], and the construction of an element with positive measure
set of points with holonomy is the same as in [16, Theorem 6.1]. When constructing
the commutator subgroup, we have to restrict to using the direct sum of symmetricgroups instead of the direct product to ensure that the action of the commutator in
our family has no essential holonomy (in fact, no points with non-trivial holonomy atall), and consequently the group we construct is not finitely generated. Merging these
constructions yields the actions in Theorem 5.1.
Proof. The demonstration of Theorem 5.1 follows from Lemmas 5.2, 5.3 and 5.4
below. Denote by A` the alternating group on n` symbols.
LEMMA 5.2. Consider the direct sum
M
G=A` = {g = (g1 , g2 , . . .) | g` ∈ A` , g` = e` for all but finitely many `} ,

(14)

`≥1

where e` is the identity element in A` . Then the component-wise action of G on X
is minimal, equicontinuous, not locally quasi-analytic and has no essential holonomy.
Proof. The group G acts on the direct product space X minimally, since theaction of every A` on S` is transitive. Define the metric on X by setting dX (x, y) = 1 forall x = (x` ), y = (y` ) ∈ X if x1 6= y1 , and otherwisedX (x, y) = 2−K ,K = max{k ≥ 1 | x` = y` for all 1 ≤ ` ≤ k}.

Since G acts on each component of X by permutations, its action is equicontinuous.
If g ∈ G has a fixed point, then this point has a clopen neighborhood entirely fixedby the action of g, since g acts non-trivially on at most a finite number of factors in the
direct product X. Thus the action of G on X has no essential holonomy.
We show that the action of G is not locally quasi-analytic. Label the elements
in S` by {1, 2, 3, . . . , n` }, and let a` = (123) ∈ A` , in particular, a` fixes the symbols{4, . . . , n` } ⊂ S` . Let  > 0, and choose an open set V of diameter less than . Then
there exists mV > 0 such thatYYV ⊃{e` } ×S` .
1≤`<mV`≥mV

Choose nV > mV , and define gV ∈ G as follows: set g` = a` for mV ≤ ` ≤ nV , andotherwise set g` = e` . Then both g and g 2 are the identity on the clopen set
Y
Y
Y
W =
{e` } ×{4, . . . , n` } ×S` ⊂ V,
1≤`<mVmV ≤`≤nV`>nV

while g|V 6= g 2 |V . Since  is arbitrary, this shows that the action of G on X is not locallyquasi-analytic.
Essential holonomy of Cantor actions

15

For sequences {n` }`≥1 with the additional property that n`+1 > 2n` for ` ≥ 1 we now
realize G as the commutator of a discrete group Γ whose action has essential holonomy.
For ` ≥ 1, define a permutation γ` = (n` − 1 n` ), i.e. γ` fixes every vertex in S`
Q
except the last two, and let γ = (γ1 , γ2 , . . .) ∈ `≥1 Sym(S` ) be an element in the direct
product of symmetric groups. DefineYΓ = hγ, Gi ⊂Sym(S` ) .
(15)
`≥1

LEMMA 5.3. The commutators [G, G] = [Γ, G] = [Γ, Γ] = G.
Proof. For each ` ≥ 1, we have n` ≥ 5, hence the alternating group A` is simple
and thus perfect, and thus G is also perfect; that is, G = [G, G].
Next we show that [Γ, Γ] = G. Note that the restriction γ|S` = γ` is an odd
permutation, and for ` 6= k the actions of γ` on S` , and of γk on Sk , commute. So forany g ∈ G the commutator [γ, g]|S` is an even permutation. For g ∈ G, let m ≥ 1 besuch that g|S` = e` for ` ≥ m. Then for ` ≥ m[γ, g]|S` = γ` γ`−1 = e` ,hence [γ, g] ∈ G. It follows that G = [G, G] ⊂ [Γ, G] ⊂ [Γ, Γ] ⊂ G.

Using the metric dX , defined above, it is convenient to think of the finite products
1≤`≤k S` as sets of vertices of a rooted tree at level k ≥ 1, where the root is a single
vertex at level 0 (it is omitted from X). In such a tree, each vertex at level ` is connectedQ
to n`+1 = |S`+1 | vertices at level ` + 1, and so one can think of vertices in 1≤`≤k S` aslabeled by finite words s1 · · · sk , where s` ∈ S` . An element of X is an infinite sequenceQs1 s2 · · · , where each truncated word s1 · · · sk corresponds to a vertex in 1≤`≤k S` . An
Qelement of the direct product γ ∈ `≥1 Sym(S` ) acts on the tree so that, for a givens` ∈ S` , γ · s` = γ` · s` , and this action depends only on s` and not on the preceding or
subsequent symbols in the sequence.
Open balls of diameter 2−K in the metric dX are the sets of infinite words in X whichcoincide for their first K symbols. The measure of each such open ball is

Q

1
|

Q

1≤`≤K S` |=1.
n1 · · · nK

We now show that γ has positive measure set of points with non-trivial holonomy
by explicitly computing the Lebesgue density of this set at each fixed point of γ. The
argument is the same as in [16, Theorem 6.1] and we give it here for completeness andfor the convenience of the reader.
Q
LEMMA 5.4. Let Fix(γ) be the set of fixed points of the element γ ∈ `≥1 Sym(S` )defined above. Then every point in Fix(γ) has non-trivial holonomy, and µ(Fix(γ)) > 0.
Proof. First note that γ fixes an infinite path s = s1 s2 · · · ∈ X if and only if forall ` ≥ 1 we have s` 6= n` and s` 6= n` − 1. We claim that each such point has non-trivial

16

S. Hurder and O. Lukina

holonomy. Indeed, let V be an open neighborhood of s. Then there is an mV ≥ 1 such
that


Y
Y
Y
Y
S` ,
{s` } ×S`  ={s` } ×V ∩1≤`≤mV

1≤`≤mV`>mV`>mV

and so V contains points which are moved by the action of γ.
For each clopen ball U` around a fixed point s, the action of γ permutes two clopenballs in U` consisting of sequences starting with s1 · · · s` (n`+1 − 1) and s1 · · · s` n`+1 ,which have the total measure 2/(n1 · · · n`+1 ). Each of the remaining n`+1 − 2 clopenballs determined by words of length ` + 1 contains 2 subsets of sequences starting with
s1 · · · s`+1 (n`+2 − 1) and s1 · · · s`+1 n`+2 permuted by the action, whose total measure is2(n`+1 − 2)/(n1 · · · n`+1 n`+2 ). Continuing by induction, we compute the upper boundon the measure of the complement of the set Fix(γ):
122(n`+1 − 2) 2(n`+1 − 2)(n`+2 − 2)µ(U` − Fix(γ)) =
+
+
+ ···
n1 · · · n` n`+1
n`+1 n`+2
n`+1 n`+2 n`+3
X 1
2
<
.
n1 · · · n`
n`+i
i≥1

Since we assume that n`+i > 2n`+i−1 > 2i−1 n`+1 , we obtain thatµ(U` − Fix(γ)) <41,n1 · · · n` n`+1

and soµ(U` ∩ Fix(γ)) >41−.
n1 · · · n`
n1 · · · n`+1

It follows that for every point in Fix(γ) the Lebesgue density is 1, namely4µ(U` ∩ Fix(γ))
1 = lim 1 −
≤ lim
≤1.
`→∞
`→∞
n`+1
µ(U` )
Thus the set of points with non-trivial holonomy for γ ∈ Γ has positive measure.

(16)We have shown that the action of Γ on X has essential holonomy, while the actionof its commutator [Γ, Γ] = G has no essential holonomy, which proves the assertions ofTheorem 5.1.
Acknowledgements. The paper started in response to the question by Eduardo
Scarparo whether an action of an amenable group must have no essentially holonomy.
The authors also thank Eduardo for bringing to their attention the work by Matthieu
Joseph [22]. The authors thank the referee for carefully reading the paper and providing
valuable feedback.

Essential holonomy of Cantor actions

17

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Steven Hurder

Olga Lukina

Department of Mathematics,
University of Illinois at Chicago,
322 SEO (m/c 249),
851 S. Morgan Street, Chicago,
IL 60607-7045, USA
E-mail: hurder@uic.edu

Mathematical Institute,
Leiden University,
P.O. Box 9512,
2300 RA Leiden,
The Netherlands
E-mail: o.lukina@math.leidenuniv.nl