Measures and stabilizers of group Cantor actions

We consider a minimal equicontinuous action of a finitely generated group $G$ on a Cantor set $X$ with invariant probability measure $\mu$, and stabilizers of points for such an action. We give sufficient conditions under which there exists a subgroup $H$ of $G$ such that the set of points in $X$ whose stabilizers are conjugate to $H$ has full measure. The conditions are that the action is locally quasi-analytic and uniformly non-constant. An action is locally quasi-analytic if its elements have unique extensions on subsets of uniform diameter. The condition that the action is uniformly non-constant is introduced in this paper. We apply our results to study the properties of invariant random subgroups induced by minimal equicontinuous actions on Cantor sets and to give new examples of mean equicontinuous group actions.


1.
Introduction. Let G be a finitely generated group, and suppose G acts on a Cantor set X preserving a probability measure µ. We denote the action by (X, G, Φ, µ), where the assignment Φ : G → Homeo(X) is a homomorphism, and use the short notation g · x = Φ(g)(x) for the action of g ∈ G on x ∈ X throughout the paper. Throughout the paper, a clopen set is a closed and open set in X.
Let G x = {g ∈ G | g · x = x} be the stabilizer, or the isotropy subgroup of the action of G at x. A stabilizer G x is trivial if G x = {e}, where e is the identity in G. The action (X, G, Φ, µ) is essentially free if the subset of points with trivial For actions where every point has non-trivial stabilizer, we can generalize the concept of an essentially free action by distinguishing points with trivial or nontrivial holonomy, as we explain now.
Let g ∈ G, and let x ∈ X such that g · x = Φ(g)(x) = x. The homeomorphism Φ(g) either fixes every point in an open neighborhood of x ∈ X, or it induces a non-trivial map on any open neighborhood of x ∈ X. We define the neighborhood stabilizer of x ∈ X by The term neighborhood stabilizer comes from [40]. Note that the open set V g in (1) may depend on g ∈ G. In Definition 1.1 below, X can be any metric space equipped with an action, and the action need not be either minimal or equicontinuous or measure-preserving.
Definition 1.1. Let X be a metric space and let (X, G, Φ) be an action. 1. We say that g ∈ G has trivial holonomy at x ∈ X if g ∈ [G] x , and otherwise g has non-trivial holonomy at x. 2. We say that x ∈ X is a point with trivial holonomy, or a point without holonomy, if G x = [G] x . Otherwise x ∈ X is a point with non-trivial holonomy, or a point with holonomy.
The terminology in Definition 1.1 is standard in foliation theory, where it was developed in a more general context of actions of pseudogroups of local homeomorphisms on compact spaces. We recall a basic result by Epstein, Millett and Tischler [19] for foliated spaces, which reads for the case of actions of groups on compact spaces as follows.
Theorem 1.2. Let (X, G, Φ) be an action of a countable group on a compact Hausdorff space. The set of points without holonomy is a residual (dense G δ ) subset of X.
If (X, G, Φ) is topologically free, then points in X with trivial stabilizers have trivial holonomy. Conversely, suppose g ∈ G such that g · x = x. Since the set of points with non-trivial stabilizers has empty interior, then there is no open set U x such that g fixes every point in U . Thus for a topologically free action, a point x ∈ X has non-trivial stabilizer if and only if it has non-trivial holonomy. Then a measure-preserving topologically free action (X, G, Φ, µ) is essentially free if and only if the set of points with non-trivial holonomy in X has measure zero.
An analogue of an essentially free action in the case when no point in (X, G, Φ, µ) has trivial stabilizer is an action where the set of points with trivial holonomy has full measure in X.

MAIK GRÖGER AND OLGA LUKINA
The action (X, G, Φ) is quasi-analytic if one can choose U = X.
Thus an action is LQA if and only if for all g ∈ G the homeomorphisms Φ(g) have unique extensions on subsets of uniform diameter in X. Topologically free actions are quasi-analytic [29,Proposition 2.2]. Examples of actions which are locally quasianalytic but not quasi-analytic are easy to construct: take any quasi-analytic action, and add a finite number of elements which fix an open subset of X but act nontrivially on its complement, as in [29,Example A.4]. A minimal equicontinuous action of a finitely generated nilpotent group is always locally quasi-analytic, in fact, a similar property holds for a more general class of groups with Noetherian property [29,Theorem 1.6]. Examples of actions which are not LQA include the actions of finite index torsion-free subgroups of SL(n, Z), for n ≥ 3, in [30].
The idea to use Lebesgue density to study the measure of points with non-trivial holonomy was introduced by Hurder and Katok [27], for linear holonomy in C 1 foliations of a smooth manifold M . In the setting of [27], the differentiability of a foliation implies that for each holonomy linear map L, the Lebesgue density of the set of all fixed points of L at fixed points with non-trivial holonomy is bounded away from 1. Then the Lebesgue density theorem implies that the set of points with nontrivial linear holonomy must have zero measure in the transverse Euclidean section to the foliation.
In absence of differentiability for actions on Cantor sets, we introduce the notion of a locally non-degenerate action. It will follow that if an action is locally nondegenerate, then for any g ∈ G the Lebesgue density of the set of fixed points of g at points with non-trivial holonomy is bounded away from 1. This condition on the action is combinatorial and it is defined using a representation of (X, G, Φ, µ) onto the boundary of a spherically homogeneous tree T , which can be considered as a choice of coordinates for (X, G, Φ, µ). We discuss such representations in more detail below and in Section 3. We will show that the property of local non-degeneracy is invariant under conjugacy of actions, and so it is an invariant of an equivalence class of representations, associated to (X, G, Φ, µ).
Local non-degeneracy of an action implies the property of Lebesgue densities, discussed above, but we do not know at the moment if the converse statement is true. Since the precise relationship between Lebesgue densities and the notion of local non-degeneracy is not clear at the moment, we are not able to give an intrinsic, coordinate-free definition of a locally non-degenerate action. A similar state of affairs is not uncommon in the study of foliated spaces, where often the only way to study an object is in terms of equivalence classes of coordinate charts representing it. We discuss this more below.
We introduce the concept of a locally non-degenerate action for an action of a subgroup G ⊂ Aut(T ) on a spherically homogeneous tree T first, see Section 3 for more details about actions on trees.
Let V = ≥0 V be the vertex set of a tree T , where V is the -th level of T . A tree T is rooted if V 0 is a singleton. A tree T is spherically homogeneous if for every ≥ 1 there is a positive integer n ≥ 2 such that every vertex in V −1 is joined by edges to n vertices in V . The sequence n = (n 1 , n 2 , . . .) is called the spherical index of T . A spherically homogeneous tree T is d-ary if there is a positive integer d ≥ 2 such that n = d for all ≥ 1.
The boundary ∂T is the set of all connected paths in T . With product topology, it is a Cantor set. For v ∈ V , we denote by T v the subtree through the vertex v, and by ∂T v the boundary of T v . Then ∂T v is identified with a clopen subset of ∂T . Denote by µ ∂T the counting measure on ∂T . An automorphism g ∈ Aut(T ) naturally induces a homeomorphism of ∂T , thus defining the homomorphism Φ. So if G ⊂ Aut(T ), then we write (∂T, G, µ ∂T ) for the action on the tree, omitting Φ from the notation. Definition 1.4. Let T be a spherically homogeneous tree, and let G ⊂ Aut(T ). The action of g ∈ G on ∂T is locally non-degenerate if there exists 0 < α g ≤ 1 such that for any vertex v ∈ V , if g fixes v and g|∂T v = id, then The action (∂T, G, µ ∂T ) is locally non-degenerate if and only if the action of every g ∈ G on ∂T is locally non-degenerate.
If an action of g ∈ G, or the action (∂T, G, µ ∂T ) is not locally non-degenerate, then it is locally degenerate.
In particular, the action of g ∈ G is locally non-degenerate if, for any decreasing sequence of nested clopen sets {∂T v } v ∈V , ≥1 , where g|∂T v = id, the ratio of the measure of the set of points in ∂T v , not fixed by the action of g, to the measure of ∂T v has a lower bound. Intuitively, we want to avoid actions, where for some {∂T v } v ∈V , ≥1 , the set of fixed points of g occupies a larger and larger proper subset of ∂T v , as → ∞. Another way to phrase that is by saying that the proportion of points in an open ball, moved by the action of g ∈ G, is at least α g for any open ball on which g acts non-trivially.
As discussed in Section 3, every minimal equicontinuous action (X, G, Φ) on a Cantor set X has a representation as an action on the boundary ∂T of a rooted spherically homogeneous tree T . More precisely, there exists a (non-unique) tree T and a homeomorphism φ : X → ∂T , such that for any g ∈ G the composition φ • g • φ −1 is an automorphism of T . The homeomorphism φ is determined by a choice of a descending sequence {U } ≥0 , U 0 = X, of clopen subsets of X such that for each U the translates U = {g · U } g∈G form a finite partition of X. Under φ, distinct sets in U correspond to clopen sets {∂T v } v∈V in the tree. The group G permutes the sets in the partition U , and so defines a permutation of the vertex set V , for ≥ 1. Inclusions of sets in U +1 into the sets in U correspond to edges between vertices in T , and so G acts on T by automorphisms. It follows that there is an induced injective map φ * : Homeo(X) → Homeo(∂T ), and the pair of maps gives a representation of (X, G, Φ) onto the boundary of the tree T . Any two such representations are conjugate to (X, G, Φ), and so to each other. The invariant ergodic measure µ on X pushes forward to the counting measure µ ∂T on ∂T . The representation (φ, φ * ) is not unique, and should be viewed as a choice of coordinates for the action. Thus associated to a minimal equicontinuous action (X, G, Φ, µ) there is an equivalence class [(∂T, φ * (Φ(G)), µ ∂T )] of actions on trees, where the equivalence relation is given by conjugacy of the induced actions on the boundary of the tree.
Definition 1.5. A minimal equicontinuous action (X, G, Φ, µ) is locally non-degenerate if there exists a representation (φ, φ * ) of the action as in (4), such that the action of φ * (Φ(G)) on ∂T is locally non-degenerate with respect to the counting measure µ ∂T on ∂T .
Actions which are locally non-degenerate are common in the study of actions on Cantor sets. We show in Section 4.1 that actions on d-ary trees generated by finite automata are locally non-degenerate. We expect that any action on a Cantor set which is conjugate to a holonomy action on a transverse section of a subset of a foliated C 1 manifold is locally non-degenerate. We give examples of actions which are locally degenerate in Section 4.2. It is not clear at the moment what type of actions is prevalent among all possible minimal equicontinuous group actions on Cantor sets.
We now can state our main theorem, which specifies conditions under which X contains a set of points with conjugate stabilizers which is both topologically and measure-theoretically large. Theorem 1.6. Let X be a Cantor set, let G be a finitely generated group, and let (X, G, Φ, µ) be a locally quasi-analytic minimal equicontinuous action. Then the following is true.
1. There exists a subgroup H ⊂ G such that the set of points with stabilizers conjugate to H is residual in X. 2. In addition, suppose (X, G, Φ, µ) is locally non-degenerate. Then the set of points with stabilizers conjugate to H has full measure in X.
If (X, G, Φ, µ) is essentially free, then both statements of Theorem 1.6 hold with H the trivial subgroup.
The proof of Theorem 1.6 consists of two ingredients, which are of interest in their own right. Below we discuss these ingredients and their applications.
To prove Theorem 1.6, we study the properties of points with trivial or nontrivial holonomy in X. Throughout the paper, we denote the set of points with trivial holonomy by Theorem 1.7. Let (X, G, Φ) be a minimal equicontinuous action of a finitely generated group G on a Cantor set X. Then the set {G x | x ∈ X 0 } of stabilizers of points with trivial holonomy is a finite set of conjugate subgroups if and only if the action (X, G, Φ) is locally quasi-analytic.
The set of stabilizers of points with non-trivial holonomy may be infinite even for quasi-analytic actions, see for instance, Example 1. The next step in the proof of Theorem 1.6 is the sufficient condition under which the set of points with trivial holonomy has full measure in X. Theorem 1.8. Let (X, G, Φ, µ) be a minimal equicontinuous action of a finitely generated group G on a Cantor set X. Suppose (X, G, Φ, µ) is locally non-degenerate. Then the set X 0 of points with trivial holonomy has full measure with respect to µ.
The proof of Theorem 1.8 uses the technique similar to that of Hurder and Katok [27,Proposition 7.1], who studied linear holonomy in C 1 foliations of smooth manifolds M . Our setting is very different from the setting of [27], as for actions on Cantor sets there are no differentials and, in fact, the notion of linear holonomy does not make sense. The condition that the action (X, G, Φ, µ) is locally nondegenerate imposes a degree of regularity which in the smooth setting comes from differentiability.
Bergeron and Gaboriau [8], and also Abért and Elek [1], gave examples of group actions on Cantor sets which are topologically free and not essentially free. By the discussion after Theorem 1.2, in these examples the set of points with non-trivial holonomy has full measure. We give more examples of this type in Section 6. As a corollary to Theorem 1.8 we obtain a criterion when a topologically free minimal equicontinuous action is essentially free. Corollary 1. If a minimal equicontinuous action (X, G, Φ, µ) is locally non-degenerate and topologically free then it is essentially free.
Kambites, Silva and Steinberg [31] showed that an action on the boundary of a rooted d-ary tree of a group, generated by finite automata, is topologically free if and only if it is essentially free. We show in Section 4 that actions generated by finite automata are locally non-degenerate, and recover the result of [31] as a consequence of Theorem 1.8.
To finish, we outline some applications of our results, described in more detail in Section 8.
Let Sub(G) be the set of all closed subgroups of a finitely generated discrete group G with the Chaubaty-Fell topology, see Section 8.1. Then Sub(G) is compact and totally disconnected.
For an action (X, G, Φ, µ) consider the mapping which assigns to each x ∈ X its stabilizer. The group G acts on Sub(G) by conjugation. Stabilizers of points in an orbit of x ∈ X are conjugate, so the mapping St maps the orbit of x ∈ X onto the orbit of G x in Sub(G) under conjugation. The pushforward ν = St * µ of the invariant ergodic measure µ is an ergodic invariant measure on Sub(G) [5]. This measure is said to be an invariant random subgroup, or simply IRS, following the terminology of [2,9]. The mapping (6) need not be injective, and may have points of discontinuity. We discuss in more detail in Section 8.1 some results about IRS for groups acting on the boundary of trees available in the literature. We use now Theorems 1.7 and 1.8 to give a partial answer to two open questions posed by Grigorchuk in [24].
Namely, for the set X 0 of points without holonomy, defined by (5), and called the set of G-typical points in [24], consider the topological closure On [24, p.123], Grigorchuk asked when the IRS ν is supported on the set Z. Also, [24,Problem 8.2] asks under what condition on the action the system (X, G, Φ, µ) is measure-theoretically isomorphic to the system (Z, G, ν)? We give partial answers to these questions below.
Theorem 1.9. Let X be a Cantor set, let G be a finitely generated group, and let (X, G, Φ, µ) be a minimal equicontinuous action. Suppose (X, G, Φ, µ) is locally non-degenerate. Then the following holds. 1. The IRS ν = St * µ is supported on Z.
2. The action (X, G, Φ, µ) is not LQA if and only if ν is non-atomic.
3. If (X, G, Φ, µ) is not LQA and the restriction St : X 0 → Sub(G) is injective, then St provides a measure-theoretical isomorphism between (X, G, Φ, µ) and (Z, G, ν). Theorem 1.9 describes the properties of the IRS ν induced on Sub(G) via the mapping (6) in the case when the set of points with trivial holonomy for the action (X, G, Φ, µ) has full measure. If an action (X, G, Φ, µ) is locally degenerate, then the set of points with non-trivial holonomy in X may have positive measure, and the following problem is natural.
Problem. Let (X, G, Φ, µ) be a minimal equicontinuous action on a Cantor set X, and suppose the set of points with non-trivial holonomy in X has positive measure. Describe the properties of the IRS ν = St * µ induced on Sub(G) via the map (6).
Related to the construction of the IRS above is the following construction. Define and denote by Φ the product action of G by Φ in the first coordinate and by conjugation in the second coordinate. Then the projection η : X → X to the first coordinate satisfies g · η(x, G x ) = η(g · x, gG x g −1 ). By [23, Proposition 1.2], we know that ( X, G, Φ) is minimal and the set { x ∈ X | |η −1 (η( x))| = 1} of singleton fibers is dense in X. Thus the system ( X, G, Φ) is an almost one-to-one extension of (X, G, Φ). As a consequence of Theorem 1.7, we show that for LQA actions such an extension is trivial in the following sense.
Theorem 1.10. Let (X, G, Φ) be a minimal equicontinuous action. Then (X, G, Φ) is locally quasi-analytic if and only if the almost one-to-one extension η : X → X is a conjugacy.
The rest of the paper is organized as follows. In Section 2 we recall the necessary background on the properties of equicontinuous actions. In Section 3 we discuss representations of minimal equicontinuous actions onto the boundaries of rooted spherically homogeneous trees. We discuss locally non-degenerate actions in Section 4, giving examples which illustrate the definition. We also state sufficient conditions for an action on a rooted tree to be locally non-degenerate in terms of the geometry of the tree. Theorem 1.8 which states that for a locally non-degenerate action the set of points with trivial holonomy has full measure and its corollaries are proved in Section 5. In Section 6 we give an example of an action where the set of points with trivial holonomy has zero measure. Theorem 1.7 and the main Theorem 1.6 are proved in Section 7. In Section 8 we discuss applications of our results proving Theorems 1.9 and 1.10.
2. Equicontinuous actions. We recall basic properties of equicontinuous group actions on Cantor sets. Sections 2.1-2.3 discuss topological properties of such actions, so we omit the measure from the notation in these sections.
An action Φ : G → Homeo(X) of a finitely generated group G on a Cantor set X is equicontinuous if X admits a metric d such that for all > 0 there is δ > 0 such that for any g ∈ G and any x, y ∈ X with d(x, y) < δ we have d(g · x, g · y) < . An action (X, G, Φ) is minimal if the orbit G(x) = {g · x | g ∈ G} of every x ∈ X is dense in X. In this paper, (X, G, Φ) is always minimal and equicontinuous.

Group chains.
Our main tool in working with group actions are group chains. In this section we explain how a group chain defines a minimal equicontinuous action on a Cantor set. Actions on Cantor sets in terms of group chains were studied in [16,20,13,12] and other works.
For each ≥ 0 the coset space X = G/G is finite, and there are inclusion maps induced by the inclusion of groups G +1 ⊂ G . The inverse limit space is a Cantor set in the relative topology, induced from the product (Tychonoff) topology on ≥0 X . We denote by (g G ) ≥0 sequences of cosets in X ∞ . Basic sets in X ∞ are given by The group G acts on the left on every coset space X , permuting the cosets in X , and there is an induced action by homeomorphisms on the inverse limit Since the action of G on every coset space X , ≥ 0, is transitive, the action (11) on X ∞ is minimal. A standard way to define an ultrametric on the space X ∞ is by This metric measures for how long sequences of cosets (g G ) ≥0 and (h G ) ≥0 coincide. Since G acts on each coset space X , ≥ 0, by permutations, it acts on X ∞ by isometries relative to the metric D, and so its action is equicontinuous.

2.2.
Adapted clopen sets and group chain models. In this section, we show how to associate a group chain to a minimal equicontinuous action (X, G, Φ), where X is a Cantor set, and G is a finitely generated discrete group. Let CO(X) denote the collection of all clopen (closed and open) subsets of X, which form a basis for the topology of X. For φ ∈ Homeo(X) and U ∈ CO(X), the image φ(U ) ∈ CO(X). The following result is folklore, and a proof is given in [29, Proposition 3.1]. Proposition 1. A minimal Cantor action (X, G, Φ) is equicontinuous if and only if, for the induced action Φ * : G×CO(X) → CO(X), the orbit of every U ∈ CO(X) is finite.
Definition 2.2. Let (X, G, Φ) be a minimal equicontinuous action. A clopen set U ∈ CO(X) is adapted to the action Φ if U is non-empty and for any g ∈ G, if That is, the translates of an adapted set by the action of G form a partition of the Cantor set X. Definition 2.3. Let (X, G, Φ) be a minimal equicontinuous action and x ∈ X. A properly descending chain of clopen sets U The following result is folklore, a proof can be found in [16, Appendix A].
Proposition 2. Let (X, G, Φ) be a minimal equicontinuous action. Given x ∈ X, there exists an adapted neighborhood basis U x at x for the action Φ.
For an adapted set U ⊂ X, consider the collection of elements in G which preserve U , that is, Then G U is a subgroup of G, called the stabilizer of U , or the isotropy subgroup of the action of G at U . Since the orbit of U under the action of G is finite, G U has finite index in G. Fix x ∈ X, and let U x = {U } ≥0 be an adapted neighborhood basis at x ∈ X. Denote by G = G U the stabilizer of U . Since U +1 ⊂ U , every element in G which stabilizers U +1 also stabilizers U , and so there are inclusions G +1 ⊂ G . Thus associated to an adapted neighborhood basis U x at x ∈ X, there is an infinite The maps κ are equivariant with respect to the action of G on X and X . Taking the inverse limit with respect to the coset inclusions X +1 → X , we obtain a homeomorphism which conjugates the actions of G on X and X ∞ , the latter being given by (11). By construction, we have κ ∞ (x) = (eG ), where eG is the coset of the identity e in G/G . For a given action (X, G, Φ) the choice of an adapted neighborhood system U x is not unique, and so the choice of a group chain G x associated to the action is not unique. The relations between the group chains representing the same conjugacy class of minimal equicontinuous actions were studied in detail in [20,16], where the following result was proved.

Proposition 3. [16]
Let G be a finitely generated group, and let G = {G } ≥0 and G = {G } ≥0 be group chains with G = G 0 = G 0 . Let (X ∞ , G) and (X ∞ , G) be minimal equicontinuous actions associated to G and G by (9) and (11). Then (X ∞ , G) and (X ∞ , G) are conjugate by a homeomorphism φ : X ∞ → X ∞ if and only if there exists a sequence of elements {g } ≥0 ⊂ G such that for any n ≥ 0 and any > n, g n G n = g G n , and, possibly after passing to a subsequence, there are inclusions In addition, φ is pointed, that is, φ(eG ) = (eG ) if and only if one can choose g = e in (15) for all ≥ 0.
The condition g n G n = g G n for > n in Proposition 3 ensures that the chain {g G g −1 } ≥0 is a descending group chain as in Definition 2.1. (15) impose restrictions on the indices of subgroups in the chains G and G . Indeed, if G ⊃ g G g −1 then the index |G : G | divides the index |G : g G g −1 |. In particular, if |G : G | = p 1 p 2 · · · p , where p = |G −1 : G |, ≥ 1, are distinct primes, and |G : G | = d for some integer d ≥ 2, then the actions defined by the group chains G and G are not conjugate, since the set of primes which divide d is finite.

Remark 1. The inclusions
2.3. Stabilizers of points and group chains. Let (X, G, Φ) be a minimal equicontinuous system, let U = {U } ≥0 be an adapted neighborhood basis at x ∈ X, and let G x = {G } ≥0 be an associated group chain. The kernel of the group chain G x is the subgroup G of elements in G which fix x, so the kernel of the group chain at x is the stabilizer If y ∈ X is another point, then for every ≥ 0 there is g ∈ G such that y ∈ Φ(g )(U ). It follows that G y = {g G g −1 } ≥0 is a group chain at y, and we can compute the stabilizer at y If y is in the orbit of x, that is, y = h · x for some h ∈ G, then we can choose g = h, and in this case the stabilizers G x and G y are conjugate subgroups of G. If y is not in the orbit of x, then the stabilizers G x and G y need not be isomorphic, as the following example shows.
where e denotes the identity in G, let G = a 2 , b for > 0, and G 0 = G. This example is very well-studied. For instance, the action defined by this group chain is conjugate to the action of the iterated monodromy group associated to the quadratic Chebyshev polynomial, see [33]. Group chains representing this action were studied in [20] and [16,Example 7.5].
Consider the dynamical system associated to the group chain {G } ≥0 by the construction in Section 2.1. One sees that for all ≥ 1 the cosets in X = G/G are represented by the powers of a, and so for each x ∈ X, the orbit of x is given by The isotropy groups of the points in the orbit G(x) are conjugate to G x , more precisely, G a n ·x = a n ba −n .
If y ∈ X ∞ is not in the orbit of x, then by [20,16] the stabilizer of y is trivial, G y = {e}. In particular, this means that the action of G on X ∞ defined by G x is topologically free.
We show that the stabilizers of the points in G(x) are pairwise distinct subgroups of G. We argue by contradiction. Suppose that G x = G a n ·x . Then a n ba −n = b, which implies that which contradicts the fact that G is an infinite group. So G x = G a n ·x .

2.4.
Counting measure on X. Let (X, G, Φ) be a minimal equicontinuous action, then the closure E = Φ(G) ⊂ Homeo(G) in the uniform topology is a profinite compact group, called the Ellis, or enveloping group [4,18]. The group E acts on X, the isotropy group E x = { g ∈ E | g(x) = x} of its action at x is a closed subgroup of E, and we have X = E/E x . An element g ∈ G acts on E via group multiplication by Φ(g). The Haar measure µ on E is invariant with respect to this action and ergodic. The measure µ on E pushes down to a probability measure µ on X, and with this measure (X, G, Φ, µ ) is uniquely ergodic [13].
Given a group chain G x = {G } ≥0 associated to the action (X, G, Φ) at a point x ∈ X, one defines a counting measure µ on the space X ∞ in (9) by setting for every basic set U g,m , defined by (10), where |G : G m | is the index of G m in G. This measure is easily seen to be invariant under the action of G. It is immediate that the pullback of µ to X along the conjugating map (14) coincides with µ . By a slight abuse of notation we denote µ and µ by the same symbol µ in the rest of the paper.
3. Actions on trees. In this section, we represent a minimal equicontinuous action as an action on the boundary of a spherically homogeneous tree T . Actions on trees, especially self-similar actions on d-ary trees, d ≥ 2, are an active topic in Geometric Group Theory, see [33,24,25] for surveys. A tree T consists of a set of vertices V = ≥0 V , where V is a finite vertex set at level , and of edges joining vertices in V +1 and V , for all ≥ 0, such that every vertex in V +1 is joined by an edge to a single vertex in V . A tree is rooted if |V 0 | = 1. A tree T is spherically homogeneous if there is a sequence n = (n 1 , n 2 , . . .), called the spherical index of T , such that for every ≥ 1 a vertex in V −1 is joined by edges to precisely n vertices in V . We assume that n ≥ 2 for ≥ 1.
A spherically homogeneous tree T is d-ary if its spherical index n = (n 1 , n 2 , . . .) is constant, that is, n = d for some positive integer d. If d = 2, then T is called a binary tree.
The spherical index n = (n 1 , n 2 , . . .) of a tree T is bounded, if there is M > 0 such that n ≤ M for ≥ 1. A spherical index n is unbounded if it is not bounded.
Let (X, G, Φ) be a minimal equicontinuous action, U x = {U } ≥0 be a neighborhood basis at x ∈ X, and let G x = {G } ≥0 be an associated group chain as in Section 2. For ≥ 0, identify the vertex set V with the coset space X = G/G . Join v ∈ V and v +1 ∈ V +1 by an edge if and only if v +1 ⊂ v as cosets. The obtained tree is spherically homogeneous, with entries n = |G −1 : G | in the spherical index n, for ≥ 1.
An infinite path in T consists of a sequence of vertices (v ) ≥0 such that v +1 and v are joined by an edge, for ≥ 0. The boundary ∂T of T is the collection of all infinite paths in T , and so it is the subspace and v are joined by an edge}.
The space ∂T is a Cantor set with the relative topology from the product topology on ≥0 V . It is immediate that the identification of vertex sets V with coset spaces X induces an identification of ∂T with the inverse limit space X ∞ defined by (9), with points in X ∞ corresponding to infinite paths in ∂T , and clopen sets U g,m defined by (10) corresponding to cylinder sets where v m is the vertex in V m identified with the coset gG m in X m . The counting measure (16) pushes forward to the measure on ∂T which we denote by µ ∂T . The group G acts on vertex levels V = X , ≥ 0 by permutations. Since the action of G preserves the containment of cosets, the action preserves the connectedness of the tree T , that is, the vertices v ∈ V and v +1 ∈ V +1 are joined by an edge if and only if for any g ∈ G the images g · v ∈ V and g · v +1 ∈ V +1 are joined by an edge. Thus every g ∈ G defines an automorphism of the tree T , and we can consider G as a subgroup of the group of tree automorphisms Aut(T ).
The composition of the map (14) with the identification X ∞ → ∂T is a homeomorphism φ : X → ∂T . The action of G on vertex levels V , ≥ 0 induces an action of G on ∂T by left translations, defined by (11). Thus there is the induced map φ * : Φ(G) → Homeo(∂T ), and Φ(G) is identified with a subgroup of Homeo(∂T ). The action (∂T, Φ(G)) is minimal equicontinuous, with the unique ergodic invariant measure µ ∂T defined by the pushforward of (16). A pair of maps is called a tree representation of (X, G, Φ).

Remark 2.
The choice of a group chain associated to an action (X, G, Φ) is not unique, and consequently the choice of a tree representation (18) is not unique. By an argument similar to the one in Remark 1 one can obtain a necessary condition under which trees T and T with respective spherical indices n = (n 1 , n 2 , . . .) and n = (n 1 , n 2 , . . .) admit conjugate actions of the same group. Namely, there must exist subsequences {i } ≥0 and {i } ≥0 such that for all ≥ 0 the product n 1 n 2 · · · n i divides the product n 1 n 2 · · · n i , and n 1 n 2 · · · n i divides n 1 n 2 · · · n i +1 . For instance, let n = (p 1 , p 2 , . . .) where {p 1 , p 2 , . . .} are distinct primes, and let n be bounded, that is, there is a constant m ≥ 0 such that for all ≥ 1 we have n ≤ m. Then the set of divisors {k ∈ N | k|n for some ≥ 1} is finite, and so there is no map between T and T which preserves the tree structure.

4.
Locally non-degenerate actions on trees. In this section we discuss the notion of a locally non-degenerate action, which we introduced in Section 1. We give a criterion for an action on a tree to be locally non-degenerate in terms of the geometry of the tree. We show that actions on rooted d-ary trees generated by finite automata are locally non-degenerate. We also give examples of actions on trees with bounded or unbounded spherical index which are locally degenerate.
We use the notation of Section 3. In particular, a clopen subset ∂T v defined in (17), is a subset of the boundary ∂T of the tree T containing all infinite paths passing through the vertex v ∈ V . We start by restating Definition 1.4 below for the convenience of the reader.
Definition 4.1. Let T be a spherically homogeneous tree, and let G ⊂ Aut(T ). The action of g ∈ G on ∂T is locally non-degenerate if there exists 0 < α g ≤ 1 such that for any vertex v ∈ V , if g fixes v and g|∂T v = id, then

MAIK GRÖGER AND OLGA LUKINA
The action (∂T, G, µ ∂T ) is locally non-degenerate if and only if the action of every g ∈ G is locally non-degenerate.
If an action of g ∈ G, or the action (∂T, G, µ ∂T ) is not locally non-degenerate, then it is locally degenerate.
It is useful to have a criterion for when a homeomorphism of a tree is locally non-degenerate in terms of the geometry of the tree T . This criterion is given in Proposition 4 below.
Proposition 4. Let T be a spherically homogeneous tree with spherical index n = (n 1 , n 2 , . . .), and let g ∈ Aut(T ). Suppose the following two conditions are satisfied: 1. The spherical index of T is bounded, that is, there is M > 0 such that n ≤ M for ≥ 1. 2. There is an integer K g > 0 such that for any ≥ 0 and any vertex v ∈ V , if g fixes v and g|∂T v = id, then there exists m ≥ and a vertex w m ∈ V m ∩T v such that g · w m = w m and m − ≤ K g . Then the action of g is locally non-degenerate with respect to the probability measure µ ∂T .
Since n ≤ M for all ≥ 1, we obtain that and g is locally non-degenerate with α g = 1/M Kg .
Proposition 4 gives sufficient conditions for an action to be locally non-degenerate. This conditions are not necessary, for instance, it is possible to have a locally nondegenerate action on a tree with strictly increasing spherical index.
Example 2. Let G be a finitely generated group, acting freely on the boundary ∂T of a spherically homogeneous tree T with any spherical index. That is, for any g ∈ G and any u ∈ ∂T if g · u = u then g = id, the identity element in G. Such an action is trivially locally non-degenerate, since no element has fixed points.

A class of locally non-degenerate actions.
We show that actions on d-ary trees of groups generated by finite automata are locally non-degenerate. Many wellknown groups belong to this class, including the Basilica group and the Grigorchuk group, as well as iterated monodromy groups associated to quadratic post-critically finite polynomials, see [33] for more specific examples and detailed discussions. If v ∈ V +1 and w ∈ V are joined by an edge, and v = s 1 s 2 . . . s +1 , then w = s 1 s 2 . . . s . It follows that every element of the boundary ∂T can be uniquely represented by an infinite sequence s 1 s 2 · · · , where s ∈ {0, 1, . . . , d − 1}, ≥ 1. More precisely, a path s = s 1 s 2 . . . passes through the vertex labelled by s 1 in V 1 , by s 1 s 2 in V 2 and, inductively, s passes through the vertex labelled by s 1 s 2 · · · s in V for ≥ 1.
Let w = w 1 w 2 · · · w be a word of length . Using the word notation, a clopen subset ∂T w of ∂T , defined in (17) is given by Since for all ≥ 1 the labels w take values in the same set {0, 1, . . . , d − 1}, for each w ∈ V there is a homeomorphism Note that ∂T = ∂T 0 ∪ ∂T 1 ∪ · · · ∪ ∂T d−1 . We use the maps (20) to define sections as in [33, Section 1.3.1], and then to recursively define elements in Aut(T ) as in [33, Section 1.4.2] (although we compose maps in the recursive formula in a different order than in [33]).
Let h ∈ Aut(T ), and let σ h,1 be the permutation of V 1 = {0, 1, . . . , d −1} induced by the action of h on T . For k ∈ V 1 , the restriction of h to T k gives the map h : ∂T k → ∂T σ h,1 (k) . Using the identification (20) we obtain a homeomorphism h| k : ∂T → ∂T which we call a section of h at k. Thus h| k ∈ Aut(T ) is the map uniquely defined by the concatenation of sequences h(k s 2 s 3 . . .) = σ h,1 (k) h| k (s 2 s 3 . . .).
Formally, for 0 ≤ k ≤ d − 1, to define h| k ∈ Aut(T ) we set for every infinite path s ∈ ∂T Then we can write the element h as the composition where for each 0 ≤ k ≤ d − 1 we write just h| σ −1 h,1 (k) instead of ψ −1 k • h| σ −1 h,1 (k) • ψ k , suppressing the notation for ψ k for simplicity.
Intuitively, (22) splits the action of h on s into two stages: first we apply the permutation σ h,1 to V 1 , and then a suitable automorphism to each subtree T k , for 0 ≤ k ≤ d − 1.

4.1.2.
Actions generated by finite automata are locally non-degenerate. Suppose T is a d-ary tree, and G ⊂ Aut(T ). Suppose the element g ∈ Aut(T ) can be computed by a finite automaton, which is equivalent to the set of sections S g = {g| v | v ∈ V , ≥ 0} being finite [33, Section 1.3].
Number the elements in S g , that is, S g = {h 1 , . . . , h k } for some k ≥ 1. For each 1 ≤ i ≤ k such that h i = id, there exists i ≥ 0 and a vertex w i ∈ V i such that h i (w i ) = w i . Then h i acts non-trivially on any path in the clopen set ∂T w i . Let and Now, suppose g ∈ G fixes a vertex v ∈ V . Then g|∂T v = g| v = h i for some 1 ≤ i ≤ k, and Definition 4.1 is satisfied with constant α g defined by (24). Therefore, the action of g ∈ G is locally non-degenerate.

4.2.
Actions which are locally degenerate.
We give examples of actions on rooted trees which are locally degenerate. Example 3 is that of an action on a spherically homogeneous tree with strictly increasing spherical index, that is, hypothesis (1) in Proposition 4 does not hold for this example. Example 4 is an example of an action on a d-ary tree for which hypothesis (2) in Proposition 4 does not hold.
Example 3. Let T be a tree with spherical index n = (n 1 , n 2 , . . .), where 2 ≤ n 1 < n 2 < · · · is an increasing sequence of integers. Let H ⊂ Aut(T ) be a group which acts minimally and equicontinuously on ∂T . We define an element c ∈ Aut(T ) whose action on ∂T is locally degenerate. Then the action of G = c, H on ∂T is minimal, equicontinuous and locally degenerate. As before, we denote by µ ∂T the counting measure on ∂T defined in Section 3.
To this end, let v = (v ) ≥0 be a path in T . For ≥ 0, choose a vertex w +1 ∈ V +1 ∩ T v such that w +1 = v +1 , and a vertex z +2 ∈ V +2 such that w +1 and z +2 are joined by an edge. Let c act non-trivially on every path in the clopen basic set ∂T z +2 , for ≥ 1, for example, by applying a cyclic permutation to the vertices in V +3 joined to z +2 . Let c act as the identity map outside of the set ≥1 ∂T z +2 . Then for each clopen set ∂T w +1 , for ≥ 1, we have Since the sequence {n } ≥1 is increasing, the action of c is locally degenerate.
We now modify Example 3 to build an action on a d-ary tree which is locally degenerate.
Example 4. Let T be a d-ary tree, for d ≥ 2, and let H ⊂ Aut(T ) be a group which acts minimally and equicontinuously on ∂T . We define an element c ∈ Aut(T ) whose action on ∂T is locally degenerate. Then the action of G = c, H on ∂T is locally degenerate.
Let v = (v ) ≥0 be a path in T . For k ≥ 1, let m k = 2 k . Let w m k be a vertex in V m k which is joined by an edge to v m k −1 and is distinct from v m k . Let z m k+1 be a vertex in V m k+1 joined by a path to w m k . This path consists of 2 k edges. Define c so that it acts non-trivially on every path in the clopen set ∂T zm k+1 , and trivially outside of the set ≥1 ∂T zm +1 . Note that by construction for every k ≥ 1 we have Then for each clopen set ∂T wm k , k ≥ 1, we have Since d −2 k → k→∞ 0 then the action of c on ∂T is locally degenerate.

5.
Sets of points without holonomy of full measure. In this section we prove Theorem 1.8. This theorem gives necessary conditions for the set of points with trivial holonomy for the action (X, G, Φ, µ) to have full measure. This theorem is one of the main ingredients of the proof of Theorem 1.6.
5.1. Lebesgue density. Let (X, G, Φ, µ) be a minimal equicontinuous action. A choice of a tree representation for the action (X, G, Φ, µ) gives X an ultrametric D (12). A Cantor set X is a Polish space, and so the following result of Miller [32] applies. Denote by B(x, ) = {y ∈ X | D(x, y) < } an open ball around x of radius > 0.
Theorem 5.1. [32, Proposition 2.10] Let X be a Polish space, and suppose X has an ultrametric D compatible with its topology. Let µ be a probability measure on X, and let A be a Borel set of positive measure. Then the Lebesgue density of A at x, given by exists and is equal to 1 for µ-almost every x ∈ A.

5.2.
Proof of Theorem 1.8. For the convenience of the reader we first restate Theorem 1.8 below.
Theorem 5.2. Let X be a Cantor set, let G be a finitely generated group, and let (X, G, Φ, µ) be a minimal equicontinuous action. Suppose (X, G, Φ, µ) is locally non-degenerate. Then the set X 0 of points with trivial holonomy has full measure with respect to µ.
Proof. Given g ∈ G, let Fix(g) = {x ∈ X | g · x = x} be the set of fixed points of g. Note that Fix(g) is always a closed subset of X, and hence is Borel. The set Fix(g) may have positive measure, or measure zero. Assume that Fix(g) has positive measure. Suppose g · x = x. Recall from Definition 1.1 that g has trivial holonomy at x ∈ X if g fixes every point in an open neighborhood of x, and g has non-trivial holonomy at x otherwise. We will show that under the hypothesis of the theorem the subset {x ∈ Fix(g) | g has trivial holonomy at x ∈ X} has positive measure in X, while the subset {x ∈ Fix(g) | g has non-trivial holonomy at x ∈ X} has zero measure in X.
If g has trivial holonomy at x ∈ Fix(g), then it fixes every point in B(x, ) for some > 0. Then the Lebesgue density of x in Fix(g) given by (25) exists and is equal to 1.
Next, suppose g has non-trivial holonomy at x ∈ Fix(g). We will show that if the Lebesgue density of Fix(g) at x exists, then it must be bounded away from 1.
Let (φ, φ * ) : (X, Φ(G)) → (∂T, Homeo(∂T )) be a tree representation for the action, such that the action of Φ(G) on ∂T is locally non-degenerate, and let φ(x) = (v ) ≥0 . For ≥ 0 let U = ∂T v be the clopen neighborhood of (v ) ≥0 consisting of all paths passing through the vertex v , and let be the subset of points in U which are not fixed by the action of g. By the hypothesis the action of g is locally non-degenerate, so there is a constant 0 < α g ≤ 1 such that Then we have It follows that if the Lebesgue density of Fix(g) at φ(x) exists, then it is bounded away from 1. By Theorem 5.1 the subset of Fix(g) of points where the Lebesgue density does not exist or it exists but is not 1 has measure zero in ∂T . Therefore, the subset of Fix(g) of points with non-trivial holonomy has measure zero. Since the group G is countable, the union g∈G {x ∈ Fix(g) | g has non-trivial holonomy at x ∈ X} is a countable union of zero measure sets, and so has zero measure.
Corollary 1 is a reformulation of Theorem 1.8 for the case of topologically free actions. Indeed, if an action (X, G, Φ, µ) is topologically free, then x ∈ X has non-trivial stabilizer if and only if it has non-trivial holonomy. By Theorem 1.8 if (X, G, Φ, µ) is topologically free and locally non-degenerate, then it is essentially free.
Kambites, Silva and Steinberg [31] studied topologically and essentially free actions on d-ary rooted trees generated by finite automata. We recover their result as a consequence of Corollary 1. Proof. If a minimal action (∂T, G, µ ∂T ) is essentially free, then it contains a dense orbit of points with trivial stabilizer. Since the action is by homeomorphisms, for each g ∈ G the set K g = {x ∈ ∂T | g · x = x} is an open dense subset of ∂T . Then since G is countable, the set g∈G K g of points with trivial stabilizer is a residual subset of ∂T . Thus (∂T, G, µ ∂T ) is topologically free.
For the converse, let (∂T, G, µ ∂T ) be topologically free. We showed in Section 4.1 that a group action generated by finite automata is locally non-degenerate. Then by Corollary 1 (∂T, G, µ ∂T ) is essentially free.
6. Sets of points with holonomy of full measure. Bergeron and Gaboriau [8] and Abért and Elek [1] gave examples of group actions on Cantor sets which are topologically free and not essentially free. By the discussion after Theorem 1.2, in these examples the set of points with non-trivial holonomy has full measure. We now describe another class of examples where the set of points with non-trivial holonomy has full measure, whose construction is somewhat easier than in [1,8].
Define a homeomorphism b of ∂T as follows. Suppose that in the spherical index n = (n 1 , n 2 , . . .) we have n ≥ 3 for ≥ 1. Recall that T v denotes the subtree of T containing all paths through a given vertex v = w 1 · · · w . Then the boundary ∂T v of T v consists of all infinite sequences starting with the finite word v = w 1 · · · w .
The root v 0 is joined by edges to n 1 ≥ 3 vertices in V 1 , labelled by 0, 1, . . . , n 1 −1. Define b to fix the vertices 0, 1, . . . , n 1 − 3, and interchange the vertices n 1 − 2 and n 1 − 1. Define the action of b on the rest of the tree by induction as follows.
Suppose the action of b on V is defined. Let v ∈ V , then v is joined by edges to n +1 vertices in V +1 which are labelled by words of length + 1, k, that is, the action of b fixes the ( + 1)-st entry in the sequence and only changes some of the preceding entries.
that is, b interchanges v (n +1 − 2) and v (n +1 − 1), and fixes other vertices in V +1 joined to v . The set Fix(b) ⊂ ∂T is non-empty. Indeed, let s = s 1 s 2 · · · be a sequence. By definition of b we have that b · s = s if and only if s = n − 2 or s = n − 1 for some ≥ 1. Therefore, every infinite sequence s = s 1 s 2 · · · such that s ≤ n − 3 for all ≥ 1 is a fixed point of b.
Also, b is not the identity on any open set. Indeed, let s = s 1 s 2 · · · be a fixed point of b, and let U ⊂ ∂T be an open neighborhood of s. Then U contains a basic clopen set ∂T s1···s , for some ≥ 1. Since s is a fixed point of b, s +1 = n +1 − 2 and s +1 = n +1 − 1. However, the open set ∂T s1···s contains the union of clopen sets ∂T s1···s (n +1 −2) ∪ ∂T s1···s (n +1 −1) on which b acts non-trivially by (26). Therefore, b is not the identity on U . Since U is an arbitrary neighborhood of s, s is a point with non-trivial holonomy. We have shown that b has non-trivial holonomy at every point in Fix(b). Theorem 6.1. Let T be a spherically homogeneous tree with spherical index n = (n 1 , n 2 , . . .) such that n +1 > 2n . Let H be a group acting minimally and equicontinuously on ∂T , and let G = H, b ⊂ Aut(T ) where b is defined as above by (26). Let µ ∂T be the counting measure on ∂T . Then the following holds: 1. The action (∂T, G, µ ∂T ) is minimal and equicontinuous. 2. The set of points with non-trivial holonomy has full measure in ∂T .
Proof. By assumption the orbits of points in ∂T under the action of H are dense in ∂T , so the action of G = H, b on ∂T is minimal. Since H and b act by permutations on each level V , ≥ 0, the action of G on ∂T is equicontinuous.
Let ∂T be given an ultrametric (12). We show that the Lebesgue density of Fix(b) at every point in Fix(b) is 1, and so Fix(b) must have positive measure. It follows that the set {v ∈ ∂T | [G] v = G v } of points with non-trivial holonomy has full measure, since it is invariant under the action of G.
Inductively, we obtain that By assumption n +k > 2 k−1 n +1 for k ≥ 2, therefore, and so It follows that which implies that Since n → ∞ as → ∞, we obtain that the Lebesgue density of the set Fix(b) at (v ) ≥0 is 1, and the statement of the theorem follows. 7. Conjugate stabilizer subgroups. In this section we prove Theorem 1.7 and the main Theorem 1.6. 7.1. Proof of Theorem 1.7. We restate the theorem in Theorem 7.1 below in a slightly different form, which will be useful in the proof of Theorem 1.9.
Recall from the Introduction that a point x ∈ X has trivial holonomy if for any g ∈ G x , where G x is the stabilizer of the action at x, there exists an open set U g x such that g| Ug = id. We denote by X 0 the set of all points with trivial holonomy in X.
Theorem 7.1. Let (X, G, Φ) be a minimal equicontinuous action of a finitely generated group G on a Cantor set X. Let x ∈ X 0 be a point with trivial holonomy. Then the set of stabilizers of the points in the orbit of x is finite if and only if (X, G, Φ) is LQA. Moreover, if the action (X, G, Φ) is LQA, then for any other point with trivial holonomy y ∈ X 0 the stabilizer G y is conjugate to the subgroups in (27).
Remark 4. The restriction to points with trivial holonomy in Theorem 7.1 is necessary. Indeed, if x ∈ X is a point with non-trivial holonomy, then the set {G g·x | g ∈ G} may be infinite even if (X, G, Φ) is topologically free (and so LQA), see Example 1.
Proof. Let x ∈ X 0 be a point without holonomy, and let U x = {U } ≥0 , U 0 = X, be an adapted neighborhood system at x, see Section 2.2 for terminology. Then G = {g ∈ G | g · U = U } is a group for any ≥ 0, with G 0 = G. Denote by Φ : G → Homeo(U ) or by (U , G , Φ ) the induced action of G on U .
Suppose (X, G, Φ) is LQA with > 0, and let ≥ 0 be such that diam(U ) < . Then for any open set W ⊂ U , if g|W = id|W , then g|U = id|U . In particular, if y ∈ U is without holonomy, then every element which fixes every point in an open neighborhood of y in U must fix every point in U . It follows that all points without holonomy in U have equal stabilizers, that is, for all y ∈ X 0 ∩ U we have G y = ker{Φ : G → Homeo(U )}. The group ker(Φ ) is a normal subgroup in G , but it need not be normal in G.
Let g / ∈ G , then x = g · x ∈ g · U , with g · U ∩ U = ∅. Moreover, G x = gG x g −1 , and h ∈ G fixes an open neighborhood of x if and only if ghg −1 fixes an open neighborhood of x, so x is a point without holonomy in g · U .
Let y ∈ g · U ∩ X 0 be another point without holonomy. Then there exists a point y ∈ U ∩ X 0 without holonomy such that y = g · y. Then we have so for all y ∈ g · U ∩ X 0 the stabilizers are equal, G y = g ker(Φ )g −1 . Since the orbit of U under the action of G is finite, ker(Φ ) has a finite number of distinct conjugates in G, and the set {G x | x ∈ X 0 } is finite. This proves that if (X, G, Φ) is LQA, then for any x ∈ X 0 the set {G g·x | g ∈ G} is a finite set of conjugate subgroups, and the stabilizer of any point y ∈ X 0 without holonomy is conjugate to the subgroups in {G g·x | g ∈ G}.
We prove the converse by showing that if (X, G, Φ) is not LQA then the set Suppose that the action (X, G, Φ) is not LQA. Given x ∈ X 0 , we show that the set {G g·x | g ∈ G} is infinite by induction. Namely, we will construct an increasing collection of finite subsets Y n , n ≥ 1, of the orbit G(x) = {z ∈ X 0 | z = g ·x, g ∈ G}, such that for n ≥ 1 the cardinality of Y n is 2 n , and all points Y n have pairwise distinct stabilizers. Points in Y n will be labelled by words k 1 · · · k n of length n, where k i ∈ {0, 1} for 1 ≤ i ≤ n. The points will be chosen so that y 0 = x, and for n ≥ 1 y k1···kn0 = y k1···kn .
We now start the construction of the subsets Y n , n ≥ 1. As in the first part of the proof, U x = {U } ≥0 , U 0 = X, is an adapted neighborhood system at x. We first construct Y 1 .
Since (X, G, Φ) is not LQA, there exists an element g 0 ∈ G which satisfies g 0 |U 1 = id, and such that g 0 |(X − U 1 ) is not the identity. Choose z ∈ X − U 1 such that g 0 · z = z. By continuity there is an open neighborhood O z such that g 0 fixes no point in O. Choose an index s 1 ≥ 1 large enough so that for some h s1 ∈ G we have z ∈ h s1 · U s1 ⊂ O, then for any z ∈ h s1 · U s1 we have g 0 · z = z . Set W 0 = U s1 , and Then for any z ∈ W 0 we have g 0 ∈ G z , in particular, for z = x. So we choose y 0 = x. For any z ∈ W 1 we have g 0 / ∈ G z . Since the action is minimal and the set W 1 is open, we can choose y 1 ∈ G(x) ∩ W 1 . Since x ∈ X 0 and y 1 is in the orbit of x, then y 1 ∈ X 0 . We set Y 1 = {y 0 , y 1 }. Now suppose we are given a finite set of points labelled by words of length n, namely, Y n = {y k1···kn | k i ∈ {0, 1}, 1 ≤ i ≤ n}, and a finite collection of clopen sets, labelled by words of length i, for 1 ≤ i ≤ n, namely, We assume that these collections of sets have the following properties: 1. For every 1 ≤ i < n and every word k 1 · · · k i+1 we have an inclusion W k1···ki+1 ⊂ W k1···ki . That is, every set W k1···ki labelled by a word of length i contains precisely two clopen sets labelled by words of length i + 1, W k1···ki0 and W k1···ki1 . 2. For every 1 ≤ i < n and every word k 1 k 2 · · · k i there is an element g k1k2···ki0 ∈ G such that the restriction g k1k2···ki0 |W k1k2···ki0 is the identity, and for every z ∈ W k1k2···ki1 we have g k1k2···ki0 · z = z. 3. For every 1 ≤ i ≤ n and every W k1k2...ki we have y k1k2...ki ∈ W k1k2...ki , where y k1k2...ki ∈ Y i ⊂ G(x). Also, y 0 = x, and for 1 ≤ i < n we have y k1···ki0 = y k1···ki , so Y i ⊂ Y i+1 . In particular, we have y 0···0 = x for any word of zeros of length i. We claim that for any two distinct points y j1j2...jn , y k1k2...kn ∈ Y n the stabilizers G yj 1 j 2 ...jn and G y k 1 k 2 ...kn are distinct. Indeed, consider the words j 1 j 2 . . . j n and k 1 k 2 . . . k n , and let s be the first digit such that j s = k s . Without loss of generality, we can assume that j s = 0 and k s = 1. Then by (1) we have that y j1j2...jn ∈ W j1j2...jn ⊂ W j1j2...js−10 , and then by (2) we have g j1j2...js−10 ∈ G yj 1 j 2 ...jn . Similarly, by (1) we have y k1k2...kn ∈ W k1k2...kn ⊂ W j1j2...js−11 , and then by (2) we have g j1j2...js−10 / ∈ G y k 1 k 2 ...kn . Then G yj 1 j 2 ...jn = G y k 1 k 2 ...kn . We now implement the inductive step and construct Y n+1 , W k1···kn+1 and g k1···kn0 , where k i ∈ {0, 1} for 1 ≤ i ≤ n.
For each k 1 k 2 . . . k n we have y k1k2...kn ∈ W k1k2...kn , where y k1k2...kn ∈ G(x). Since the action is not LQA, there exists a clopen neighborhood W of y k1k2...kn properly contained in W k1k2...kn , and an element g k1k2...kn0 ∈ G such that the restriction g k1k2...kn0 |W is the identity, while the restriction g k1k2...kn0 |(W k1k2...kn − W ) to the complement of W in W k1k2...kn is not the identity. Let z ∈ W k1k2...kn − W be so that g k1k2...kn0 (z) = z. By continuity there is a neighborhood W z such that for every z ∈ W we have g k1k2...kn0 (z ) = z .
For n ≥ 1, we have #Y n = #{y k1k2...kn | k i ∈ {0, 1}, 1 ≤ i ≤ n} = 2 n , and all points in this set are contained in the orbit G(x) = G(y 0 ) and have distinct stabilizers. It follows {G g·x | g ∈ G} is infinite. This finishes the proof of the theorem.

7.2.
Proof of Theorem 1.6. We restate the theorem for the convenience of the reader below.
Theorem 7.2. Let X be a Cantor set, let G be a finitely generated group, and let (X, G, Φ, µ) be a locally quasi-analytic minimal equicontinuous action. Then the following is true.
1. There exists a subgroup H ⊂ G such that the set of points with stabilizers conjugate to H is residual in X. 2. Suppose in addition that (X, G, Φ, µ) is locally non-degenerate. Then the set of points with stabilizers conjugate to H has full measure in X.
Proof. By Theorem 1.2 the set X 0 of points with trivial holonomy is residual in X. Choose x ∈ X 0 and let H = G x , where G x is the stabilizer of x. By Theorem 1.7, if the action is LQA then the stabilizer of any other point in X 0 is conjugate to H. This proves the first statement. For the second statement, assume in addition that the action is locally nondegenerate. Then by Theorem 1.8 the set X 0 has full measure, which completes the proof.
8. Applications. The goal of this section is to prove Theorems 1.9 and 1.10. We start by recalling some background on the invariant random subgroups, as needed for the proof of Theorem 1.9.
8.1. Preliminaries on invariant random subgroups. We denote by Sub(G) the space of closed subgroups of a finitely generated group G. The space Sub(G) is equipped with the Chaubaty-Fell topology. Open sets in this topology are given by [5,2,40,7] where A and B are finite sets. The space Sub(G) is a compact totally disconnected space, and G acts on Sub(G) by conjugation. Let (X, G, Φ, µ) be a minimal equicontinuous action, and consider the mapping which assigns to each x ∈ X its stabilizer. Stabilizers of points in the same orbit in (X, G, Φ, µ) are conjugate, so (30) maps the orbit of x in X onto the orbit of G x in Sub(G). The map (30) need not be injective. For instance, if (X, G, Φ, µ) is a free action, which means that for any x ∈ X we have G x = {e} where e is the identity in G, then the image of (30) is the trivial subgroup. The properties of the mapping (30) were studied in many works. We recall the following result.
For the action (X, G, Φ, µ) the measure µ pushes forward along (30) to the ergodic IRS ν = St * µ. For instance, if (X, G, Φ, µ) is a free action then ν is an atomic measure supported on a single point in Sub(G). At the other extreme, if X = ∂T is a boundary of a d-ary tree T , and G is a weakly branch group, then the stabilizers of all points in X are pairwise distinct [7,Proposition 8], and ν is non-atomic.
By Lemma 8.2 the set X 0 = {x ∈ X | G x = [G] x } defined in (5) contains all points at which the mapping (30) is continuous. Recall from (7) that we denote While X is a Cantor set, since the map (30) may be discontinuous, the set {G x | x ∈ X} ⊂ Sub(G) is only a Polish space and may contain isolated points. For example, if G is the Grigorchuk group, then stabilizers of points with non-trivial holonomy are isolated points in Sub(G) [40]. Thus the closed set Z need not be a subset of {G x | x ∈ X} ⊂ Sub(G) but of course it is a subset of {G x | x ∈ X} ⊂ Sub(G). More precisely, we have the following statement. The space Sub(G) is metrizable. One way to define a metric on this space is by a pullback from a metric on the space of Schreier graphs of subgroups of G, see [2,Section 3] or [24,10].
Let S be a finite symmetric generating set for G. Given a subgroup H ⊂ G, construct the Schreier graph Γ G/H as follows: the cosets in G/H are the vertices of Γ G/H , and two vertices hH and gH are joined by an edge, directed from hH to gH and labeled by s ∈ S if and only if gH = shH. Edges in the graph are assigned unit length, and Γ G/H has a length metric D Γ G/H , that is, the distance between two points in Γ G/H is the length of the shortest path between these points. The graph Γ G/H has a distinguished vertex which is the coset of the identity eH, so Γ G/H is a pointed metric space. Denote by B Γ G/H (r) a metric ball of radius r in Γ G/H centered at eH. Given H 1 , H 2 ∈ Sub(G), the metric balls B Γ G/H 1 (r) and B Γ G/H 2 (r) are isomorphic if and only if there exists an isometry f : B Γ G/H 1 (r) → B Γ G/H 2 (r) which preserves the labelling of edges and such that f (eH 1 ) = eH 2 .
Denote the set of all Schreier graphs associated to closed subgroups of G by Sch(G, S) = {Γ G/H | H ∈ Sub(G)}.
We define a metric on Sch(G, S) by setting k = max{r ≥ 0 | B Γ G/H 1 (r) and B Γ G/H 2 (r) are isomorphic}.
The metric space Sch(G, S) is compact. The action of G on Sch(G, S) is defined by setting g · Γ G/H = Γ G/gHg −1 , for any g ∈ G. The graphs Γ G/H and Γ G/gHg −1 are isomorphic as metric spaces, but not necessarily as pointed metric spaces. We think about the action of G on Sch(G, S) as moving the distinguished vertex in Γ G/H from eH to the coset of gH. It is immediate that the map studied IRS's for the full group of minimal Z d -actions on Cantor sets, and for branch groups. Thomas and Tucker-Drob in [36] and [37] classified IRS's for diagonal inductive limits of respectively finite symmetric groups and finite alternating groups. Dudko and Medynets [15] studied IRS's of full groups with associated Bratteli diagrams admitting finite number of ergodic measures.
In this paper, we consider the IRS's defined by actions of finitely generated groups on rooted spherically homogeneous trees. We allow the tree to have any spherical index. For the class of locally non-degenerate actions, we study the support of the IRS depending on whether the action is locally quasi-analytic. 8.2. Proof of Theorem 1.9. We obtain Theorem 1.9 as a direct consequence of Theorem 8.5.
Recall that two measure-preserving systems (X, G, Φ, µ) and (Y, G, Ψ, ν) are isomorphic in the measure-theoretical sense if there are invariant sets of full measure S X ⊂ X and S Y ⊂ Y , and a bi-measurable bijection ψ : S X → S Y such that µ(ψ −1 (A)) = ν(A) for all measurable A ⊂ S Y , and ψ conjugates the action of G on S X and S Y , that is, ψ(Φ(g) · x) = Ψ(g) · ψ(x) for all x ∈ S X .
In this case we call ψ a measure-theoretical isomorphism.
Recall that X 0 denotes the subset of X consisting of points without holonomy, and Z = {G x | x ∈ X 0 }. Theorem 8.5. Let X be a Cantor set, let G be a finitely generated group, and let (X, G, Φ, µ) be a minimal equicontinuous action. Suppose the set X 0 of points without holonomy has full measure in X. Then the following holds: 1. The IRS ν = St * µ is supported on Z. Proof. The first statement is a direct consequence of the assumption that the set X 0 of points without holonomy has full measure in X.
If the action (X, G, Φ, µ) is locally quasi-analytic (LQA), then by Theorem 1.7 Z is finite. Since Z is the support of ν, then in this case ν must be atomic. So if ν is non-atomic, then (X, G, Φ, µ) is not LQA. If (X, G, Φ, µ) is not LQA, then by Theorem 1.7 for every point without holonomy x ∈ X 0 the set O Gx = {G g·x | g ∈ G} = {gG x g −1 | g ∈ G} is infinite. This set is the orbit of G x in Sub(G) under the action of G by conjugation. By Lemma 8.3 Z is minimal, so O Gx is dense in Z.
Since O Gx is infinite, the closure Z of O Gx is strictly larger than O Gx , containing limit points of O Gx . By Lemma 8.3 the orbit of every limit point is dense in Z, and it follows that Z is perfect. Then Z is a Cantor set. Since the measure ν supported on Z is finite, it is non-atomic. This proves statement (2).
In the case when (X, G, Φ, µ) is not LQA under the additional assumption that the restriction St| X0 is injective, the third statement follows from the fact that ν is a push-forward measure.
Proof. (of Theorem 1.9) Since (X, G, Φ, µ) is locally non-degenerate, by Theorem 1.8 the set X 0 of points without holonomy has full measure in X. Then the statement follows by Theorem 8.5.

8.3.
Almost one-to-one extensions. Recall that in (8) we defined the set and the factor map η : X → X is almost one-to-one. Moreover, recall that any continuous action has a unique (up to conjugacy) maximal equicontinuous factor, in the sense that any other equicontinuous factor is also a factor of this maximal one, see for instance [4].
We prove Theorem 1.10 which provides the following alternative characterization of LQA actions.
Theorem 8.6. Let X be a Cantor set, and let G be a finitely generated group. Let (X, G, Φ) be a minimal equicontinuous action. Then (X, G, Φ) is locally quasianalytic if and only if η : X → X is a conjugacy.
Proof. First, assume that (X, G, Φ) is LQA. By Theorem 1.7, Z is finite, and the action of G on Z by conjugation is periodic, and so equicontinuous. Then the product action of G on X × Z is also equicontinuous, see [4,Lemma 4 in Chapter 2]. This in turn implies that the orbit closure of every point in X × Z is minimal, see [4, Lemma 3 in Chapter 2]. Now, using that X is the unique minimal subset in {(x, G x ) | x ∈ X} ⊂ X × Sub(G) by [23, Proposition 1.2 (3)], we get that X = X × Z. Finally, since η : X → X is almost one-to-one, we have that (X, G, Φ) is the maximal equicontinuous factor of ( X, G,Φ) by Proposition 5, but this immediately implies that η must be a conjugacy.
For the opposite direction, assume that η is a conjugacy. Then the action of G on X is equicontinuous. Moreover, since Z is a factor system of X (simply by projecting in the second coordinate), we get that the action of G on Z by conjugation is equicontinuous as well, see [4, Corollary 6 in Chapter 2]. Hence by Lemma 8.4, Z is finite and so, by Theorem 1.7, the action (X, G, Φ) is LQA.