Prime spectrum and dynamics for nilpotent Cantor actions

A minimal equicontinuous action by homeomorphisms of a discrete group $\Gamma$ on a Cantor set $X$ is locally quasi-analytic, if each homeomorphism has a unique extension from small open sets to open sets of uniform diameter on $X$. A minimal action is stable, if the actions of $\Gamma$ and of the closure of $\Gamma$ in the group of homeomorphisms of $X$, are both locally quasi-analytic. When $\Gamma$ is virtually nilpotent, we say that $\Phi \colon \Gamma \times \mathfrak{X} \to \mathfrak{X}$ is a nilpotent Cantor action. We show that a nilpotent Cantor action with finite prime spectrum must be stable. We also prove there exist uncountably many distinct Cantor actions of the Heisenberg group, necessarily with infinite prime spectrum, which are not stable.


Introduction
A minimal equicontinuous action : × X → X of a countable group on a Cantor space X is called a generalized odometer [9; 14].When = ‫,ޚ‬ this is just the abstract form of a traditional odometer action of the integers.For = ‫ޚ‬ n with n ≥ 2, one obtains a more complex class of actions, whose classification becomes increasingly intractable as n increases [27], even while the dynamical properties of minimal equicontinuous Cantor actions by ‫ޚ‬ n are well behaved.For in general, we simply refer to these as Cantor actions, which will always be assumed minimal and equicontinuous.
It is a classical result that a ‫-ޚ‬odometer is classified by its Steinitz order, which is calculated using a representation of the action as an inverse limit of actions on finite cyclic groups.One can also associate to a Cantor action by ‫ޚ‬ n its Steinitz order and also a collection of types, called its typeset, which consists of equivalence classes of Steinitz orders of individual elements of ‫ޚ‬ n .As discussed by Thomas [28,Section 4], the additional data of the typeset is still not sufficient to reduce the classification problem for Cantor actions by ‫ޚ‬ n to a standard Borel equivalence relation.
In the authors' work [20], we associate the type and typeset invariants to a Cantor action (X, , ) for an arbitrary countable group .The type τ [X, , ] is the asymptotic equivalence class of the Steinitz order ξ(X, , ) of a presentation of the action as an inverse limit of actions of on finite sets.
Associated to the type τ [X, , ] is an even more basic invariant, the prime spectrum π[X, , ], which consists of the set of primes which appear in a Steinitz order ξ(X, , ) representing the type τ [X, , ]; see Definition 2.14.The prime spectrum decomposes into two parts, where the infinite prime spectrum π ∞ [X, , ] consists of the primes that occur with infinite multiplicity in ξ(X, , ) and the finite prime spectrum π f [X, , ] consists of the primes that occur with finite multiplicity.The prime spectrum and the finite prime spectrum are only well defined modulo finite subsets of π f [X, , ]. Definition 1.1.A Cantor action (X, , ) has finite spectrum if the prime spectrum π [X, , ] is a finite set and is said to have infinite spectrum otherwise.
The classification of Cantor actions for is, in general, intractable and one seeks invariants for Cantor actions which at least distinguish between particular classes of actions.The authors' works [15; 16; 17; 18] study dynamical properties which yield invariants of Cantor actions.In particular, one of the most basic invariants is the property that the action is either stable or wild.The purpose of this note is to give a relation between the prime spectrum of a Cantor action and the wild property.
As explained in detail in Section 2E below, the property that the action (X, , ) is stable is a property of the action of the completion G( ) = ( ) ⊂ Homeo(X), which is a profinite group naturally acting on X.The property that the action (X, , ) is locally quasianalytic is defined in Definition 2.10, and (X, , ) is stable if the action of G( ) on X is also locally quasianalytic.If (X, , ) is stable, then (X, , ) is locally quasianalytic.The converse need not hold even for actions of nilpotent groups, as we show later.
A Cantor action (X, , ) is said to be nilpotent if contains a finitely generated nilpotent subgroup with finite index.This class of group actions is particularly interesting, as it has the natural next level of complexity after the abelian Cantor actions.We show the following three results for nilpotent Cantor actions.Theorem 1.2.Let (X, , ) be a nilpotent Cantor action.If the prime spectrum π [X, , ] is finite, then the action is stable.Theorem 1.2 does not have a converse.We show that every collection of primes, finite or infinite, can be realized as the prime spectrum of a stable nilpotent Cantor action.
Theorem 1.3.Let π f and π ∞ be two distinct sets of primes, where π f is a finite set and π ∞ is a nonempty finite or infinite set.Then there exists a stable nilpotent Cantor action (X, , ) such that π ∞ [X, , ] = π ∞ and π f [X, , ] = π f .Let (X, , ) be an abelian Cantor action.If the action is effective, then it is free, and the action of the closure G( ) is also free, which implies that the action is stable.An effective nilpotent Cantor action need not be free and may even have elements which fix every point in a clopen subset of the Cantor set X.The authors showed in their work [16] that nilpotent Cantor actions are locally quasianalytic, which means that such subsets of fixed points cannot be arbitrarily small, as their diameter has lower bound which is uniform over the Cantor set X.It is then surprising to discover that if one allows G( ) to have infinite prime spectrum then one can construct wild nilpotent actions, for which the action of the closure G( ) is not locally quasianalytic, as shown in Theorem 1.4.In addition, Theorem 1.4 is a realization result, which shows that every infinite set of primes can be realized as the prime spectrum of a wild nilpotent Cantor action.
Theorem 1.4.Given any two distinct sets π f and π ∞ of primes, where π f is infinite and π ∞ is any (possibly empty) set, there is a minimal equicontinuous action (X, , ) of the Heisenberg group such that π Moreover, there exists an uncountable number of nilpotent Cantor actions (X, , ) of the Heisenberg group with infinite prime spectra such that (1) each (X, , ) is topologically free, (2) each (X, , ) is wild, (3) the prime spectra of such actions are pairwise distinct.
The notion of return equivalence for Cantor actions and its relationship with conjugacy of action is explained in Section 2D.The result of Corollary 1.5 below follows from the result that the prime spectrum of the action is an invariant of its return equivalence class; see Theorem 2.16.

Corollary 1.5.
There exists an uncountable number of nilpotent Cantor actions (X, , ) of the Heisenberg group which are not return equivalent and therefore not conjugate.
The conclusion of Theorem 1.4 is used in [19] for the calculation of the mapping class groups of solenoidal manifolds whose base is a nil-manifold.
We note that for more general groups , an analog of Theorem 1.2 need not hold.For example, a weakly branch group, as studied in [3; 5; 6; 25], acts on the boundary of a d-regular rooted tree, and so has finite prime spectrum {d}, but the dynamics of the action on the Cantor boundary are wild.
Question 1.6.Let (X, , ) be a Cantor action.For which classes of groups does the finiteness of the prime spectrum of the action imply that the action is stable?
The paper is organized as follows.In Section 2A we recall basic properties of minimal equicontinuous group actions on Cantor sets.In particular, the definition of the prime spectrum of a minimal equicontinuous action is given in Definition 2.14.We prove Theorem 1.2 in Section 3, and give basic examples of nilpotent Cantor actions in Section 4. In Section 5 we construct examples of stable and wild actions of the Heisenberg group with prescribed prime spectrum, proving Theorems 1.3 and 1.4, from which we deduce Corollary 1.5.

Cantor actions
We recall some of the basic properties of Cantor actions, as required for the proofs of Theorems 1.2 and 1.4.More complete discussions of the properties of equicontinuous Cantor actions are given in the text by Auslander [1], the papers by Cortez and Petite [9], Cortez and Medynets [8], and the authors' works, in particular [10;11;17,Section 3].
2A. Basic concepts.Let (X, , ) denote an action : ×X → X.We write g • x for (g)(x) when appropriate.The orbit of x ∈ X is the subset The action is minimal if for all x ∈ X, its orbit O(x) is dense in X.
An action (X, , ) is equicontinuous with respect to a metric d X on X if for all ε > 0 there exists δ > 0 such that for all x, y ∈ X and g ∈ we have that d X (x, y) < δ implies d X (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.Now assume that X is a Cantor space.Let CO(X) denote the collection of all clopen (closed and open) subsets of X, which forms a basis for the topology of X.For φ ∈ Homeo(X) and U ∈ CO(X), the image φ(U ) belongs to CO(X).The next result is folklore, and a proof is given in [16,Proposition 3.1].
Proposition 2.1.For X a Cantor space, a minimal action : × X → X is equicontinuous if and only if the -orbit of every U ∈ CO(X) is finite for the induced action * : × CO(X) → CO(X).Definition 2.2.We say that U ⊂ X is adapted to the action (X, , ) if U is a nonempty clopen subset, and for any g ∈ , g The proof of [16,Proposition 3.1] shows that given x ∈ X and clopen set x ∈ W , there is an adapted clopen set U with x ∈ U ⊂ W .
For an adapted set U , the set of "return times" to U , (1) is a subgroup of , called the stabilizer of U .Then for g, g ′ ∈ with g•U ∩g ′ •U ̸ = ∅ we have g −1 g ′ • U = U , and hence g −1 g ′ ∈ U .Thus, the translates {g • U | g ∈ } form a finite clopen partition of X and are in one-to-one correspondence with the quotient space X U = / U .Then acts by permutations of the finite set X U and so the stabilizer group U ⊂ G has finite index.Note that this implies that if V ⊂ U is a proper inclusion of adapted sets, then the inclusion V ⊂ U is also proper.Definition 2.3.Let (X, , ) be a Cantor action.A properly descending chain of clopen sets U = {U ℓ ⊂ X | ℓ ≥ 0} is said to be an adapted neighborhood basis at x ∈ X for the action if x ∈ U ℓ+1 ⊂ U ℓ is a proper inclusion for all ℓ > 0, with U 0 = X, ℓ>0 U ℓ = {x}, and each U ℓ is adapted to the action .Given x ∈ X and ε > 0, Proposition 2.1 implies there exists an adapted clopen set U ∈ CO(X) with x ∈ U and diam(U ) < ε.Thus, one can choose a descending chain U of adapted sets in CO(X) whose intersection is x, from which the next result follows: Proposition 2.4.Let (X, , ) be a Cantor action.Given x ∈ X, there exists an adapted neighborhood basis U at x for the action .
Combining the above remarks, we have: Corollary 2.5.Let (X, , ) be a Cantor action and U be an adapted neighborhood basis.Set ℓ = U ℓ , with 0 = .Then there is a nested chain of finite index subgroups 2B. Profinite completion.Let ( ) ⊂ Homeo(X) denote the image subgroup for an action (X, , ).When the action is equicontinuous, the closure ( ) ⊂ Homeo(X) in the uniform topology of maps is a separable profinite group.We adopt the notation G( ) ≡ ( ).
Let : G( ) × X → X denote the induced action of G( ) on X.For ĝ ∈ G( ), we write its action on X by ĝ • x = ( ĝ)(x).Since the action (X, , ) is minimal, the action of on X is transitive; that is, for all x ∈ X, the orbit which is a closed subgroup of G( ), and thus is either finite or is an infinite profinite group.As the action : G( )×X → X is transitive, the conjugacy class of D( , x) in G( ) is independent of the choice of x, and by abuse of notation we omit the subscript x.The group D( ) is called the discriminant of the action (X, , ) in [11; 15; 17] and is called a parabolic subgroup (of the profinite completion of a countable group) in the works by Bartholdi and Grigorchuk [4;5].
2C. Algebraic Cantor actions.We next describe the algebraic construction of Cantor actions, starting with a group chain in a given group , and then deriving the Cantor action from this data.This is often the most versatile method of constructing examples of Cantor actions with specific properties.
Let G = { = 0 ⊃ 1 ⊃ 2 ⊃ • • • } be a properly descending chain of finite index subgroups.Let X ℓ = / ℓ and note that acts transitively on the left on the finite set X ℓ .The inclusion ℓ+1 ⊂ ℓ induces a natural -invariant quotient map p ℓ+1 : X ℓ+1 → X ℓ .Introduce the inverse limit Then X ∞ is a Cantor space with the Tychonoff topology, where the actions of on the factors X ℓ induce a minimal equicontinuous action ∞ : × X ∞ → X ∞ .There is a natural basepoint x ∞ ∈ X ∞ given by the cosets of the identity element e ∈ , so x ∞ = (e ℓ ).An adapted neighborhood basis of x ∞ is given by the clopen sets (4) There is a tautological identity ℓ = V ℓ where V ℓ is the isotropy group as defined by Corollary 2.5.Given a minimal equicontinuous Cantor action : × X → X and an adapted neighborhood basis U We can then associate to this group chain an algebraic action ∞ : × X ∞ → X ∞ as above.
For each ℓ ≥ 0, we have the "partition coding map" ℓ : X → X ℓ which isequivariant.The maps { ℓ } are compatible with the map on quotients in (3), and so they induce a limit map x : X → X ∞ .The fact that the diameters of the clopen sets The following is folklore: The action (X ∞ , , ∞ ) is called the odometer model centered at x for the action (X, , ).The dependence of the model on the choices of a base point x ∈ X and adapted neighborhood basis U is discussed in detail in the works [10; 12; 15; 17].
Next, we develop the algebraic model for the profinite action : G( ) × X → X of the completion G( ) ≡ ( ) ⊂ Homeo(X).Choose a group chain { ℓ | ℓ ≥ 0} as above, which provides an algebraic model for the action (X, , ).
For each ℓ ≥ 1, let C ℓ ⊂ ℓ denote the core of ℓ , i.e., the largest normal subgroup of ℓ in .So ( 5) As ℓ has finite index in , the same holds for C ℓ .Observe that for all ℓ ≥ 0, we have Introduce the quotient group Q ℓ = /C ℓ with identity element e ℓ ∈ Q ℓ .There are natural quotient maps q ℓ+1 : Q ℓ+1 → Q ℓ , and we can form the inverse limit group which is a Cantor space with the Tychonoff topology.The left actions of on the spaces X ℓ = / ℓ induce a minimal equicontinuous action of ∞ on X ∞ , again denoted by : ∞ × X ∞ → X ∞ .Note that the isotropy group of the action of There is a natural basepoint ê∞ ∈ ∞ given by the cosets of the identity element e ∈ , so ê∞ = (e ℓ ) where For each ℓ ≥ 0, let ℓ : ∞ → Q ℓ denote the projection onto the ℓ-th factor in (6), so in the coordinates of (7), we have ℓ ( ĝ) = g ℓ ∈ Q ℓ .The maps ℓ are continuous for the profinite topology on ∞ , so the preimages of points in Q ℓ are clopen subsets.In particular, the fiber of ℓ : ∞ → Q ℓ over e ℓ is the normal subgroup ( 8) There is an isomorphism τ : G( ) → ∞ such that τ conjugates the profinite action (X, G( ), ) with the profinite action (X ∞ , ∞ , ∞ ).In particular, τ identifies the isotropy group D( ) with the inverse limit subgroup The maps q ℓ+1 in the formula (9) need not be surjections, and thus the calculation of the inverse limit D ∞ can involve some subtleties.For example, it is possible that each group Q ℓ is nontrivial for ℓ > 0, and yet D ∞ is the trivial group.
2D. Equivalence of Cantor actions.We next recall the notions of equivalence of Cantor actions.The first and strongest is that of isomorphism, which is a generalization of the notion of conjugacy of topological actions.For = ‫,ޚ‬ isomorphism corresponds to the notion of "flip conjugacy" introduced in the work of Boyle and Tomiyama [7].The definition below also appears in the papers [8; 15; 23].Definition 2.8.Cantor actions (X 1 , 1 , 1 ) and (X 2 , 2 , 2 ) are said to be isomorphic if there is a homeomorphism h : X 1 → X 2 and a group isomorphism The notion of return equivalence for Cantor actions is weaker than isomorphism and is natural when considering the dynamical properties of Cantor systems which should be independent of the restriction of the action to a clopen cross-section.
Given a minimal equicontinuous Cantor action (X, , ) and an adapted set U ⊂ X, recall that U denotes the isotropy group for U , as in (1).By a small abuse of notation, we use U to denote both the restricted action U : U × U → U and the induced quotient action U : Then (U, H U , U ) is called the holonomy action for .Definition 2.9.Two minimal equicontinuous Cantor actions (X 1 , 1 , 1 ) and (X 2 , 2 , 2 ) are return equivalent if there exists an adapted set U 1 ⊂ X 1 for the action 1 and an adapted set U 2 ⊂ X 2 for the action 2 , such that the holonomy actions If the actions 1 and 2 are isomorphic in the sense of Definition 2.8, then they are return equivalent with U 1 = X 1 and U 2 = X 2 .However, the notion of return equivalence is weaker even for this case, as the conjugacy is between the holonomy groups H 1,X 1 and H 2,X 2 , and not the groups 1 and 2 .A topological action (X, , ) on a metric Cantor space X is locally quasianalytic (LQA) if there exists ε > 0 such that for any nonempty open set U ⊂ X with diam(U ) < ε, and for any nonempty open subset V ⊂ U , and elements g 1 , g 2 ∈ , (11) if The action is said to be quasianalytic if (11) holds for U = X.
In other words, (X, , ) is locally quasianalytic if for every g ∈ , the homeomorphism (g) has unique extensions on the sets of diameter ε > 0 in X, with ε uniform over X.We note that for a countable group , an effective action (X, , ) is topologically free if and only if it is quasianalytic.
Recall that a group is Noetherian [2] if every increasing chain of subgroups has a maximal element.Equivalently, a group is Noetherian if every subgroup of is finitely generated.A group is topologically Noetherian if every increasing chain of closed subgroups has a maximal element; see Section 3 for details.
A finitely generated nilpotent group is Noetherian, so as a corollary we obtain that all Cantor actions by finitely generated nilpotent groups are locally quasianalytic.
The notion of a locally quasianalytic Cantor action extends to the case of a profinite group action : G × X → X. Definition 2.12.Let (X, , ) be a Cantor action and : G × X → X the induced profinite action.We say that the action is stable if the induced profinite action (X, G( ), ) is locally quasianalytic, and we say it is wild otherwise.
A profinite completion G of a Noetherian group need not be Noetherian, as can be seen for the example of = ‫,ޚ‬ where G is the full profinite completion of ‫.ޚ‬More generally, a finitely generated nilpotent group is always Noetherian, while Proposition 3.4 gives an "if and only if" condition for a profinite completion G of to be topologically Noetherian.
2F. Type and typeset for Cantor actions.A Steinitz number ξ can be written uniquely as the formal product over the set of primes : where the characteristic function χ ξ : → {0, 1, . . ., ∞} counts the multiplicity with which a prime p appears in the infinite product ξ .In terms of their characteristic functions χ 1 , χ 2 , we have ξ a ∼ ξ ′ if and only if the following conditions are satisfied: • χ 1 ( p) = χ 2 ( p) for all but finitely many primes p ∈ .
Next, we define the type of a Cantor action (X ∞ , , ∞ ) defined by a chain of finite index subgroups, Definition 2.15.Let (X ∞ , , ) be a minimal equicontinuous Cantor action defined by a group chain G.The type τ [X ∞ , , ∞ ] of the action is the equivalence class of the Steinitz order Finally, we note the following result: Theorem 2.16 [20, Theorem 1.9].Let (X, , ) be a Cantor action.The Steinitz order ξ(X, , ) is defined as the Steinitz order for an algebraic model (X ∞ , , ∞ ) of the action, which does not depend upon the choice of an algebraic model.The type τ [X, , ] depends only on the return equivalence class of the action.
2G. Type for profinite groups.The Steinitz order [G] of a profinite group G is defined by the supernatural number associated to a presentation of G as an inverse limit of finite groups (see [ Definition 2.17.Let (X, , ) be a minimal equicontinuous Cantor action, with choice of a basepoint x ∈ X.The Steinitz orders of the action are defined as The Steinitz orders satisfy the Lagrange identity, where the multiplication is taken in the sense of supernatural numbers, (14) ξ(G( )) = ξ(G( ) : D( )) • ξ(D( )).

Nilpotent actions
We apply the notion of the Steinitz order of a nilpotent Cantor action to the study of its dynamical properties.The proof of Theorem 1.2 is based on the special properties of the profinite completions of nilpotent groups, in particular the uniqueness of their Sylow p-subgroups, and on the relation of this algebraic property with the dynamics of the action.
3A. Noetherian groups.A countable group is said to be Noetherian [2] The closed subgroups of ‫ޚ‬ p are given by p i • ‫ޚ‬ p for some fixed i > 0, and hence satisfy the ascending chain property in Definition 3.1.A profinite group G is pro-nilpotent if it is the inverse limit of finite nilpotent groups.For example, if G is a profinite completion of a nilpotent group , then G is pro-nilpotent.
The group G is topologically finitely generated if it contains a dense subgroup ⊂ G where is finitely generated.The completion G( ) associated to a Cantor action (X, , ) with finitely generated is topologically finitely generated.
Assume that G is pro-nilpotent.Then for each prime p, there is a unique Sylow p-subgroup of G, which is normal in G (see [29,Proposition 2.4.3]).Denote this group by G ( p) .Also, G ( p) is nontrivial if and only if p ∈ π(ξ(G)).We use the following result for pro-nilpotent groups, which is a consequence of [29,Proposition 2.4.3].Proposition 3.5.Let G be a profinite completion of a finitely generated nilpotent group .Then there is a topological isomorphism From the isomorphism ( 16) it follows immediately that if the prime spectrum π(ξ(G)) is infinite, then G is not topologically Noetherian.To see this, list π(ξ(G)) = { p i | i = 1, 2, . ..}.Then we obtain an infinite strictly increasing chain of closed subgroups If the prime spectrum π(ξ(G)) is finite, then the isomorphism ( 16) reduces the proof that G is topologically Noetherian to the case of showing that if G is topologically finitely generated, then each of its Sylow p-subgroups is Noetherian.The group G ( p) is nilpotent and topologically finitely generated, so we can use the lower central series for G ( p) and induction to reduce to the case where H is a topologically finitely generated abelian pro-p-group, and so is isomorphic to a finite product of p-completions of ‫,ޚ‬ which are topologically Noetherian.
Observe that a profinite completion G of a finitely generated nilpotent group is a topologically finitely generated nilpotent group, and we apply the above remarks.□ Corollary 3.6.Let be a virtually nilpotent group; that is, there exists a finitely generated nilpotent subgroup 0 ⊂ of finite index.Then a profinite completion G of is topologically Noetherian if and only if its prime spectrum π(ξ(G)) is finite.
Proof.We can assume that 0 is a normal subgroup of .Thus, its closure G 0 ⊂ G satisfies the hypotheses of Proposition 3. We follow the outline of its proof in [16].
Proposition 3.7.Let G be a topologically Noetherian group.Then a minimal equicontinuous action (X, G, ) on a Cantor space X is locally quasianalytic.
Proof.The closure G( ) is contained in Homeo(X), so the action of G( ) is effective.Suppose that the action is not locally quasianalytic.Then there exists an infinite properly decreasing chain of clopen subsets of X, {U 1 ⊃ U 2 ⊃ • • • }, which satisfy, for all ℓ ≥ 1, the properties • U ℓ is adapted to the action with isotropy subgroup G U ℓ ⊂ G; • there is a closed subgroup K ℓ ⊂ G U ℓ+1 whose restricted action to U ℓ+1 is trivial, but the restricted action of K ℓ to U ℓ is effective.
Hence, we obtain a properly increasing chain of closed subgroups Let (X, , ) be a nilpotent Cantor action, and we are given that the prime spectrum π(ξ(G( ))) is finite.Then there exists a finitely generated nilpotent subgroup 0 ⊂ of finite index, and we can assume without loss of generality that 0 is normal.Let G( ) 0 be the closure of 0 in G( ).The group G( ) has finite prime spectrum implies that the group G( ) 0 has finite prime spectrum, and thus by Proposition 3.4 the group G( ) 0 is topologically Noetherian.Let x ∈ X.Then it suffices to show that the action of 0 on the orbit X 0 = G( ) 0 • x is stable.This reduces the proof to showing the claim when is nilpotent.Then the profinite closure G( ) is also nilpotent, and we have a profinite action (X, G( ), ).
Suppose that the action is not locally quasianalytic.Then there exists an increasing chain of closed subgroups K ℓ ⊂ D( ) where K ℓ acts trivially on the clopen subset U ℓ ⊂ X.As D( ) is a closed subgroup of G( ), the increasing chain {K ℓ | ℓ > 0} consists of closed subgroups of G( ).This contradicts the fact that G( ) is topologically Noetherian.Hence, the action must be locally quasianalytic.That is, the action (X, , ) is stable.□

Basic examples
We construct two basic examples of nilpotent Cantor actions.These examples illustrate the principles behind the subsequent more complex constructions in Section 5, which are used to prove Theorems 1.3 and 1.4.
The integer Heisenberg group is the simplest nonabelian nilpotent group, and it can be represented as the upper triangular matrices in GL(3, ‫.)ޚ‬That is, We denote a 3×3 matrix in by the coordinates as (a, b, c).
Example 4.1.A renormalizable Cantor action, as defined in [21], can be constructed from the group chain defined by a proper self-embedding of a group into itself.For a prime p ≥ 2, define the self-embedding ϕ p : → by ϕ(a, b, c) = ( pa, pb, p 2 c).Then define a group chain in by setting For ℓ > 0, the normal core for ℓ is given by and so the quotient group is given by The profinite group ∞ is the inverse limit of the quotient groups Q ℓ so we have Even though the quotient groups ℓ /C ℓ are all nontrivial, for this action the inverse limit D ∞ is the trivial group.This follows from the fact that there are inclusions The triviality of D ∞ implies that there is an equivalent group chain for the action [10] which can be chosen so that every subgroup in the chain is normal in .
Example 4.2.For distinct primes p, q ≥ 2, define the self-embedding ϕ p,q : → by ϕ(a, b, c) = ( pa, qb, pqc).Then define a group chain in by setting For ℓ > 0, the normal core for ℓ is given by and so the quotient group is given by The profinite group ∞ is the inverse limit of the quotient groups Q ℓ , so we have , and D ∞ is the inverse limit of the finite groups ℓ /C ℓ by ( 9), so D ∞ ∼ = ‫ޚ‬ q × ‫ޚ‬ p .

Nilpotent actions with prescribed spectrum
We construct stable actions of the discrete Heisenberg group with prescribed prime spectrum, proving Theorem 1.3.Then we construct examples of wild nilpotent Cantor actions, proving Theorem 1.4, from which we deduce Corollary 1.5.For simplicity, our examples all use the Heisenberg group represented by 3×3 matrices.Of course, these examples can be generalized to the integer upper triangular matrices in all dimensions, where there is much more freedom in the choices made in the construction.The calculations become correspondingly more tedious, but yield analogous results.It seems reasonable to expect that similar constructions can be made for any finitely generated torsion-free nilpotent (nonabelian) group .That is, there are always group chains in which yield wild nilpotent Cantor actions.
Let ⊂ GL(3, ‫)ޚ‬ denote the discrete Heisenberg group, given by formula (17).The basis for the constructions below is the structure theory for nilpotent group completions in Proposition 3.5, in particular the formula (16).Given sets of primes π f and π ∞ , we embed an infinite product of finite actions, as in Section 5A, into a profinite completion ∞ of , and thus obtain a nilpotent Cantor action 5A. Basic components of the construction.Fix a prime p ≥ 2.
For n ≥ 1 and 0 ≤ k < n, we have the finite groups That is, the adjoint action of x on the "plane" in the ( ȳ, z)-coordinates is a "shear" action along the z-axis, and the adjoint action of x on the z-axis fixes all points on the z-axis.Enumerate π f = {q 1 , q 2 , . . ., q m }, and then choose integers 1 ≤ r i ≤ n i for 1 ≤ i ≤ m.
Enumerate π ∞ = { p 1 , p 2 , . ..} with the convention (for notational convenience) that if ℓ is greater than the number of primes in π ∞ then we set p ℓ = 1.For each ℓ ≥ 1, define the integers For all ℓ ≥ 1, observe that M ℓ divides N ℓ .Define a subgroup of the Heisenberg group , in the coordinates in (17), By Proposition 3.5, and in the notation of Section 5A, we have for Then the Cantor space X ∞ = ∞ /D ∞ associated to the group chain { ℓ | ℓ ≥ 1} is given by ( 23) In particular, as the first factor in ( 23) is a finite product of finite sets, the second factor defines an open neighborhood where x i ∈ X q i ,n i ,k i is the basepoint given by the coset of the identity element.That is, U is a clopen neighborhood of the basepoint in X ∞ .The isotropy group of U is given by ( 24) The restriction of ∞ |U to U is isomorphic to the subgroup ( 25) where ēi ∈ G q i ,n i is the identity element.The group K |U acts freely on U , and thus the action of ∞ on X ∞ is locally quasianalytic.The prime spectrum of the action of on X ∞ is the union π = π f ∪π ∞ = π(ξ( ∞ )).If π ∞ is infinite, then the prime spectrum of the action is infinite.Note that the group embeds into ∞ , since the integers M ℓ and N ℓ tend to infinity with ℓ.This completes the proof of Theorem 1.3.
5C. Wild nilpotent actions with infinite prime spectrum.We prove Theorem 1.4.We must show that every infinite set of primes can be realized as the prime spectrum of a wild action of the Heisenberg group , as defined by (17).Let π f and π ∞ be disjoint collections of primes, with π f an infinite set and π ∞ arbitrary, possibly empty.Enumerate π f = {q 1 , q 2 , . ..} and choose integers 1 ≤ r i < n i for 1 ≤ i < ∞.Enumerate π ∞ = { p 1 , p 2 , . ..}, again with the convention that if ℓ is greater than the number of primes in π ∞ then we set p ℓ = 1.
As in Section 5B, for each ℓ ≥ 1, define the integers For ℓ ≥ 1, define a subgroup of the Heisenberg group , in the coordinates in (17), The first factor in ( 23) is an infinite product of finite sets, so fixing the first ℓ coordinates in this product determines a clopen subset of X ∞ .Let x i ∈ X q i ,n i ,k i denote the coset of the identity element, which is the basepoint in X q i ,n i ,k i .Then for each ℓ ≥ 1, we define a clopen set in X ∞ by ( 29) By calculations in Section 5A, the subgroup H q i ,n i ,k i is the isotropy group of the basepoint x i ∈ X q i ,n i ,k i .Thus, the isotropy subgroup of U ℓ for the ∞ -action is given by For j ̸ = i, the subgroup H q i ,n i ,k i acts as the identity on the factors X q j ,n j ,k j in (28).Thus, the image of ∞ | U ℓ in Homeo(U ℓ ) is isomorphic to the subgroup where ēi ∈ G q i ,n i is the identity element.We next show that this action is not stable; that is, for any ℓ > 0 there exists a clopen subset V ⊂ U ℓ and nontrivial ĝ ∈ Z ℓ so that the action of ∞ restricts to the identity map on V .
We can assume without loss of generality that V = U ℓ ′ for some ℓ ′ > ℓ.Consider the restriction map for the isotropy subgroup of Z ℓ to U ℓ ′ which is given by We must show that there exists ℓ ′ > ℓ such that this map has a nontrivial kernel.Calculate this map in terms of the product representations above: For ℓ < i ≤ ℓ ′ , the group H q i ,n i ,k i fixes the point ℓ ′ i=1 {x i }, and acts trivially on ∞ i=ℓ ′ +1 X q i ,n i ,k i .Thus, the kernel of the restriction map contains the second factor in (32): (33) As this group is nontrivial for all ℓ ′ > ℓ, the action of ∞ on X ∞ is not locally quasianalytic, and hence the action of on X ∞ is wild.Also, the prime spectrum of the action of on X ∞ equals the union π = π f ∪ π ∞ .
We now prove the second part of Theorem 1.4, showing that choices in the construction above can be made in such a way that the action of on a Cantor set is topologically free while the action of ∞ is wild, and the prime spectrum is prescribed.
Choose the constants as in Section 5A, with n i = 2 and k i = 1 for all i ≥ 1. Define the Cantor space X ∞ by (28), where the second factor is trivial; that is, a point.The action of ∞ is wild by the calculations in formulas (30) to (33).
We claim that the action of on X ∞ is topologically free.If not, then there exists an open set U ⊂ X ∞ and g ∈ such that the action of ∞ (g) is nontrivial on X ∞ but leaves the set U invariant and restricts to the identity action on U .The action of on X ∞ is minimal, so there exists h ∈ with h • x ∞ ∈ U .Then ∞ (h −1 gh)(x ∞ ) = x ∞ and the action ∞ (h −1 gh) fixes an open neighborhood of x ∞ .Replacing g with h −1 gh we can assume that ∞ (g)(x ∞ ) = x ∞ ∈ U .From the definition (29), the clopen sets (34) form a neighborhood basis at x ∞ , and thus there exists ℓ > 0 such that U ℓ ⊂ U .The group embeds into ∞ along the diagonal in the product (16).That is, we can write g = (g, g, . ..) ∈ ∞ i=1 G q i ,2 .The action of ∞ (g) is factorwise, and The assumption that ∞ (g) fixes the points in U implies that it acts trivially on each factor X q i ,2,1 for i > ℓ.As each factor H q i ,2,1 acts effectively on X q i ,2,1 this implies that the projection of g to the i-th factor group H q i ,2,1 is the identity for i > ℓ.This implies that every entry above the diagonal in the matrix representation of g in ( 17) is divisible by an infinite number of distinct primes {q i | i ≥ ℓ}, so by the prime factorization theorem the matrix g is the identity.
Alternatively, observe that we have g ∈ ℓ i=1 H q i ,2,1 .This is a finite product of finite groups, which implies that g ∈ is a torsion element.However, the Heisenberg group is torsion-free, and hence g must be the identity.Thus, the action of on X ∞ must be topologically free.
Finally, the above construction allows the choice of any infinite subset π f of distinct primes, and there are an uncountable number such choices which are distinct.Thus, by Theorem 1.9 in [20] there are an uncountable number of topologically free, wild nilpotent Cantor actions with distinct prime spectrum.This completes the proof of Theorem 1.4.
5D. Proof of Corollary 1.5.Consider the family of wild topologically free actions on the Heisenberg group with infinite distinct prime spectrum, as constructed at the end of Section 5C.We show that the uncountable number of infinite choices of π f in this family can be made so that the actions have pairwise disjoint types.
By Definition 2.13, for two Steinitz numbers ξ and ξ ′ we have that their types are equal, τ (ξ ) = τ (ξ ′ ), if and only if there exist integers m, m ′ such that m • ξ = m ′ • ξ ′ .Thus two actions with prime spectra π f and π ′ f have distinct types if and only if π f and π ′ f differ by an infinite number of entries.This happens, for instance, if π f and π ′ f are almost disjoint infinite sets, i.e., they are infinite sets with finite intersection.The set of prime numbers is countable, so the family of infinite almost disjoint subsets of prime numbers is uncountable if and only if the family of infinite almost disjoint subsets of natural numbers is uncountable.The family of almost disjoint subsets of natural numbers is uncountable by [13,Corollary 2.3].Since the set of finite subsets of natural numbers is countable, the set of almost disjoint infinite subsets of natural numbers is uncountable.
It follows that the prime spectra of the uncountable family of actions of the Heisenberg group in Theorem 1.4 can be chosen so that they form a family of almost disjoint infinite sets.Then their types are pairwise distinct, and by Theorem 2.16 these actions of the Heisenberg group are pairwise not return equivalent.Therefore, they are pairwise not conjugate.

2E.
Locally quasianalytic.The quasianalytic property for Cantor actions was introduced by Álvarez López and Candel in [24, Definition 9.4] as a generalization of the notion of a quasianalytic action studied by Haefliger for actions of pseudogroups of real-analytic diffeomorphisms.The authors introduced a local form of the quasianalytic property in [11; 15]: Definition 2.10 [15, Definition 2.1].

Definition 2 . 13 .
Two Steinitz numbers ξ and ξ ′ are said to be asymptotically equivalent if there exists finite integers m, m ′ ≥ 1 such that m • ξ = m ′ • ξ ′ , and we then write ξ a ∼ ξ ′ .A type is an asymptotic equivalence class of Steinitz numbers.The type associated to a Steinitz number ξ is denoted by τ [ξ ].
Definition 3.1[29, page 153].A profinite group G is said to be topologically Noetherian if every increasing chain of closed subgroups {H i | i ≥ 1} of G has a maximal element H i 0 .Let ‫ޚ‬ p denote the p-adic integers, for p a prime.That is, ‫ޚ‬ p is the completion of ‫ޚ‬ with respect to the chain of subgroups G if every increasing chain of subgroups {H i | i ≥ 1} of has a maximal element H i 0 .The group ‫ޚ‬ is Noetherian; a finite product of Noetherian groups is Noetherian; and a subgroup and quotient group of a Noetherian group is Noetherian.Thus, a finitely generated nilpotent group is Noetherian.The notion of a Noetherian group has a generalization which is useful for the study of actions of profinite groups.
be the direct sum of the first ℓ factors.Then {H ℓ | ℓ ≥ 1} is an increasing chain of subgroups of ‫ޚ‬ π which does not stabilize, so ‫ޚ‬ π is not topologically Noetherian.These two examples illustrate the idea behind the proof of the following result.For a prime p, a finite group H is a p-group if every element of H has order a power of p.A profinite group H is a pro-p-group if H is the inverse limit of finite p-groups.A Sylow p-subgroup H ⊂ G is a maximal pro-p-subgroup [29, Definition 2.2.1].
4, and the Steinitz orders satisfy ξ(G 0 ) As G 0 is topologically Noetherian if and only if G is topologically Noetherian, the claim follows.□ 3B.Dynamics of Noetherian groups.We relate the topologically Noetherian property of a profinite group with the dynamics of a Cantor action of the group to obtain the proof of Theorem 1.2.We first give the profinite analog of [16, Theorem 1.6].
Set X p,n,k = G p,n /H p,n,k .Then the isotropy group of the action of G p,n on X p,n,k at the coset H p,n,k of the identity element is H p,n,k .The core subgroup C p,n,k ⊂ H p,n,k contains elements in H p,n,k which fix every point in X p,n,k .The action of x ∈ H p,n,k on the coset space X p,n,k satisfies (18)∞ ( x)( ȳ H p,n,k ) = ȳ z H p,n,k , so the identity is the only element in G p,n which acts trivially on every coset in X p,n,k , so C p,n,k is the trivial group.Then D p,n,k = H p,n,k /C p,n,k = H p,n,k , and for each g ∈ H p,n,k its action fixes the cosets of the multiples of z. 5B.Stable nilpotent actions with finite or infinite prime spectrum.We now prove Theorem 1.3 by constructing a family of stable examples with prescribed prime spectra.Let π f and π ∞ be two disjoint collections of primes, with π f a finite set and π ∞ a nonempty set.