# Prime spectrum and dynamics for nilpotent Cantor actions

Steven Hurder  Steven Hurder, Department of Mathematics, University of
Illinois at Chicago, 322 SEO (m/c 249), 851 S. Morgan Street, Chicago, IL
60607-7045<EMAIL_ADDRESS>and  Olga Lukina  Olga Lukina, Mathematical
Institute, Leiden University, PO Box 9512, 2300 RA Leiden, The Netherlands
<EMAIL_ADDRESS>
###### Abstract.

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.

Version date: May 1, 2023

2020 Mathematics Subject Classification. Primary: 20E18, 37B05, 37B45;
Secondary: 57S10

Keywords: odometers, Cantor actions, profinite groups, Steinitz numbers,
Heisenberg group

##  1\. Introduction

A minimal equicontinuous action
$\Phi\colon\Gamma\times{\mathfrak{X}}\to{\mathfrak{X}}$ of a countable group
$\Gamma$ on a Cantor space ${\mathfrak{X}}$ is called a _generalized odometer_
[9, 15]. When $\Gamma={\mathbb{Z}}$, this is just the abstract form of a
traditional odometer action of the integers. For $\Gamma={\mathbb{Z}}^{n}$
with $n\geq 2$, 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
${\mathbb{Z}}^{n}$ are well-behaved. For $\Gamma$ 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 ${\mathbb{Z}}$-odometer 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 ${\mathbb{Z}}^{n}$ its Steinitz order, and also a collection
of types, called its _typeset_ , which consists of equivalence classes of
Steinitz orders of individual elements of ${\mathbb{Z}}^{n}$. 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
${\mathbb{Z}}^{n}$ to a standard Borel equivalence relation.

In the authors’ work [20], we associate the _type_ and _typeset_ invariants to
a Cantor action $({\mathfrak{X}},\Gamma,\Phi)$ for arbitrary countable group
$\Gamma$. The type $\tau[{\mathfrak{X}},\Gamma,\Phi]$ is the asymptotic
equivalence class of the Steinitz order $\xi({\mathfrak{X}},\Gamma,\Phi)$ of a
presentation of the action as an inverse limit of actions of $\Gamma$ on
finite sets.

Associated to the type $\tau[{\mathfrak{X}},\Gamma,\Phi]$ is an even more
basic invariant, the _prime spectrum_ $\pi[{\mathfrak{X}},\Gamma,\Phi]$, which
consists of the set of primes which appear in a Steinitz order
$\xi({\mathfrak{X}},\Gamma,\Phi)$ representing the type
$\tau[{\mathfrak{X}},\Gamma,\Phi]$, see Section 2.14. The prime spectrum
decomposes into two parts,

$\pi[{\mathfrak{X}},\Gamma,\Phi]=\pi_{\infty}[{\mathfrak{X}},\Gamma,\Phi]\cup\pi_{f}[{\mathfrak{X}},\Gamma,\Phi],$

where the _infinite prime spectrum_ $\pi_{\infty}[{\mathfrak{X}},\Gamma,\Phi]$
consists of the primes which occur with infinite multiplicity in
$\xi({\mathfrak{X}},\Gamma,\Phi)$, and the _finite prime spectrum_
$\pi_{f}[{\mathfrak{X}},\Gamma,\Phi]$ which 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
$\pi_{f}[{\mathfrak{X}},\Gamma,\Phi]$.

######  DEFINITION 1.1.

A Cantor action $({\mathfrak{X}},\Gamma,\Phi)$ has _finite spectrum_ if the
prime spectrum $\pi[{\mathfrak{X}},\Gamma,\Phi]$ is a finite set, and is said
to have _infinite spectrum_ otherwise.

The classification of Cantor actions for $\Gamma$ is, in general, intractable
and one seeks invariants for Cantor actions which at least distinguish between
particular classes of actions. The authors’ works [16, 17, 18, 19] study
dynamical properties which yield invariants of Cantor actions. In particular,
one of the most basic invariants, is the property that the action is _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 2.5 below, the property that the action
$({\mathfrak{X}},\Gamma,\Phi)$ is stable is a property of the action of the
completion ${\mathfrak{G}}(\Phi)=\overline{\Phi(\Gamma)}\subset{\rm
Homeo}({\mathfrak{X}})$, which is a profinite group naturally acting on
${\mathfrak{X}}$. The property that the action $({\mathfrak{X}},\Gamma,\Phi)$
is locally quasi-analytic is defined in Definition 2.10, and
$({\mathfrak{X}},\Gamma,\Phi)$ is _stable_ if the action of
${\mathfrak{G}}(\Phi)$ on ${\mathfrak{X}}$ is also locally quasi-analytic. If
$({\mathfrak{X}},\Gamma,\Phi)$ is stable, then $({\mathfrak{X}},\Gamma,\Phi)$
is locally quasi-analytic. The converse need not hold even for actions of
nilpotent groups, as we see later.

A Cantor action $({\mathfrak{X}},\Gamma,\Phi)$ is said to be _nilpotent_ if
$\Gamma$ 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 $({\mathfrak{X}},\Gamma,\Phi)$ be a nilpotent Cantor action. If the prime
spectrum $\pi[{\mathfrak{X}},\Gamma,\Phi]$ 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 $\pi_{f}$ and $\pi_{\infty}$ be two distinct sets of primes, where
$\pi_{f}$ is a finite set, and $\pi_{\infty}$ is a non-empty finite or
infinite set. Then there exists a stable nilpotent Cantor action
$({\mathfrak{X}},\Gamma,\Phi)$ such that
$\pi_{\infty}[{\mathfrak{X}},\Gamma,\Phi]=\pi_{\infty}$ and
$\pi_{f}[{\mathfrak{X}},\Gamma,\Phi]=\pi_{f}$.

Let $({\mathfrak{X}},\Gamma,\Phi)$ be an Cantor action. If the action is
effective, then it is free, and the action of the closure
${\mathfrak{G}}(\Phi)$ 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
${\mathfrak{X}}$. The authors showed in their work [17] that nilpotent Cantor
actions are locally quasi-analytic, 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 ${\mathfrak{X}}$. It is then surprising to
discover that if one allows ${\mathfrak{G}}(\Phi)$ to have infinite prime
spectrum then one can construct wild nilpotent actions, for which the action
of the closure ${\mathfrak{G}}(\Phi)$ is not locally quasi-analytic, 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 $\pi_{f}$ and $\pi_{\infty}$ of primes, where
$\pi_{f}$ is infinite and $\pi_{\infty}$ is any (possibly empty) set, there is
a minimal equicontinuous action $({\mathfrak{X}},\Gamma,\Phi)$ of the
Heisenberg group, such that $\pi_{f}[{\mathfrak{X}},\Gamma,\Phi]=\pi_{f}$ and
$\pi_{\infty}[{\mathfrak{X}},\Gamma,\Phi]=\pi_{\infty}$.

Moreover, there exists an uncountable number of nilpotent Cantor actions
$({\mathfrak{X}},\Gamma,\Phi)$ of the Heisenberg group $\Gamma$ with infinite
prime spectra such that:

  1. (1)

Each $({\mathfrak{X}},\Gamma,\Phi)$ is topologically free,

  2. (2)

Each $({\mathfrak{X}},\Gamma,\Phi)$ is wild,

  3. (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 2.4. 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
$({\mathfrak{X}},\Gamma,\Phi)$ of the Heisenberg group $\Gamma$ which are not
return equivalent, and therefore not conjugate.

The conclusion of Theorem 1.4 is used in [21] for the calculation of the
mapping class groups of solenoidal manifolds whose base is a nil-manifold.

We note that for more general groups $\Gamma$, an analog of Theorem 1.2 need
not hold. For example, a weakly branch group, as studied in [4, 6, 7, 25],
acts on the boundary of a $d$-regular rooted tree, and so has finite prime
spectrum $\\{d\\}$, but the dynamic of the actions on the Cantor boundary are
wild.

######  QUESTION 1.6.

Let $({\mathfrak{X}},\Gamma,\Phi)$ be a Cantor action. For which classes of
groups $\Gamma$ does the finiteness of the prime spectrum of the action imply
that the action is stable?

The paper is organized as follows. In Section 2.1 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.

##  2\. 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 [2], the
papers by Cortez and Petite [9], Cortez and Medynets [10], and the authors’
works, in particular [11, 12] and [18, Section 3].

###  2.1. Basic concepts

Let $({\mathfrak{X}},\Gamma,\Phi)$ denote an action
$\Phi\colon\Gamma\times{\mathfrak{X}}\to{\mathfrak{X}}$. We write $g\cdot x$
for $\Phi(g)(x)$ when appropriate. The orbit of $x\in{\mathfrak{X}}$ is the
subset ${\mathcal{O}}(x)=\\{g\cdot x\mid g\in\Gamma\\}$. The action is
_minimal_ if for all $x\in{\mathfrak{X}}$, its orbit ${\mathcal{O}}(x)$ is
dense in ${\mathfrak{X}}$.

An action $({\mathfrak{X}},\Gamma,\Phi)$ is _equicontinuous_ with respect to a
metric $d_{{\mathfrak{X}}}$ on ${\mathfrak{X}}$, if for all ${\varepsilon}>0$
there exists $\delta>0$, such that for all $x,y\in{\mathfrak{X}}$ and
$g\in\Gamma$ we have that $\displaystyle d_{{\mathfrak{X}}}(x,y)<\delta$
implies $d_{{\mathfrak{X}}}(g\cdot x,g\cdot y)<{\varepsilon}$. The property of
being equicontinuous is independent of the choice of the metric on
${\mathfrak{X}}$ which is compatible with the topology of ${\mathfrak{X}}$.

Now assume that ${\mathfrak{X}}$ is a Cantor space. Let ${\rm
CO}({\mathfrak{X}})$ denote the collection of all clopen (closed and open)
subsets of ${\mathfrak{X}}$, which forms a basis for the topology of
${\mathfrak{X}}$. For $\phi\in{\rm Homeo}({\mathfrak{X}})$ and $U\in{\rm
CO}({\mathfrak{X}})$, the image $\phi(U)\in{\rm CO}({\mathfrak{X}})$. The
following result is folklore, and a proof is given in [17, Proposition 3.1].

######  PROPOSITION 2.1.

For ${\mathfrak{X}}$ a Cantor space, a minimal action
$\Phi\colon\Gamma\times{\mathfrak{X}}\to{\mathfrak{X}}$ is equicontinuous if
and only if the $\Gamma$-orbit of every $U\in{\rm CO}({\mathfrak{X}})$ is
finite for the induced action $\Phi_{*}\colon\Gamma\times{\rm
CO}({\mathfrak{X}})\to{\rm CO}({\mathfrak{X}})$.

######  DEFINITION 2.2.

We say that $U\subset{\mathfrak{X}}$ is _adapted_ to the action
$({\mathfrak{X}},\Gamma,\Phi)$ if $U$ is a _non-empty clopen_ subset, and for
any $g\in\Gamma$, if $\Phi(g)(U)\cap U\neq\emptyset$ implies that
$\Phi(g)(U)=U$.

The proof of [17, Proposition 3.1] shows that given $x\in{\mathfrak{X}}$ and
clopen set $x\in W$, there is an adapted clopen set $U$ with $x\in U\subset
W$.

For an adapted set $U$, the set of “return times” to $U$,

(1) $\Gamma_{U}=\left\\{g\in\Gamma\mid g\cdot U\cap U\neq\emptyset\right\\}$

is a subgroup of $\Gamma$, called the _stabilizer_ of $U$. Then for
$g,g^{\prime}\in\Gamma$ with $g\cdot U\cap g^{\prime}\cdot U\neq\emptyset$ we
have $g^{-1}\,g^{\prime}\cdot U=U$, hence $g^{-1}\,g^{\prime}\in\Gamma_{U}$.
Thus, the translates $\\{g\cdot U\mid g\in\Gamma\\}$ form a finite clopen
partition of ${\mathfrak{X}}$, and are in 1-1 correspondence with the quotient
space $X_{U}=\Gamma/\Gamma_{U}$. Then $\Gamma$ acts by permutations of the
finite set $X_{U}$ and so the stabilizer group $\Gamma_{U}\subset G$ has
finite index. Note that this implies that if $V\subset U$ is a proper
inclusion of adapted sets, then the inclusion $\Gamma_{V}\subset\Gamma_{U}$ is
also proper.

######  DEFINITION 2.3.

Let $({\mathfrak{X}},\Gamma,\Phi)$ be a Cantor action. A properly descending
chain of clopen sets
${\mathcal{U}}=\\{U_{\ell}\subset{\mathfrak{X}}\mid\ell>0\\}$ is said to be an
_adapted neighborhood basis_ at $x\in{\mathfrak{X}}$ for the action $\Phi$, if
$x\in U_{\ell+1}\subset U_{\ell}$ is a proper inclusion for all $\ell>0$, with
$\cap_{\ell>0}\ U_{\ell}=\\{x\\}$, and each $U_{\ell}$ is adapted to the
action $\Phi$.

Given $x\in{\mathfrak{X}}$ and ${\varepsilon}>0$, Proposition 2.1 implies
there exists an adapted clopen set $U\in{\rm CO}({\mathfrak{X}})$ with $x\in
U$ and ${\rm diam}(U)<{\varepsilon}$. Thus, one can choose a descending chain
${\mathcal{U}}$ of adapted sets in ${\rm CO}({\mathfrak{X}})$ whose
intersection is $x$, from which the following result follows:

######  PROPOSITION 2.4.

Let $({\mathfrak{X}},\Gamma,\Phi)$ be a Cantor action. Given
$x\in{\mathfrak{X}}$, there exists an adapted neighborhood basis
${\mathcal{U}}$ at $x$ for the action $\Phi$.

Combining the above remarks, we have:

######  COROLLARY 2.5.

Let $({\mathfrak{X}},\Gamma,\Phi)$ be a Cantor action, and ${\mathcal{U}}$ be
an adapted neighborhood basis. Set $\Gamma_{\ell}=\Gamma_{U_{\ell}}$, with
$\Gamma_{0}=\Gamma$, then there is a nested chain of finite index subgroups,
$\displaystyle{\mathcal{G}}_{{\mathcal{U}}}=\\{\Gamma_{0}\supset\Gamma_{1}\supset\cdots\\}$.

###  2.2. Profinite completion

Let $\Phi(\Gamma)\subset{\rm Homeo}({\mathfrak{X}})$ denote the image subgroup
for an action $({\mathfrak{X}},\Gamma,\Phi)$. When the action is
equicontinuous, the closure $\overline{\Phi(\Gamma)}\subset{\rm
Homeo}({\mathfrak{X}})$ in the _uniform topology of maps_ is a separable
profinite group. We adopt the notation
${\mathfrak{G}}(\Phi)\equiv\overline{\Phi(\Gamma)}$.

Let
$\widehat{\Phi}\colon{\mathfrak{G}}(\Phi)\times{\mathfrak{X}}\to{\mathfrak{X}}$
denote the induced action of ${\mathfrak{G}}(\Phi)$ on ${\mathfrak{X}}$, which
is transitive as the action $({\mathfrak{X}},\Gamma,\Phi)$ is minimal. For
$\widehat{g}\in{\mathfrak{G}}(\Phi)$, we write its action on ${\mathfrak{X}}$
by $\widehat{g}\,x=\widehat{\Phi}(\widehat{g})(x)$. Given
$x\in{\mathfrak{X}}$, introduce the isotropy group,

(2)
$\displaystyle{\mathfrak{D}}(\Phi,x)=\\{\widehat{g}\in{\mathfrak{G}}(\Phi)\mid\widehat{g}\,x=x\\}\subset{\rm
Homeo}({\mathfrak{X}})\ ,$

which is a closed subgroup of ${\mathfrak{G}}(\Phi)$, and thus is either
finite, or is an infinite profinite group. As the action
$\widehat{\Phi}\colon{\mathfrak{G}}(\Phi)\times{\mathfrak{X}}\to{\mathfrak{X}}$
is transitive, the conjugacy class of ${\mathfrak{D}}(\Phi,x)$ in
${\mathfrak{G}}(\Phi)$ is independent of the choice of $x$, and by abuse of
notation we omit the subscript $x$. The group ${\mathfrak{D}}(\Phi)$ is called
the _discriminant_ of the action $({\mathfrak{X}},\Gamma,\Phi)$ in [12, 16,
18], and is called a _parabolic_ subgroup (of the profinite completion of a
countable group) in the works by Bartholdi and Grigorchuk [5, 6].

###  2.3. Algebraic Cantor actions

Let
${\mathcal{G}}=\\{\Gamma=\Gamma_{0}\supset\Gamma_{1}\supset\Gamma_{2}\supset\cdots\\}$
be a descending chain of finite index subgroups. Let
$X_{\ell}=\Gamma/\Gamma_{\ell}$ and note that $\Gamma$ acts transitively on
the left on the finite set $X_{\ell}$. The inclusion
$\Gamma_{\ell+1}\subset\Gamma_{\ell}$ induces a natural $\Gamma$-invariant
quotient map $p_{\ell+1}\colon X_{\ell+1}\to X_{\ell}$. Introduce the inverse
limit

$\displaystyle X_{\infty}$ $\displaystyle\equiv$
$\displaystyle\lim_{\longleftarrow}~{}\\{p_{\ell+1}\colon X_{\ell+1}\to
X_{\ell}\mid\ell\geq 0\\}$ $\displaystyle=$
$\displaystyle\\{(x_{0},x_{1},\ldots)\in X_{\infty}\mid
p_{\ell+1}(x_{\ell+1})=x_{\ell}~{}{\rm for~{}all}~{}\ell\geq
0~{}\\}~{}\subset\prod_{\ell\geq 0}~{}X_{\ell}\ .$

Then $X_{\infty}$ is a Cantor space with the Tychonoff topology, where the
actions of $\Gamma$ on the factors $X_{\ell}$ induce a minimal equicontinuous
action denoted by $\Phi\colon\Gamma\times X_{\infty}\to X_{\infty}$. There is
a natural basepoint $x_{\infty}\in X_{\infty}$ given by the cosets of the
identity element $e\in\Gamma$, so $x_{\infty}=(e\Gamma_{\ell})$. An adapted
neighborhood basis of $x_{\infty}$ is given by the clopen sets

(4) $U_{\ell}=\left\\{x=(x_{i})\in X_{\infty}\mid x_{i}=e\Gamma_{i}\in
X_{i}~{},~{}0\leq i\leq\ell~{}\right\\}\subset X_{\infty}\ .$

There is a tautological identity $\Gamma_{\ell}=\Gamma_{U_{\ell}}$.

For each $\ell\geq 0$, we have the “partition coding map”
$\Theta_{\ell}\colon{\mathfrak{X}}\to X_{\ell}$ which is $\Gamma$-equivariant.
The maps $\\{\Theta_{\ell}\\}$ are compatible with the map on quotients in
(2.3), and so they induce a limit map $\Theta_{x}\colon{\mathfrak{X}}\to
X_{\infty}$. The fact that the diameters of the clopen sets $\\{U_{\ell}\\}$
tend to zero, implies that $\Theta_{x}$ is a homeomorphism. Moreover,
$\Theta_{x}(x)=x_{\infty}\in X_{\infty}$ where $\\{x\\}=\cap_{\ell>0}\
U_{\ell}$. The following is folklore:

######  THEOREM 2.6.

[11, Appendix A] The map $\Theta_{x}\colon{\mathfrak{X}}\to X_{\infty}$
induces an isomorphism of the Cantor actions $({\mathfrak{X}},\Gamma,\Phi)$
and $(X_{\infty},\Gamma,\Phi_{x})$.

The action $(X_{\infty},\Gamma,\Phi_{x})$ is called the _odometer model_
centered at $x$ for the action $({\mathfrak{X}},\Gamma,\Phi)$. The dependence
of the model on the choices of a base point $x\in{\mathfrak{X}}$ and adapted
neighborhood basis ${\mathcal{U}}$ is discussed in detail in the works [11,
13, 16, 18]. Again, we abuse notation in the following and omit the subscript
“$x$”.

Next, we develop the algebraic model for the profinite action
$\widehat{\Phi}\colon{\mathfrak{G}}(\Phi)\times{\mathfrak{X}}\to{\mathfrak{X}}$
of the completion
${\mathfrak{G}}(\Phi)\equiv\overline{\Phi(\Gamma)}\subset{\rm
Homeo}({\mathfrak{X}})$. Fix a choice of group chain
$\\{\Gamma_{\ell}\mid\ell\geq 0\\}$ as above, which provides an algebraic
model for the action $({\mathfrak{X}},\Gamma,\Phi)$.

For each $\ell\geq 1$, let $C_{\ell}\subset\Gamma_{\ell}$ denote the _core_ of
$\Gamma_{\ell}$, i.e. the largest normal subgroup of $\Gamma_{\ell}$ in
$\Gamma$. So

(5) $C_{\ell}~{}={\rm Core}(\Gamma_{\ell})~{}=~{}\bigcap_{g\in\Gamma}~{}g\
\Gamma_{\ell}\ g^{-1}~{}\subset\Gamma_{\ell}~{}.$

As $\Gamma_{\ell}$ has finite index in $\Gamma$, the same holds for
$C_{\ell}$. Observe that for all $\ell\geq 0$, we have $C_{\ell+1}\subset
C_{\ell}$.

Introduce the quotient group $Q_{\ell}=\Gamma/C_{\ell}$ with identity element
$e_{\ell}\in Q_{\ell}$. There are natural quotient maps $q_{\ell+1}\colon
Q_{\ell+1}\to Q_{\ell}$, and we can form the inverse limit group

(6) $\displaystyle\widehat{\Gamma}_{\infty}$ $\displaystyle\equiv$
$\displaystyle\lim_{\longleftarrow}~{}\\{q_{\ell+1}\colon Q_{\ell+1}\to
Q_{\ell}\mid\ell\geq 0\\}$ (7) $\displaystyle=$
$\displaystyle\\{(g_{\ell})=(g_{0},g_{1},\ldots)\mid g_{\ell}\in
Q_{\ell}~{},~{}q_{\ell+1}(g_{\ell+1})=g_{\ell}~{}{\rm for~{}all}~{}\ell\geq
0~{}\\}~{}\subset\prod_{\ell\geq 0}~{}\Gamma_{\ell}~{},$

which is a Cantor space with the Tychonoff topology. The left actions of
$\Gamma$ on the spaces $X_{\ell}=\Gamma/\Gamma_{\ell}$ induce a minimal
equicontinuous action of $\widehat{\Gamma}_{\infty}$ on $X_{\infty}$, again
denoted by $\widehat{\Phi}\colon\widehat{\Gamma}_{\infty}\times X_{\infty}\to
X_{\infty}$. Note that the isotropy group of the action of
$Q_{\ell}=\Gamma_{\ell}/C_{\ell}$ at the identity coset in
$X_{\ell}=\Gamma/\Gamma_{\ell}$ is the subgroup
$D_{\ell}=\Gamma_{\ell}/C_{\ell}$.

Denote the points in $\widehat{\Gamma}_{\infty}$ by
$\widehat{g}=(g_{\ell})\in\widehat{\Gamma}_{\infty}$ where $g_{\ell}\in
Q_{\ell}$. There is a natural basepoint
$\widehat{e}_{\infty}\in\widehat{\Gamma}_{\infty}$ given by the cosets of the
identity element $e\in\Gamma$, so $\widehat{e}_{\infty}=(e_{\ell})$ where
$e_{\ell}=eC_{\ell}\in Q_{\ell}$ is the identity element in $Q_{\ell}$.

For each $\ell\geq 0$, let $\Pi_{\ell}\colon\widehat{\Gamma}_{\infty}\to
Q_{\ell}$ denote the projection onto the $\ell$-th factor in (6), so in the
coordinates of (7), we have $\Pi_{\ell}(\widehat{g})=g_{\ell}\in Q_{\ell}$.
The maps $\Pi_{\ell}$ are continuous for the profinite topology on
$\widehat{\Gamma}_{\infty}$, so the pre-images of points in $Q_{\ell}$ are
clopen subsets. In particular, the fiber of
$\Pi_{\ell}\colon\widehat{\Gamma}_{\infty}\to Q_{\ell}$ over $e_{\ell}$ is the
normal subgroup

(8)
$\widehat{C}_{\ell}=\Pi_{\ell}^{-1}(e_{\ell})=\\{(g_{i})\in\widehat{\Gamma}_{\infty}\mid
g_{i}\in C_{i}~{},~{}0\leq i\leq\ell\\}\ .$

The collection $\\{\widehat{C}_{\ell}\mid\ell\geq 1\\}$ forms a basis of
clopen neighborhoods of $\widehat{e}_{\infty}\in\widehat{\Gamma}_{\infty}$.
That is, for each clopen set $\widehat{U}\subset\widehat{\Gamma}_{\infty}$
with $\widehat{e}_{\infty}\in\widehat{U}$, there exists $\ell_{0}>0$ such that
$\widehat{C}_{\ell}\subset\widehat{U}$ for all $\ell\geq\ell_{0}$.

######  THEOREM 2.7.

[11, Theorem 4.4] There is an isomorphism
${\widehat{\tau}}\colon{\mathfrak{G}}(\Phi)\to\widehat{\Gamma}_{\infty}$ which
conjugates the profinite action
$({\mathfrak{X}},{\mathfrak{G}}(\Phi),\widehat{\Phi})$ with the profinite
action $(X_{\infty},\widehat{\Gamma}_{\infty},\widehat{\Phi})$. In particular,
${\widehat{\tau}}$ identifies the isotropy group ${\mathfrak{D}}(\Phi)$ with
the inverse limit subgroup

(9) $D_{\infty}=\varprojlim\
\\{q_{\ell+1}\colon\Gamma_{\ell+1}/C_{\ell+1}\to\Gamma_{\ell}/C_{\ell}\mid\ell\geq
0\\}\subset\widehat{\Gamma}_{\infty}~{}.$

The maps $q_{\ell+1}$ in the formula (9) need not be surjections, and thus the
calculation of the inverse limit $D_{\infty}$ can involve some subtleties. For
example, it is possible that each group $Q_{\ell}$ is non-trivial for
$\ell>0$, and yet $D_{\infty}$ is the trivial group.

###  2.4. 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 $\Gamma={\mathbb{Z}}$, isomorphism
corresponds to the notion of “flip conjugacy” introduced in the work of Boyle
and Tomiyama [8]. The definition below also appears in the papers [10, 16,
24].

######  DEFINITION 2.8.

Cantor actions $({\mathfrak{X}}_{1},\Gamma_{1},\Phi_{1})$ and
$({\mathfrak{X}}_{2},\Gamma_{2},\Phi_{2})$ are said to be _isomorphic_ if
there is a homeomorphism $h\colon{\mathfrak{X}}_{1}\to{\mathfrak{X}}_{2}$ and
a group isomorphism $\Theta\colon\Gamma_{1}\to\Gamma_{2}$ so that

(10) $\Phi_{1}(g)=h^{-1}\circ\Phi_{2}(\Theta(g))\circ h\in{\rm
Homeo}({\mathfrak{X}}_{1})\ \textrm{for all}\ g\in\Gamma_{1}\ .$

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.

For a minimal equicontinuous Cantor action $({\mathfrak{X}},\Gamma,\Phi)$ and
an adapted set $U\subset{\mathfrak{X}}$, by a small abuse of notation, we use
$\Phi_{U}$ to denote both the restricted action
$\Phi_{U}\colon\Gamma_{U}\times U\to U$ and the induced quotient action
$\Phi_{U}\colon{\mathcal{H}}_{U}\times U\to U$ for
${\mathcal{H}}_{U}=\Phi(G_{U})\subset{\rm Homeo}(U)$. Then
$(U,{\mathcal{H}}_{U},\Phi_{U})$ is called the _holonomy action_ for $\Phi$.

######  DEFINITION 2.9.

Two minimal equicontinuous Cantor actions
$({\mathfrak{X}}_{1},\Gamma_{1},\Phi_{1})$ and
$({\mathfrak{X}}_{2},\Gamma_{2},\Phi_{2})$ are _return equivalent_ if there
exists an adapted set $U_{1}\subset{\mathfrak{X}}_{1}$ for the action
$\Phi_{1}$ and an adapted set $U_{2}\subset{\mathfrak{X}}_{2}$ for the action
$\Phi_{2}$, such that the restricted actions
$(U_{1},{\mathcal{H}}_{1,U_{1}},\Phi_{1,U_{1}})$ and
$(U_{2},{\mathcal{H}}_{2,U_{2}},\Phi_{2,U_{2}})$ are isomorphic.

If the actions $\Phi_{1}$ and $\Phi_{2}$ are isomorphic in the sense of
Definition 2.8, then they are return equivalent with
$U_{1}={\mathfrak{X}}_{1}$ and $U_{2}={\mathfrak{X}}_{2}$. However, the notion
of return equivalence is weaker even for this case, as the conjugacy is
between the holonomy groups ${\mathcal{H}}_{1,{\mathfrak{X}}_{1}}$ and
${\mathcal{H}}_{2,{\mathfrak{X}}_{2}}$, and not the groups $\Gamma_{1}$ and
$\Gamma_{2}$.

###  2.5. Locally quasi-analytic

The quasi-analytic property for Cantor actions was introduced by Álvarez López
and Candel in [1, Definition 9.4] as a generalization of the notion of a
_quasi-analytic action_ studied by Haefliger for actions of pseudogroups of
real-analytic diffeomorphisms. The authors introduced a local form of the
quasi-analytic property in [12, 16]:

######  DEFINITION 2.10.

[16, Definition 2.1] A topological action $({\mathfrak{X}},\Gamma,\Phi)$ on a
metric Cantor space ${\mathfrak{X}}$, is _locally quasi-analytic_ if there
exists ${\varepsilon}>0$ such that for any non-empty open set
$U\subset{\mathfrak{X}}$ with ${\rm diam}(U)<{\varepsilon}$, and for any non-
empty open subset $V\subset U$, and elements $g_{1},g_{2}\in\Gamma$

(11) $\text{if the restrictions}~{}~{}\Phi(g_{1})|V=\Phi(g_{2})|V,~{}\text{
then}~{}~{}\Phi(g_{1})|U=\Phi(g_{2})|U.$

The action is said to be _quasi-analytic_ if (11) holds for
$U={\mathfrak{X}}$.

In other words, $({\mathfrak{X}},\Gamma,\Phi)$ is locally quasi-analytic if
for every $g\in\Gamma$, the homeomorphism $\Phi(g)$ has unique extensions on
the sets of diameter ${\varepsilon}>0$ in ${\mathfrak{X}}$, with
${\varepsilon}$ uniform over ${\mathfrak{X}}$. We note that an effective
action $({\mathfrak{X}},\Gamma,\Phi)$, for a countable group $\Gamma$, is
topologically free if and only if it is quasi-analytic.

Recall that a group $\Gamma$ is _Noetherian_ [3] if every increasing chain of
subgroups has a maximal element. Equivalently, a group is Noetherian if every
subgroup of $\Gamma$ is finitely generated.

######  THEOREM 2.11.

[17, Theorem 1.6] Let $\Gamma$ be a Noetherian group. Then a minimal
equicontinuous Cantor action $({\mathfrak{X}},\Gamma,\Phi)$ is locally quasi-
analytic.

A finitely-generated nilpotent group is Noetherian, so as a corollary we
obtain that all Cantor actions by finitely-generated nilpotent groups are
locally quasi-analytic.

The notion of an LQA Cantor action extends to the case of a profinite group
action
$\widehat{\Phi}\colon{\mathfrak{G}}\times{\mathfrak{X}}\to{\mathfrak{X}}$.

######  DEFINITION 2.12.

Let $({\mathfrak{X}},\Gamma,\Phi)$ be a Cantor action, and
$\widehat{\Phi}\colon{\mathfrak{G}}\times{\mathfrak{X}}\to{\mathfrak{X}}$ the
induced profinite action. We say that the action is _stable_ if the induced
profinite action $({\mathfrak{X}},{\mathfrak{G}}(\Phi),\widehat{\Phi})$ is
locally quasi-analytic, and is said to be _wild_ otherwise.

A profinite completion ${\mathfrak{G}}$ of a Noetherian group $\Gamma$ need
not be Noetherian, as can be seen for the example of $\Gamma={\mathbb{Z}}$,
and $\widehat{\Gamma}$ the full profinite completion of ${\mathbb{Z}}$. In
particular, the assumption that the action $({\mathfrak{X}},\Gamma,\Phi)$ is
locally quasi-analytic does not imply that the action is stable.

###  2.6. Type and typeset for Cantor actions

A Steinitz number $\xi$ can be written uniquely as the formal product over the
set of primes $\Pi$,

(12) $\xi=\prod_{p\in\Pi}~{}p^{\chi_{\xi}(p)}$

where the _characteristic function_
$\chi_{\xi}\colon\Pi\to\\{0,1,\ldots,\infty\\}$ counts the multiplicity with
which a prime $p$ appears in the infinite product $\xi$.

######  DEFINITION 2.13.

Two Steinitz numbers $\xi$ and $\xi^{\prime}$ are said to be _asymptotically
equivalent_ if there exists finite integers $m,m^{\prime}\geq 1$ such that
$m\cdot\xi=m^{\prime}\cdot\xi^{\prime}$, and we then write
$\xi\mathrel{\stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{\tiny
a}}}}{{\sim}}}\xi^{\prime}$.

A _type_ is an asymptotic equivalence class of Steinitz numbers. The type
associated to a Steinitz number $\xi$ is denoted by $\tau[\xi]$.

In terms of their characteristic functions $\chi_{1},\chi_{2}$, we have
$\xi\mathrel{\stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{\tiny
a}}}}{{\sim}}}\xi^{\prime}$ if and only if the following conditions are
satisfied:

  * •

$\chi_{1}(p)=\chi_{2}(p)$ for all but finitely many primes $p\in\Pi$,

  * •

$\chi_{1}(p)=\infty$ if and only iff $\chi_{1}(p)=\infty$ for all primes
$p\in\Pi$.

Given two types, $\tau$ and $\tau^{\prime}$ we write $\tau\leq\tau^{\prime}$,
if there exists representatives $\xi\in\tau$ and
$\xi^{\prime}\in\tau^{\prime}$ such that their characteristic functions
satisfy $\chi_{\xi}(p)\leq\chi_{\xi^{\prime}}(p)$ for all primes $p\in\Pi$.

######  DEFINITION 2.14.

Let $\pi$ denote the set of primes. Given $\xi=\prod_{p\in\pi}\
p^{\chi_{\xi}(p)}$, define:

$\displaystyle\pi(\xi)~{}$ $\displaystyle=$
$\displaystyle~{}\\{p\in\pi\mid\chi_{\xi}(p)>0\\}\ ,~{}\emph{the prime
spectrum of}\ \xi,$ $\displaystyle\pi_{f}(\xi)~{}$ $\displaystyle=$
$\displaystyle~{}\\{p\in\pi\mid 0<\chi_{\xi}(p)<\infty\\}\ ,~{}\emph{the
finite prime spectrum of}\ \xi,$ $\displaystyle\pi_{\infty}(\xi)~{}$
$\displaystyle=$ $\displaystyle~{}\\{p\in\pi\mid\chi_{\xi}(p)=\infty\\}\
,~{}\emph{the infinite prime spectrum of}\ \xi\ .$

Note that if $\xi\mathrel{\stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{\tiny
a}}}}{{\sim}}}\xi^{\prime}$, then
$\pi_{\infty}(\xi)=\pi_{\infty}(\xi^{\prime})$. The property that
$\pi_{f}(\xi)$ is an _infinite_ set is also preserved by asymptotic
equivalence of Steinitz numbers.

We define the type of a Cantor action $(X_{\infty},\Gamma,\Phi)$ defined by a
chain of finite index subgroups,
${\mathcal{G}}=\\{\Gamma=\Gamma_{0}\supset\Gamma_{1}\supset\cdots\\}$. Let
$C_{\ell}\subset\Gamma_{\ell}$ denote the normal core of $\Gamma_{\ell}$ in
$\Gamma$.

######  DEFINITION 2.15.

Let $(X_{\infty},\Gamma,\Phi)$ be a minimal equicontinuous Cantor action
defined by a group chain ${\mathcal{G}}$. The type
$\tau[X_{\infty},\Gamma,\Phi]$ of the action is the equivalence class of the
Steinitz order

(13) $\xi(X_{\infty},\Gamma,\Phi)={\rm
lcm}\\{\\#X_{\ell}=\\#(\Gamma/\Gamma_{\ell})\mid\ell>0\\}\ .$

Finally, we note the following result:

######  THEOREM 2.16.

[20, Theorem 1.9] Let $({\mathfrak{X}},\Gamma,\Phi)$ be a Cantor action. The
Steinitz order $\xi({\mathfrak{X}},\Gamma,\Phi)$ is defined to be the Steinitz
order for an algebraic model $(X_{\infty},\Gamma,\Phi)$ of the action, which
does not depend upon the choice of an algebraic model. Moreover, the type
$\tau[{\mathfrak{X}},\Gamma,\Phi]$ depends only on the return equivalence
class of the action.

###  2.7. Type for profinite groups

The _Steinitz order_ $\Pi[{\mathfrak{G}}]$ of a profinite group
${\mathfrak{G}}$ is defined by the supernatural number associated to a
presentation of ${\mathfrak{G}}$ as an inverse limit of finite groups (see
[29, Chapter 2] or [26, Chapter 2.3]). The Steinitz order appears in the study
of analytic representations of profinite groups associated to groups acting on
rooted trees, see for example [23].

Recall that for a profinite group ${\mathfrak{G}}$, an open subgroup
${\mathfrak{U}}\subset{\mathfrak{G}}$ has finite index [26, Lemma 2.1.2].

######  DEFINITION 2.17.

Let $({\mathfrak{X}},\Gamma,\Phi)$ be a minimal equicontinuous Cantor action,
with choice of a basepoint $x\in{\mathfrak{X}}$. The _Steinitz orders_ of the
action are defined as follows:

  1. (1)

$\xi({\mathfrak{G}}(\Phi))={\rm lcm}\\{\\#\
{\mathfrak{G}}(\Phi)/{\mathfrak{N}}\mid{\mathfrak{N}}\subset{\mathfrak{G}}(\Phi)~{}\text{open
normal subgroup}\\}$,

  2. (2)

$\xi({\mathfrak{D}}(\Phi))={\rm lcm}\\{\\#\
{\mathfrak{D}}(\Phi)/({\mathfrak{N}}\cap{\mathfrak{D}}(\Phi))\mid{\mathfrak{N}}\subset{\mathfrak{G}}(\Phi)~{}\text{open
normal subgroup}\\}$,

  3. (3)

$\xi({\mathfrak{G}}(\Phi):{\mathfrak{D}}(\Phi))={\rm lcm}\\{\\#\
{\mathfrak{G}}(\Phi)/({\mathfrak{N}}\cdot{\mathfrak{D}}(\Phi))\mid{\mathfrak{N}}\subset{\mathfrak{G}}(\Phi)~{}\text{open
normal subgroup}\\}$.

The Steinitz orders satisfy the Lagrange identity, where the multiplication is
taken in the sense of supernatural numbers,

(14)
$\xi({\mathfrak{G}}(\Phi))=\xi({\mathfrak{G}}(\Phi):{\mathfrak{D}}(\Phi))\cdot\xi({\mathfrak{D}}(\Phi))\
.$

and thus we always have
$\tau[{\mathfrak{D}}(\Phi)]\leq\tau[{\mathfrak{G}}(\Phi)]$. The following is a
direct consequence of the definitions:

######  THEOREM 2.18.

Let $({\mathfrak{X}},\Gamma,\Phi)$ be a Cantor action. Then there is equality
of Steinitz orders,
$\xi({\mathfrak{X}},\Gamma,\Phi)=\xi({\mathfrak{G}}(\Phi):{\mathfrak{D}}(\Phi))$.

##  3\. Nilpotent actions

In this section, 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.

###  3.1. Noetherian groups

A countable group $\Gamma$ is said to be _Noetherian_ [3] if every increasing
chain of subgroups $\\{H_{i}\mid i\geq 1\\}$ of $\Gamma$ has a maximal element
$H_{i_{0}}$. The group ${\mathbb{Z}}$ 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.

######  DEFINITION 3.1.

[29, page 153] A profinite group ${\mathfrak{G}}$ is said to be _topologically
Noetherian_ if every increasing chain of _closed_ subgroups
$\\{{\mathfrak{H}}_{i}\mid i\geq 1\\}$ of ${\mathfrak{G}}$ has a maximal
element ${\mathfrak{H}}_{i_{0}}$.

We illustrate this concept with two canonical examples of profinite
completions of ${\mathbb{Z}}$.

######  EXAMPLE 3.2.

Let $\widehat{{\mathbb{Z}}}_{p}$ denote the $p$-adic integers, for $p$ a
prime. That is, $\widehat{{\mathbb{Z}}}_{p}$ is the completion of
${\mathbb{Z}}$ with respect to the chain of subgroups
${\mathcal{G}}=\\{\Gamma_{\ell}=p^{\ell}{\mathbb{Z}}\mid\ell\geq 1\\}$. The
closed subgroups of $\widehat{{\mathbb{Z}}}_{p}$ are given by
$p^{i}\cdot\widehat{{\mathbb{Z}}}_{p}$ for some fixed $i>0$, hence satisfy the
ascending chain property in Definition 3.1.

######  EXAMPLE 3.3.

Let $\widehat{\pi}=\\{p_{i}\mid i\geq 1\\}$ be an infinite collection of
distinct primes. Define an increasing chain of subgroups of ${\mathbb{Z}}$,
where ${\mathcal{G}}_{\widehat{\pi}}=\\{\Gamma_{\ell}=p_{1}p_{2}\cdots
p_{\ell}{\mathbb{Z}}\mid\ell\geq 1\\}$. Let
$\widehat{{\mathbb{Z}}}_{\widehat{\pi}}$ be the completion of ${\mathbb{Z}}$
with respect to the chain ${\mathcal{G}}_{\widehat{\pi}}$. Then we have a
topological isomorphism

(15) $\widehat{{\mathbb{Z}}}_{\widehat{\pi}}\cong\prod_{i\geq 1}\
{\mathbb{Z}}/p_{i}{\mathbb{Z}}\ .$

Let
$H_{\ell}={\mathbb{Z}}/p_{1}{\mathbb{Z}}\oplus\cdots\oplus{\mathbb{Z}}/p_{\ell}{\mathbb{Z}}$
be the direct sum of the first $\ell$-factors. Then $\\{H_{\ell}\mid\ell\geq
1\\}$ is an increasing chain of subgroups of
$\widehat{{\mathbb{Z}}}_{\widehat{\pi}}$ which does not stabilize, so
$\widehat{{\mathbb{Z}}}_{\widehat{\pi}}$ is not topologically Noetherian.

These two examples illustrate the idea behind the proof of the following
result.

######  PROPOSITION 3.4.

Let $\Gamma$ be a finitely generated nilpotent group, and let
$\widehat{\Gamma}$ be a profinite completion of $\Gamma$. Then
$\widehat{\Gamma}$ is topologically Noetherian if and only if the prime
spectrum $\pi(\xi(\widehat{\Gamma}))$ is finite.

###### Proof.

First, recall some basic facts about profinite groups. (See for example, [29,
Chapter 2].) 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 ${\mathfrak{H}}$ is
a pro-$p$-group if ${\mathfrak{H}}$ is the inverse limit of finite $p$-groups.
A Sylow $p$-subgroup ${\mathfrak{H}}\subset{\mathfrak{G}}$ is a maximal
pro-$p$-subgroup [29, Definition 2.2.1].

A profinite group ${\mathfrak{G}}$ is _pro-nilpotent_ if it is the inverse
limit of finite nilpotent groups. For example, if ${\mathfrak{G}}$ is a
profinite completion of a nilpotent group $\Gamma$, then ${\mathfrak{G}}$ is
pro-nilpotent.

The group ${\mathfrak{G}}$ is topologically finitely generated if it contains
a dense subgroup $\Gamma\subset{\mathfrak{G}}$ where $\Gamma$ is finitely
generated. The completion ${\mathfrak{G}}(\Phi)$ associated to a Cantor action
$({\mathfrak{X}},\Gamma,\Phi)$ with $\Gamma$ finitely generated is
topologically finitely generated.

Assume that ${\mathfrak{G}}$ is pro-nilpotent, then for each prime $p$, there
is a unique Sylow $p$-subgroup of ${\mathfrak{G}}$, which is normal in
${\mathfrak{G}}$ (see [29, Proposition 2.4.3]). Denote this group by
${\mathfrak{G}}_{(p)}$. Moreover, ${\mathfrak{G}}_{(p)}$ is non-trivial if and
only if $p\in\pi(\xi({\mathfrak{G}}))$. We use the following result for pro-
nilpotent groups, which is a consequence of [29, Proposition 2.4.3].

######  PROPOSITION 3.5.

Let ${\mathfrak{G}}$ be a profinite completion of a finitely-generated
nilpotent group $\Gamma$. Then there is a topological isomorphism

(16) ${\mathfrak{G}}\cong\prod_{p\in\pi(\xi({\mathfrak{G}}))}\
{\mathfrak{G}}_{(p)}\ .$

From the isomorphism (16) it follows immediately that if the prime spectrum
$\pi(\xi({\mathfrak{G}}))$ is infinite, then ${\mathfrak{G}}$ is not
topologically Noetherian. To see this, list
$\pi(\xi({\mathfrak{G}}))=\\{p_{i}\mid i=1,2,\ldots\\}$, then we obtain an
infinite strictly increasing chain of closed subgroups,

${\mathfrak{H}}_{\ell}=\prod_{i=1}^{\ell}\ {\mathfrak{G}}_{(p_{i})}\ .$

If the prime spectrum $\pi(\xi({\mathfrak{G}}))$ is finite, then the
isomorphism (16) reduces the proof that ${\mathfrak{G}}$ is topologically
Noetherian to the case of showing that if ${\mathfrak{G}}$ is topologically
finitely generated, then each of its Sylow $p$-subgroups is Noetherian. The
group ${\mathfrak{G}}_{(p)}$ is nilpotent and topologically finitely
generated, so we can use the lower central series for ${\mathfrak{G}}_{(p)}$
and induction to reduce to the case where ${\mathfrak{H}}$ is a topologically
finitely-generated abelian pro-$p$-group, and so is isomorphic to a finite
product of $p$-completions of ${\mathbb{Z}}$, which are topologically
Noetherian.

Observe that a profinite completion ${\mathfrak{G}}$ of a finitely generated
nilpotent group $\Gamma$ is a topologically finitely-generated nilpotent
group, and we apply the above remarks. ∎

######  COROLLARY 3.6.

Let $\Gamma$ be a virtually nilpotent group; that, is there exists a finitely-
generated nilpotent subgroup $\Gamma_{0}\subset\Gamma$ of finite index. Then a
profinite completion ${\mathfrak{G}}$ of $\Gamma$ is topologically Noetherian
if and only if its prime spectrum $\pi(\xi({\mathfrak{G}}))$ is finite.

###### Proof.

We can assume that $\Gamma_{0}$ is a normal subgroup of $\Gamma$, then its
closure ${\mathfrak{G}}_{0}\subset{\mathfrak{G}}$ satisfies the hypotheses of
Proposition 3.4, and the Steinitz orders satisfy
$\xi({\mathfrak{G}}_{0})\mathrel{\stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{\tiny
a}}}}{{\sim}}}\xi({\mathfrak{G}})$. As ${\mathfrak{G}}_{0}$ is topologically
Noetherian if and only if ${\mathfrak{G}}$ is topologically Noetherian, the
claim follows. ∎

###  3.2. 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 [17, Theorem 1.6]. We follow the outline
of its proof in [17].

######  PROPOSITION 3.7.

Let ${\mathfrak{G}}$ be a topologically Noetherian group. Then a minimal
equicontinuous action $({\mathfrak{X}},{\mathfrak{G}},\widehat{\Phi})$ on a
Cantor space ${\mathfrak{X}}$ is locally quasi-analytic.

###### Proof.

The closure ${\mathfrak{G}}(\Phi)\subset{\rm Homeo}({\mathfrak{X}})$, so the
action $\widehat{\Phi}$ of ${\mathfrak{G}}(\Phi)$ is effective. Suppose that
the action $\widehat{\Phi}$ is not locally quasi-analytic, then there exists
an infinite properly decreasing chain of clopen subsets of ${\mathfrak{X}}$,
$\\{U_{1}\supset U_{2}\supset\cdots\\}$, which satisfy the following
properties, for all $\ell\geq 1$:

  * •

$U_{\ell}$ is adapted to the action $\widehat{\Phi}$ with isotropy subgroup
${\mathfrak{G}}_{U_{\ell}}\subset{\mathfrak{G}}$;

  * •

there is a closed subgroup $K_{\ell}\subset{\mathfrak{G}}_{U_{\ell+1}}$ whose
restricted action to $U_{\ell+1}$ is trivial, but the restricted action of
$K_{\ell}$ to $U_{\ell}$ is effective.

It follows that we obtain a properly increasing chain of closed subgroups
$\\{K_{1}\subset K_{2}\subset\cdots\\}$ in ${\mathfrak{G}}$, which contradicts
the assumption that ${\mathfrak{G}}$ is topologically Noetherian. ∎

###### Proof of Theorem 1.2.

Let $({\mathfrak{X}},\Gamma,\Phi)$ be a nilpotent Cantor action, and we are
given that the prime spectrum $\pi(\xi({\mathfrak{G}}(\Phi)))$ is finite. Then
there exists a finitely-generated nilpotent subgroup $\Gamma_{0}\subset\Gamma$
of finite index, and we can assume without loss of generality that
$\Gamma_{0}$ is normal. Let ${\mathfrak{G}}(\Phi)_{0}$ be the closure of
$\Gamma_{0}$ in ${\mathfrak{G}}(\Phi)$. The group ${\mathfrak{G}}(\Phi)$ has
finite prime spectrum implies that the group ${\mathfrak{G}}(\Phi)_{0}$ has
finite prime spectrum, and thus by Proposition 3.4 the group
${\mathfrak{G}}(\Phi)_{0}$ is topologically Noetherian. Let
$x\in{\mathfrak{X}}$, then it suffices to show that the action of $\Gamma_{0}$
on the orbit ${\mathfrak{X}}_{0}={\mathfrak{G}}(\Phi)_{0}\cdot x$ is stable.
This reduces the proof to showing the claim when $\Gamma$ is nilpotent. Then
the profinite closure ${\mathfrak{G}}(\Phi)$ is also nilpotent, and we have a
profinite action $({\mathfrak{X}},{\mathfrak{G}}(\Phi),\widehat{\Phi})$.

Suppose that the action $\widehat{\Phi}$ is not locally quasi-analytic, then
there exists an increasing chain of closed subgroups
$K_{\ell}\subset{\mathfrak{D}}(\Phi)$ where $K_{\ell}$ acts trivially on the
clopen subset $U_{\ell}\subset{\mathfrak{X}}$. As ${\mathfrak{D}}(\Phi)$ is a
closed subgroup of ${\mathfrak{G}}(\Phi)$, the increasing chain
$\\{K_{\ell}\mid\ell>0\\}$ consists of closed subgroups of
${\mathfrak{G}}(\Phi)$. This contradicts the fact that ${\mathfrak{G}}(\Phi)$
is topologically Noetherian. Hence, the action $\widehat{\Phi}$ must be
locally quasi-analytic. That is, the action $({\mathfrak{X}},\Gamma,\Phi)$ is
stable. ∎

##  4\. Basic examples

In this section, 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 non-abelian nilpotent group, and
it can be represented as the upper triangular matrices in ${\rm
GL(3,{\mathbb{Z}})}$. That is,

(17) $\Gamma=\left\\{\left[{\begin{array}[]{ccc}1&a&c\\\ 0&1&b\\\ 0&0&1\\\
\end{array}}\right]\mid a,b,c\in{\mathbb{Z}}\right\\}.$

We denote a $3\times 3$ matrix in $\Gamma$ by the coordinates as $(a,b,c)$.

######  EXAMPLE 4.1.

A _renormalizable Cantor action_ , as defined in [22], can be constructed from
the group chain defined by a proper self-embedding of a non-abelian group
$\Gamma$ into itself.

For a prime $p\geq 2$, define the self-embedding
${\varphi}_{p}\colon\Gamma\to\Gamma$ by ${\varphi}(a,b,c)=(pa,pb,p^{2}c)$.
Then define a group chain in $\Gamma$ by setting

$\Gamma_{\ell}={\varphi}_{p}^{\ell}(\Gamma)=\\{(p^{\ell}a,p^{\ell}b,p^{2\ell}c)\mid
a,b,c\in{\mathbb{Z}}\\}\quad,\quad\bigcap_{\ell>0}\ \Gamma_{\ell}=\\{e\\}\ .$

For $\ell>0$, the normal core for $\Gamma_{\ell}$ is given by $C_{\ell}={\rm
core}(\Gamma_{\ell})=\\{(p^{2\ell}a,p^{2\ell}b,p^{2\ell}c)\mid
a,b,c\in{\mathbb{Z}}\\}$, and so the quotient group
$Q_{\ell}=\Gamma/C_{\ell}\cong\\{(\overline{a},\overline{b},\overline{c})\mid\overline{a},\overline{b},\overline{c}\in{\mathbb{Z}}/p^{2\ell}{\mathbb{Z}}\\}$.
The profinite group $\widehat{\Gamma}_{\infty}$ is the inverse limit of the
quotient groups $Q_{\ell}$ so we have
$\displaystyle\widehat{\Gamma}_{\infty}=\\{(\widehat{a},\widehat{b},\widehat{c})\mid\widehat{a},\widehat{b},\widehat{c}\in\widehat{{\mathbb{Z}}}_{p^{2}}\\}$.
Thus, $\xi(\widehat{\Gamma})=\\{p^{\infty}\\}$. Even though the quotient
groups $\Gamma_{\ell}/C_{\ell}$ are all non-trivial, for this action the
inverse limit ${\mathfrak{D}}(\Phi)$ is the trivial group. This follows from
the fact that there are inclusions

$\Gamma_{2\ell}=\\{(p^{2\ell}a,p^{2\ell}b,p^{4\ell}c)\mid
a,b,c,\in{\mathbb{Z}}\\}\subset
C_{\ell}=\\{(p^{2\ell}a,p^{2\ell}b,p^{2\ell}c)\mid a,b,c\in{\mathbb{Z}}\\}\ .$

The triviality of ${\mathfrak{D}}(\Phi)$ implies that there is an equivalent
group chain for the action [11] which can be chosen so that every subgroup in
the chain is normal in $\Gamma$.

######  EXAMPLE 4.2.

For distinct primes $p,q\geq 2$, define the self-embedding
${\varphi}_{p,q}\colon\Gamma\to\Gamma$ by ${\varphi}(a,b,c)=(pa,qb,pqc)$. Then
define a group chain in $\Gamma$ by setting

$\Gamma_{\ell}={\varphi}_{p,q}^{\ell}(\Gamma)=\\{(p^{\ell}a,q^{\ell}b,(pq)^{\ell}c)\mid
a,b,c\in{\mathbb{Z}}\\}\quad,\quad\bigcap_{\ell>0}\ \Gamma_{\ell}=\\{e\\}\ .$

For $\ell>0$, the normal core for $\Gamma_{\ell}$ is given by $C_{\ell}={\rm
core}(\Gamma_{\ell})=\\{((pq)^{\ell}a,(pq)^{\ell}b,(pq)^{\ell}c)\mid
a,b,c\in{\mathbb{Z}}\\}$, and so the quotient group
$Q_{\ell}=\Gamma/C_{\ell}\cong\\{(\overline{a},\overline{b},\overline{c})\mid\overline{a},\overline{b},\overline{c}\in{\mathbb{Z}}/(pq)^{\ell}{\mathbb{Z}}\\}$.
The profinite group $\widehat{\Gamma}_{\infty}$ is the inverse limit of the
quotient groups $Q_{\ell}$ so we have
$\displaystyle\widehat{\Gamma}_{\infty}=\\{(\widehat{a},\widehat{b},\widehat{c})\mid\widehat{a},\widehat{b},\widehat{c}\in\widehat{{\mathbb{Z}}}_{pq}\\}$.
Thus, $\xi(\widehat{\Gamma}_{\infty})=\\{p^{\infty},q^{\infty}\\}$, and
$D_{\infty}$ is the inverse limit of the finite groups
$\Gamma_{\ell}/C_{\ell}$ by (9), so
$D_{\infty}\cong\widehat{{\mathbb{Z}}}_{q}\times\widehat{{\mathbb{Z}}}_{p}$.

##  5\. Nilpotent actions with prescribed spectrum

In this section, 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 $3\times 3$ 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 (non-abelian) group $\Gamma$. That
is, there are always group chains in $\Gamma$ which yield wild nilpotent
Cantor actions.

Let $\Gamma\subset{\rm GL}(3,\mathbb{Z})$ 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 $\pi_{f}$ and
$\pi_{\infty}$, we embed an infinite product of finite actions, as in Example
5.1, into a profinite completion $\widehat{\Gamma}_{\infty}$ of $\Gamma$, and
thus obtain a nilpotent Cantor action $(X_{\infty},\Gamma,\Phi)$ on the
quotient space $X_{\infty}=\widehat{\Gamma}_{\infty}/D_{\infty}$.

###  5.1. Basic components of the construction.

Fix a prime $p\geq 2$.

For $n\geq 1$ and $0\leq k<n$, we have the following finite groups:

(18)
$G_{p,n}=\left\\{\left[{\begin{array}[]{ccc}1&\overline{a}&\overline{c}\\\
0&1&\overline{b}\\\ 0&0&1\\\
\end{array}}\right]\mid\overline{a},\overline{b},\overline{c}\in{\mathbb{Z}}/p^{n}{\mathbb{Z}}\right\\}~{},~{}H_{p,n,k}=\left\\{\left[{\begin{array}[]{ccc}1&p^{k}\overline{a}&0\\\
0&1&0\\\ 0&0&1\\\
\end{array}}\right]\mid\overline{a}\in{\mathbb{Z}}/p^{n}{\mathbb{Z}}\right\\}$

Note that $\\#[G_{p,n}]=p^{3n}$ and $\\#[H_{p,n,k}]=p^{n-k}$.

Let $\overline{x}=(1,0,0),\overline{y}=(0,1,0),\overline{z}=(0,0,1)\in
G_{p,n}$, then
$\overline{x}\cdot\overline{y}\cdot\overline{x}^{-1}=\overline{y}\overline{z}$
and $\overline{x}\cdot\overline{z}\cdot\overline{x}^{-1}=\overline{z}$. That
is, the adjoint action of $\overline{x}$ on the “plane” in the
$(\overline{y},\overline{z})$-coordinates is a “shear” action along the
$\overline{z}$-axis, and the adjoint action of $\overline{x}$ on the
$\overline{z}$-axis fixes all points on the $\overline{z}$-axis.

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 $C_{p,n,k}\subset H_{p,n,k}$ contains elements
in $H_{p,n,k}$ which fix every point in $X_{p,n,k}$. The action of
$\overline{x}\in H_{p,n,k}$ on the coset space $X_{p,n,k}$ satisfies

$\Phi(\overline{x})(\overline{y}\,H_{p,n,k})=\overline{y}\overline{z}\,H_{p,n,k},$

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 $g\in H_{p,n,k}$ its
action fixes the cosets of the multiples of $\overline{z}$.

###  5.2. 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 $\pi_{f}$ and $\pi_{\infty}$ be two disjoint collections of primes, with
$\pi_{f}$ a finite set, and $\pi_{\infty}$ a non-empty set.

Enumerate $\pi_{f}=\\{q_{1},q_{2},\ldots,q_{m}\\}$, and then choose integers
$1\leq r_{i}\leq n_{i}$ for $1\leq i\leq m$.

Enumerate $\pi_{\infty}=\\{p_{1},p_{2},\ldots\\}$ with the convention (for
notational convenience) that if $\ell$ is greater than the number of primes in
$\pi_{\infty}$ then we set $p_{\ell}=1$. For each $\ell\geq 1$, define the
integers

(19) $\displaystyle M_{\ell}$ $\displaystyle=$ $\displaystyle
q_{1}^{r_{1}}q_{2}^{r_{2}}\cdots q_{m}^{r_{m}}\cdot
p_{1}^{\ell}p_{2}^{\ell}\cdots p_{\ell}^{\ell}\ ,$ (20) $\displaystyle
N_{\ell}$ $\displaystyle=$ $\displaystyle q_{1}^{n_{1}}q_{2}^{n_{2}}\cdots
q_{m}^{n_{m}}\cdot p_{1}^{\ell}p_{2}^{\ell}\cdots p_{\ell}^{\ell}\ .$

For all $\ell\geq 1$, observe that $M_{\ell}$ divides $N_{\ell}$.

Define a subgroup of the Heisenberg group $\Gamma$, in the coordinates above,

$\Gamma_{\ell}=\\{(aM_{\ell},bN_{\ell},cN_{\ell})\mid
a,b,c\in{\mathbb{Z}}\\}.$

Its core subgroup is given by $C_{\ell}=\\{(aN_{\ell},bN_{\ell},cN_{\ell})\mid
a,b,c\in{\mathbb{Z}}\\}$. Observe that

${\mathbb{Z}}/N_{\ell}{\mathbb{Z}}\cong{\mathbb{Z}}/q_{1}^{n_{1}}{\mathbb{Z}}\oplus\cdots\oplus{\mathbb{Z}}/q_{m}^{n_{m}}{\mathbb{Z}}\oplus{\mathbb{Z}}/p_{1}^{\ell}{\mathbb{Z}}\oplus\cdots\oplus{\mathbb{Z}}/p_{\ell}^{\ell}{\mathbb{Z}}\
.$

By Proposition 3.5, and in the notation of Example 5.1, we have for
$k_{i}=n_{i}-r_{i}$ that

(21)
$\widehat{\Gamma}_{\infty}~{}=~{}\lim_{\longleftarrow}\\{\Gamma/C_{\ell}\to\Gamma/C_{\ell-1}\mid\ell\geq
1\\}~{}\cong~{}\prod_{i=1}^{m}\
G_{q_{i},n_{i}}~{}\cdot~{}\prod_{j=1}^{\infty}\ \widehat{\Gamma}_{(p_{j})},$
(22) $\quad
D_{\infty}~{}=~{}\lim_{\longleftarrow}\\{\Gamma_{\ell}/C_{\ell}\to\Gamma_{\ell-1}/C_{\ell-1}\mid\ell\geq
1\\}~{}\cong~{}\prod_{i=1}^{m}\ H_{q_{i},n_{i},k_{i}}\ .$

Then the Cantor space $X_{\infty}=\widehat{\Gamma}_{\infty}/D_{\infty}$
associated to the group chain $\\{\Gamma_{\ell}\mid\ell\geq 1\\}$ is given by

(23) $X_{\infty}~{}\cong~{}\prod_{i=1}^{m}\
X_{q_{i},n_{i},k_{i}}~{}\times~{}\prod_{j=1}^{\infty}\
\widehat{\Gamma}_{(p_{j})}\ .$

In particular, as the first factor in (23) is a finite product of finite sets,
the second factor defines an open neighborhood

$U=\prod_{i=1}^{m}\ \\{x_{i}\\}~{}\times~{}\prod_{j=1}^{\infty}\
\widehat{\Gamma}_{(p_{j})}$

where $x_{i}\in X_{q_{i},n_{i},k_{i}}$ is the basepoint given by the coset of
the identity element. That is, $U$ is a clopen neighborhood of the basepoint
in $X_{\infty}$. The isotropy group of $U$ is given by

(24) $\widehat{\Gamma}_{\infty}|U~{}=~{}\prod_{i=1}^{m}\
H_{q_{i},n_{i},k_{i}}~{}\times~{}\prod_{j=1}^{\infty}\
\widehat{\Gamma}_{(p_{j})}\ .$

The restriction of $\widehat{\Gamma}_{\infty}|U$ to $U$ is isomorphic to the
subgroup

(25) $K|U~{}=~{}\prod_{i=1}^{m}\
\\{\overline{e}_{i}\\}~{}\times~{}\prod_{j=1}^{\infty}\
\widehat{\Gamma}_{(p_{j})}~{}\subset~{}{\rm Homeo}(U)\ ,$

where $\overline{e}_{i}\in G_{q_{i},n_{i}}$ is the identity element. The group
$K|U$ acts freely on $U$, and thus the action of $\widehat{\Gamma}_{\infty}$
on $X_{\infty}$ is locally quasi-analytic. Moreover, the prime spectrum of the
action of $\Gamma$ on $X_{\infty}$ is the union
$\widehat{\pi}=\pi_{f}\cup\pi_{\infty}=\pi(\xi(\widehat{\Gamma}_{\infty}))$.
If $\pi_{\infty}$ is infinite, then the prime spectrum of the action is
infinite. Note that the group $\Gamma$ embeds into
$\widehat{\Gamma}_{\infty}$, since the integers $M_{\ell}$ and $N_{\ell}$ tend
to infinity with $\ell$. This completes the proof of Theorem 1.3.

###  5.3. Wild nilpotent actions with infinite prime spectrum

We now 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
$\Gamma$, as defined by (17). Let $\pi_{f}$ and $\pi_{\infty}$ be disjoint
collections of primes, with $\pi_{f}$ an infinite set and $\pi_{\infty}$
arbitrary, possibly empty.

Enumerate $\pi_{f}=\\{q_{1},q_{2},\ldots\\}$ and choose integers $1\leq
r_{i}<n_{i}$ for $1\leq i<\infty$.

Enumerate $\pi_{\infty}=\\{p_{1},p_{2},\ldots\\}$, again with the convention
that if $\ell$ is greater than the number of primes in $\pi_{\infty}$ then we
set $p_{\ell}=1$.

As in Section 5.2, for each $\ell\geq 1$, define the integers

$\displaystyle M_{\ell}$ $\displaystyle=$ $\displaystyle
q_{1}^{r_{1}}q_{2}^{r_{2}}\cdots q_{\ell}^{r_{\ell}}\cdot
p_{1}^{\ell}p_{2}^{\ell}\cdots p_{\ell}^{\ell}\ ,$ $\displaystyle N_{\ell}$
$\displaystyle=$ $\displaystyle q_{1}^{n_{1}}q_{2}^{n_{2}}\cdots
q_{\ell}^{n_{\ell}}\cdot p_{1}^{\ell}p_{2}^{\ell}\cdots p_{\ell}^{\ell}\ .$

For $\ell\geq 1$, define a subgroup of the Heisenberg group $\Gamma$, in the
coordinates above,

(26) $\Gamma_{\ell}=\\{(aM_{\ell},bN_{\ell},cN_{\ell})\mid
a,b,c\in{\mathbb{Z}}\\}\ ,$

Its core subgroup is given by $C_{\ell}=\\{(aN_{\ell},bN_{\ell},cN_{\ell})\mid
a,b,c\in{\mathbb{Z}}\\}$. For $k_{i}=n_{i}-r_{i}$ we then have

(27) $\widehat{\Gamma}_{\infty}~{}\cong~{}\prod_{i=1}^{\infty}\
G_{q_{i},n_{i}}~{}\cdot~{}\prod_{j=1}^{\infty}\
\widehat{\Gamma}_{(p_{j})}\quad,\quad
D_{\infty}~{}\cong~{}\prod_{i=1}^{\infty}\ H_{q_{i},n_{i},k_{i}}\ .$

The Cantor space $X_{\infty}=\widehat{\Gamma}_{\infty}/D_{\infty}$ associated
to the group chain $\\{\Gamma_{\ell}\mid\ell\geq 1\\}$ is given by

(28) $X_{\infty}~{}\cong~{}\prod_{i=1}^{\infty}\
X_{q_{i},n_{i},k_{i}}~{}\times~{}\prod_{j=1}^{\infty}\
\widehat{\Gamma}_{(p_{j})}\ .$

The first factor in (23) is an infinite product of finite sets, so fixing the
first $\ell$-coordinates in this product determines a clopen subset of
$X_{\infty}$. Let $x_{i}\in X_{q_{i},n_{i},k_{i}}$ denote the coset of the
identity element, which is the basepoint in $X_{q_{i},n_{i},k_{i}}$. Then for
each $\ell\geq 1$, we define a clopen set in $X_{\infty}$

(29) $U_{\ell}=\prod_{i=1}^{\ell}\
\\{x_{i}\\}~{}\times~{}\prod_{i=\ell+1}^{\infty}\
X_{q_{i},n_{i},k_{i}}~{}\times~{}\prod_{j=1}^{\infty}\
\widehat{\Gamma}_{(p_{j})}\ .$

Recalling the calculations in Example 5.1, the subgroup
$H_{q_{i},n_{i},k_{i}}$ is the isotropy group of the basepoint $x_{i}\in
X_{q_{i},n_{i},k_{i}}$. Thus, the isotropy subgroup of $U_{\ell}$ for the
$\widehat{\Gamma}_{\infty}$-action is given by the product

(30) $\widehat{\Gamma}_{\infty}|_{U_{\ell}}~{}=~{}\prod_{i=1}^{\ell}\
H_{q_{i},n_{i},k_{i}}~{}\times~{}\prod_{i=\ell+1}^{\infty}\
G_{q_{i},n_{i}}~{}\times~{}\prod_{j=1}^{\infty}\ \widehat{\Gamma}_{(p_{j})}\
.$

For $j\neq i$, 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
$\widehat{\Gamma}_{\infty}|_{U_{\ell}}$ in ${\rm Homeo}(U_{\ell})$ is
isomorphic to the subgroup

(31)
$Z_{\ell}~{}=~{}\widehat{\Gamma}_{\infty}|U_{\ell}~{}=~{}\prod_{i=1}^{\ell}\
\\{\overline{e}_{i}\\}~{}\times~{}\prod_{i=\ell+1}^{\infty}\
G_{q_{i},n_{i}}~{}\times~{}\prod_{j=1}^{\infty}\
\widehat{\Gamma}_{(p_{j})}~{}\subset~{}{\rm Homeo}(U_{\ell})\ ,$

where $\overline{e}_{i}\in G_{q_{i},n_{i}}$ is the identity element.

We next show that this action is not stable; that is, for any $\ell>0$ there
exists a clopen subset $V\subset U_{\ell}$ and non-trivial $\widehat{g}\in
Z_{\ell}$ so that the action of $\widehat{\Gamma}_{\infty}$ restricts to the
identity map on $V$.

We can assume without loss of generality that $V=U_{\ell^{\prime}}$ for some
$\ell^{\prime}>\ell$. Consider the restriction map for the isotropy subgroup
of $Z_{\ell}$ to $U_{\ell^{\prime}}$ which is given by

$\rho_{\ell,\ell^{\prime}}\colon Z_{\ell}|_{U_{\ell^{\prime}}}\to
Z_{\ell^{\prime}}\subset{\rm Homeo}(U_{\ell^{\prime}})\ .$

We must show that there exists $\ell^{\prime}>\ell$ such that this map has a
non-trivial kernel. Calculate this map in terms of the product representations
above,

(32) $Z_{\ell}|_{U_{\ell^{\prime}}}~{}=~{}\prod_{i=1}^{\ell}\
\\{\overline{e}_{i}\\}~{}\times~{}\prod_{i=\ell+1}^{\ell^{\prime}}\
H_{q_{i},n_{i},k_{i}}~{}\times~{}\prod_{i=\ell^{\prime}+1}^{\infty}\
G_{q_{i},n_{i}}~{}\times~{}\prod_{j=1}^{\infty}\ \widehat{\Gamma}_{(p_{j})}\
.$

For $\ell<i\leq\ell^{\prime}$, the group $H_{q_{i},n_{i},k_{i}}$ fixes the
point $\prod_{i=1}^{\ell^{\prime}}\ \\{x_{i}\\}$, and acts trivially on
$\prod_{i=\ell^{\prime}+1}^{\infty}\ X_{q_{i},n_{i},k_{i}}$. Thus, the kernel
of the restriction map contains the second factor in (32),

(33) $\prod_{i=\ell+1}^{\ell^{\prime}}\
H_{q_{i},n_{i},k_{i}}~{}\subset~{}\ker\left\\{\rho_{\ell,\ell^{\prime}}\colon
Z_{\ell}|_{U_{\ell^{\prime}}}\to{\rm Homeo}(U_{\ell^{\prime}})\right\\}\ .$

As this group is non-trivial for all $\ell^{\prime}>\ell$, the action of
$\widehat{\Gamma}_{\infty}$ on $X_{\infty}$ is not locally quasi-analytic,
hence the action of $\Gamma$ on $X_{\infty}$ is wild. Moreover, the prime
spectrum of the action of $\Gamma$ on $X_{\infty}$ equals the union
$\widehat{\pi}=\pi_{f}\cup\pi_{\infty}$.

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 $\Gamma$ on a
Cantor set is topologically free while the action of
$\widehat{\Gamma}_{\infty}$ is wild, and the prime spectrum is prescribed.

Choose an infinite set of distinct primes $\pi_{f}=\\{q_{1},q_{2},\ldots\\}$,
and let $\pi_{\infty}$ be empty.

Choose the constants as in Section 5.1, with $n_{i}=2$ and $k_{i}=1$ for all
$i\geq 1$.

Define the Cantor space $X_{\infty}$ by (28), where the second factor is
trivial; that is a point. The action of $\widehat{\Gamma}_{\infty}$ is wild by
the calculations in formulas (30) to (33).

We claim that the action of $\Gamma$ on $X_{\infty}$ is topologically free.
Suppose not, then there exists an open set $U\subset X_{\infty}$ and
$g\in\Gamma$ such that the action of $\Phi(g)$ is non-trivial on $X_{\infty}$
but leaves the set $U$ invariant, and restricts to the identity action on $U$.
The action of $\Gamma$ on $X_{\infty}$ is minimal, so there exists
$h\in\Gamma$ with $h\cdot x_{\infty}\in U$. Then
$\Phi(h^{-1}gh)(x_{\infty})=x_{\infty}$ and the action $\Phi(h^{-1}gh)$ fixes
an open neighborhood of $x_{\infty}$. Replacing $g$ with $h^{-1}gh$ we can
assume that $\Phi(g)(x_{\infty})=x_{\infty}\in U$. From the definition (29),
the clopen sets

(34) $U_{\ell}=\prod_{i=1}^{\ell}\
\\{x_{i}\\}~{}\times~{}\prod_{i=\ell+1}^{\infty}\ X_{q_{i},2,1}$

form a neighborhood basis at $x_{\infty}$, and thus there exists $\ell>0$ such
that $U_{\ell}\subset U$.

The group $\Gamma$ embeds into $\widehat{\Gamma}_{\infty}$ along the diagonal
in the product (16). That is, we can write $\displaystyle
g=(g,g,\ldots,g)\in\prod_{i=1}^{\infty}\ G_{q_{i},2}$. The action of $\Phi(g)$
is factorwise, and $\Phi(g)(x_{\infty})=x_{\infty}$ implies that
$\displaystyle g\in D_{\infty}\cong\prod_{i=1}^{\infty}\
H_{q_{i},n_{i},k_{i}}$. The assumption that $\Phi(g)$ fixes the points in $U$
implies that it acts trivially on each factor $X_{q_{i},2,1}$ for $i>\ell$. 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 $i>\ell$. 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 $\\{q_{i}\mid i\geq\ell\\}$, so by the Prime Factorization
Theorem the matrix $g$ is the identity.

Alternatively, observe that we have $\displaystyle g\in\prod_{i=1}^{\ell}\
H_{q_{i},2,1}$. This is a finite product of finite groups, which implies that
$g\in\Gamma$ is a torsion element. However, the Heisenberg group $\Gamma$ is
torsion-free, hence $g$ must be the identity. Thus, the action of $\Gamma$ on
$X_{\infty}$ must be topologically free.

Finally, the above construction allows the choice of any infinite subset
$\pi_{f}$ 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.

###  5.4. Proof of Corollary 1.5

Consider the family of wild topologically free actions on the Heisenberg group
$\Gamma$ with infinite distinct prime spectrum, as constructed at the end of
Section 5.3. We show that the uncountable number of infinite choices of
$\pi_{f}$ in this family can be made so that the actions have pairwise
disjoint types.

By Definition 2.13, for two Steinitz numbers $\xi$ and $\xi^{\prime}$ we have
that their types are equal, $\tau(\xi)=\tau(\xi^{\prime})$, if and only if
there exist integers $m,m^{\prime}$ such that
$m\cdot\xi=m^{\prime}\cdot\xi^{\prime}$. Thus two actions with prime spectra
$\pi_{f}$ and $\pi_{f}^{\prime}$ have distinct types if and only if $\pi_{f}$
and $\pi_{f}^{\prime}$ differ by an infinite number of entries. This happens,
for instance, if $\pi_{f}$ and $\pi_{f}^{\prime}$ 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 [14, 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.

## References

  * [1] J. Álvarez López and A. Candel,  Equicontinuous foliated spaces,  Math. Z., 263:725–774, 2009.
  * [2] J. Auslander,  Minimal flows and their extensions,  North-Holland Mathematics Studies, Vol. 153, North-Holland Publishing Co., Amsterdam, 1988.
  * [3] R. Baer,  Noethersche Gruppen,  Math. Z., 66:269–288, 1956.
  * [4] L. Bartholdi,  Endomorphic presentations of branch groups,  J. Algebra, 268:419–443, 2003.
  * [5] L. Bartholdi and R.I. Grigorchuk,  On the spectrum of Hecke type operators related to some fractal groups,  Tr. Mat. Inst. Steklova, 231:5–45, 2000.
  * [6] L. Bartholdi and R.I. Grigorchuk,  On parabolic subgroups and Hecke algebras of some fractal groups,  Serdica Math. J., 28:47–90, 2002.
  * [7] L. Bartholdi, R. Grigorchuk and Z. Šuniḱ,  Branch groups,  Handbook of algebra, Vol. 3, Elsevier/North-Holland, Amsterdam, 2012, pages 989–1112.
  * [8] M. Boyle and J. Tomiyama,  Bounded topological orbit equivalence and $C^{*}$-algebras,  J. Math. Soc. Japan, 50:317–329, 1998.
  * [9] M.-I. Cortez and S. Petite,  $G$-odometers and their almost one-to-one extensions,  J. London Math. Soc., 78(2):1–20, 2008.
  * [10] M.I. Cortez and K. Medynets,  Orbit equivalence rigidity of equicontinuous systems,  J. Lond. Math. Soc. (2), 94:545–556, 2016.
  * [11] J. Dyer, S. Hurder and O. Lukina,  The discriminant invariant of Cantor group actions,  Topology Appl., 208: 64–92, 2016.
  * [12] J. Dyer, S. Hurder and O. Lukina,  Molino theory for matchbox manifolds,  Pacific J. Math., 289:91-151, 2017.
  * [13] R. Fokkink and L. Oversteegen,  Homogeneous weak solenoids,  Trans. Amer. Math. Soc., 354(9):3743–3755, 2002.
  * [14] S. Geschke,  Almost disjoint and independent families,  RIMS Kôkyûroku (Kyoto University research information repository), 1790:1-9, 2012.
  * [15] T. Giordano, I. Putman and C. Skau,  $\mathbb{Z}^{d}$-odometers and cohomology,  Groups Geom. Dyn., 13:909–938, 2019.
  * [16] S. Hurder and O. Lukina,  Wild solenoids,  Transactions A.M.S., 371:4493-4533, 2019; arXiv:1702.03032.
  * [17] S. Hurder and O. Lukina,  Orbit equivalence and classification of weak solenoids,  Indiana Univ. Math. J., Vol. 69:2339–2363, 2020; arXiv:1803.02098.
  * [18] S. Hurder and O. Lukina,  Limit group invariants for non-free Cantor actions,  Ergodic Theory Dynam. Systems, Vol. 41:1751–1794, 2021; arXiv:1904.11072.
  * [19] S. Hurder and O. Lukina,  Nilpotent Cantor actions,  Proceedings A.M.S., Vol. 150:289–304, 2022; arXiv:1905.07740.
  * [20] S. Hurder and O. Lukina,  Type invariants for solenoidal manifolds,  preprint, 2023.
  * [21] S. Hurder and O. Lukina,  Mapping Class Groups of solenoidal manifolds,  preprint, 2023.
  * [22] S. Hurder, O. Lukina and W. van Limbeek,  Cantor dynamics of renormalizable groups,  Groups, Geometry, and Dynamics, 15:1449–1487, 2021; arXiv:2002.01565.
  * [23] S. Kionke,  Groups acting on rooted trees and their representations on the boundary,  J. Algebra, 528:260–284, 2019.
  * [24] X. Li,  Continuous orbit equivalence rigidity,  Ergodic Theory Dynam. Systems, 38:1543–1563, 2018.
  * [25] V. Nekrashevych,  Self-similar groups,  Mathematical Surveys and Monographs, Vol. 117,  American Mathematical Society, Providence, RI, 2005.
  * [26] L. Ribes and P. Zalesskii,  Profinite groups, Springer-Verlag, Berlin 2000.
  * [27] S. Thomas,  The classification problem for torsion-free abelian groups of finite rank,  J. Amer. Math. Soc., 16:233–258, 2003.
  * [28] S. Thomas,  The classification problem for finite rank Butler groups,  in Models, modules and abelian groups, Walter de Gruyter, Berlin, 2008, 329–338.
  * [29] J.S. Wilson,  Profinite groups,  London Mathematical Society Monographs. New Series, Vol. 19,  The Clarendon Press, Oxford University Press, New York, 1998.