% Group Theory Working Seminar
% Purdue University
% April 16, 2024
 
 
% slides for talk "Applications of the Contraction Mapping Theorem: Actions on Cantor sets"
 
 
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\date[April 17, 2024, 2023]


  
\title{Applications of the Contraction Mapping Theorem: Actions on Cantor sets}


\author{Steve Hurder\\ \small University of Illinois at Chicago\\April 17, 2024} 
 
 
\begin{document}
 
   \frame{\titlepage} % # 1

 
\frame
{
  \frametitle{}

  \section{Prelude}
 
 
$\bullet$~  A    self-covering map   $\phi \colon M \to M$ of a compact manifold without boundary is \emph{expanding} if there exists a Riemannian metric on $TM$ and $\lambda > 1$ such that for all $\vec{v} \in TM$, $\|D\phi (\vec{v})\| \geq \lambda \cdot \|\vec{v}\|$.


\bigskip


{\bf Theorem.} [Franks 1968] Let $M$ be a compact manifold without boundary that admits an expanding map. Then the fundamental group $\G = \pi_1(M,x)$ has polynomial growth, and hence contains a nilpotent subgroup of finite index.

\medskip

 
$\star$~  {Michael Shub}, {\it Expanding maps}, {\bf Global {A}nalysis ({P}roc. {S}ympos. {P}ure {M}ath., {V}ols.
              {XIV}, {XV}, {XVI}, {B}erkeley, {C}alif., 1968)}, {Amer. Math. Soc., Providence, RI}, 1970, pages 273--276.


$\star$~  {M.~Gromov}, 
  {\it Groups of polynomial growth and expanding maps}, 
  {\bf Inst. Hautes \'{E}tudes Sci. Publ. Math.}, 53:53--73,  1981.


 \vfill   
}
 
 \section{Self-embeddings}

 
 
\frame
{
  \frametitle{}
  
  Let $\phi \colon M \to M$ be an expanding map with a fixed point $x \in M$.
  
  Set $\G = \pi_1(M, x)$, then get embedding 
  $\vp = \phi_{\#} \colon \G \to \G$.
  
  Set $\G_0 = \G$  and $\G_{\ell+1} = \vp(\G_{\ell})$ for all $\ell \geq 0$.
  
  As $\phi$ is expanding, we have $ \cap_{\ell > 0} \ \G_{\ell} ~ = ~ \{0\}$. 
  
  \bigskip
  
  {\bf Definition:} Let $\G$ be a finitely generated group. A proper inclusion $\vp \colon \G \to \G$ with finite index image is called a \underline{renormalization} of $\G$. 

\medskip

The terminology is motivated by the case when $\G = \mZ^n$.

{\bf Example.} $\G = \mZ^2$ and choose $p, q >1$ integers. Then   $\vp(a,b) = (p a, qb)$ is a renormalization.
\medskip

 

 \vfill   
}
 
 
 
\frame
{
  \frametitle{}
  
  {\bf Definition:}  $\G$ is \emph{non co-Hopfian} if it admits a renormalization.

\medskip

Benjamini, and Nekrashevych and Pete introduced the terminology

 {\bf Definition:}  $\G$ is \emph{strongly scale invariant} if there is a renormalization $\vp \colon \G \to \G$  such that, for $\G_{\ell} = \vp^{\ell}(\G)$, 
 $$K(\vp) = \bigcap_{\ell > 0} ~ \G_{\ell} \quad {\rm is \ a \ finite \ group}$$
 
 
{\bf Conjecture:}   Strongly scale-invariant group $\G$  is virtually nilpotent.

 \medskip
 
 $\star$~  {V.~Nekrashevych and G.~Pete},
 {\it Scale-invariant groups},
 {\bf Groups Geom. Dyn.} 5:139--167, 2011.
 \vfill   
}
 
 
 
\frame
{
  \frametitle{}
  
  Franks-Gromov theorem shows that the conjecture is true if $\G$ is the fundamental group of a compact manifold with an expanding map.

\bigskip

Another special case of the conjecture is known:

  {\bf Theorem:} Let $\G$ be a strongly scale-invariant group, with a renormalization $\vp \colon \G \to \G$ such that  $\G_{\ell} = \vp^{\ell}(\G)$ is \underline{normal} in $\G$.
    Then $\G/K(\vp)$ is abelian.

\medskip

$\star$~{\it W. Van Limbeek}, {\bf Towers of regular self-covers and linear endomorphisms of tori}, {Geom. Topol.}, 2018.

\bigskip

It is not usual that the subgroups $\G_{\ell}$ are normal in $\G$.

 \vfill   
}
 
 \section{Heisenberg}
 
 
\frame
{
  \frametitle{}
  
  Let     $\cH$ denote the $3$-dimensional Heisenberg group represented as $(\mZ^3, *)$ with the group operation $*$, so  for $x, u,y, v,z, w  \in \mZ$, 
$$
(x,y,z)*(u,v,w)=(x+u,y+v,z+w+xv)$$
$$(x,y,z)^{-1} = (-x, -y, -z +xy)  
$$
This is equivalent to the upper triangular representation in $GL(\mZ^3)$.  In particular,  
$$
(x,y,z) * (u,v,w)*(x,y,z)^{-1}=(u,v,w + xv -yu)  
$$

For integers $p,q > 1$ let $\vp \colon \cH \to \cH$ by  
$\vp(x,y,z) = (px, qy, pqz)$.   
$$\cH_{\ell} = \vp^{\ell}(\cH) = \{(p^{\ell} x, q^{\ell}y, (pq)^{\ell}z) \mid x,y,z \in \mZ\}$$
$$\bigcap_{\ell > 0} \ \cH_{\ell} = \{e\}$$

 \vfill   
}
 
 
 
\frame
{
  \frametitle{}
Assume that $p,q > 1$ are distinct prime numbers. 

The \emph{normal core} for $\cH_{\ell}$ is  the largest normal subgroup of $\cH_{\ell}$. Then 
$$C_{\ell} = {\rm core}(\cH_{\ell})  = \{((pq)^{\ell} x, (pq)^{\ell}y, (pq)^{\ell}z) \mid x,y,z \in \mZ\}$$
is a proper subgroup of $\cH_{\ell}$, with quotient group 
 $$
 \cH_{\ell}/C_{\ell} \cong  \{( x,  y,  0) \mid x  \in \mZ/q^{\ell}\mZ ~ , ~  y  \in \mZ/p^{\ell}\mZ \}  
$$

So as $\ell \to \infty$, the groups $\G_{\ell}$ become  less and less normal.

 \vfill   
}
 
\section{Group chains} 
 
\frame
{
  \frametitle{}
  
Let $\G$ be a finitely generated group.

{\bf Definition.} A \emph{group chain}  is a descending chain $\cG = \{\G_{\ell} \mid \ell \geq 0\}$   where $\G_0 = \G$ and $\G_{\ell+1} \subset \G_{\ell}$ is a proper subgroup of finite index.

The quotient   $X_{\ell} = \G/\G_{\ell}$ is a finite set with transitive left $\G$-action.


Inclusion $\G_{\ell +1} \subset \G_{\ell}$ induces a surjection $p_{\ell+1} \colon X_{\ell+1} \to X_{\ell}$. Define
$$\fX ~ \equiv ~ \varprojlim \ \{p_{\ell +1} \colon X_{\ell +1} \to X_{\ell} \mid \ell \geq 0\} \subset \prod_{\ell \geq 0} \ X_{\ell} \ .$$


The product of finite sets is given the Tychonoff topology - cylinder sets generate the topology.

Then $\fX$ is a closed subset, so is a Cantor space with left $\G$-action. 

A point in $\fX$ is a sequence $x = (x_0, x_1, \ldots)$ where $p_{\ell +1}(x_{\ell +1}) = x_{\ell}$.

 For $\gamma \in \G$ we have $\Phi(\gamma)(x) = \gamma \cdot x = (\gamma \cdot x_0, \gamma \cdot x_1, \ldots)$. 
 \vfill   
}
 
 
  \frame % 1
{
  \frametitle{}



 $\fX$ has an  ultrametric $\dX$:   
 
 for $x=(x_0, x_1, \ldots)$ and $y = (y_0, y_1, \ldots)$
 $$\dX(x,y) = 2^{-m}\quad {\rm where} \quad m = \min \{\ell \mid x_{\ell} \ne y_{\ell} \}$$
 
 This satisfies the \emph{ultrametric} triangle inequality, for $x,y,z \in \fX$, 
 $$\dX(x,y) \leq \min \{ \dX(x,z) , \dX(z,y)\}$$
 
 Then $\dX(\gamma \cdot x, \gamma \cdot y) = \dX(x,y)$ so $\dX$ is $\G$-invariant ultrametric.
 
 \bigskip
 
The action $\Phi \colon \G \times \fX \to \fX$ is a \emph{generalized odometer}. 
 
$\star$~  {M.-I.~Cortez and S.~Petite},
  {\it $G$-odometers and their almost one-to-one extensions},
  {\bf J. London Math. Soc.}, 78(2):1--20, 2008. 

 \vfill   
}
 
 
  \frame % 1
{
  \frametitle{}

 
 A Cantor action $\Phi \colon \G  \times \fX \to \fX$ is \emph{equicontinuous} if for some metric $\dX$ on $\fX$,   for every $\e > 0$ there exists $\delta > 0$ such that 
 $$\dX(x,y) < \delta ~ \Longrightarrow ~ \dX(\Phi(g)(x) , \Phi(g)(y)) < \e \quad {\rm for ~ all } ~ g \in \G   .$$
   
 

 The action $\Phi$ is  isometric for the  ultrametric the $\dX$  on $\fX$,  so  

$\bullet$~ $(\fX, \G, \Phi)$ is an \emph{equicontinuous Cantor action}.



\bigskip
{\bf Remark:} A minimal equicontinuous Cantor action can also be viewed as a group action  on a rooted tree, aka an \emph{arboreal action}.

  \medskip

 $\star$~  {R.I.~Grigorchuk},
  {\it Some problems of the dynamics of group actions on rooted trees},
{\bf Proc. Steklov Institute of Math.}, 273: 64--175, 2011.

 \vfill   

}


   
    \frame % 1
{
  \frametitle{}

   Let $\Phi \colon \G \times \fX \to \fX$ be an equicontinuous Cantor action.

 The action is a homomorphism $\Phi  \colon \G \to \Homeo(\fX)$, and let

\smallskip

   $\whGamma = \overline{\Phi(\G)} \subset \Homeo(\fX)$ denote the closure in uniform topology

\medskip


{\bf Proposition:} [Ellis, 1969] $\Phi$ equicontinuous implies that   $\whGamma$ is a  profinite group, that is,  \emph{compact} and t\emph{otally disconnected}.

\medskip
 
 {\bf Lemma:}   $\Phi$ minimal action implies that  $\whGamma$ acts transitively       on $\fX$.
 \medskip
 
  For  $x \in \fX$, 
define the isotropy subgroup 
$$\cD_x = \{\whgamma \in \whGamma \mid \whPhi(\whgamma)(x) = x\}$$  
The isomorphism class of $\cD_x$ is independent of choice of $x$, and 
$\fX \cong \whGamma/\cD_x$ as left $\G$-spaces.


\bigskip

 
 \vfill   

}
  
   
 
\frame
{
  \frametitle{}
  
  $\whGamma$ has a discrete description analogous to that for $\fX$.
  
  The action of $\G$ on $X_{\ell} = \G/\G_{\ell}$ permutes the cosets, where $\G_{\ell}$ is the isotropy of the basepoint $e \cdot \G_{\ell}$.
  
  The kernel of the action $\G \to {\bf Perm}(X_{\ell})$ is the core normal subgroup of $\G_{\ell}$,
  $$C_{\ell} = \bigcap_{\gamma \in \G} \ \gamma \cdot \G_{\ell} \cdot \gamma^{-1}$$
 Then there is an isomorphism
 $$\whGamma = \varprojlim \{ \widehat{p}_{\ell+1} \colon \G/C_{\ell+1} \to \G/C_{\ell} \mid \ell \geq 0\}$$
 and the isotropy subgroup is 
 $$\cD_x = \varprojlim \{ \widehat{p}_{\ell+1} \colon \G_{\ell+1}/C_{\ell+1} \to \G_{\ell}/C_{\ell} \mid \ell \geq 0\}$$

 
 \vfill   

}
  
    
 
\frame
{
  \frametitle{}
  
Now consider a  renormalization $\vp \colon \G \to \G$ and define $\G_{\ell} = \vp^{\ell}(\G)$ with associated 
 inverse limit space $\fX_{\vp} = \whGamma_{\vp}/\cD_{\vp}$ and   $\G$-action $\Phi_{\vp}$.
 
 {\bf Proposition.}  $\vp$ induces a contraction   $\lambda_{\vp} \colon \fX_{\vp} \to \fX_{\vp}$.
 
 {\it Proof.} $\vp$ induces a map of quotients $\ovp \colon \G/\G_{\ell} \to \G_1/\G_{\ell+1}$. This induces the \underline{right} shift map $\lambda_{\vp} \colon \fX_{\vp} \to U_1 \subset \fX_{\vp}$, 
 $$\lambda_{\vp}(x_0,x_1, \ldots) = (e, \vp(x_0), \vp(x_1), \ldots)$$
The contraction has a unique fixed-point $x_{\vp} \in \fX_{\vp}$.

\bigskip

How does this help with showing that $\G$ is virtually nilpotent?

 What we need is a contraction
$\whvp \colon \whGamma_{\vp} \to \whGamma_{\vp}$
 with open image. 
 
 We take  a detour into Cantor dynamics.



 \vfill   

}
     \section{Dynamics}

       
    \frame % 1
{
  \frametitle{}

  
Let  $\Phi \colon \G \times \fX \to \fX$ be a Cantor action of a countable group $\G$.
 
The topology of Cantor space $\fX$ is generated by   clopen subsets:\\ U is   closed and open.

A non-empty clopen   $U \subset \fX$ is \underline{adapted} if the return times to $U$ form a subgroup:
$$\G_U = \{ g \in \G \mid \Phi(g)(U) = U  \} \subset \G$$


{\bf Lemma:} Let $\Phi \colon \G \times \fX \to \fX$ be an equicontinuos  Cantor action. For   $x \in \fX$ and   $V$ open with   $x \in V$, there is an adapted set $U$ with $x \in U \subset V$.


 \vfill   

}

    \frame % 1
{
  \frametitle{}
  
  {\bf Definition:}    A Cantor action $\Phi \colon H \times \fX \to \fX$  is  \underline{quasi-analytic}  if for each adapted clopen set $U \subset \fX$ and   $g \in H$ we have
  
 \smallskip
 
 
  $\bullet$~ if   $\Phi(g)(U) = U$ and the restriction $\Phi(g) | U$ is the identity map on $U$, 
  then $\Phi(g)$ acts as the identity on all of $\fX$.  

\medskip
For $H$ a countable group, this is equivalent to topologically free.

\medskip

 Here is our key technical result:
 
 {\bf Theorem 1.}  The  action $\whPhi_{\vp} \colon \whGamma_{\vp} \times \fX_{\vp} \to \fX_{\vp}$  is quasi-analytic.

\smallskip
 {\bf Corollary.}  Let $\whgamma \in \whGamma_{\vp}$.
   The homeomorphism $\whPhi_{\vp} (\whgamma) \colon \fX_{\vp} \to \fX_{\vp}$ is uniquely determined by its restriction to an adapted subset of $\fX$.


 \vfill   

}
  
 
  
 \frame % 1
{
  \frametitle{}

 Theorem~1 is the key to the proof of the next two results:
 
 
\medskip


{\bf Theorem 2.} A renormalization map $\vp$ induces a contraction map on the closure,
$\whvp \colon \whGamma_{\vp} \to \whGamma_{\vp}$ with open image.

  \bigskip

{\bf Theorem 3.} $\ds \cD_{\vp} =   \bigcap_{n > 0} \ \whvp^n(\whGamma_{\vp}) \subset \whGamma_{\vp}$
 
 This connects discriminant invariants for Cantor actions, with invariants for contraction profinite groups.

\medskip

For the proof see

$\star$~  {S.~Hurder, O.~Lukina and W.~Van Limbeek},
 {\it Cantor dynamics of renormalizable groups},
 {\bf Groups, Geometry, and Dynamics},    15:1449-1487, 2021.

 
  \vfill   

}


   \frame % 1
{
  \frametitle{}
      
   
  \bigskip
  
 The proof of Theorem~2 has a key point.
  
  \medskip
  
The renormalization  $\vp \colon \G \to \G$ naturally induces a   map
 $\whvp \colon \whGamma_{\vp} \to \Homeo(U_1)$ where $U = {\rm image}(\lambda_{\vp}) \subset \fX_{\vp}$.
  
  \medskip
  
We need to show that the maps in the image of $\whvp$ have unique extensions to $\Homeo(\fX_{\vp})$.

\medskip

This is exactly what Theorem~1 says is true.


 \vfill   

}
 
\section{Heisenberg}

\frame
{
  \frametitle{}


  Let     $\cH$ denote the $3$-dimensional Heisenberg group. Assume that $p,q > 1$ are distinct prime numbers, and define 
 $\vp \colon \cH \to \cH$ by  $\vp(x,y,z) = (px, qy, pqz)$.   Then 
$$\cH_{\ell} = \vp^{\ell}(\cH) = \{(p^{\ell} x, q^{\ell}y, (pq)^{\ell}z) \mid x,y,z \in \mZ\}$$
$$C_{\ell} = {\rm core}(\cH_{\ell})  = \{((pq)^{\ell} x, (pq)^{\ell}y, (pq)^{\ell}z) \mid x,y,z \in \mZ\}$$
 $$ \cH_{\ell}/C_{\ell} \cong  \{( x,  y,  0) \mid x  \in \mZ/q^{\ell}\mZ ~ , ~  y  \in \mZ/p^{\ell}\mZ \}  $$
Then taking the inverse limit of $\widehat{p}_{\ell+1} \colon \G/C_{\ell+1} \to \G/C_{\ell}$
 $$\whGamma \cong \{ (a,b,c) \mid a,b,c \in \widehat{\mZ}_{pq} \}$$
 and the   inverse limit of $\widehat{p}_{\ell+1} \colon \G_{\ell+1}/C_{\ell+1} \to \G_{\ell}/C_{\ell}$
$$\cD_{\vp} = \{(a,b,0) \mid a \in \widehat{\mZ}_q , b \in \widehat{\mZ}_p\} \cong \widehat{\mZ}_q \times \widehat{\mZ}_p$$
 $\vp$ maps $\cD_{\vp}$ to itself and is multiplication by $p$ and $q$, respectively.
 \vfill   
}
 
  
\section{Profinite}

\frame
{
  \frametitle{}

Theorem 2 gives us a contraction mapping $\whvp \colon \whGamma_{\vp} \to \whGamma_{\vp}$ with open image, where $\whGamma_{\vp}$ is a Cantor group, so is a complete metric space.
\bigskip




 There is an extensive literature on the structure of profinite groups with an open contraction mapping. 
 We use  results from:

 \medskip
 
$\star$~  {U.~Baumgartner and G.~Willis}, {\it Contraction groups and scales of automorphisms of totally disconnected locally compact groups}, {\bf  Israel J. Math.}, 142:221--248,  2004.

\medskip

$\star$~  {C.~Reid},
  {\it Endomorphisms of profinite groups},
  {\bf Groups Geom. Dyn.},   8:553--564, 2014.   


 \vfill   

}
 
\frame
{
  \frametitle{}
  
  
 {\bf Theorem.}  
 Let $\whvp \colon \whGamma_{\vp} \to \whGamma_{\vp}$ be a  contraction map with open image.  Then there is an isomorphism with a semi-direct product
  $$\whGamma_{\vp} \cong \cN_{\vp} \rtimes \cD_{\vp}$$
\begin{eqnarray*}
\cN_{\vp} & = &  \{ \whg \in \whGamma_{\vp} \mid \lim_{\ell \to \infty} \ \whvp^{\ell}(\whg) = \whe \} \\
\cD_{\vp} & = &  \bigcap_{n > 0} \ \whvp^n(\whGamma_{\vp}) \subset \whGamma_{\vp}
\end{eqnarray*}
Moreover,  the contraction factor $\cN_{\vp}$ is pro-nilpotent. 

That is, $\cN_{\vp}$ is an inverse limit of nilpotent groups.



 \vfill   

}
 
\frame
{
  \frametitle{}
  
  We use this structure theorem for contraction maps to show:
 \medskip
 
 {\bf Theorem.}
  Let $\vp$ be a renormalization of the finitely generated group $\G$. Suppose that  
$$K(\vp) = \bigcap_{\ell > 0} \vp^{\ell}(\G)  \subset \G \quad , \quad  \cD_{\vp} =   \bigcap_{n > 0} \ \whvp_0^n(\whGamma_{\vp}) \subset \whGamma_{\vp}$$
are both finite groups,  then 

$\bullet$~ $\G$ is  virtually nilpotent, 

$\bullet$~ If both groups  are trivial, then $\G$ is nilpotent.


\bigskip

{\bf Remark:} The normality assumption in van Limbeek's Theorem is replaced by the assumption that $\cD_{\vp}$ is a finite group.

\medskip

 {\bf Basic Problem:}  Show that $\cD_{\vp}$ is  virtually nilpotent.
 
 


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 The Conjecture has been shown for another special case:
 
 \medskip
  
 {\bf Theorem.} Let $\G$ be a finitely generated strongly scale invariant group, and assume that $\G$ is virtually polycyclic. Then  $\G$ is virtually nilpotent.
 
 
 $\star$~ {J.~ Der\'{e}}, {\it Strongly scale-invariant virtually polycyclic groups},
 {\bf Groups Geom. Dyn.},  16:985--1004, 2022.
  


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Let $\G$ be a residually finite finitely generated group.

Then $\G$ has at least one group chain $\cG = \{\G_{\ell} \mid \ell \geq 0\}$.

We then obtain an action $\whPhi \colon \G \times \fX \to \fX$, where $\fX \cong \whGamma/\cD_x$.

\bigskip

{\bf Problem.} How are the dynamical properties of this Cantor action related to the group structure of $\G$?
 \bigskip

 One motivation for the problem is the observation that associated to the action  $(\fX, \G, \Phi)$ is a cross-product $C^*$-algebra 
 $C^*(\fX, \G, \Phi)$. This construction of $C^*$-algebras appears in many contexts.

\medskip

{\bf Problem.} How is the structure of $C^*(\fX, \G, \Phi)$ related to the dynamical properties of the Cantor action $(\fX, \G, \Phi)$?



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 {\small
  {\bf Cantor action bibliography.}
 }
 
 \medskip
 
 
{\tiny 

  $\star$~ {S.~Hurder and O.~Lukina}, {\it Wild solenoids},
 {\bf Transactions A.M.S.},  371:4493--4533, 2019.

  $\star$~ {O.~Lukina}, {\it Arboreal {C}antor actions},
 {\bf J. Lond. Math. Soc. (2)},  99:678--706, 2019.

 

 $\star$~ {S.~Hurder and O.~Lukina}, {\it Orbit equivalence and classification of weak solenoids},
 {\bf Indiana Univ. Math. J.},  69:2339--2363, 2020.

$\star$~ {O.~Lukina}, {\it Galois Groups and Cantor actions},
 {\bf Transactions A.M.S.},  374:1579--1621, 2021.

$\star$~ {M.~Gr\"{o}ger and O.~Lukina}, {\it Measures and regularity of group Cantor actions},
 {\bf Discrete Contin. Dyn. Syst. Ser. A.},  41:2001--2029, 2021.

 $\star$~ {S.~Hurder and O.~Lukina}, {\it Limit group invariants for non-free Cantor actions},
 {\bf Ergodic Theory Dynamical Systems}, 41:1751--1794, 2021.

 $\star$~ {S.~Hurder, O.~Lukina and W.~van Limbeek}, {\it Cantor dynamics of renormalizable groups},
 {\bf Groups, Geometry, and Dynamics},  15:1449--1487, 2021.
 
  $\star$~ {J.~Alvarez Lopez, R.~Barral Lijo, O.~Lukina and H.~Nozawa}, {\it Wild Cantor actions},
 {\bf J. Math. Soc. Japan}, 474:447--472, 2022.

 $\star$~ {M.I. Cortez and O.~Lukina}, {\it Settled elements in profinite groups},
 {\bf Advances in Mathematics}, 404, 2022.

 

  $\star$~ {S.~Hurder and O.~Lukina}, {\it Nilpotent Cantor actions},
 {\bf Proceedings A.M.S.}, 150:289--304, 2022.


 $\star$~ {S.~Hurder and O.~Lukina}, {\it Essential holonomy of Cantor actions},
 {\bf Journal Math. Society Japan},  to appear. {arxiv:2205.06285}

 $\star$~ {S.~Hurder and O.~Lukina}, {\it Type invariants for solenoidal manifolds},
 {\bf submitted}, {arXiv:2305.00863}.

 $\star$~ {S.~Hurder and O.~Lukina}, {\it Prime spectrum and dynamics for nilpotent Cantor actions},
 {\bf Pacific Journal Mathematics}, to appear. {arXiv:2305.00896}

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