% Resistência Dinâmica Zoom Seminar
% March 12, 2021
 
 
% slides for talk "The dynamical classification of arboreal actions"
 
 
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\date[March 12, 2021]


  
\title{The dynamical classification of arboreal actions}


\author{Steve Hurder, University of Illinois at Chicago\\     joint work with   \\ Olga Lukina, University of Vienna} 
 
 
\begin{document}
 
   \frame{\titlepage} % # 1

 
       \frame % 1
{
  \frametitle{}
In this talk we consider:
\bigskip

 $\star$~   classification, up to   \underline{return equivalence},  minimal equicontinuous actions of a finitely generated group $\G$ on a Cantor space $\fX$.
  
  \medskip
  
$\star$~     a new approach, based on the \underline{Steinitz orders} of   profinite groups associated to the group action.
  
   \medskip
  
$\star$~     two new classes of actions which are invariants of return equivalence - \underline{frothy} and \underline{turbulent} actions.
  
 \medskip
   First,  we discuss the   
motivation for the study of return equivalence.

 \vfill   

}
 
   

       \frame % 1
{
  \frametitle{}

 
 Consider  
 $\cP = \{\, q_{\ell} \colon  M_{\ell} \to M_{\ell -1} \mid  \ell \geq 1\}$, 
 where each $M_{\ell}$ is a compact connected manifold without boundary of dimension $n \geq 1$, and $q_{\ell}$ is a proper covering map.
The inverse limit     
$$
\cS_{\cP} \equiv \lim_{\longleftarrow} ~ \{ q_{\ell } \colon M_{\ell } \to M_{\ell -1}\} ~ \subset \prod_{\ell \geq 0} ~ M_{\ell} ~  
$$
is   the  \emph{solenoidal manifold}   associated to $\cP$. It is compact and connected but not locally connected, has a fibration map $p_0 \colon \cS_{\cP} \to M_0$ and foliated by leaves which cover $M_0$. 

\medskip
For $x_0 \in M_0$, the fiber $\fX_0 = p_0^{-1}(x_0)$ is a Cantor set, and the monodromy along leaves gives an action of $\G = \pi_1(M_0, x_0)$ on $\fX_0$.

\bigskip
{\bf  Problem:} Classify the solenoidal manifolds up to homeomorphism.


  \vfill   

}
 
 
       \frame % 1
{
  \frametitle{}

{\bf Vietoris - van Dantzig Solenoid:}
$$\cS(\vec{m}) =  \underleftarrow{\lim} ~ \{\, q_{\ell}:\mS^1 \to \mS^1 \mid  \ell \geq 1\} 
$$
where $q_{\ell}$ is  a covering map  of the circle $\mS^1$   of degree  $m_{\ell} > 1$. Let $\vec{m} = (m_1, m_2,\dots)$  be the collection of covering degrees, then the \underline{Steinitz degree} of the covering map $q_0 \colon \cS(\vec{m}) \to \mS^1$ is the   product
$$\Pi[\vec{m}] = m_1 \cdot m_2 \cdots m_i \cdots  = \prod_{p \in \pi} \ p^{n(p)} \quad , \quad 0 \leq n(p) \leq \infty$$
When $m_i =2$ for all $i \geq 1$ we get the Smale attractor:
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.35\textwidth]{Smale-Williams_Solenoid_Large}
\label{figure-originalhirsch}
\end{center}
\end{figure}

 
 \vfill   

}
 
   

       \frame % 1
{
  \frametitle{}
  
Steinitz numbers are \underline{asymptotically equivalent}, or $\Pi[\vec{m}] \mor \Pi[\vec{n}]$,\\ if  there exists $m, n \geq 1$ with 
$m \cdot \Pi[\vec{m}] = n\cdot \Pi[\vec{n}]$.

\medskip

[Bing, 1960] and [McCord, 1965] showed the following:

{\bf Theorem:} Solenoids $\cS(\vec{m}) $ and $\cS(\vec{n}) $ are homeomorphic  if and only if ~ $\Pi[\vec{m}] \mor \Pi[\vec{n}]$. 

\bigskip

 

{\bf Question:} Does such a result hold for the case of solenoidal manifolds with dimension greater than 1?

\medskip

$\bullet$~   If solenoidal manifolds $\cS_{\cP}$ and $\cS_{\cP'}$ are homeomorphic, then their monodromy Cantor actions   are \emph{return equivalent}.

\medskip
$\star$~  Clark - Hurder - Lukina, ``Classifying matchbox manifolds'',  Geom \& Top,  23, 2019;  {arXiv:1311.0226}.

  
  \vfill   

}
 
 
       \frame % 1
{
  \frametitle{}
{\bf Minimal equicontinuous Cantor actions:}
\medskip

$\bullet$~ $\G$ is a finitely generated group, 

\smallskip

$\bullet$~ $\Phi \colon \G \times \fX \to \fX$ is a topological action.

\smallskip

 $\bullet$~ $(\fX,\G,\Phi)$ is minimal if every orbit $\cO(x) = \{gx \mid g \in \G\}$ is dense.
 
 \smallskip

 $\bullet$~ $(\fX,\G,\Phi)$ is \emph{equicontinuous} with respect to a metric $\dX$ on $\fX$, \\ if for all $\e >0$ there exists $\delta > 0$, such that for all $x , y \in \fX$ and $g \in \G$,  we have  that 
 $\ds  \dX(x,y) < \delta$ implies   $\dX(g   x, g   y) < \e$.
 
 \medskip
 
  $\bullet$~ $\fX$ Cantor space then the \emph{clopen} (closed and open) subsets $\CO(\fX)$ form a basis for the topology.
  
  \medskip
  
{\bf Fact:}     $\fX$ a Cantor space, a minimal   action   $\Phi \colon \G \times \fX \to \fX$  is  equicontinuous  if and only if  the $\G$-orbit of every $U \in \CO(\fX)$ is finite for the induced action $\Phi_* \colon \G \times \CO(\fX) \to \CO(\fX)$.

  \vfill   

}
 
 
       \frame % 1
{
  \frametitle{}
 
 {\bf Two Models}
 
 \bigskip
 
 The \underline{tree model}, or arboreal actions, where $\G$ acts on a tree $\cT$ preserving a root vertex $v$, then $\fX$ is identified with the ends of $\cT$
 
 \bigskip
 
 The \underline{group chain model}, where $\G$ acts on the coset spaces $X_{\ell} = \G/\G_{\ell}$ where 
 $\G_1 \supset \G_2 \supset \cdots $ is a descending chain of proper subgroups, 
  so $\G$ acts on the inverse limit   space
 $$\fX = \underleftarrow{\lim} \ \{X_{1} \leftarrow X_2 \leftarrow \cdots \}$$

\bigskip

{\bf Fact:} The two models are equivalent. 

\medskip

$\bullet$~ Group chain model yields solenoidal manifolds directly.
       
   \vfill   

}
 
 
       \frame % 1
{
  \frametitle{}

 
{\bf Definition:} $U \subset \fX$  is \emph{adapted} to the action $(\fX,\G,\Phi)$ if $U$ is a   \emph{non-empty clopen} subset, and for any $g \in \G$, 
if $\Phi(g)(U) \cap U \ne \emptyset$ implies that  $\Phi(g)(U) = U$.   

\medskip
$\bullet$ ~ Given  $x \in \fX$ and clopen set $x \in W$, there is an adapted clopen set $U$ with $x \in U \subset W$. 

\medskip
$\bullet$ ~ For   $U$ adapted,   the set of ``return times'' to $U$, 
$$
\G_U = \left\{g \in \G \mid g \cdot U  \cap U \ne \emptyset  \right\}  
$$
is a subgroup of   $\G$, called the \emph{stabilizer} of $U$.      

\bigskip

$\bullet$ ~ $\cH_U = \Phi(\G_U) \subset \Homeo(U)$ is the monodromy of $\G_U$

 \vfill   

}
 


       \frame % 1
{
  \frametitle{}


{\bf Definition:} 
Minimal equicontinuous Cantor  actions $(\fX_1, \G_1, \Phi_1)$ and $(\fX_2, \G_2, \Phi_2)$ are  \underline{return equivalent} if there exists 
  an adapted set $U_1 \subset \fX_1$ for the action $\Phi_1$   and  
  an adapted set $U_2 \subset \fX_2$ for the action $\Phi_2$,
and a homeomoprhism $h \colon U_1 \to U_2$ which induces an isomorphism of the monodromy groups $\cH_{U_1}$ with $\cH_{U_2}$.

\bigskip
{\bf Basic Problem:} Classify the minimal equicontinuous Cantor actions up to return equivalence.

{ \centering $\Uparrow$\par} 

 
{\bf Easier Problem:} Find properties of minimal equicontinuous Cantor actions which are return invariant.

{ \centering $\Uparrow$\par} 

{\bf Fun Problem:} Find interesting examples of minimal equicontinuous Cantor actions.

 \vfill   

}
 


       \frame % 1
{
  \frametitle{}
 
$\bullet$~  {\bf Properties}
 
\quad ~ $\star$~ Steinitz orders

\medskip

\quad ~ $\star$~ Stable \& Wild

\medskip

\quad ~ $\star$~ Frothy \& Turbulent

\bigskip


 
 
 $\bullet$~ {\bf Examples}
 
 \medskip

\quad ~ $\star$~ Odometer Actions

 \medskip

\quad ~ $\star$~ Heisenberg Actions

 
  \vfill   

}
 



       \frame % 1
{
  \frametitle{}
 
$(\fX, \G, \Phi)$ minimal equicontinuous Cantor action

$\Longrightarrow$ ~ $\Phi(\G) \subset \Homeo(\fX)$ is equicontinuous subgroup, 

$\Longrightarrow$ ~ closure $\fG(\Phi) = \overline{\Phi(\G)} \subset \Homeo(\fX)$ is profinite group.

 $\Longrightarrow$ ~ $\fG(\Phi)$ acts transitively on $\fX$, so have $\whPhi \colon \fG(\Phi) \times \fX \to \fX$
 
  $\Longrightarrow$ ~ Isotropy subgroup $\fD(\Phi , x) = \{\whg \in \fG(\Phi) \mid \whPhi(\whg)(x) = x\}$
 
 \quad $\star$~ $\fD(\Phi , x)$ is finite, or Cantor group
 
  \quad $\star$~ $\fD(\Phi , x) \sim \fD(\Phi , y)$ for $x, y \in \fX$
  
  \quad $\star$~ $\fX \cong \fG(\Phi)/\fD(\Phi , x)$
 
 \medskip
 
 The closure $\overline{\Phi(\G)}$ is also called the Ellis group of the action.
 
 \bigskip
  
 {\bf Problem:} How does dynamics of action $(\fX, \G, \Phi)$ depend on subgroup $\fD(\Phi , x)$?
 Or more precisely, on the left (adjoint) action of $\fD(\Phi , x)$ on $\fG(\Phi)/\fD(\Phi , x)$?
  \vfill   

}
 



       \frame % 1
{
  \frametitle{}
  {\bf Steinitz numbers:}
  
  
{\bf Example:}  Suppose $a$ and $b$ are Steinitz numbers, with
 $$a = \prod_{p \in \pi} p^{n(p)} \quad , \quad b = \prod_{p \in \pi} p^{m(p)}$$ 
 where $\pi$ is the set of distinct prime numbers.   
  $$LCM(a,b) = \prod_{p \in \pi} p^{\max\{n(p),m(p)\}}.$$

  
 \medskip
 
  {\bf Definition:}    $\cN = \{n_i \mid i \in \cI\}$ collection of positive integers.   
$$ LCM(\cN) = \prod_{p \in \pi} \ p^{n(p)} \quad , \quad 0 \leq n(p) \leq \infty$$ 
is \emph{least common multiple} as Steinitz number 

   \bigskip
 
    

  \vfill   

}
 


       \frame % 1
{
  \frametitle{}

   {\bf Steinitz order:}


 $\fG$ a profinite group
 
  $\fN \subset \fG$ open normal subgroup then $\fG/\fN$ is finite group.
 
 
{\bf Definition:}
  $\fH \subset \fG$ be a closed subgroup  of the profinite group $\fG$.    
$$
\Pi[\fG : \fH] =   LCM \{\# \{ \fG/(\fN \cdot \fH)\}  \mid \fN \subset \fG~ \text{clopen normal subgroup}\}  
$$
is the \underline{relative Steinitz order} of $\fH$ in $\fG$. 

\medskip

  $\bullet$~   \underline{Steinitz order} of $\fG$ is  $\Pi[\fG] = \Pi[\fG : \{\whe\}]$. 
  
  \bigskip
  
 $\bullet$~  Steinitz numbers $\Pi_1 \mor \Pi_2$ (\underline{asymptotic equivalence}) 
  
\quad   $\Longleftrightarrow$   $m \cdot \Pi_1 = n \cdot \Pi_2$ for integers $m, n \geq 1$
 
 \bigskip
 
 $\star$~ J.S.~Wilson,  Chapter 2, {\bf  Profinite groups},
  {London Mathematical Society Monographs. New Series}, Vol. 19,   1998.  

  \vfill   

}
 


       \frame % 1
{
  \frametitle{}
 

{\bf Theorem:}   
$(\fX, \G, \Phi)$ minimal equicontinuous Cantor action, then the asymptotic   relative Steinitz order 
$ \Pi_a[\fG(\Phi) : \fD(\Phi)] $ 
is an invariant of return equivalence class of the action.
 
 {\bf Corollary:} Asymptotic Steintiz order of tower of coverings is an invariant of the homeomorphism class of solenoidal manifold.
 \medskip
 
 {\bf Definition:} Prime spectrum of $\fG$ is the collection 
$$ \pi(\Pi[\fG]) = \{ p ~ {\rm prime} \mid p ~ {\rm divides} ~ \Pi[\fG] \}$$

{\bf Theorem:}  $(\fX, \G, \Phi)$ minimal equicontinuous Cantor action, then the   prime spectra
  $\pi(\Pi[\fG(\Phi)])$ and  $\pi(\Pi[\fD(\Phi)])$
are invariants of return equivalence of the action, modulo finite sets of primes.

\medskip

{\bf Remark:} Classification problem can be considered in terms of prime spectra of actions.

 
 \vfill   

}
 


       \frame % 1
{
  \frametitle{}
{\bf Regularity properties of Cantor actions:}

\medskip

 Here are alternate versions of \emph{topologically free} actions\\
 which are valid for $\G$ profinite group.
 
 \medskip
 
$\bullet$ ~ $(\fX, \G, \Phi)$  is \underline{quasi-analytic} $\Longleftrightarrow$

 for  any clopen subset $U \subset \fX$ $\&$ any $g \in \G$,  if $\Phi(g)(U) = U$ and  $\Phi(g)|U$ is the identity, then $\Phi(g)$ is the identity on all of $\fX$.
 
 \bigskip
 
$\bullet$ ~   $(\fX, \G, \Phi)$  is \underline{locally quasi-analytic}  $\Longleftrightarrow$

if there exists $\e > 0$ such that for any adapted subset $U \subset \fX$ with $\diam(U) < \e$ $\&$  any $g \in \G$ with $\Phi(g)(U) = U$, if there exists clopen $V \subset U$ with $\Phi(V) = V$  and the restriction $\Phi(g)|V$ is the identity, then $\Phi(g)|U$ is the identity on all of $U$.
 

 \vfill   

}
 

       \frame % 1
{
  \frametitle{}
  
    
{\bf Definition:}   $(\fX, \G, \Phi)$ is   \underline{stable} if  the \emph{profinite} action
$\whPhi \colon \fG(\Phi) \times \fX \to \fX$ is locally quasi-analytic.

\medskip

{\bf Theorem:} Stable property is an invariant of return equivalence.

  \medskip
  
  {\bf Remark:} The classification problem for stable actions essentially reduces to a problem in algebra.
  
  \bigskip
  
 $\star$~ Cortez  - Medynets, ``Orbit equivalence rigidity of equicontinuous systems'',  Journal Lond. Math. Soc. (2), 94, 2016.


 $\star$~ Hurder - Lukina, ``Orbit equivalence and   classification of weak solenoids'', Indiana Univ. Math. Journal,  Vol. 69, 2020; {arXiv:1803.02098}.
 
 $\star$~ Hurder - Lukina, ``Nilpotent   Cantor actions''; {arXiv:1905.07740}.

  
 
 \vfill   

}
   
  
       \frame % 1
{
  \frametitle{}
  
   
{\bf Definition:} $(\fX, \G, \Phi)$ is   \underline{wild} if  the \emph{profinite} action
$\whPhi \colon \fG(\Phi) \times \fX \to \fX$ is not locally quasi-analytic.

\medskip

   
Wild Cantor actions include:

$\bullet$~    actions of  weakly  branch groups on their boundaries 

$\star$~  Bartholdi - Grigorchuk -  \v{S}uni\'{k}, ``Branch groups'',  {\bf {Handbook of algebra, {V}ol. 3}},   2012. 


$\bullet$~   actions of higher rank arithmetic lattices on quotients of their profinite completions

$\star$~   Hurder- Lukina, ``Wild solenoids'', Transactions A.M.S., 371,  2019; {arXiv:1702.03032}.


$\bullet$~  subgroups of wreath product groups acting on trees  

$\star$~ {\'A}lvarez L{\'o}pez - Barral Lij\'{o} - Lukina  - Nozawa, ``Wild {C}antor actions'', J. Math. Soc. Japan,  to appear;  {arXiv:2010.00498}.

 
 \vfill   

}
   


  \frame % 1
{
  \frametitle{}
  {\bf Classifying nilpotent Cantor actions:}
  
  \medskip
  
  $(\fX, \G, \Phi)$ is a nilpotent Cantor action $\Leftrightarrow$
  
  $\bullet$~   minimal \& equicontinuous, 
  
  $\bullet$~  $\G$ contains a finitely-generated nilpotent subgroup of finite index.
  
 \medskip
 {\bf Question:} How do the dynamical properties of nilpotent Cantor actions differ from those of $\mZ^n$-odometers?
 
  \bigskip
  
  
{\bf Theorem:} [Hurder - Lukina, 2021]  Let  $(\fX, \G, \Phi)$ be a nilpotent  Cantor action.  Then
 $$ \text{prime spectrum} ~ \pi(\Pi[\fG(\Phi)]) ~ \text{is finite } \Longrightarrow   \text{ action is stable}$$

 

{\bf Problem:} Show there exist    wild nilpotent Cantor actions.
   
 \vfill   

}
   
  

  \frame % 1
{
  \frametitle{}
 
 {\bf Two more properties:}
 
 \bigskip
 
 {\bf Definition:} A  wild Cantor action $(\fX, \G, \Phi)$ is said to be \underline{frothy} if   
 $\displaystyle \fD(\Phi) \cong \prod_{i=1}^{\infty} ~  H_i$ ~ , 
where each $H_i$ is a finite group.

 \bigskip
 
 {\bf Definition:} A  wild Cantor action $(\fX, \G, \Phi)$ is said to be \underline{turbulent} 
 if the set of points with non-trivial holonomy   has full   measure.

\medskip

This notion has applications to the study of I.R.S.'s

$\star$~ Gr\"oger -  Lukina, ``Measures and regularity of group Cantor actions'',   Discrete Contin. Dynam. Sys.-A, 41(5) 2021;  {arXiv:1911.00680}. 


   
 \vfill   

}
   
  


  \frame % 1
{
  \frametitle{}
 
 {\bf Examples:}

$\bullet$~ Toroidal Actions

\bigskip

$\bullet$~ Heisenberg (nilpotent) Actions

\medskip

\quad $\star$~ Stable

\medskip

\quad $\star$~ Wild

\medskip

\quad $\star$~ Frothy

\medskip

\quad $\star$~ Turbulent

   
 \vfill   

}
   
  


  \frame % 1
{
  \frametitle{}
 


{\bf Classic odometers:} 
 Choose two disjoint sets of distinct primes,
$$\pi_f = \{q_1 , q_2, \ldots \} \quad , \quad \pi_{\infty} = \{p_1 , p_2, \ldots\}$$
where $\pi_f$ and $\pi_{\infty}$ can be chosen to be finite or infinite sets. 

Choose  multiplicities $n(q_i) \geq 1$ for the primes in $\pi_f$.

For each $\ell > 0$, define a subgroup of $\G = \mZ$ by 
$$\G_{\ell} = \{q_1^{n(q_1)} q_2^{n(q_2)} \cdots q_{\ell}^{n(q_{\ell})} \cdot p_1^{\ell} p_2^{\ell} \cdots p_{\ell}^{\ell} \cdot n \mid n \in \mZ \} \ , $$
The completion $\whGamma$ of $\mZ$ with respect to this group chain  admits a product decomposition into its Sylow $p$-subgroups
$$
\whGamma ~ \cong ~ \prod_{i =1}^{\infty} \  \mZ/q_i^{n(q_i)} \mZ ~   \cdot ~ \prod_{p \in \pi_{\infty}} ~  \whmZ_{(p)} \quad , ~ \pi(\Pi[\whGamma]) = \pi_f \cup \pi_{\infty}
$$
$\mZ$-action on $\fX = \whGamma$ is free, so certainly topologically free \& stable. 
 \vfill   

}
   
  

  \frame % 1
{
  \frametitle{}

 {\bf Heisenberg odometers:}
  $\cH \subset {\rm GL(\mZ^3)}$

\begin{equation}\label{eq-cH}
\cH =   \left\{  \left[ {\begin{array}{ccc}
   1 & a & c\\
   0 & 1 & b\\
  0 & 0 & 1\\
  \end{array} } \right] \mid a,b,c  \in \mZ\right\} .
\end{equation}
The group operation $*$ in coordinates $(a,b,c) , (a',b',c') \in \mZ^3$, 
$$
(a,b,c)*(a',b',c')=(a+a',b+b',c+c'+ab')$$
 
\medskip

The normal subgroups and representations of $\cH$ are described in 

$\star$~ Lightwood -  {\c{S}}ahin -  Ugarcovici, ``The structure and spectrum of {H}eisenberg odometers'',
  Proc. Amer. Math. Soc., 142(7), 2014.

$\star$~  Danilenko - Lema\'{n}czyk, ``Odometer actions of the {H}eisenberg group'',  J. Anal. Math., 128, 2016.

 \vfill   

}


  \frame % 1
{
  \frametitle{}

Our interest is in group chains in $\cH$ which are   \underline{not normal}.

Here is a very useful    result:

{\bf Theorem:}
Let $\whGamma$ be a profinite completion of a finitely-generated nilpotent group $\G$. Then there is a topological isomorphism
$$
\whGamma \cong \prod_{p \in \pi(\Pi[\whGamma])} \ \whGamma_{(p)} \ ,
$$
where $\whGamma_{(p)} \subset \whGamma$ denotes the Sylow $p$-subgroup of $\whGamma$ for a prime $p$.
 
\bigskip

Thus the action of $\whGamma$ can be analyzed for each prime.

 Conversely, actions of $\cH$ can be constructed prime by prime.
  
  \vfill   

}

 

  \frame % 1
{
  \frametitle{}
 {\bf A model action of a finite $p$-group:}
 
  Fix a prime $p \geq 2$. 
  
  For $n \geq 1$ and  $0 \leq k < n$, we have the following finite groups: 
$$
G_{p,n} =   \left\{  \left[ {\begin{array}{ccc}
   1 & \oa & \oc\\
   0 & 1 & \ob\\
  0 & 0 & 1\\
  \end{array} } \right] \mid \oa,\ob,\oc  \in \mZ/p^{n}\mZ\right\}$$
  
$$  H_{p,n,k} =   \left\{  \left[ {\begin{array}{ccc}
   1 & p^k \oa & 0\\
   0 & 1 & 0\\
  0 & 0 & 1\\
  \end{array} } \right] \mid \oa  \in \mZ/p^{n}\mZ\right\}
$$

$$X_{p,n,k} = G_{p,n}/H_{p,n,k}$$
The isotropy group of the action of $G_{p,n}$ on $X_{p,n,k}$ at the coset $eH_{p,n,k}$ of the identity element is $H_{p,n,k}$. 


 \vfill   

}

  
       
  \frame % 1
{
  \frametitle{}
 
 {\bf Construction of a wild example:}
 
 Let $\pi_f $ and  $\pi_{\infty}$ be two 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 $1 \leq r_i \leq n_i$ 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 $p_{\ell} = 1$. 


For  each $\ell \geq 1$,  define the integers
\begin{eqnarray*}
M_{\ell}    & =   &  q_1^{r_1} q_2^{r_2} \cdots q_{\ell}^{r_{\ell}} \cdot p_1^{\ell} p_2^{\ell} \cdots p_{\ell}^{\ell} \  , \\
N_{\ell}    & =   &  q_1^{n_1} q_2^{n_2} \cdots q_{\ell}^{n_{\ell}} \cdot p_1^{\ell} p_2^{\ell} \cdots p_{\ell}^{\ell} \  .
\end{eqnarray*}

 
 \vfill   

}

    
     
\frame % 1
{
  \frametitle{}

 For $\ell \geq 1$, define a subgroup of $\cH$, in the coordinates above, 
$$
\cH_{\ell} = \{ (a M_{\ell},b N_{\ell} ,c N_{\ell}) \mid a,b,c \in \mZ \} \ , 
$$
 Its core  subgroup is given by 
$C_{\ell} =  \{ (a N_{\ell},b N_{\ell} ,c N_{\ell}) \mid a,b,c \in \mZ \}$. 

For   $k_i = n_i - r_i$ we then have
$$
 \whcH_{\infty} ~ \cong ~ \prod_{i=1}^{\infty} \ G_{q_i, n_i} ~ \cdot ~ \prod_{j=1}^{\infty} \ \whcH_{(p_j)} \quad , \quad D_{\infty} ~ \cong ~  \prod_{i=1}^{\infty} \ H_{q_i, n_i, k_i} \ . 
$$
  
 The   Cantor space  $X_{\infty} = \whcH_{\infty}/D_{\infty}$ associated to the group chain $\{\cH_{\ell} \mid \ell \geq 1\}$ is given by
$$
X_{\infty} ~ \cong ~ \prod_{i=1}^{\infty} \ X_{q_i, n_i, k_i} ~ \times ~ \prod_{j=1}^{\infty} \ \whcH_{(p_j)}  \ . 
$$


\vfill   

}


 
    
\frame % 1
{
  \frametitle{}
Let $x_i \in X_{q_i, n_i, k_i}$ denote the  coset of the identity element.

For   each $\ell \geq 1$, we define a clopen set in $X_{\infty}$ 
$$
U_{\ell} = \prod_{i=1}^{\ell} \ \{x_i\} ~ \times ~  \prod_{i=\ell+1}^{\infty} \ X_{q_i, n_i, k_i} ~ \times ~ \prod_{j=1}^{\infty} \ \whcH_{(p_j)} \ .
$$

$\bullet$~ This action is wild.

\medskip

$\bullet$~  If the set $\pi_{\infty}$ is empty, then the action is frothy as well.

\medskip

$\bullet$~ With proper choices of  integers $1 \leq r_i \leq n_i$ for $1 \leq i < \infty$, the action will be turbulent.
  
  \medskip
  
  Details of calculations and more examples are in the paper
  
   $\star$~ Hurder-Lukina, ``The prime spectrum of solenoidal manifolds'', 2021; arXiv:2103.06825


\vfill   

}


 
  


\end{document}

