% Operator Algebras Seminar
% University of Iowa
% February 18, 2020 
 
 
% slides for talk "Stable Cantor dynamics"
 
 
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\date[February 18, 2020]


  
\title{Stable Cantor dynamics}


\author{Steve Hurder, University of Illinois at Chicago\\   {\footnotesize  Joint work with}  \\ Olga Lukina, University of Vienna} 

 
 
\begin{document}
 
  \frame{\titlepage} % # 1

 \section{Intro}
    
  \frame % 1
{
  \frametitle{}

\bigskip
 Three approaches to the same subject:
 
 \medskip
 
$\bullet$~ Equicontinuous Cantor actions

\qquad Dynamical systems approach

\medskip

$\bullet$~ Group actions on rooted trees

\qquad Geometric group theory approach

\medskip

$\bullet$~ Clopen subset chains for profinite groups

\qquad Descriptive set theory approach

 \bigskip

Each approach has its own language. 

The approach of Lukina \& myself is a sort of  ``creole'', combining language and techniques from each of these three areas.
 \vfill   

}

  \frame % 1
{
  \frametitle{}

Cantor actions

\medskip

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

$\bullet$~ $\fX$ is a Cantor space

$\bullet$~ $\Phi \colon \G \times \fX \to \fX$ is a \underline{minimal} continuous action
 
\bigskip

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

 
 \vfill   

}

 
 \frame % 1
{
  \frametitle{}

When $G = \mZ$ then a minimal equicontinuous Cantor action $\Phi \colon \G  \times \fX \to \fX$ is conjugate to a classical odometer. 

\medskip
For a general group $\G$,  the action is called a (sub) odometer in  

$\bullet$ ~ [Cortez \&   Petite, {\it J. London Math Soc.}, 2008]

\medskip

$\bullet$ ~ [Cortez \&   Medynets, {\it J. London Math Soc.}, 2016]

\medskip

$\bullet$ ~ [Li, {\it Ergodic Theory Dynamical Systems},  2018]

  
 \vfill   

}

  
 
 \frame % 1
{
  \frametitle{}

General approach via group chains:  $\cG = \{ \G_{\ell} \mid \ell \geq 0\}$
 
 \medskip
  
  $\G_{\ell+1} \subset \G_{\ell}$ is proper inclusion with finite index.
   $$\G = \G_0 \supset  \G_1 \supset \G_2 \supset \G_3  \cdots $$
    Each quotient $X_{\ell} = \G/\G_{\ell}$ is a finite set with left $\G$-action.

   The Cantor space is 
$$\fX = \varprojlim \, \left\{ \G_0/\G_{\ell +1} \longrightarrow \G_0/\G_{\ell} \right\} ~ \subset ~ \prod_{\ell > 0} X_{\ell} $$
  
 Induces left $\G$-action $\Phi$ on  $\fX$ which   is minimal and equicontinuous.
 
 \bigskip
 
 {\bf Problem:} How to organize and classify Cantor actions?
 
 \vfill   

}
  
  \section{Equivalences}
  
 \frame % 1
{
  \frametitle{}

Topology of Cantor space $\fX$ is generated by   clopen subsets: \\U is   closed and open.
Non-empty clopen   $U \subset \fX$ is \underline{adapted} if the return times to $U$ form a subgroup 
$$\G_U = \{ g \in \G \mid \Phi(g)(U) = U  \} \subset \G$$
 
 \medskip
 
{\bf Definition:}
   $(\fX,\G,\Phi)$     a Cantor    action.
A properly descending chain of clopen sets $\cU = \{U_{\ell} \subset \fX  \mid \ell \geq 0\}$ is said to be an \underline{adapted neighborhood basis} at $x \in \fX$ for the action $\Phi$,   if
    $x \in U_{\ell +1} \subset U_{\ell}$ for all $ \ell \geq 0$ with     $\cap_{\ell > 0}  \ U_{\ell} = \{x\}$, and  each $U_{\ell}$ is adapted to the action $\Phi$.
 
 \medskip
 
{\bf Proposition:}
Let  $(\fX,\G,\Phi)$   be a Cantor    action. Given $x \in \fX$, there exists an adapted neighborhood basis $\cU$ at $x$ for the action $\Phi$.
 
 The defining group chain is given by $\cG = \{\G_{\ell} \equiv \G_{U_{\ell}} \mid \ell \geq 0\}$.


 \vfill   

}

 
 \frame % 1
{
  \frametitle{}

{\bf Isomorphism} (Rigidity Theory)
\bigskip

 {\bf Definition:} 
Cantor actions $(\fX_1, \G_1, \Phi_1)$ and $(\fX_2, \G_2, \Phi_2)$ are said to be \underline{isomorphic}  if there is a homeomorphism $h \colon \fX_1 \to \fX_2$ and group isomorphism $\Theta \colon \G_1 \to \G_2$ so that 
\begin{equation}\label{eq-isomorphism}
\Phi_1(g) = h^{-1} \circ \Phi_1(\Theta(g)) \circ h   \in   \Homeo(\fX_1) \  {\rm for \ all} \ g \in \G_1 \ .
\end{equation}
 
   
\vfill   

}

  
  
  
 \frame % 1
{
  \frametitle{}

{\bf Return equivalence} (Dynamical systems)

\bigskip
{\bf Definition:} Equicontinuous Cantor actions $\Phi_1 \colon \G_1 \times \fX_1 \to \fX_1$ and  $\Phi_2  \colon \G_2 \times \fX_2 \to \fX_2$ 
 are \underline{return equivalent} if there exist adapted clopen subsets $U \subset \fX_1$ and $V \subset \fX_2$,  such that the restricted actions $\Phi_{1,U} \colon \G_{1,U} \times U \to U$ and $\Phi_{2,V} \colon \G_{2,V} \times V \to V$ are isomorphic. 

\bigskip

This is weaker than isomorphism, even for $U = \fX$.

It loses information about the kernel of the action map
$\Phi_0 \colon \G \to \Homeo(\fX)$
  
\vfill   

}

  
  
  
 \frame % 1
{
  \frametitle{}

{\bf Continuous orbit equivalence} ($C^*$-algebras)


{\bf Definition:}
Let $(\fX_1, \G_1, \Phi_1)$ and $(\fX_2, \G_2, \Phi_2)$ be      Cantor actions.\\
A \underline{continuous orbit equivalence}  between the actions  is a homeomorphism    $h \colon \fX_1 \to \fX_2$ which is an orbit equivalence, and satisfies the locally constant properties:      
\begin{enumerate}
\item  for each $x \in \fX_1$ and $g \in \G_1$, there exists $\alpha(g,x) \in \G_2$ and an open set $x \in U_x \subset \fX_1$ s.t.
$ \Phi_2(\alpha(g,x))  \circ h | U_x = h \circ \Phi_1(g) | U_x$; \\
\smallskip
\item for each $y \in \fX_2$ and $k \in \G_2$, there exists $\beta(k,y) \in \G_1$ and an open set $y \in V_y \subset \fX_2$ s.t. 
$ \Phi_1(\beta(k,y))  \circ h | V_y = h \circ \Phi_2(k) | V_y$.
\end{enumerate}
 
The functions $\alpha \colon \G_1 \times \fX \to \G_2$ and $\beta \colon \G_2 \times \fX_2 \to \G_1$ are continuous, as the groups $\G_1$ and $\G_2$ have the discrete topology, but need not be cocycles over the actions.

  

\medskip

 $\bullet$~ Renault showed that    COE   is basic notion for isomorphism of  cross-product $C^*$ -algebras with Cartan subalgebra.
    
 \vfill   
  
}




\section{Examples}


 \frame % 1
{
  \frametitle{}
{\bf Diagonal Odometers}

 \medskip
  
  $\bullet$~ $\G = \mZ^k$ for $k \geq 1$
  
  $\bullet$~ Choose sequence of integer vectors $\vec{n}^{\ell} = (n_1^{\ell}, \ldots , n_k^{\ell})$,  $n^{\ell}_i > 1$
  
    $\bullet$~ Set $m^{\ell}_i = n^{\ell}_1 \cdot n^{\ell}_2 \cdots n^{\ell}_k$
    
   $\bullet$~ $\G_{\ell} = \{(m^{\ell}_1 n_1 , \ldots , m^{\ell}_k n_k) \mid (n_1, \ldots n_k) \in \mZ^k\}$

\bigskip

{\bf Random Odometers}

 \medskip
  
  $\bullet$~ $\G = \mZ^k$ for $k \geq 1$
  
 $\bullet$~ Choose sequence of integer matrices $A_{\ell} \in {\bf GL(k,\mZ)}$,  $\det A_{\ell} > 1$
  
     
   $\bullet$~ $\G_{\ell} = \{A_{\ell} A_{\ell-1} \cdots A_1 \cdot \vec{n} \mid \vec{n}  \in \mZ^k\}$

\medskip

In both cases, the inverse limit  $\fX$ is profinite group.

For $k =1,2$   classified by Giordano, Putnam \& Skau [2017]

 \vfill   

}
  
   \frame % 1
{
  \frametitle{}
{\bf Renormalizable}

 \medskip
  A countable group  $\G$   is \underline{finitely non-co-Hopfian} if there exists   a self-embedding $\vp \colon \G \to \G$ whose image is a proper subgroup of  finite index.
  $\vp \colon \G \to \G$   is called a \underline{renormalization} of $\G$
  
  If $\G$ admits a renormalization, then  it is  said to be  \underline{renormalizable}.
\medskip

The renormalization group chain $\cG_{\vp} = \{\G_{\ell} = \vp^{\ell}(\G) \mid \ell \geq 0\}$.

  Yields equicontinuous Cantor action $\Phi_{\vp} \colon \G \times X_{\vp} \to X_{\vp}$
  
$\cG_{\vp}$ is a scale for $\G$ if $\ds K(\cG_{\vp}) \ \equiv \ \bigcap_{\ell > 0} \ \G_{\ell}$ \quad is a finite group.

\medskip

{\bf Conjecture:} [Benjamini, Nekrashevych \& Pete] ~ If $\G$ admits a renormalization which defines a scale, then $\G$ is virtually nilpotent.
 \vfill   

}
 
   
   \frame % 1
{
  \frametitle{}
{\bf Heisenberg group:}

$$\G  \ = \ \left\{ [x,y,z] = \left[ \begin{array}{ccc} 1 & x & z \\ 0 & 1 & y \\ 0 & 0 & 1 \end{array}\right] \mid x,y,z \in \mZ \right\}$$
 {\bf Example 1:}
 
 $\bullet$~  Let $m > 1$. Define 
$\vp_1[x,y,z] = [mx,my,m^2z]$

$\bullet$~  $X_{\vp_1}$ is a profinite Heisenberg group, 

$\bullet$~ $\Phi_{\vp_1}$ is left multiplication by $\G$.
\bigskip

 {\bf Example 2:}
 
 $\bullet$~   $p,q > 1$   distinct primes. Define  $\vp_2[x,y,z] = [px,qy,pqz]$

$\bullet$~  $\ds X_{\vp_2}  \cong    \{[x,y,z] \mid x \in \widehat{\mZ}_{p} \ , \ y \in \widehat{\mZ}_{q} \ ,  \ z \in \widehat{\mZ}_{pq} \}$.


$\bullet$~ $\Phi_{\vp_2}$ is left multiplication by $\G$.
\bigskip


 
 \vfill   

}

\section{Profinite actions}

    \frame % 1
{
  \frametitle{}

$\bullet$~  Equicontinuous Cantor action $\Phi \colon \G \times \fX \to \fX$,  same  as homomorphism $\Phi_0 \colon \G \to \Homeo(\fX)$

$\bullet$~  $\whGamma = \overline{\Phi_0(\G)} \subset \Homeo(\fX)$ - closure in uniform topology

$\bullet$~  $\whGamma$ is separable profinite group, compact and totally disconnected

$\bullet$~ $\whPhi \colon \whGamma \times \fX \to \fX$ is transitive equicontinuous action

$\bullet$~ For $x \in \fX$, define $\cD = \{\whg \in \whGamma \mid \whPhi(\whg)(x) = x\}$

$\bullet$~ $\cD$ is called the \underline{discriminant} of the action $\Phi$

$\bullet$~ Isomorphism class of $\cD$ is independent of choice of $x$ and invariant of isomorphism of actions.

 

 \vfill   

}
  
 
  
 \frame % 1
{
  \frametitle{}
  $\bullet$~ If $\cD$ is the trivial group, then $\fX \cong \whGamma$ is a profinite group, and the action of $\Phi$ on $\fX$  is given by left multiplication on $\whGamma$.
  
  \medskip
  
  $\bullet$~ For $\G$ abelian, $\cD$ is always trivial.
  
  \qquad  Dynamics of action are studied using structures of profinite groups. If action is effective then it is fixed-point free.
  
  \bigskip
  
  $\bullet$~ There are fixed-point free actions for which $\cD$ is non-trivial.
  
  
   
  
   $\bullet$~  Study properties of equicontinuous actions with $\cD$ non-trivial.
  
  
  
  
$\bullet$~  $\cD$ acts via adjoint on $\whGamma$: ${\bf Ad}(\whh)(\whg) = \whh^{-1} \ \whg \ \whh$, $\whh \in \cD$, $\whg \in \whGamma$

\medskip

{\bf Problem:} Study properties of adjoint ${\bf Ad} \colon \cD \times \whGamma \to \whGamma$.

 \medskip
  
  
  

 \vfill   

}
  
 
  
 \frame % 1
{
  \frametitle{}
For an equicontinuous Cantor action $\Phi \colon \G \times \fX \to \fX$

 \begin{enumerate}
\item (Dyer Thesis 2016) $\cD$ can be a Cantor group for a Cantor action $(\fX, \G, \Phi)$ when  $\G$ is 3-dimensional Heisenberg group.
 \smallskip
\item ([DHL2016])  Every \underline{finite group} and every \underline{separable profinite group} can be realized as  $\cD$ for a Cantor action by   a torsion-free, finite index subgroup of  $ {\bf SL}(n,\mZ)$, $n \geq 3$.
 \smallskip
\item  ([Lukina2018a,Lukina2018b]) $\cD$ can be wide-ranging for arboreal representations of absolute Galois groups of number fields and function fields.
 \end{enumerate}

   \vfill   

}

\section{Regularity}


    \frame % 1
{
  \frametitle{}

 Cantor action $\Phi \colon \G \times \fX \to \fX$

$\bullet$~ is   \underline{effective}, or \emph{faithful},  if   $\Phi_0 \colon \G \to \Homeo(\fX)$ has  trivial kernel.
   
$\bullet$~ is \underline{free} if for all $x \in \fX$ and $g \in \G$,   $g \cdot x = x$ implies that $g = e$

$\bullet$~ \underline {isotropy group} of $x \in \fX$ is  $\G_x = \{ g \in \G \mid g \cdot x = x\}$


$\bullet$~  $\Fix(g) = \{x \in \fX \mid g \cdot x = x \}$, and   \underline{isotropy set}
$$ \Iso(\Phi) = \{ x \in \fX \mid \exists ~ g \in \G ~ , ~ g \ne id ~, ~g \cdot x = x    \} = \bigcup_{e \ne g \in \G} \ \Fix(g) $$
 
   
 
$\bullet$~  is \underline{topologically free}  if  $\Iso(\Phi) $ is meager in $\fX$. 
 
 \smallskip
\qquad If   $\Iso(\Phi)$   meager, then $\Iso(\Phi)$ has empty interior. 





 \medskip
  
  
  

 \vfill   

}
  

   \frame % 1
{
  \frametitle{}
{\bf Quasi-analytic} (General topological actions)

\medskip
  
  {\bf Definition:}    An action $\Phi \colon H \times \fX \to \fX$, where 
    
 $\bullet$~    $H$ is a topological group and  
 
 $\bullet$~  $\fX$ is a Cantor space   
 
 is  \underline{quasi-analytic}  if for each clopen set $U \subset \fX$,  $g \in H$ 
 
  $\bullet$~ 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.
 \vfill   

}
  

   \frame % 1
{
  \frametitle{}

{\bf Locally quasi-analytic} (Equicontinuous Cantor actions)

\medskip

{\bf Definition:}  
   An action $\Phi \colon H \times \fX \to \fX$, where 
   
    $\bullet$~    $H$ is a topological group and  
 
 $\bullet$~  $\fX$ is a Cantor space   
 
 is  \underline{locally quasi-analytic}  if   if there exists $\e > 0$ such that 
 
 $\bullet$~ for any adapted   set $U \subset \fX$ with $\diam (U) < \e$,  
 
  $\bullet$~  for any adapted subset $V \subset U$,  
  
   $\bullet$~ $g \in H$  satisfies $\Phi(g)(V) = V$ and the restriction $\Phi(g) | V$ is the identity map on $V$,  
     then $\Phi(g)$ acts as the identity on all of $U$.  
 
 \medskip
 
{\bf Definition:}  An equicontinuous  Cantor action $\Phi \colon H \times \fX \to \fX$ is   \underline{stable} if the associated profinite action  $\whPhi \colon \whGamma \times \fX \to \fX$ is locally quasi-analytic. The action  is said to be \underline{wild} otherwise. 

 

 \vfill   

}
  
\section{Adjoints}

   \frame % 1
{
  \frametitle{}


 

Equicontinuous Cantor action $\Phi \colon \G \times \fX \to \fX$ defined by 
group chain  $\cG = \{ \G_{\ell} \mid \ell \geq 0\}$, 
$\fX = \varprojlim \, \left\{ \G_0/\G_{\ell +1} \longrightarrow \G_0/\G_{\ell} \right\}$. 


 
   For  $k \geq 0$, define 
  \begin{eqnarray*} 
  U_k & = &  \{(g_0,g_1,g_2,\ldots) \in \fX \mid g_i=e \textrm{ for } 0 \leq i \leq k \} \\
  & =&  \varprojlim \ \{p_{\ell +1} \colon \G_k/\G_{\ell +1} \to \G_k/\G_{\ell} \mid \ell \geq k\} ~ , 
\end{eqnarray*}
which is a clopen subset of $X$ adapted to the action $\Phi$, with    stabilizer subgroup     $\G_{U_k} = \G_k$. Define
  $$\whU_k = \{\whg \in \whGamma \mid \whPhi(\whg)(U_k) = U_k$$
which is a clopen set in $\whGamma$, and $\ds \cD = \bigcap_{\ell> 0} \ \whU_k$.
  

 \vfill   

}
  
 
   \frame % 1
{
  \frametitle{}

$$\cK^{(k)} = \ker \ \left\{ \whPhi^{(k)} \colon \cD \to \cD^{(k)} \subset \Homeo(U_k) \right\}$$
  $$ C(\cG)  = \cK^{(0)} \subset \cK^{(1)} \subset \cK^{(2)} \subset \cK^{(3)} \subset \cdots $$

\medskip
  
  {\bf Proposition:}  $\Phi$ is stable iff the chain has an upper bound;  
  
  i.e. there exists $k_0 \geq 0$ such that $k > k_0$ implies $\cK^{(k)} = \cK^{(k_0)}$.
   
   \bigskip
   
   {\bf Definition:} If $\Phi$ is stable, then $\cD^s \equiv \whPhi^{(k)}(\cD) \cong \cD/\cK_{k}$ for $k \geq k_0$. This is called the \underline{stable discriminant} for the action.
   

 \vfill   

}
  
 
   \frame % 1
{
  \frametitle{}

\medskip

{\bf Definition:} An equicontinuous   Cantor action $\Phi \colon \G \times \fX \to \fX$ is \underline{wild} if the chain $\cK = \{\cK_0 \subset \cK_1 \subset \cK_2 \subset \cdots \}$ does not stabilize.

\medskip
$\bullet$~  The   examples of group actions on trees generated by automata studied by Nekrashevych, Bartholdi, Grigorchuk \emph{et al}  typically induce wild actions on the boundary of the trees.


\medskip
$\bullet$~ A  torsion-free, finite index subgroup  $ \G \subset {\bf SL}(n,\mZ)$, $n \geq 3$ has uncountably many wild equicontinuous Cantor actions which are pairwise not return equivalent - see  [HL2017].

Idea is to use isomorphism 
$$\widehat{{\bf SL}(n,\mZ)} \cong {\bf SL}(n,\widehat{\mZ}) \cong \prod_{p ~ prime} \ {\bf SL}(n, \widehat{\mZ}_p)$$
    
 \vfill   

}
  

 
   
 \section{Results}

  
    
\frame % 1
{

$\bullet$~ The dynamical properties  of equicontinuous Cantor actions are a lot more mysterious than one might expect.

\medskip

$\bullet$~ Analyze the cases:

\qquad $\star$~ action is stable;

\qquad $\star$~ $\G$ is virtually nilpotent; 

\qquad $\star$~ $\G$ is renormalizable.


\bigskip

$\bullet$~ Classification of wild actions is mostly unknown. Closely related to presence of non-Hausdorff elements for the action of $\whGamma$ on $\fX$.


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{\bf Stable actions}

 \medskip
 
{\bf Theorem [HL2018]:} The stable property    is invariant under continuous orbit equivalence.

\bigskip
{\bf Theorem [HL2019b]:}   Let     $(\fX,\G,\Phi)$ be a nilpotent Cantor action. Then the action is stable. 

\bigskip

{\bf Theorem [HLvW2020]:} The  Cantor action  $\Phi_{\vp} \colon \G \times X_{\vp} \to X_{\vp}$ associated to a renormalization $\vp \colon \G \to \G$ is quasi-analytic.  Hence, if the action $\Phi_{\vp}$ is also effective, then it is topologically free.   

 

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{\bf Rigidity}

\medskip

 {\bf Theorem [HL2018,HL2019b]:}  Let  $\Phi_1 \colon \G_1 \times \fX_1 \to \fX_1$ be a stable equicontinuous Cantor action. 
 Let $\Phi_2 \colon \G_2 \times \fX_2 \to \fX_2$ be a  Cantor action   which is  continuously orbit equivalent to $\Phi_1$.  


Then the action $\Phi_2$ is equicontinuous and stable, and the actions  $\Phi_1$ and $\Phi_2$ are return equivalent. 

\bigskip

$\bullet$~ The dynamics of a  stable equicontinuous Cantor action is essentially preserved by continuous orbit equivalence.
\vfill   

}


  
    
 
    
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{\bf Nilpotent actions}

\medskip

A finitely generated torsion-free nilpotent group $\G$ has many special properties:

$\star$~ $\G$ is Noetherian  (ascending group chains stabilize)

$\star$~ The profinite completion $\widehat{\G}$ is nilpotent and  torsion-free.

\medskip

 {\bf Theorem [HL2019b]:}  Let    $\Phi_1 \colon \G_1 \times \fX_1 \to \fX_1$  be an equicontinuous Cantor action with $\G$ virtually nilpotent. Suppose that the action is   continuously orbit equivalent to a   Cantor action  $\Phi_2 \colon \G_2 \times \fX_2 \to \fX_2$.  Then    
 
 $\bullet$~   The actions $\Phi_1$ and $\Phi_2$ are return equivalent. 
 
  $\bullet$~ If the action $\Phi_2$ is effective, then $\G_2$ is virtually nilpotent.  

  $\bullet$~ The stable discriminants of the two actions are isomorphic. 

 
 


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}


  
    
 
    
\frame % 1
{

{\bf Renormalizable groups}

 

Recall that    $\G$   is \underline{renormalizable} if there exists   a self-embedding $\vp \colon \G \to \G$ whose image is a proper subgroup of  finite index.
 
 This induces a very strong type of self-symmetry for the action $\Phi_{\vp} \colon \G \times X_{\vp} \to X_{\vp}$.

\medskip

 {\bf Theorem [HLvL2020]:}
  Let $\vp$ be a renormalization of the finitely generated group $\G$. Suppose that  
$$K(\cG_{\vp}) = \bigcap_{\ell > 0} \vp^{\ell}(\G)  \subset \G \quad , \quad  \cD_{\vp} =   \bigcap_{n > 0} \ \whvp_0^n(\whGamma_{\vp}) \subset \whGamma_{\vp}$$
are both finite groups,  then 

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

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



\vfill   

}


  
    
 
    
\frame % 1
{


Key point in the proof is to use results by Colin Reid on self-embeddings of profinite groups:

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

\bigskip

Extensive literature of profinite subgroups with self-embeddings.

One expects there are many more ideas in these works to exploit.
\bigskip

{\bf Question:} What are the implications of the above results for the $C^*$-algebras associated to equicontinuous Cantor actions?
 

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\frame % 1
{
  \frametitle{References}
 {\scriptsize    
 
  

  
 [DHL2015] {J.~Dyer, S.~Hurder and O.~Lukina},
 {\it The discriminant invariant of Cantor group actions},
 {\bf Topology and Its Applications},  208 (2017), 64--92; {arXiv:1509.06227}.
 
  
[DHL2016] {J.~Dyer, S.~Hurder and O.~Lukina},
 {\it Molino theory for matchbox manifolds},
 {\bf Pac. Jour. Math.},  289 (2017), 91--151; {arXiv:1610.03896}. 
 
 [HL2017]  {S.~Hurder and O.~Lukina},
 {\it Wild solenoids},
 {\bf Trans. A.M.S.},  371 (2019), 4493--4533; {arXiv:1702.03032}
 
 
[HL2018] {S.~Hurder and O.~Lukina},
 {\it Orbit equivalence and classification of weak solenoids},
 {\bf Indiana Univ. Math. J.}, to appear; {arXiv:1803.02098}.

[HL2019a] {S.~Hurder and O.~Lukina},
 {\it Limit group invariants for non-free Cantor actions},
 {\bf Ergodic Theory Dynamical Systems}, to appear; {arXiv:1904.11072}.

[HL2019b]  {S.~Hurder and O.~Lukina},
 {\it Classification of nilpotent Cantor actions},
 {\bf under revision}; {arXiv:1905.07740}.

[HLvL2020] {S.~Hurder, O.~Lukina and W.~Van Limbeek},
 {\it Cantor dynamics of renormalizable groups},
 {\bf submitted}; {arXiv:2002.01565}.

[Lukina2018a] {O.~Lukina}, {\it Arboreal Cantor actions}, 
{\bf J. Lond. Math. Soc. (2)}, 99:678--706, 2019; {arXiv:1801.01440}.

     
[Lukina2018b] {O.~Lukina}, {\it Galois groups and {C}antor actions},
 {submitted}; 2018, {arXiv:1809.08475}.
 
}  

 
 

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