% 2020 Vision for Dynamics
% https://www.impan.pl/en/activities/banach-center/conferences/19-vision2020
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% August 12-17, 2019 (Monday - Friday) 
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% slides for talk "Rigidity for Cantor Actions"
 
 
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\date[April 12, 2018]


  
\title[Cantor Actions]{Rigidity and Classification of Cantor Actions}


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

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

  \section{Intro}


 \frame % 1
{
  \frametitle{}
\begin{center}
KatokFest 2004 - Berkeley
 \medskip
 
 \includegraphics[width=0.8\textwidth]{KatokFest2004}  
 \end{center}
 
 \vfill   

}

     
 
\frame % 1
{
  \frametitle{}

In this talk, describe program of research that combines ideas from the programs at the M.S.R.I. during the \emph{excellent} years 1983-85.

\bigskip

  {\bf Problem:} Classify weak solenoids, up to homeomorphism.
     
\medskip

{\bf Solution:} Classify arboreal actions of finitely-generated groups, \\ up to return equivalence.

\medskip

$\bullet$ ~ If $G$ is Noetherian group $\Rightarrow$ \emph{Rigidity Property}.

$\bullet$ ~ If $G$ admits uncountably many subgroups $\Rightarrow$  \emph{Wild Actions}.


$\bullet$ ~ Related to properties of \emph{Invariant Random Subgroups}.

$\bullet$ ~ Applications to \emph{Arithmetic Number Theory}.

$\bullet$ ~ Variety of \emph{Open Problems}.
 \vfill   

}

\section{Solenoids}

 \frame % 1
{
  \frametitle{}

  
 $M_0$   a compact manifold without boundary.
 
 Given a sequence of proper non-trivial coverings, set
 \begin{eqnarray*}
 \lefteqn{ \fM = \varprojlim ~ \{ p_{\ell+1}^{\ell}\colon M_{\ell+1} \to M_{\ell}  \mid \ell \geq 0 \}}  \\
& =&  \{ (y_0, y_1, y_2, \ldots) \mid p_{\ell+1}^{\ell}(y_{\ell+1}) = y_{\ell}  \mid \ell \geq 0\} \\
&  \subset & \prod_{\ell \geq 0} ~ M_{\ell}
\end{eqnarray*}
  $\bullet$ ~ $\fM$ is a  \emph{(weak) solenoid}, with foliation $\F_{\fM}$. 
  
   $\bullet$ ~ Leaves are the path connected components of $\fM$, which are non-compact covering spaces of $M_0$
 
 $\bullet$ ~  $(\fM, \F_{\fM})$ is a \emph{generalized lamination} $\equiv$ \emph{foliated space} with Cantor transversals.
     

 \vfill   

}


 \frame % 1
{
  \frametitle{}

 Everybody's favorite example:  Vietoris solenoids
 
    \begin{align*}  \xymatrix{
 \mS^1   &   \ar[l]_{m_1} \mS^1    &  \ar[l]_{m_2} \mS^1   & \ar[l]_{m_3} \mS^1  & \ar[l]_{m_4} \mS^1    \cdots     
 } 
\end{align*}

 
Each $m_{\ell} \colon \mS^1 \to \mS^1$ is an $m_{\ell}$-fold covering map, $m_{\ell} > 1$.

  
  \medskip

 \hspace{60pt} \includegraphics[width=0.5\textwidth]{Smale-Williams_Solenoid_Large} 
 

  

 \vfill   

}
 

  
 \frame % 1
{
  \frametitle{}

 \medskip
  
 $\bullet$ ~ $G = \pi_1(M,x_0)$ is finitely generated group.
 $$M_0 \stackrel{p_1}{\longleftarrow} M_1 \stackrel{p_2}{\longleftarrow} M_2 \stackrel{p_3}{\longleftarrow} M_3  \cdots $$
 Choose $x_{\ell} \in M_{\ell}$ with $p_{\ell}(x_{\ell}) = x_{\ell -1}$,   set $G_{\ell} = \pi_1(M_{\ell}, x_{\ell})$
 
Inclusion maps $q_{\ell +1} \colon G_{\ell +1} \subset G_{\ell}$, descending chain of groups
  $$G = G_0 \stackrel{q_1}{\longleftarrow} G_1 \stackrel{q_2}{\longleftarrow} G_2 \stackrel{q_3}{\longleftarrow} G_3  \cdots $$
$$\fX = \varprojlim \, \left\{ G_0/G_{\ell +1} \longrightarrow G_0/G_{\ell} \right\} $$
 
 The left $G$ action $\Phi$ on Cantor set $\fX$ is conjugate to monodromy action on transversal in $\fM$.
 Action is minimal and equicontinuous.
 \vfill   

}
  
   
  
   
 \frame % 1
{
  \frametitle{}

$\bullet$ ~   $(\fX, G, \Phi)$ is a Cantor action if $\fX$ is Cantor set, and action is minimal and equicontinuous. This is equivalent to an action of $G$ on a pointed tree, or arboreal action.

  $\bullet$ ~ Clopen set $U \subset \fX$ is adapted if the stabilizer is a subgroup  
  $$G_U = \{ g \in G \mid \vp(g)(U) = U\}$$
  $\bullet$ ~ $G_U$ has finite index, and acts transitively on the finite set of translates $\{g \cdot U \mid g \in G\}$ (the vertices in a tree model)
 
  $\bullet$ ~ $\cH^{\Phi}_U \equiv \left\{\Phi(g)|U \mid g \in G_U \right\} \subset \Homeo(U)$.
  
 \medskip
   
   {\bf Definition:} 
  Equicontinuous Cantor actions $(\fX, G, \Phi)$  and $(\fY, H, \Psi)$ are \emph{return equivalent} if there exists adapted sets $U \subset \fX$ and $V \subset \fY$ and a homeomorphism $h \colon U \to V$ which conjugates the groups $\cH^{\Phi}_U \subset \Homeo(U)$ and $\cH^{\Psi}_V \subset \Homeo(V)$.

 


  \vfill   

}
  
   
   

   
 \frame % 1
{
  \frametitle{}
 
{\bf Theorem:} Given weak solenoids $\fM$ and $\fM'$   with monodromy actions $(\fX, G, \Phi)$  and $(\fY, H, \Psi)$, if $\fM$ and $\fM'$ are homeomorphic, then their monodromy actions are return equivalent.

\bigskip

There is a not a converse to this, in general, except in special cases:

{\bf Definition:} $\fM$ is a nil-solenoid if the base manifold $M_0$ is a compact nil-manifold, and  $\F_{\fM}$ has a simply connected leaf.

\bigskip

We then have a generalization of the classification result for $1$-dimensional solenoids by  Aarts and Fokkink.

\medskip

{\bf Theorem:} Let $\fM$ and $\fM'$ be nil-solenoids. If their monodromy actions are return equivalent, then the spaces are homeomorphic.



 \vfill   
  
}

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

$\bullet$ ~ Classify   Cantor actions up to return equivalence.

$\bullet$ ~ Give   invariants of return equivalence.
 
 \bigskip
 
 
 
 {\bf Definition:}   $\Phi$ is \emph{locally quasi-analytic}   (LQA)  if there exists $\e > 0$ so that if $U$ adapted and $\diam_{\fX}(U) < \e$,   for all clopen $V \subset U$,   
 $$\Phi(g) | V = Id ~ \Longrightarrow ~  \Phi(g) | U = Id \quad , \quad {\rm for ~ all} ~  g \in G_U$$
 That is, the action of $\cH^{\Phi}_U$ on $U$ is  \emph{topologically free}.
 

 \vfill   
  
}

  
 \frame % 1
{
  \frametitle{}


{\bf Definition:} Cantor actions $(\fX, G, \Phi)$  and $(\fY, H, \Psi)$ are \emph{continuously orbit equivalent} (COE)  if there exists a homeomorphism $h \colon \fX \to \fY$ and continuous functions

\medskip

 $\alpha \colon G \times \fX \to H$,  $h(\Phi(g)(x)) = \Psi(\alpha(g , x),  h(x))$,   $g \in G, x \in \fX$ 

\medskip

 $\beta \colon  H \times \fY \to G$,  $h^{-1}(\Psi(g, y)) = \Phi(\beta(g, y), h^{-1}(y))$, $g \in H, y \in \fY$
 
 \medskip

  Actions are   \emph{locally continuously orbit equivalent} (LCOE) if there exists adapted   subsets $U \subset \fX$ and $V \subset \fY$ such that the restricted actions  are continuously orbit equivalent.

\bigskip

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


  
 \frame % 1
{
  \frametitle{}
 
$\bullet$~ Extend Cortez \& Medynets \emph{rigidity theorem} for free actions:
 
\medskip

  {\bf Theorem:}  Let $(\fX,G,\Phi)$ and $(\fY,H,\Psi)$ be LQA   Cantor actions.   
\begin{center}
Locally   Continuously   Orbit   Equivalent~  $\Leftrightarrow$ ~  Return ~ Equivalent
\end{center}
  
   
   
  \medskip
  
   {\bf Definition:} $G$ is \emph{Noetherian} if every subgroup $G' \subset H$ is finitely generated.  
  
   \medskip
   
     {\bf Theorem:}      $G$    Noetherian $\Rightarrow$ every   Cantor action of $G$ is   LQA.  

 \medskip
   
     {\bf Corollary:} Cantor  actions by Noetherian groups satisfy orbit equivalence rigidity.
     
  \vfill   

}

   
\section{Profinite models}


  \frame % 1
{
  \frametitle{ }

   
  
  
 {\bf Definition:}  $\oG = \overline{H_{\Phi}}$    is   closure of $H_{\Phi} = \Phi(G) \subset \Homeo(\fX)$   \\
   in the \emph{uniform topology on maps}.  
   
   $\oG$ is a profinite group acting transitively on $\fX$.


\medskip

  $\cD_x = \oG_x = \{ \whh \in \overline{H_{\Phi}} \mid \whh \cdot x = x \}$     (isotropy group of $x \in \fX$)
 
 
  $\cD_x \subset \oG$   is independent of the choice of basepoint $x$, up to topological isomorphism. 
  
   
  
  $\cD_x$ acts effectively on $\fX$.

 \medskip
 
  
The action $(\fX, \oG, \whPhi)$ is called the  \underline{profinite model} for $(\fX, G, \Phi)$. 
 
      \vfill   

}

  
 
  \frame % 1
{
  \frametitle{ }

 
 
  
{\bf Proposition:} $\cD_x$ is \emph{totally not normal}: for any $\whh \in \cD_x$ there exists $\whg \in \oG$ such that $\whg^{-1} \ \whh \ \whg \not\in \cD_x$.

 
  
 
\medskip
  For group chain $\cG = \{G_{\ell} \mid \ell \geq 0\}$    the  \underline{normal core} of $G_{\ell}$ in $G$
     $$ C_{\ell} =  \bigcap_{g \in G} gG_{\ell} g^{-1}  \subset G_{\ell}$$

 
    
  {\bf Theorem [Dyer-Hurder-Lukina, 2016].}  
   \begin{equation*} 
  \cD_x \cong \varprojlim ~ \{\pi_{\ell+1} \colon  G_{\ell+1}/C_{\ell +1 } \to G_{\ell}/C_{\ell} \mid \ell \geq 0\} ~ .
\end{equation*}

\medskip

$\bullet$ ~ This result is   abstract, but in practice is often effective for calculating the discriminant group $\cD_x$ for a given group chain.

\vfill   

}

 
  
 \frame % 1
{
  \frametitle{}
 
{\bf Recipe for Cantor actions:} 

\medskip

$\star$ ~ Take one finitely-generated group $G$.

\medskip

$\star$ ~  Choose  a profinite completion $\oG$ of $G$.

\medskip

$\star$ ~ Choose totally not normal closed subgroup $\cD \subset \oG$.

\medskip

 Then action of $G$ on $\fX \equiv \oG/\cD$ is minimal and equicontinuous. 
 


\vfill   

}

 
  
 \frame % 1
{
  \frametitle{}

 \begin{enumerate}
\item $\cD_x$ is trivial for   Cantor action $(\fX, G, \Phi)$   with $G$ abelian.
\smallskip
\item $\cD_x$ can be a Cantor group for a Cantor action $(\fX, G, \Phi)$ when  $G$ is 3-dimensional Heisenberg group.
 \smallskip
\item  Every \underline{finite group} and every \underline{separable profinite group} can be realized as  $\cD_x$ for a Cantor action by   a torsion-free, finite index subgroup of  $ {\bf SL}(n,\mZ)$.
 \smallskip
\item  $\cD_x$ can be wide-ranging for arboreal representations of absolute Galois groups of number fields and function fields.
 \smallskip
\item Every Cantor action by a finitely generated group $G$ can be realized as the monodromy of a weak solenoid.
\end{enumerate}

\medskip

The proof of $3$ uses ideas of   {\bf Lubotzky} on torsion elements in the profinite completion of torsion free subgroups of ${\bf SL}(n,\mZ)$, and a construction due to {\bf Lenstra}.

 
  \vfill   

}

 \section{Non-LQA} 
   
 
   




 \frame % 1
{
  \frametitle{}

{\bf Problem:} If $G$ is not Noetherian, then how to classify its actions?

\medskip

Let   $(\fX, G, \Phi)$ be a Cantor action which is not LQA.

  Choose an \emph{adapted  neighborhood basis} $\{U_{\ell} \mid \ell \geq 1\}$ of   $x \in \fX$.
 
Set  $G_0 = G$, $G_{\ell} = G_{U_{\ell}}$ for $\ell \geq 1$.
 
 Then $\cG = \{G_{\ell} \mid \ell \geq 0\}$ defines $\fX$ as inverse limit space.
 
 
  Set $K_{\ell} = \left\{ g \in G \mid  \Phi(g) | U_{\ell} = Id \right\}$.
  
 $K_{\ell} \subset K_{\ell +1}$ so  $\fK^x = \{K_{\ell} \mid \ell \geq 1\}$ is increasing chain.
  
  
   \medskip
   
   {\bf Theorem:}   $(\fX, G, \Phi)$  is LQA $\Leftrightarrow$ $\{K_{\ell} \mid \ell \geq 1\}$ is bounded.
    
   \vfill   

}

 
 \frame % 1
{
  \frametitle{}

{\bf Theorem:} If Cantor action $(\fX, G, \Phi)$  is not LQA, then $G$ admits uncountably many subgroups. 

\medskip

{\it Sketch of   proof:}

Let $\fG = \{U_{\ell} \mid \ell \geq 1\}$ be the adapted  neighborhood basis of   $x \in \fX$.

For any $y \in \fX$, there is $\whg \in \oG$ with $\whg \cdot x = y$.

$\whg = (g_0, g_1 , g_2, \ldots)$ with $g_{\ell} \in G$ and  $g_{\ell} C_{\ell} = g_{\ell+1}C_{\ell}$

$C_{\ell}    \subset G_{\ell}$ is the normal core subgroup.

Then $\fG^y =  \{g_{\ell} \cdot U_{\ell} \mid \ell \geq 1\}$ is adapted neighborhood basis of $y$.

$\fK^y = \{K^y_{\ell} = g_{\ell} \ K_{\ell} \ g_{\ell}^{-1} \mid \ell \geq 1\}$ is increasing subgroup chain.

$K_{\infty}^y = \bigcup_{\ell \geq 1} \ K^y_{\ell}$ is infinitely generated if $\fK^x$ is not bounded, and 

  $K_{\infty}^y \ne K_{\infty}^z$ if  $y \ne z$.

   \vfill   

}

 
 \frame % 1
{
  \frametitle{}


 {\bf Bartholdi, Grigorchuk, Nekrashevych, et al:} ~ Examples of weakly branch group actions on trees       induce non-LQA actions on the Cantor boundary of a $d$-ary tree, $d \geq 2$.

\bigskip
 
  

 {\bf  Lukina:}
Let $f(x)$ be a quadratic polynomial with critical point $c$.
  If the post-critical set $P_C$ contains at least 3 points, then the action of ${\rm Gal}_{\rm geom}(f)$  and   ${\rm Gal}_{\rm arith}(f)$  on the boundary of the  tree formed by iterated  solutions is non-LQA.

\bigskip

{\bf  Groeger \& Lukina:} If Cantor action $(\fX, G, \Phi)$  is not LQA, then the push-forward of an ergodic measure on $\fX$ via the mapping $x \mapsto G_x$ is a continuous (non-atomic) I.R.S. This is a broader class of examples than just weakly branch actions.

 

\medskip

$\bullet$ ~Thus, can use \emph{Ergodic Theory}/\emph{Descriptive Set Theory} to classify non-LQA Cantor actions.

   \vfill   

}

  \section{Wild models} 


 \frame % 1
{
  \frametitle{}

There is another approach to the study of non-LQA Cantor actions.

Recall:  $\oG = \overline{H_{\Phi}}  \subset \Homeo(\fX)$   
 
 $\cD_x = \oG_x$     (isotropy group of $x \in \fX$)
 
 $\cD_x \subset U$ for all adapted $x \in U \subset \fX$ 
 
 $\fG = \{U_{\ell} \mid \ell \geq 1\}$   adapted  neighborhood basis of   $x \in \fX$.
 
$\Longrightarrow$ ~    $\cD_x \subset U_{\ell}$ for all $\ell \geq 1$.
 
 
 
$\whPhi_{\ell} \colon G_{\ell} \times U_{\ell} \to U_{\ell}$ induces a local   action map   
$$\rho_{\ell} \colon \cD_x \to \Homeo(U_{\ell})$$

\medskip

 Set $\whK_{\ell} \equiv \ker \{\rho_{\ell}\} \subset \oG$ for $\ell \geq 1$. Then $\whK_1 \subset \whK_2 \subset \cdots$

  \vfill   

}

    
  
 \frame % 1
{
  \frametitle{}

 
 

 
 {\bf Theorem:} The isomorphism class of the direct limit group 
 $$\Upsilon(\Phi) = \varinjlim \ \{ \whK_{\ell} \subset \whK_{\ell+1} \mid \ell \geq 1\}$$
 is a conjugacy invariant of a Cantor action  $(\fX, G, \Phi)$.
 
\medskip
 A Cantor action  $(\fX, G, \Phi)$ is:
 
 $\bullet$ ~  \underline{stable} if the  chain $\{\whK_{\ell} \mid \ell \geq 1\}$  is bounded.
 
That is,  if there exists $\ell_0$ so that $\whK_{\ell} = \whK_{\ell+1}$ for $\ell \geq \ell_0$. 

 

 $\bullet$ ~    \underline{wild} if the  chain $\{\whK_{\ell} \mid \ell \geq 1\}$  is unbounded.  
 
 \medskip
 
 {\bf Theorem:} The property that a Cantor action is wild, is    a locally continuous orbit equivalence invariant.


  \vfill   

}

    
  
 \frame % 1
{
  \frametitle{}

The class of wild actions can be divided into two subclasses.

\medskip

Let $\whU_{\ell} = \{ \whg \in \oG \mid \whPhi(\whg) \cdot U_{\ell} = U_{\ell} \}$

The collection $\{\whU_{\ell} \mid \ell \geq 1\}$ is a neighborhood basis of clopen sets about the identity $\whe \in \oG$.
Consider the restricted Adjoint action 
$$\whrho_{\ell} \colon \cD_x \to \Homeo(\whU_{\ell})$$

\medskip

Set $\whK^a_{\ell} \equiv \ker \{\whrho_{\ell}\} \subset \oG$ for $\ell \geq 1$.
Then $\whK^a_1 \subset \whK^a_2 \subset \cdots$

Observe that $\whK^a_{\ell} \subset  \whK_{\ell}$ for all $\ell \geq 1$.

\medskip

 Analogous to homogeneous space $M = G/K$ where geometry of model $M$ is studied via adjoint representation of $K$ on the Lie algebra $\mathfrak{g}$ of $G$. Local model for Cantor action may be unstable. 


  \vfill   

}

    
  
 \frame % 1
{
  \frametitle{}

{\bf Definition:} A wild Cantor action $(\fX, G, \Phi)$ is:
 
  
 $\bullet$ ~    \underline{flat wild} if   $\whK^a_{\ell} = \whK_{\ell}$  for $\ell$ sufficiently large.  

$\bullet$ ~    \underline{dynamically wild} if    $\whK^a_{\ell} \ne  \whK_{\ell}$   for $\ell$ sufficiently large.

\bigskip

 $\bullet$ ~ The Cantor actions associated to weakly branched groups are \emph{dynamically wild}.

\medskip

 $\bullet$ ~ There exists  lattices   $G \subset {\bf SL}_N(\mZ)$ with actions on a Cantor space $\fX$ that are \emph{flat wild}.
 
 \medskip

 $\bullet$ ~ In the dynamically wild case, it is not known if the chain  $\{\whK^a_{\ell} \mid \ell \geq 1\}$ must be unbounded as well.
  
  \vfill   

}


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

 
  {\bf Question 1:} How to classify Cantor actions of  a finitely-generated nilpotent group $G$?
  
  Use invariants of the associated cross-product $C^*$-algebra? \\ In terms of the representations of $G$?
  
 
\medskip
{\bf Question 2:} If an action is wild, when is the action    non-LQA?
 
  
\medskip


 {\bf Question 3:} For which numbers fields and polynomials $f$  is the action of the absolute Galois group ${\rm Gal}_{\rm arith}(f)$,  on the boundary of the    tree  of iterated solutions, non-LQA?
 
\medskip


 

 {\bf Question 4:} If $G$ is a higher rank lattice and the action is effective, must it be stable?
 
 
\medskip


 

 {\bf Question 5:} If $G$ is a higher rank lattice and the action is wild, must it be flat wild?
 
 



  \vfill   

}



   

   
    
\frame % 1
{
  \frametitle{References}
 {\tiny    
 
 
{N.~Boston and R.~Jones},
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}  
}

 \section{HBD}
 
 

 
  


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