% Colloquium - IUPUI
% Date: October 19, 2018
% Place: Indianapolis, IN

% slides for talk "Geometry and Dynamics of Solenoids"
 
 
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\date[April 12, 2018]


  
\title[Wild things]{Geometry and Dynamics of Solenoids}


\author{Steve Hurder\\   {\scriptsize joint work with  Olga Lukina}  \\ \footnotesize {University of Illinois at Chicago} }

 
 
\begin{document}

\frame{\titlepage} % # 1


  
 
  
  
 \frame % 1
{
  \frametitle{}

 \medskip
 
This is a story of how an example leads to new discoveries. 

\medskip

$\bullet$ ~ The  Smale/Vietoris  solenoid, and its properties.

\medskip

$\bullet$ ~ Generalizations of   (1-dimensional) solenoids \\
\quad \quad (coverings of the Klein bottle)

\medskip

$\bullet$ ~Modeling a solenoid by group actions on rooted trees \\
\quad \quad (generalized odometers)

 \medskip

$\bullet$ ~ Local invariants of odometers  
\\
\quad \quad (non-Hausdorff elements)

 \medskip

$\bullet$ ~Applications to number theory
\\
\quad \quad (wild Galois groups)

 \vfill   

}

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

\begin{center}
 Visualize an embedding of a solid torus into itself
 \end{center}
 
 \bigskip
 \bigskip
 
 \hspace{60pt} \includegraphics[width=0.6\textwidth]{Smale_Solenoid} 
 

  

 \vfill   

}
 

 \frame % 1
{
  \frametitle{}

 \begin{center}
 After a few repetitions we get something like this
 \end{center}
 
 \bigskip
 \bigskip

 \hspace{60pt} \includegraphics[width=0.6\textwidth]{Smale-Williams_Solenoid_Large} 
 

  

 \vfill   

}
 
 
 \frame % 1
{
  \frametitle{}


\begin{center}
  This is modeled  by a sequence of coverings of the circle
   \end{center}

      
    \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  & \ar[l]_{m_5} \cdots     
 } 
\end{align*}

$\star$ ~ each $m_{\ell} > 1$ is an integer 

$\star$ ~ $m_{\ell} \colon \mS^1 \to \mS^1$ is an $m_{\ell}$-fold covering map.
  
  \bigskip
  
  In the embedding construction, each $m_{\ell} = 2$, but no reason to restrict the degrees this way.
   
 \vfill   
  
}
  
   
 \frame % 1
{
  \frametitle{}


\begin{center}
 Repeat this covering construction infinitely often.\\ The inverse limit space is a Vietoris solenoid (1927)
   \end{center}
 \begin{eqnarray*}
\cS   & \equiv & \varprojlim \,\{\, m_{\ell+1} \colon  \mS^1 \to \mS^1 \mid     \ell \geq 0\}\\
& & \\
& \equiv & \{(x_0, x_1, \ldots ) \in \cS   \mid m_{\ell+1}(x_{\ell + 1}) =  x_{\ell} ~ {\rm for ~ all} ~ \ell \geq 0 ~\} \\
& & \\
&  \subset & \prod_{\ell \geq 0} ~ \mS^1
\end{eqnarray*}

$\star$ ~ Give the product space the product or Tychonoff topology;

$\star$ ~  $\cS$ has the restricted topology;

$\star$ ~ $\cS$ is a compact, connected metric space, so a continuum.

 \vfill   
  
}


   
 \frame % 1
{
  \frametitle{}

\begin{center}
Properties of the space $\cS$
   \end{center}

 
$\star$~ $\cS$ is an uncountable union of lines, each dense in $\cS$! 

\quad    $\cS$ can be thought of as a bundle of wires, tightly coiled


 \bigskip
 
 \hspace{60pt} \includegraphics[width=0.6\textwidth]{coil} 

 \vfill   
  
}

  
 \frame % 1
{
  \frametitle{}

\begin{center}
Properties of the space $\cS$
   \end{center}

 
   
 
 $\star$~ Whatever  stage $k \geq 0$ you start at, the limit is the same:
$$\cS  \cong  \varprojlim \,\{\, m_{\ell+1} \colon  \mS^1 \to \mS^1 \mid     \ell \geq k\}$$

 \smallskip
 
 $\star$~ $\cS$ is homogeneous - 
  for any pair of points $x,y \in \cS$ there is a homeomorphism $h \colon \cS \to \cS$ with $h(x) = y$.
 
  \bigskip
  $\star$~  Lastly,   recall  results  of
 
 $\bullet$~ R.H.~Bing,    Canad. J. Math., 1960\\
 $\bullet$~ M.C.~McCord, Transactions A.M.S, 1965


 \vfill   
  
}

  
 \frame % 1
{
  \frametitle{}

  {\bf Theorem:}   Let $\vec{p} = \{p_1 , p_2, p_3, \ldots \}$ and $\vec{q} = \{q_1 , q_2, q_3, \ldots \}$ be sequences of prime numbers. Say that  $\vec{p} \sim  \vec{q}$ if there is $\ell_p , \ell_q \geq 0$ so that the two sets 
 $$\{ p_{\ell} \mid \ell \geq \ell_p \} = \{ q_{\ell} \mid \ell \geq \ell_q \}$$
 Then the solenoids they generate, call them $\cS(\vec{p})$ and $\cS(\vec{q})$, are homeomorphic if and only if $\vec{p} \sim  \vec{q}$.
 
 
 \bigskip
 
 That is pretty much the end of the story - for 1-dimensional spaces!

\bigskip
{\bf Question:} Which of these properties are valid, if we repeat this procedure for coverings of   manifolds with dimension $> 1$?
  
 \vfill   
  
}


 \frame % 1
{
  \frametitle{}

\bigskip

Here is a nice   example due to [Rogers \& Tollefson, 1971/72], obtained by coverings of the Klein bottle.
\bigskip

\pgfputat{\pgfxy(-1.6, -3.4)}{\pgfbox[left,base]{\pgfimage[width=7.0cm]{Kleinbottle.png}}}\pgfputat{\pgfxy(4, 0)}{\pgfbox[left,base]{Let $\mathbb{T}^2 = \mathbb{S}^1 \times \mathbb{S}^1$, and consider an involution}}\pgfputat{\pgfxy(4, -0.6)}{\pgfbox[left,base]{\phantom{---------}$r \times i (x,y) = ( x+ \frac{1}{2}, -y).$}} \pgfputat{\pgfxy(4, -1.2)}{\pgfbox[left,base]{The quotient $K = \mathbb{T}^2/ (x,y) \sim r \times i(x,y)$}}\pgfputat{\pgfxy(4, -1.8)}{\pgfbox[left,base]{ \phantom{---------------------}is the Klein bottle. }} \pgfputat{\pgfxy(4, -2.4)}{\pgfbox[left,base]{ The double cover $L\colon \mathbb{T}^2 \to \mathbb{T}^2\colon (x,y) \mapsto (x,2y)$ }} \pgfputat{\pgfxy(4, -3.0)}{\pgfbox[left,base]{ induces a double cover $p: K \to K$.}} 

\vspace*{4cm}
Define $K_\infty$ to be the inverse limit of the iterations of $p: K \to K$.

\medskip

Since $ i \circ L =  p \circ i$,  there is a double cover $i_\infty: \mathbb{T}_\infty \to K_\infty$.

\medskip

Space $K_\infty$ cannot be homogeneous, as there is exactly one leaf with $\mZ/2\mZ$ as isotropy group for the monodromy action.
 

\vfill   

}


  

   
 \frame % 1
{
  \frametitle{}


 
Let $M_0$ be a connected closed manifold, and  let $f^{\ell}_{\ell+1}: M_{\ell +1} \to M_{\ell}$ be a sequence of finite-to-one proper covering maps. 

Compose these coverings  to get
$$f^0_{\ell+1} \colon  M_{\ell +1} \to M_0$$
Pick basepoints $x_{\ell+1} \in M_{\ell +1}$ with $x_{\ell} = f^{\ell}_{\ell+1}(x_{\ell +1})$.
Then 
$$G_{\ell} = {\rm Image} \left\{ (f^0_{\ell+1})_{\#}  \colon  \pi_1(M_{\ell +1}, x_{\ell+1}) \to \pi_1(M_0 , x_0) \equiv G_0  \right\}$$
is the subgroup for which $M_{\ell+1}$ is the associated covering space.

{\bf Basic Fact:} The Galois action on the fibers $(f^0_{\ell+1})^{-1}(x_0)$ is transitive iff $G_{\ell+1}$ is a normal subgroup of $G_0$.

$\star$ ~ For  $G_0$   abelian,   every subgroup is normal. 

$\star$ ~ For $G_0$ not abelian, ``normality'' becomes a central question.

\vfill   

}


  

   
 \frame % 1
{
  \frametitle{}

Now take the next step, and form the inverse limit space:
\begin{eqnarray*}
 \lefteqn{ M_{\infty} = \varprojlim ~ \{ f_{\ell+1}^{\ell}\colon M_{\ell+1} \to M_{\ell}  \mid \ell \geq 0 \}}  \\
& =&  \{ (y_0, y_1, y_2, \ldots) \mid f_{\ell+1}^{\ell}(y_{\ell+1}) = y_{\ell}  \mid \ell \geq 0\} \\
&  \subset & \prod_{\ell \geq 0} ~ M_{\ell}
\end{eqnarray*}
is a compact connected metrizable space called a \emph{(weak) solenoid}.

For each $\ell > 0$, there is a fibration map $\Pi_{\ell} \colon M_{\infty} \to M_{\ell}$.
\medskip

For fixed $x_{\ell} \in M_{\ell}$ the fiber $\fX_{\ell} = \Pi_{\ell}^{-1}(x_{\ell}) \subset \fX_0$ is a Cantor space.

\medskip

The path connected components of $M_{\infty}$ are manifolds, which are non-compact covering spaces for $M_0$, so that    $M_{\infty}$ is a generalized lamination, or foliated space with Cantor transversals.
     

 \vfill   

}
 
  
  
    
  
 \frame % 1
{
  \frametitle{}

  
 The fundamental group $G_0 = \pi_1(M_0,x_0)$ acts on the fiber $\fX_0$ via lifts of paths in $M_0$ to the leaves of $\F_{\fM}$  giving  the monodromy action on the fiber,  $\Phi \colon G_0 \times \fX_0 \to \fX_0$. The action is minimal, the orbit of each point is dense in $\fX_0$.

\bigskip
The monodromy  is    an \emph{equicontinuous  group action}:

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


 \vfill   

}
 


  
 \frame % 1
{
  \frametitle{}

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


When the action of $G$ is free - there are no fixed points for the action of any $g \in G$ except the identity - then the action is called a (strong) odometer in the works

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

\medskip

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

 
 \vfill   

}

  
 \frame % 1
{
  \frametitle{}

\bigskip

Our interest is in the general case, when there is no assumption that the action is free, such as for the coverings of the Klein bottle, and other even more interesting constructions. These actions are also called \emph{subodometers} by Cortez \& Petite.

\bigskip

Two dimensional laminations formed by branched covers of $\mC$ in works of Lyubich \& Minsky, and Nekrashevych, are not exactly the same as weak solenoids, but  their monodromy group actions are. 

 \vfill   

}

  
 \frame % 1
{
  \frametitle{}


{\bf Research Focus:}

\bigskip

$\star$ ~   What can one say about the geometry of the leaves of the foliation $\F_{\fM}$ of  $M_{\infty}$ ?

\bigskip
   
$\star$ ~ When is the space $M_{\infty}$ homogeneous? 

\bigskip
   
$\star$ ~ Give criteria sufficient to imply that   two weak solenoids $M_{\infty}$ and $N_{\infty}$ are homeomorphic.
   
\bigskip
   
     

 \vfill   

}
 
 \frame % 1
{
  \frametitle{}


In a Cantor group action $\vp \colon G \times \fX \to \fX$, a clopen subset $V\subset X$ is \emph{adapted} if the restricted action to $V$ is that of a subgroup $G_V\subset G$.
 
 \medskip
 
{\bf Definition:} Minimal equicontinuous group actions $\vp_0 , \vp_1  \colon G \times \fX \to \fX$ 
 are \emph{return equivalent} if there exist adapted clopen subsets $V, U \subset \fX$, and subgroups $G_V, H_U$ such that the restricted actions $\vp_0 \colon G_V \times V \to V$ and $\vp_1 \colon G_U \times U \to U$ are conjugate. 
 
\bigskip
{\bf Theorem:} If two weak solenoids are homeomorphic then their monodromy actions are return equivalent.

 In fact, with stronger assumptions, one gets:
 
$\bullet$ ~ Return equivalent $\Longleftrightarrow$  homeomorphic weak solenoids.  
 
 
\vfill   

}
 
 \frame % 1
{
  \frametitle{}

{\bf Summary:}

\medskip

$\bullet$ ~  Smale solenoids are simplest example of 1-dimensional solenoids

\medskip


$\bullet$ ~  Essentially all questions about 1-dim solenoids are answered.

\medskip

$\bullet$ ~  Weak solenoids form  huge class of generalizations.
  
  \medskip


$\bullet$ ~    Classification of weak solenoids leads to study of minimal equicontinuous Cantor actions.

 \medskip

   $\bullet$ ~  Need to understand return equivalence for minimal equicontinuous Cantor actions.

 \medskip




$\bullet$ ~ Need more techniques to answer questions for weak solenoids.


\vfill   

}

 
\section{Models}


  \frame % 1
{
  \frametitle{ }

  
\medskip

  \underline{Profinite model} for a Cantor action is powerful, limited usefulness.
 

 {\bf Definition:} The  closure $E(\Phi)$ of    $H_{\Phi} = \Phi(G) \subset   {\Homeo}(\fX_{\infty})$, \\ 
 in the topology of pointwise convergence on maps, is called the   \emph{Ellis (enveloping) semigroup}.
 
   \medskip
   
 {\bf Proposition:} Let $\Phi$ be an equicontinuous    Cantor action. Then $E(\Phi) = \overline{H_{\Phi}} = $     closure of $H_{\Phi}$   
   in the \emph{uniform topology on maps}. 


\medskip

  
 For $x \in \fX_0$ let $\cD_x = \{ h \in \overline{H_{\Phi}} \mid h(x) = x \}$ be its isotropy group. 
 
  
 
 {\bf Lemma:} The left action of $\overline{H_{\Phi}}$ on $\fX_{\infty}$ is transitive,  hence  $\fX_{\infty} \cong \overline{H_{\Phi}}/\cD_x$. This is    the \emph{profinite model} for $\fX$. 
 
 \medskip
The closed subgroup $\cD_x \subset \overline{H_{\Phi}}$   is independent of the choice of basepoint $x$, up to topological isomorphism. 

   
  
{\bf Theorem:} If $\cD_x$ is trivial, then   $M_{\infty}$ is homogeneous.
 
 \medskip
 
  $\cD_x \cap \Phi(G) = \{id\}$ is possible with $\cD_x \ne \{id\}$.


\vfill   

}

 
    
    
 
 
   
 \frame % 1
{
  \frametitle{ }
\underline{Group chain model} for a Cantor action   is clumsy, but very useful.
 
 \medskip
 
Given a descending chain of proper subgroups of finite index
$$\cG \equiv \{G_0 \supset G_1 \supset \ldots \supset G_{\ell} \supset \cdots \}$$
we obtain a Cantor set 
 $  G_{\infty} = \varprojlim ~ \{G/G_{\ell} \to G/G_{\ell-1}\}$, with a minimal left $G$-action $\Phi_0 \colon G \times G_{\infty} \to G_{\infty}$.
 
 \bigskip
 
 {\bf Theorem:} [folklore]  Given a minimal equicontinuous action  $\vp \colon G  \times \fX \to \fX$, there exists a group chain $\cG$ such that the action  $\Phi_0 \colon G  \times G_{\infty} \to G_{\infty}$ is conjugate to $\vp$.
 

 \vfill   

}



 
 \frame % 1
{
  \frametitle{ }

  
  Relating the profinite and group chain models:
  
  \medskip
  
 
   $\ds C_{\ell} =  \bigcap_{g \in G} gG_{\ell} g^{-1}  \subset G_{\ell}$ is the  normal core of $G_{\ell}$ in $G$.   
   
     $C_{\ell}$ has finite index in $G$, so   define the profinite group
 $$C_{\infty} \equiv  \lim_{\longleftarrow} ~  \{q_{\ell +1} \colon  G/C_{\ell +1} \to G/C_{\ell} \mid \ell > 0\}~ .$$

Each group $G/C_{\ell}$ acts on the finite set $X_{\ell}  = G/G_{\ell}$, so there is an induced action   $\whPhi_{\infty} \colon C_{\infty} \to {\Homeo}(C_{\infty})$.
 
 \bigskip
  
   
  {\bf Theorem [Dyer-Hurder-Lukina, 2016].}    $\overline{H_{\Phi}} = \whPhi_{\infty}(G_{\infty})$, and  
   \begin{equation*}\label{eq-discriminant}
  \cD_{\infty} \equiv \varprojlim ~ \{\pi_{\ell+1} \colon  G_{\ell+1}/C_{\ell +1 } \to G_{\ell}/C_{\ell} \mid \ell \geq 0\}~ \cong \cD_x ~ .
\end{equation*}
 
  \vfill   

}

 


 
 \frame % 1
{
  \frametitle{}

 
The fundamental group of the Klein bottle is the dihedral group 
$$G_0 = \pi_1(K,0) = \langle a,b \mid bab = a^{-1}, b^2 = 1 \rangle.$$

For the cover $p^{\ell} \colon K \to K$,   $\ell  \geq 1$, 
$$G_1 = p_*\pi_1(K,0) = \langle a^2,b   \rangle ~ , ~  G_{\ell} = (p^{\ell})_* \pi_1(K,0) = \langle a^{2^{\ell}},b   \rangle $$

The cosets of $G/G_{\ell}$ are represented by $a^i G_{\ell}$, $i = 0,\ldots, 2^{\ell}-1$,
$$C_{\ell} = \bigcap_{g \in G} gG_{\ell}g^{-1} = \langle a^{2^{\ell}}  \rangle ~ .$$
Then $G_{\ell}/C_{\ell} = \{C_{\ell}, b C_{\ell}\}$, and so $\mathcal{D}_x  \cong \mZ/2\mZ$.


\vfill   

}


   
     
\frame % 1
{

\underline{Tree model} for a group chain action is geometric.


 
 
 \pgfputat{\pgfxy(6.6, -2.2)}{\pgfbox[left,base]{\pgfimage[width=4.8cm]{D-ary-tree-2}}} 

\medskip
Let $\cT$ be a tree with vertex set $V$ and edge set $E$:

\bigskip
\bigskip
\bigskip
$V_0 = \{v_0\}$ is root vertex at level 0.

The vertices at level $\ell$ given by $V_{\ell} = G_0/G_{\ell}$.
 
 
Edges between $V_{\ell}$ and $V_{\ell+1}$ correspond to elements of $G_{\ell}/G_{\ell+1}$.  

\bigskip

  Let $\cP(\cT)$ be the space of semi-infinite rays in $\cT$.
  
Give $\cP(\cT)$ the cylinder topology, then it  is a Cantor set.

\medskip 
  $G$ acts on $\cT$ by permuting vertices at each level so that the paths in  $\cP(\cT)$ are preserved. 
  The arboreal action $\Phi \colon G \times \cP(\cT) \to \cP(\cT)$ is conjugate to the group model action  $\Phi_0 \colon G  \times G_{\infty} \to G_{\infty}$.

 
 
 
\vfill   

}
 
 \section{Local invariants} 
   

 \frame % 1
{
  \frametitle{}

\underline{Return equivalence for   arboreal actions}:

  $\Phi \colon G \times \cP(\cT) \to \cP(\cT)$  
  
\pgfputat{\pgfxy(6.5, -.5)}{\pgfbox[left,base]{\pgfimage[width=5cm]{D-ary-tree-branch}}}   

Let $x = (v_{k}) \in V$ be a vertex point.
 
\medskip
Let $U(v_{k}) \subset \cP(\cT)$ be the set of all paths through $v_{k}$.

\medskip
Then $U(v_{k})$ is a basic clopen subset of $\cP(\cT)$.

\medskip 
The restriction of the action of $G$ to  $U(v_{k})$ is given by the 
action of  the subgroup $G_{k} \subset G$, so have $\Phi_{k} \colon G_{k} \to {\bf Homeo}(U_{k})$.

\medskip

 For each $\ell \geq k$, the normal core of $G_{\ell}$ in $G_{k}$ is
   $$C_{k,\ell} = \bigcap_{g \in G_k} gG_{\ell} g^{-1}.  \quad {\rm  Observe ~ that } ~ C_{k,\ell} \supset C_{\ell}.$$


\vfill   

}
 
  
   

 \frame % 1
{
  \frametitle{}

  The isotropy group of the action of $\overline{\Phi(G_k)}$ at $x \in \cP(\cT)$
    $$\mathcal{D}_{x,k} = \varprojlim ~ \{G_{\ell}/C_{k,\ell} \to G_{\ell-1}/C_{k,\ell-1} \mid \ell \geq k \}$$
  is   the \emph{discriminant group}  of the action $\Phi_k$.  

We have coset inclusions $G_i/C_i \to G_i/ C_{k,i}$.

{\bf Theorem 1:} [Dyer-Hurder-Lukina, 2017]
Let $\Phi_0 \colon G  \times G_{\infty} \to G_{\infty}$ be the Cantor group action for the group chain $\{G_i\}_{i \geq 0}$, let $x \in G_{\infty}$ be a point. Then for any $k>j \geq 0$ there is a well-defined surjective homomorphism
  $\Lambda^{k,j}: \mathcal{D}_{x,j} \to \mathcal{D}_{x,k}$. 
  
  \bigskip
  The action is called \underline{stable}, if there exists $j_0$, such that for all $k>j \geq j_0$ the homomorphism $\Lambda_{k,j}$ is an isomorphism.
  
\medskip
If such   $j_0$ does not exist, then the action is called   \underline{wild}.



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}



 \frame % 1
{
  \frametitle{}


{\bf Definition:} [Hurder-Lukina, 2017] 
The \emph{asymptotic discriminant} of  $\Phi_0 \colon G  \times G_{\infty} \to G_{\infty}$ is the equivalence class (with respect to   tail equivalence)  of the chain of surjective group homomorphisms
   $$\mathcal{D}_{x,0} \to \mathcal{D}_{x,1} \to \mathcal{D}_{x,2} \to \cdots .$$
 

 
The asymptotic discriminant is an invariant of Cantor actions.

{\bf Theorem:}  [Hurder-Lukina, 2017] 
The asymptotic discriminant of a Cantor group action $\Phi_0 \colon G  \times G_{\infty} \to G_{\infty}$ is invariant under the return equivalence of actions.
 
\bigskip


In   the tree model,   choose a ray to infinity, $x \in \cP(\cT)$, and form the chain of discriminant groups for the isotropy action at each vertex along the ray. This is a \emph{germinal} invariant of the action.



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}



 \frame % 1
{
  \frametitle{}

 \begin{enumerate}
\item All equicontinuous actions of \underline{abelian groups }on Cantor sets \underline{are stable}, with trivial discriminant group $\mathcal{D}_x$.
\smallskip
\item If a discriminant is \underline{finite}, then it \underline{is stable}. The action of the dihedral group defined earlier is stable with discriminant $\mathbb{Z}_2$.
\smallskip
\item  Every \underline{finite group} and every \underline{separable profinite group} can be realized as the stable discriminant group of an action of a torsion-free finite index subgroup of  $ {\bf SL}(n,\mathbb{Z})$ on a Cantor set.
\smallskip
\item 
There exists \underline{uncountably many wild actions} of torsion-free finite index subgroups of ${\bf SL}(n,\mathbb{Z})$ with distinct asymptotic discriminants.
\end{enumerate}

The proof of $3$ and $4$ uses ideas of {\bf Lubotzky 1993} on torsion elements in the profinite completion of torsion free subgroups of ${\bf SL}(n,\mathbb{Z})$, and a construction due to {\bf Lenstra}.

 
  \vfill   

}


 \frame % 1
{
  \frametitle{}

{\bf Theorem:} [folklore]
Let $G$ be a finitely-generated group, and $\vp \colon G \times \fX \to \fX$ be an equicontinuous minimal Cantor action. Then there is a weak solenoid $M_{\infty}$ whose monodromy action is conjugate to $\vp$. Moreover, this can be achieved for a 2-dimensional solenoid, whose base manifold $M_0$ is a closed surface.

\medskip

{\bf Corollary:} Every   group listed in the previous Theorem is realized as the discriminant invariant for a weak solenoid.
 
 \bigskip
 
$\bullet$ ~ Upon the passage from 1-dimensional solenoids  to higher dimensional solenoids, you enter a wild world of possible behaviors for the dynamics of their monodromy actions.
 
  \vfill   

}


 \frame % 1
{
  \frametitle{}

{\bf Summary:}

\medskip

$\bullet$ ~ Three models for minimal equicontinuous Cantor actions: \

\medskip


$\bullet$ ~ profinite model defines isotropy/discriminant group of action
  
  \medskip


$\bullet$ ~ group chain model calculates discriminants 

 \medskip

$\bullet$ ~ arboreal model suggest local invariants at ends of branches

\medskip

$\bullet$ ~ all three models yield constructions of examples

\medskip

$\bullet$ ~Need to understand meaning of these invariants


\vfill   

}


  

  \section{Non-Hausdorff}
 
   
 \frame % 1
{
  \frametitle{}
 
 
There is a geometric interpretation of the stable/wild property.

\medskip
The notion of LQA actions was introduced by {\bf \'Alvarez L\'opez and Candel 2009} as a generalization to topological spaces of the notion of quasi-analytic actions on manifolds due to {\bf Haefliger}.

{\bf Definition:} 
A topological action $\Phi \colon G \times \fX \to \fX$ is \emph{locally quasi-analytic (LQA)} if there exists $\epsilon>0$ such that for any open set $U \subset \mathfrak{X}$ with $diam (U) < \epsilon$,   and for any open $V \subset U$ and $g_1,g_2 \in G$ if
  $$ \textrm{if } \Phi(g_1)|V = \Phi(g_2)|V \textrm{ then }\Phi(g_1)|U = \Phi(g_2)|U ~ .$$
 

  
Alternatively, the action is locally quasi-analytic if and only if for all $g \in G$ if $\Phi(g)|V = id$, then $\Phi(g)|U = id$, for open sets $V \subset U$.


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}


 \frame % 1
{
  \frametitle{ }

{\bf Theorem:}  [Hurder-Lukina, 2017] 
A minimal equicontinuous action $\Phi \colon G \times \fX \to \fX$ is   stable if and only if the action of the Ellis group $\overline{\Phi} \colon  \overline{\Phi(G)} \times \fX \to \fX$ is locally quasi-analytic.
 

\medskip
{\bf Example:} Let $(\cT, v_0)$ be a rooted tree. Then the action of ${\bf Aut}(\cT, v_0)$ on $\cP(\cT)$ is not LQA. 

\medskip
Let $\cT_d$ be a $d$-ary tree, then ${\bf Aut}(\cT_d, v_0) = S_d^{\infty}$, where $S_d$ is the permutation group on $d$ elements, and $S_d^{\infty}$ denotes the infinite wreath product.

In the wreath product, the actions \\
on subtrees are independent, \\
so the restricted action \\
on $U(v_k)$ has multiple \\
extensions to $U(v_{k-1})$.

\pgfputat{\pgfxy(5.5,.3)}{\pgfbox[left,base]{\pgfimage[width=6cm]{D-ary-tree-branch}}}     


   
       
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}


 \frame % 1
{
  \frametitle{ }

Let $\fX$ be a Cantor set, and let $G \subset {\bf Homeo}(\fX)$ be a group.

\medskip
An element $g \in G$ is \emph{non-Hausdorff}  if there exists $x \in \fX$ and a collection of open sets $\{U_n\}_{n \geq 1}$ with $\bigcap U_n = \{x\}$, such that

\begin{enumerate}
\item $g(x) = x$,
\item $g|_{U_n}$ is non-trivial,
\item for $n \geq 1$, there exists an open set $O_n \subset U_n$ such that $g|_{O_n} = id$.
\end{enumerate}

{\bf Theorem:} [Winkelnkemper, 1983] 
The germinal \'etale groupoid $\Gamma(\fX,G)$ associated to the action of $G$ on $\fX$ is non-Hausdorff if and only if $G$ contains a non-Hausdorff element.
 



  
\vfill   

}


 \frame % 1
{
  \frametitle{}

{\bf Lemma:} [Hurder-Lukina, 2018]
Let $G \subset {\bf Homeo}(\fX)$ be a (countable or profinite) group acting minimally on a Cantor set $\fX$. Suppose $g \in G$ is non-Hausdorff, then the action of $G$  is non-LQA.
 

\medskip
{\bf Question:} If the action of $G \subset {\bf Homeo}(\fX)$ is non-LQA, does it have to contain a non-Hausdorff element?

\medskip
The answer is negative when $G$ is a countable group.

This example, due to Lukina, was found amongst the absolute Galois groups for function fields, and their arboreal representations.

 
  
\vfill   

}


 \frame % 1
{
  \frametitle{}

{\bf Summary:}

\medskip

$\bullet$ ~ Classification of weak solenoids leads to study of minimal equicontinuous Cantor actions.

\medskip


$\bullet$ ~ Asymptotic discriminant is invariant of return equivalence.

\medskip

$\bullet$ ~ Stable actions satisfy continuous orbit equivalence rigidity.
  
  \medskip


$\bullet$ ~ Wild actions may also have non-Hausdorff elements. 

 \medskip

$\bullet$ ~ Wild actions give rise to new classes of $C^*$-algebras.

\medskip

$\bullet$ ~ Applications to   absolute Galois groups.


\vfill   

}

  
 
 

\section{Galois groups}

  
  
 \frame % 1
{
  \frametitle{ }

   
    
 The analogy between theory of finite coverings and Galois theory of finite field extensions suggests looking for examples of   minimal Cantor actions arising from purely arithmetic constructions.
 \medskip
 
$\bullet$ [R.W.K. Odoni, 1985] began  the study of arboreal representations of absolute Galois groups on the rooted trees formed by the solutions of iterated polynomial equations.

\medskip
$\bullet$      [Rafe Jones, 2013]  gives  an introduction and survey of this program,  from the point of view of arithmetic dynamical systems and number theory.
  
   
      
\vfill   

}

 
 \frame % 1
{
  \frametitle{ }

Here is a typical result from the recent work  

$\bullet$ ~{ O.~Lukina},
 {\it Galois Groups and Cantor actions},    arXiv:1809.08475

  \medskip
   
{\bf Theorem:} [Lukina, 2018]

Let $f(x)$ be a quadratic polynomial with integer coefficients. Let $P_c$ be the orbit of the critical point $c$ of $f(x)$, and suppose $P_c$ is finite. Let $Gal_{\rm geom}(f)$ be the geometric iterated monodromy group, that is, the Galois group of the infinite field extension of $\mathbb{C}(t)$ obtained by adjoining to $\mathbb{C}(t)$ the roots of $f^n(x) - t$, for $n \geq 1$. 

If $\#P_c \geq 3$, and   the   post-critical orbit is pre-periodic, 
 the asymptotic discriminant of the action of ${\rm Gal}_{\rm geom}(f)$ is wild, 
 and ${\rm Gal}_{\rm geom}(f)$ contains \emph{non-Hausdorff elements}. 
 
 \medskip
  
\vfill   

}

 
 \frame % 1
{
  \frametitle{ }
  
  
More is true, as Lukina realizes essentially all of the phenomena discussed in this talk, as the dynamical behavior for arboreal representations of  Galois groups of    function fields.

\bigskip

 {\bf Problem:} Develop   relations between    invariants for minimal equicontinuous Cantor actions that arise from   arboreal representations of Galois groups of function fields and number fields, and properties such as the density of prime divisors for sequences such as $\{f^n(\alpha)\}_{n \geq 0}$ for an algebraic integer $\alpha$.
 
 
\vfill   

}

 
 \frame % 1
{
  \frametitle{ }

\begin{center}
Welcome to the wild world of  solenoids!
\end{center}

\bigskip
\bigskip

 \pause

\begin{center}
\emph{Thank you for your attention.}
\end{center}

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}

 
 
    
  
     

  \section{Refs}

    
\frame % 1
{
  \frametitle{References}
 {\tiny    
 
 {R.H.~Bing},
  {\it A simple closed curve is the only homogeneous bounded plane continuum that contains an arc},
 {\bf Canad. J. Math.}, 12:209--230, 1960.

{N.~Boston and R.~Jones},
 {\it Arboreal {G}alois representations},
 {\bf Geom. Ded.},  124, 2007.

{A.~Clark and S.~Hurder},
 {\it Homogeneous matchbox manifolds},
 {\bf Trans. A.M.S.},   365:3151--3191, 2013.
 
  {J.~Dyer, S.~Hurder and O.~Lukina},
 {\it Molino theory for matchbox manifolds},
 {\bf Pac. Jour. Math.},  Vol. 289:91-151, 2017.
 
{R.~Fokkink and L.~Oversteegen},
{\it Homogeneous weak solenoids},
  {\bf Trans. A.M.S}, 354(9):3743--3755, 2002.

{S.~Hurder and O.~Lukina},
 {\it Wild solenoids},
 {\bf Trans. A.M.S.},  to appear, 2018, arXiv:1702.03032.
 
{R. Jones}, {\it Galois representations from pre-image trees: an arboreal survey},
  in {\bf Actes de la {C}onf\'erence ``{T}h\'eorie des {N}ombres et
 {A}pplications''}, 107-136, 2013.

 
{A.~Lubotzky},  {\it Torsion in profinite completions of torsion-free groups},
 {\bf Quart. J. Math.} 44:327--332, 1993.


{O.~Lukina},
 {\it Arboreal {C}antor actions}, {\bf Jour. L.M.S.},  to appear,  arXiv:1801.01440.

{O.~Lukina},
 {\it Galois Groups and Cantor actions}, {preprint},   arXiv:1809.08475.

  {M.C.~McCord},
  {\it Inverse limit sequences with covering maps},
  {\bf Trans. Amer. Math. Soc.}, 114:197--209, 1965


{R.W.K. Odoni},
  {\it The {G}alois theory of iterates and composites of polynomials},
  {\bf Proc. L.M.S.}, 51:385-414, 1985.


 {R.~Schori},
  {\it Inverse limits and homogeneity},
  {\bf Transactions A.M.S.}, 124:533--539, 1966.

 {L.~ Vietoris}, {\it  \"{U}ber den h\"oheren {Z}usammenhang kompakter {R}\"aume und eine {K}lasse von zusammenhangstreuen {A}bbildungen},
 {\bf Math. Ann.}, 97:454--472, 1927.



}  
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 \section{HBD}
 
 

 
  


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