% Arithmetic and Analysis
% A conference on the occasion of Christopher Deninger's 60th birthday% 
% Dates: April 9 - April 13, 2018
% Place: Münster, Germany
% https://www.uni-muenster.de/SFB878/activities/AaA_announce.html
% talk on April 12, 2018

% slides for talk "Symmetries of laminations"
 
 
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\date[April 12, 2018]


  
\title[Wild things]{Symmetries of laminations}


\author{Steve Hurder\\   {\scriptsize joint work with  Olga Lukina}  \\ \footnotesize {University of Illinois at Chicago}\\....    \\{\footnotesize {\it Arithmetic and Analysis}\\ {\scriptsize A conference on the occasion of Christopher Deninger's 60th birthday} \\   {\scriptsize M\"unster, Germany -- April 12, 2018}}}

 
 
\begin{document}

\frame{\titlepage} % # 1


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

\bigskip
 
 
\bigskip
 
 Christopher Deninger's talk at the International Congress of Mathematicians in Berlin, 1998 highlighted some deep  analogies between number theory and dynamical systems on foliated spaces.  In particular, here is a quote from   his paper
 \emph{``On the nature of the `explicit formulas' in analytic number theory''}:


 \begin{quotation}
Our example suggests that under suitable conditions transversal index theory [for smooth foliated manifolds] generalizes to solenoids or even more general laminated spaces instead of manifolds.   
\end{quotation}
In this talk, we discuss the nature of   ``general laminated spaces'' and some of  their properties, and give a selection of examples related to group actions and number theory. 

 \vfill   

}

  
  
 \frame % 1
{
  \frametitle{}

 
 \hspace{60pt} \includegraphics[width=0.3\textwidth]{Smale-Williams_Solenoid_Large} 
 \hspace{20pt} \includegraphics[width=0.2\textwidth]{blue_matches} 

The objects of   study are called various names in the literature:
 
$\bullet$ ~ \emph{Generalized laminations}, [Deninger, Ghys, Lyubich \& Minsky]


$ \bullet$ ~ \emph{Matchbox manifolds}, [Aarts \& Martens, Clark \& Hurder]

$\bullet$ ~ \emph{Solenoidal manifolds}, [Sullivan]


All are foliated spaces as introduced in the book 

$\bullet$ ~ Moore \& Schochet, {\bf Global analysis on foliated spaces},  1988. 

  

 \vfill   

}
 

  
 
  \frame % 1
{
  \frametitle{}
 
 
  
{\bf Definition:}  A $C^r$-foliation $\F$ of   a   manifold $M$ is  a ``uniform partition'' of $M$ into submanifolds of constant dimension $p$ and codimension $q$, such that  there is a covering of $M$ by $C^r$-coordinate charts whose change of coordinate functions preserve the leaves, for $r\geq 1$.
 \begin{center}
\includegraphics[width=0.48\textwidth]{chart.pdf}
\end{center}
 
 
   \vfill   

}
 

 
\frame % 2
{
  \frametitle{}
 
  
    
\bigskip

 {\bf Definition:}     $\fM$ is   an \emph{$n$-dimensional matchbox manifold} if:

$\bullet$~ $\fM$  is a continuum $\equiv$ a compact, connected metric space; 

$\bullet$~ $\fM$  admits a covering by  foliated coordinate charts\\\quad \quad  $\ds \cU = \left\{ \vp_{i} \colon U_i \to [-1,1]^n \times \fX_i \mid 1 \leq i \leq k \right\}$; 

$\bullet$~  each  $\fX_i$ is a     \emph{clopen} subset of  a \emph{totally disconnected}   space   $\fX$; 

$\bullet$~  plaques  $\ds \cP_i(z) = \vp_{i}^{-1} ( [-1,1]^n \times \{z\})$ are connected, $z \in \fX_i$; 

$\bullet$~  for $U_i \cap U_j \ne \emptyset$, each plaque  $\ds \cP_i(z)$ intersects at most one plaque $\ds \cP_j(z')$, and changes of coordinates along intersection of plaques are smooth diffeomorphisms; 

{\bf +}~ some other technicalities.

 
 

  

 
 \vfill   

}



\frame % 1
{
  \frametitle{}

  
 The path connected components of $\fM$ are the leaves of the foliation  $\F$. To the above list, we add the condition:

$\bullet$  there is a leafwise smooth Riemannian metric on the leaves of $\F$, which is continuous in each foliation chart.

{\bf Proposition:} Each leaf of $\F$ is a complete   Riemannian manifold with bounded geometry.


It follows that associated to a matchbox manifold are many of the traditional aspects of Riemannian manifolds, such as leafwise curvature, leafwise De~Rham cohomology, leafwise operators, foliation-preserving transformation groups, and invariants constructed from these data.

{\bf Basic question:} What are these spaces? What are there properties and invariants?

     \vfill   

}


 \frame % 1
{
  \frametitle{}

{\bf Problems to investigate:}

\bigskip
 
$\bullet$ ~ Topological properties of laminated spaces. 


\hspace{4pt} $\star$~  Properties of minimal actions on Cantor spaces.

  

\bigskip

$\bullet$ ~   Lefschetz Theorems for laminated spaces.

 
\hspace{4pt} $\star$~  Extensions of Trace Formulae of Alvarez-Lopez and Kordyukov.

 \smallskip
 
\hspace{4pt} $\star$~  Distributional trace invariants associated to transverse profinite group actions.

  
 
\bigskip

$\bullet$ ~ Relations to number theory.

\hspace{4pt} $\star$~ Constructions of examples from absolute Galois groups.



 \vfill   

}
 
 \section{Solenoids}


  
 \frame % 1
{
  \frametitle{}

 The Lefschetz trace   formulae of Alvarez-Lopez and Kordyukov appearing in Deninger's works are usually shown for flows on   Riemannian foliations; that is,  a smooth foliation  of compact manifold   with a holonomy-invariant transverse Riemannian metric.  
  
The lamination analog of   a Riemannian foliation of a compact manifold $M$, is a matchbox manifold $\fM$ whose holonomy is given by a minimal equicontinuous   action on a Cantor space $\fX$:

 

$\bullet$  ~  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{}



The equicontinuous hypothesis has strong implications for the   structure of $\fM$ as we next discuss.  We make a brief detour, to introduce the notion of a  \emph{weak solenoid}.
\medskip

Let $M_0$ be a connected closed manifold, and  let $f^i_{i-1}: M_i \to M_{i-1}$ be a sequence of finite-to-one proper covering maps. Then
\begin{eqnarray}
 \lefteqn{ M_\infty = \lim_{\longleftarrow} \{ f^i_{i-1}: M_i \to M_{i-1}  \mid i \geq 1 \}} \label{invlimit}\\
& =&  \{ (y_0, y_1, y_2, \ldots) \mid f^i_{i-1}(y_i) = y_{i-1}  \mid i \geq 1\} \nonumber
\end{eqnarray}
is a compact connected metrizable space called a \emph{(weak) solenoid}.

There is a fibration map $\Pi_0 \colon M_{\infty} \to M_0$, and for $b \in M_0$ the fiber $\fX_0 = \Pi_0^{-1}(x_0)$ is a Cantor space.


 
 \bigskip
 
 {\bf Theorem [McCord, 1966].}   A solenoid is a matchbox manifold. 
    

 \vfill   

}
 
  
\frame
{
  \frametitle{}

The   classical \emph{Vietoris solenoid} is an example.

Let ${\bf P} = (p_1, p_2, \ldots)$ be an infinite sequence, where each $p_i>1$.

Let $f^i_{i-1}: \mathbb{S}^1 \to \mathbb{S}^1$ be a $p_i$-to-$1$ self-covering map of a circle.

\smallskip

  A Vietoris solenoid is the inverse limit space
  $$\Sigma_{\bf P} = \{ (y_i) = (y_0, y_1, y_2, \ldots) ~|~ f^i_{i-1}(y_i) = y_{i-1}~\} \subset \prod_{i \geq 0} \mathbb{S}^1$$
with subspace topology from the Tychonoff topology on $\prod_{i \geq 0} \mathbb{S}^1$ .

\smallskip
Let $b \in \mathbb{S}^1$, then the fibre $\fX_b = \{(b,y_1,y_2, \ldots)\} \subset \Sigma_{\bf P}$ is a Cantor section, transverse to the foliation by path-connected components.

\smallskip
The fundamental group $\pi_1(\mathbb{S}^1,b) = \mathbb{Z}$ acts on $\fX_b$ via lifts of paths in $\mathbb{S}^1$, so the monodromy action on the fiber defines a group action $\Phi \colon \mZ \times \fX_b \to \fX_b$ which is a classical odometer action. 

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

\bigskip

The following result, whose proof is surprisingly technical, shows that the study of equicontinuous matchbox manifolds reduces to the study of weak solenoids:

\bigskip

{\bf Theorem [Clark \& Hurder, 2013].} Let $\fM$ be a   matchbox manifold with equicontinuous holonomy pseudogroup action.  
Then $\fM$ is  foliated  homeomorphic to a weak solenoid.

  \vfill   

}
 


   \section{Chains}

  
 \frame % 1
{
  \frametitle{}

The   study of weak solenoids reduces to the study of Cantor actions obtained from group chains. This   approach was initiated by [Fokkink \& Oversteegen, 2002].

\medskip
 
 Let $\Pi_0 \colon M_{\infty} \to M_0$ be a weak solenoid defined by the system of maps $\{f_0^i \colon M_i \to M_0 \mid i > 0\}$, where    $f^i_0 = f^1_0 \circ \cdots \circ f^i_{i-1}$.
 
  Choose  a  basepoint  $b \in M_0$ and  basepoints $x_i \in M_i$ such that $f_0^i(x_i) = b$. Set 
  $x = \lim x_i  \in  \fX_b \equiv \Pi_0^{-1}(b)$. 
  
 \medskip


Define $G = G_0 = \pi_1(M_0, b)$, and let $G_i \subset G$ be the subgroup defined by 
 $\ds G_ i = {\rm Image} \{ (f_0^i)_{\#} \colon \pi_1(M_i , x_i) \to \pi_1(M_0 , b)\}$. 
 
 \bigskip
 
 $\{G_i \mid i \geq 0\}$ is a   descending   chain of subgroups of finite index in $G$.

The subgroups $G_i$ are not assumed to be normal in $G$. 
 \vfill   

}
 
 
   
 \frame % 1
{
  \frametitle{}

  {\bf Example 1:} Here is a simple example
\medskip

Consider the Vietoris solenoid 
$$\Sigma = \{ f^i_{i-1}: \mathbb{S}^1 \to \mathbb{S}^1\}.$$
Then $G  = \pi_1(\mathbb{Z},0) = \mathbb{Z}$, and $ G_i =  (p_1 \cdots p_i)  \, \mZ$, where $p_i$ is the degree of $f^i_{i-1}$.

\medskip
Since $\mathbb{Z}$ is abelian, each subgroup $G_i$ is normal.


 \vfill   

}
 
  
 \frame % 1
{
  \frametitle{}



Each $X_i = G_0/G_i$ is a finite set with a left action of $G$. It is a group if $G_i$ is normal in $G$. The Cantor fiber     $\fX_b$ is identified with 
$$    \fX_b \cong \fX_{\infty} \equiv \varprojlim ~ \left\{  X_i \to X_{i-1} \right\}  = \varprojlim ~ \left\{  G/G_i \to G/G_{i-1} \right\}  .$$
and has a left $G$-action   $\Phi \colon G \to {\Homeo}(X_{\infty})$.  

\medskip

The action  $(\fX_{\infty} , G, \Phi)$ is called a \emph{generalized odometer}, or also called a \emph{subodometer} by [Cortez \& Petite, 2008].
 
 \medskip
 
 The properties of a generalized odometer action are best seen by introducing the closure of the action, which is a profinite group.
 
 \vfill   

}

  
 \frame % 1
{
  \frametitle{ }

  


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

   
 \medskip
The \underline{normal core} $N$ of a subgroup $H \subset G$ is the largest subgroup $N \subset H$ which is   normal in $G$.
   
   {\bf Lemma:} The normal core of    $\cD_x$ in $\overline{H_{\Phi}}$ is trivial. 

 
\vfill   

}

 
  

 
 \frame % 1
{
  \frametitle{ }

 
 
 Let $C_{i} \subset G_{i}$ be the  normal core of $G_{i}$ in $G$, then $C_{i}$ has finite index in $G$.   Define the profinite group
 $$G_{\infty} \equiv  \lim_{\longleftarrow} ~  \{q_{i} \colon  G/C_{i +1} \to G/C_{i} \mid i > 0\}~ .$$

Each group $G/C_{i}$ acts on the finite set $X_i  = G/G_{i}$, so there is an induced action   $\whPhi_{\infty} \colon G_{\infty} \to {\Homeo}(\fX_{\infty})$.
 
 \medskip
  
   
  {\bf Theorem [Dyer-Hurder-Lukina, 2016].}    $\overline{H_{\Phi}} = \whPhi_{\infty}(G_{\infty})$, and  
   \begin{equation}\label{eq-discriminant}
  \cD_{\infty} \equiv \lim_{\longleftarrow} ~  \{\pi_{i} \colon  G_{i+1}/C_{i +1 } \to G_{i}/C_{i} \mid \ell \geq 0\}~ \cong \cD_x ~ .
\end{equation}
 
 
 The group chain model for minimal equicontinuous Cantor actions is used to construct examples with prescribed special properties.
 \vfill   

}


\frame
{
  \frametitle{}

 {\bf Example 1 continued:}  
 
 \bigskip

For the Vietoris solenoid 
$$\Sigma = \{ f^i_{i-1}: \mathbb{S}^1 \to \mathbb{S}^1\} \ ,$$
  $G  = \pi_1(\mathbb{Z},0) = \mathbb{Z}$, and $ G/G_i = \mathbb{Z}/p_1 \cdots p_i \mathbb{Z}$, where $p_i$ is the degree of $f^i_{i-1}$.

\medskip
Since $\mathbb{Z}$ is abelian, $G_i = C_i$, and so $G_i/C_i$ is a trivial group.

\medskip
Thus $C_\infty \cong \fX_b$, where $\fX_b$ is a fibre of $\Sigma \to \mS^1$, and so the discriminant group $\cD_x$ of the Vietoris solenoid is trivial.

 \vfill   

}

 



 \frame % 1
{
  \frametitle{}

\bigskip

 {\bf Example 2:}  Here is a more interesting example,  with $\cD_x$ non-trivial. It is due to [Rogers \& Tollefson, 1971/72].
\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.
 

\vfill   

}

 
 \frame % 1
{
  \frametitle{}

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

For the cover $p: K \to K$ we have 
$$p_*\pi_1(K,0) = \langle a^2,b \mid bab^{-1} = a^{-1} \rangle ,$$
 and for $p^n = p \circ \cdots \circ p : K \to K$ we have
$$G_n = (p^n)_* \pi_1(K,0) = \langle a^{2^n},b \mid bab^{-1} = a^{-1} \rangle.$$
The cosets of $G/G_n$ are represented by $a^i G_i$, $i = 0,\ldots,n-1$,
$$C_n = \bigcap_{g \in G} gG_ng^{-1} = \langle a^{2^n} \mid bab^{-1} = a^{-1} \rangle.$$
Then $G_n/C_n = \{C_n, b C_n\}$, and so $\mathcal{D}_x  \cong \mZ/2\mZ$.


\vfill   

}


 

\section{Symmetries}


 \frame % 1
{
  \frametitle{}

We next discuss some properties of weak solenoids and their associated monodromy Cantor actions.
 
The \emph{symmetries} of an action $(\fX_{\infty}, G , \Phi)$ are given by the right actions on $\fX_{\infty}$ that commute with the given  left action.

These form a group, $\Aut(\fX_{\infty}, G , \Phi)$.

If $\fX_{\infty}$ is a group, then $\fX_{\infty} \cong G_{\infty}$ and the right action of $G_{\infty}$ on $\fX_{\infty}$ is transitive.
 

{\bf Proposition:} $\Aut(\fX_{\infty}, G , \Phi)$ is  isomorphic to the commutant of $\cD_x$ in $G_{\infty}$. In particular, $\Aut(\fX_{\infty}, G , \Phi)$ acts transitively on $\fX_{\infty}$ if and only if   $\cD_x$ is the trivial group.

\medskip

However, this is not the full story for symmetries of weak solenoids.

 \vfill   

}



 \frame % 1
{
  \frametitle{}

For a weak solenoid as defined by \eqref{invlimit}, and any $k \geq 1$ we have 
 \begin{equation}
  M_\infty   \cong  \lim_{\longleftarrow} \{ f^i_{i-1}: M_i \to M_{i-1}  \mid i > k \}  \label{invlimitk}
\end{equation}
Set $f^i_0 = f^1_0 \circ \cdots \circ f^i_{i-1}$. Then 
the fiber of the projection $\Pi_k \colon M_{\infty} \to M_k$ is the clopen set 
\begin{eqnarray*}
\lefteqn{ \fX_k =  \varprojlim ~ \left\{  G_k/G_i \to G_k/G_{i-1} \mid  i > k\right\} }  \\
& = &   \{(f^i_0(x_k),  \ldots , x_k, y_{k+1}, \ldots) \in M_\infty  \}  \subset \fX_0  \  .
\end{eqnarray*}
The global monodromy with respect to $\Pi_k$ is conjugate to  the restricted action 
$\Phi_k \colon G_k \to \Homeo(\fX_k)$. 
 \vfill   

}



 \frame % 1
{
  \frametitle{}

We derive the group quotient model for the action of $G_k$ on $\fX_k$. 

The key change is we must   introduce the   groups 
 $C_{k,i} \subset G_{k}$, for $i > k$, which are the  normal cores of $G_{i}$ in $G_k$, so 
 $$C_{k,i} = \bigcap_{g \in G_k} ~ g \, G_i \, g^{-1} \supset \bigcap_{g \in G} ~ g \, G_i \, g^{-1}  = C_{0,k} = C_k .$$
 Then we have  the collection of profinite groups
 \begin{eqnarray}
G_{k,\infty} & \equiv & \lim_{\longleftarrow} ~  \{q_{i} \colon  G_k/C_{k, i +1} \to G_k/C_{k, i} \mid i > k\}\\
C_{k,\infty} & \equiv & \lim_{\longleftarrow} ~  \{q_{i} \colon  C_{k, i+1}/C_{i +1} \to C_{k, i}/C_{i} \mid i > k\}\\
\cD_{k,\infty} & \equiv & \lim_{\longleftarrow} ~  \{q_{i} \colon  G_{i+1}/C_{k,i +1} \to G_i/C_{k,i} \mid i > k\} \\
\cD_{\infty} & \equiv & \lim_{\longleftarrow} ~  \{q_{i} \colon  G_{i+1}/C_{i +1} \to G_i/C_{i} \mid i > k\}
\end{eqnarray}

 


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 \frame % 1
{
  \frametitle{}

Each $G_{k,\infty}$ acts transitively on $\fX_k$ with isotropy group $\cD_{k,\infty}$ at $x$.

Moreover,  there is an exact sequence
  $$C_{k,\infty} \longrightarrow \cD_{\infty} = \cD_{0, \infty} \longrightarrow \cD_{k, \infty} . $$

Also,   for $\ell > k \geq 0$ there is a surjective map 
\begin{equation}\label{rhokl}
\rho_{k, \ell} \colon \cD_{k,\infty} \longrightarrow \cD_{\ell, \infty}
\end{equation}
 and  $C_{k,\infty} \subset C_{\ell,\infty}$. 

\medskip

The   right action of $C_{k,\infty} \subset G_{\infty}$ on $X_{\infty} = G_{\infty}/\cD_{\infty}$ fixes all the points in the clopen set $\fX_k$ but the action is not trivial on $\fX_{\infty}$.  

 \medskip
 
 Each group  $C_{k,\infty}$ is an ``internal symmetry'' of the weak solenoid with monodromy is given by $(\fX_{\infty}, G , \Phi)$.
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   \frame % 1
{
  \frametitle{ }



{\bf Definition:} An action $(X_{\infty}, G, \Phi)$ is \emph{stable} if there exists $k_0 \geq 0$ such that for $\ell > k \geq k_0$ the map $\rho_{k, \ell}$ in \eqref{rhokl} has trivial kernel. 

The action is said to be \emph{wild} otherwise.

\bigskip

{\bf Theorem [Hurder-Lukina, 2017].} Suppose that $(X_{\infty}, G, \Phi)$ is a wild action. Then there exists a strictly increasing chain of indices $\{1 \leq k_1 < k_2 < \cdots \}$ so that the sequence of finite subgroups in the profinite group $G_{\infty}$
$$ C_{k_1,\infty} \subset  C_{k_2,\infty} \subset \cdots \subset C_{k_j , \infty} \subset \cdots \subset \cD_{\infty}$$
is strictly increasing. 

\bigskip

{\bf Corollary:} A weak solenoid with wild monodromy   admits an infinitely increasing chain of internal symmetry groups. 

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  \section{Non-Hausdorff}
 
   
 \frame % 1
{
  \frametitle{}
We next discuss the notion of an ``analytic Cantor action'', which was introduced in the works of Alvarez Lopez, and its relation to wildness and the Hausdorff property for the action. 

\medskip

  Let $U, V \subset \fX$ be clopen subsets of a Cantor space $\fX$.
 
 \medskip
 
$\bullet$  A   homeomorphism $h \colon U \to V$ is \underline{quasi-analytic} (\emph{QA}) if either $U = V$ and $h$ is the identity map, or for every \emph{clopen} subset $W \subset U$ the fixed-point set of the restriction $h|W \colon W \to h(W) \subset V$ has no interior.

 
 
$\bullet$  A   homeomorphism $h \colon U \to V$ is \underline{locally quasi-analytic} (\emph{LQA}) if for each $x \in U$ there exists a clopen neighborhood $x \in U' \subset U$ such that the restriction $h_{U'} \colon U' \to V' = H(U')$ is QA.

$\bullet$  A group action $\vp \colon G \colon \fX \to \fX$ is \emph{LQA} if  for each $x \in \fX$, there exists a clopen neighborhood $x \in U$, such that the  restrictions of elements of $G$ to $U$ are quasi-analytic.



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 \frame % 1
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  \frametitle{ }

{\bf Remarks:} 

$\bullet$ A free action $G \times \fX \to \fX$ is  quasi-analytic.

 
 $\bullet$ The   automorphism group of a spherically homogeneous rooted tree $T_d$, acting on the Cantor set of ends,  is not LQA.    
   
 \bigskip
 
    
  {\bf Theorem  [Hurder \& Lukina, 2017].}
  Let $\vp \colon G  \times \fX \to \fX$ be an equicontinuous  minimal Cantor action, where $G$ is finitely generated. Then the action $\vp$ is stable if and only if the action of the profinite group  $G_{\infty}$ on $X_{\infty}$ satisfies the LQA property. 
  
     
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 \frame % 1
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 The LQA property for a group action  $(\fX,G,\Phi)$ can be interpreted in terms of the properties of the germinal groupoid  $\cG(\fX, G, \Phi)$ associated to the action.  
   
   \bigskip
   
 Recall that for $g_1, g_2 \in G$, we say that $\Phi(g_1)$ and $\Phi(g_2)$ are \emph{germinally equivalent} at $x \in \fX$ if $\Phi(g_1) (x) = \Phi(g_2)(x)$, and there exists an open neighborhood $x \in U \subset \fX$ such that the restrictions agree, $\Phi(g_1)|U = \Phi(g_2)|U$. We then write $\Phi(g_1) \sim_x \Phi(g_2)$. 
 
 \medskip
 
 For $g \in G$, denote the equivalence class of  $\Phi(g)$ at $x$ by $[g]_x$.
 The collection of germs $\cG(\fX, G, \Phi) = \{ [g]_x \mid g \in G ~ , ~ x \in \fX\}$ is given the sheaf topology, and   forms an \emph{\'etale groupoid} modeled on $\fX$. 
 

     
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 \frame % 1
{
  \frametitle{ }


 \medskip
 
{\bf Theorem  [Hurder \& Lukina, 2017].}
If an action $(\fX,G,\Phi)$   is   locally quasi-analytic, then  $\cG(\fX, G, \Phi)$ is    Hausdorff.     

 \medskip
   
   The Hausdorff property for a germinal groupoid  $\cG(\fX, G, \Phi)$    appears in the work of [Renault, 2008] on the $C^*$-algebra associated to the action, and has been studied in various works in $C^*$-algebras. 


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\section{Examples}


\frame % 1
{
  \frametitle{}

For the remainder of this talk, we discuss a variety of more subtle constructions of weak solenoids and Cantor actions, that illustrate all of the properties introduced above.

 {\bf Example 3:} {\it  $3$-dimensional matchbox manifolds.}

\medskip
 
Let  $\wtM_0 = \mH$ be the real Heisenberg group, presented in the form $\mH = (\mR^3, *)$ with the group operation $*$ given by $(x,y,z)*(x',y',z')=(x+x',y+y',z+z'+xy')$. This operation is standard addition in the first two coordinates, and addition with a twist in the last coordinate. Let $\cH = (\mZ^3, *)$ be the integer lattice subgroup, so that $M_0 = \mH/\cH$ is a compact $3$-manifold.

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}

\frame % 1
{
  \frametitle{ }
\bigskip

Consider   subgroups of $\cH$ which can be written in the form $\Gamma=M\mZ^2\times m\mZ$ where $M = \left( \begin{array}{cc} i & j \\ k & l\end{array}\right)$ is a 2-by-2 matrix with non-negative integer entries and $m>0$ is an integer. Then $\gamma \in \Gamma$ is of the form $\gamma = (ix+jy, kx+ly, mz)$ for some $x,y,z \in \mZ$. A straightforward computation gives the following: 
  
  
{\bf Theorem [Dyer, 2015].}
Let      $A_n =  \left( \begin{array}{cc} p^n & 0 \\ 0 & q^n\end{array}\right)$,   $p$ and $q$ are distinct primes.    
Define the   group chain 
$$G_0 =\cH  \quad , \quad \{G_n\}_{n \geq 1} = \{A_n \mZ^2 \times p^n \mZ\}_{n \geq 1} .$$
 Then the discriminant $\cD_{\infty}$ for the action is a Cantor group, and the action is stable.
 
 
 \vfill   

}

\frame % 1
{
  \frametitle{ }
\bigskip

 Thus, the weak solenoid $M_{\infty}$ defined by the coverings of $M_0$ associated to this chain is not homogeneous.

\medskip

Note that the intersection $\ds \bigcap_{n\geq 0}~ G_n = \{0\}$.

This implies that the   leaf $L_x$ of   the foliation $\F$ on $M_{\infty}$ through the basepoint $x \in \fX_b$ is isometric to the real Heisenberg group $\mH$.

 
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 \frame % 1
{
  \frametitle{ }

 
  
 {\bf Example 4:}   [Schori, 1966] gave first example of a non-homogeneous weak solenoid.
 It  is obtained by taking repeated 3-fold coverings starting with a closed surface $\Sigma_2$ of genus $2$. 
 \begin{center}
\includegraphics[width=0.7\textwidth]{SchoriConstr1}
\end{center}
 {\bf Proposition [Hurder \& Lukina, 2017].} The monodromy action of  $G = \pi_1(\Sigma_2,b_0)$ on the    fiber of the solenoid over $\Sigma_2$  is wild.
  
   
   
     
 
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 \frame % 1
{
  \frametitle{ }

   
 
 {\bf Example 5:} Wild actions  of arithmetic lattices. Lubotzky [1993] showed that the profinite completions of higher rank arithmetic lattices contain   arbitrary products of finite torsion groups.

${\bf SL}_N(\mZ)$ =  $N \times N$ matrices with integer entries and determinant $1$
 
 
%  $\G_N(M) \equiv  {\rm Ker} \left\{\varphi_M \colon  {\bf SL}_M(\mZ) \to {\bf SL}_N(\mZ/M \mZ) \right\}$
   
   $\widehat{{\bf SL}_N(\mZ)}$ profinite completion of ${\bf SL}_N(\mZ)$
   
   $\cP$ = set of all primes
   \begin{equation}\label{eq-prodprimes}
\widehat{{\bf SL}_N(\mZ)} \equiv \lim_{\longleftarrow} ~ {\bf SL}_N(\mZ/M \mZ) \cong {\bf SL}_N(\widehat{\mZ}) \cong  \prod_{p \in \cP} \ {\bf SL}_N(\widehat{\mZ}_p) \ .
\end{equation}
Let $G \subset {\bf SL}_N(\mZ)$ be a finite-index, torsion free subgroup.
 
 Then $G$ is finitely generated, and   its profinite completion $\whG$ is a clopen subgroup of $\widehat{{\bf SL}_N(\mZ)}$, hence there is a cofinite   $\cP' \subset \cP$, with
 $$ \prod_{p \in \cP'} \ {\bf SL}_N(\widehat{\mZ}_{p}) \subset \whG \subset \prod_{p \in \cP} \ {\bf SL}_N(\widehat{\mZ}_p)$$
 
 
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}

 \frame % 1
{
  \frametitle{ }

  Set $\ds \whH =  \prod_{p \in \cP'} \ {\bf SL}_N(\mZ/p\mZ)$. Then there is a homomorphism with dense image $\alpha \colon G \to \whH$.
 For each $p \in \cP'$, choose $D_p \subset {\bf SL}_N(\mZ/p\mZ)$ with trivial normal core. Set $\cD = \prod_{p \in \cP'} \ D_p$.
 \medskip
 
 {\bf Theorem [Hurder \& Lukina, 2017].}  For a  closed subgroup $\cD \subset \whH$   as above,   the induced action $\vp_{\alpha, \cD}$ of $G$ on $\whH/\cD$ by $\alpha$ satisfies:
 
   
  $\bullet$ The action $\vp_{\alpha, \cD}$ is minimal and equicontinuous;  
  
  $\bullet$ The action $\vp_{\alpha, \cD}$ is wild for suitable choices of $\cD$;  
 
 $\bullet$  The actions  $\vp_{\alpha, \cD}$ for uncountably many such choices of $\cD$ yield non-homeomorphic weak solenoids. 
  
   
 
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}

 
 \frame % 1
{
  \frametitle{ }

   
 
 {\bf Example 6:}   Arboreal actions of Galois groups.
 
  \bigskip
  
 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
$\bullet$      [Jones, 2013]  gives  a nice introduction and survey of this program,  from the point of view of arithmetic dynamical systems and number theory.
  
    
   
       
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}

 
    
\frame % 1
{

The following discussion concerns  results of [Lukina, 2018].
 
 \pgfputat{\pgfxy(6, -5)}{\pgfbox[left,base]{\pgfimage[width=4.8cm]{D-ary-tree-2}}} 

\medskip
  Let $X=\mathcal{P}_d$ be the space of paths

in a spherically homogeneous rooted tree $T_d$.

\medskip
Let $G$ be any discrete group, acting on $T_d$

by permuting edges at each level

so that the paths are preserved.

The space of paths with the 

cylinder topology is a Cantor set

\medskip 
This action is equicontinuous.

 
 
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}

\frame
{

\bigskip

 

Let $f(x)$ be an irreducible polynomial of degree $d$ over a number field $K$. 
Let $\alpha \in K$, and suppose $f(x) = \alpha$ has $d$ distinct solutions.

\medskip

Identify $\alpha$ with the root of a $d$-ary tree $T_d$, and 
identify every  solution $\alpha_{11}, \alpha_{12},\ldots,\alpha_{1d}$ of $f(x) = \alpha$ with a vertex at level $1$ in the tree. \pgfputat{\pgfxy(5.6, -3.4)}{\pgfbox[left,base]{\pgfimage[width=4cm]{D-ary-tree}}} 

\medskip
${\rm Gal}(K(f^{-1}(\alpha))/K)$ is identified with a subgroup 

of the symmetric group $S_d$.

 
For every $\alpha_{1i}$, consider the equation
  $$f(x) = \alpha_{1i}, \textrm{ so }f\circ f(x) = f(\alpha_{1i}) = \alpha.$$
Suppose there are $d^2$ distinct roots. 
Identify the solutions of $f(x) = \alpha_{1i}$ with the $d$ vertices at level $2$ connected with $\alpha_{1i}$ at level $1$.


\vfill   

}

\frame
{

 \bigskip
 

The action of ${\rm Gal}(K(f^{-2}(\alpha))/K)$ preserves the structure of the tree, so 
    $${\rm Gal}(K(f^{-2}(\alpha))/K) \subseteq [S_d]^2,$$
where $[S_d]^2$ denotes the two-fold wreath product of symmetric groups $S_d$.

\medskip
Continue by induction, assuming that for each $i>0$ the polynomial $f^i(x)$ has $d^i$ distinct roots.   

\medskip
In the limit, we get a $d$-ary infinite tree $T_d$ of preimages of $\alpha$ under the iterations of $f(x)$, and the profinite group
  $${\rm Gal}_\infty(f) = \lim_{\longleftarrow}\{{\rm Gal}(K(f^{-i}(\alpha))/K) \to {\rm Gal}(K(f^{-(i-1)}(\alpha))/K)\} ,$$
 a subgroup of the infinite wreath product $Aut(T_d) = [S_d]^\infty$.

 
\vfill   

}

\frame
{

\bigskip

 
The group ${\rm Gal}_\infty(f)$ is called an  \emph{arboreal representation} of the absolute Galois group ${\rm Gal}(K^{sep}/K)$.

The representation depends on the polynomial $f$.

Thus ${\rm Gal}_\infty(f)$ is a \emph{profinite} group acting on the Cantor set of paths in the tree $T_d$. 

\medskip
{\bf Example [Odoni, 1985].}  If $K = \mathbb{Q}$, $\alpha = 0$, $f(x) = x^2-x+1$, then
  $$Gal_\infty(f) \cong  Aut(T_2) \cong [S_2]^\infty \ .$$

  
\vfill
}

    
 
\frame
{


{\bf Theorem [Lukina, 2018].}
Let $f(x)$ be a polynomial of degree $d \geq 2$ over a field $K$, suppose all roots of $f^i(x)$ are distinct and $f^i(x) - \alpha$ is irreducible for all $i \geq 0$.

Let ${\bf v}$ be a path in the space of paths $\mathcal{P}_d$ of the tree $T_d$. 

\medskip
Then there exists a countably generated group $G_0$, a homomorphism
  $\Phi: G_0 \to Homeo(\mathcal{P}_d)$
and a chain $\{G_i\}_{i \geq 0}$ of subgroups in $G_0$ such that

\medskip
(1) There is an isomorphism $\widetilde{\phi}: \overline{\Phi(G_0)} \to {\rm Gal}_\infty(f)$,

(2) There is a homeomorphism ${\displaystyle \phi: \lim_{\longleftarrow}\{G_0/G_i\} \to \mathcal{P}_d}$ with $\phi(eG_i) = {\bf v}$,

(3) For all ${\bf u} \in \mathcal{P}_d$ and ${\bf g} \in \overline{\Phi(G_0)}$ we have
   $$\widetilde{\phi}(\bf g) \cdot \phi(\bf u) = \phi({\bf g}({\bf u})).$$



\vfill
}

    
 
\frame
{

\bigskip


\bigskip

{\bf Theorem [Lukina, 2018].}
Suppose the image of an arboreal representation ${\rm Gal}_\infty(f)$ is a subgroup of finite index in $Aut(T_d)$. Then the action of the dense subgroup $G_0$ on the path space $\mathcal{P}_d$ is wild.
 

 \bigskip
 
{\bf Remark:} The proof of this result  is geometric, it uses the absence of the \emph{strong quasi-analytic property} of wild actions. 

The proof does not require an explicit description of the Galois groups involved.


\vfill
}

    
 
\frame
{

\bigskip

 
{\bf Theorem [Lukina, 2018].}
Let $p$ and $d$ be distinct odd primes, let $K=\mathbb{Q}_p$ be the field of $p$-adic numbers, and let
  $$f(x) = (x+p)^d - p.$$
Then the action of the dense subgroup group $G_0$ of the arboreal representation $Gal_\infty(f)$ is stable.

\medskip


{\bf Remark:} The group acting on $T_d$ in this result is the Baumslag-Solitar group $BS(p,1) = \{\tau, \sigma \mid \sigma \tau \sigma^{-1} = \tau^p\}$. The proof uses explicit representations of subgroups $G_i$ in terms of generators and relations.

  \medskip
  
 There are many further  constructions of arboreal actions, by [Boston \& Jones, 2007] and Nekrashevych for example. It is work in progress, to understand further the structure of arboreal  Galois actions, from the point of view of the geometry and dynamics of the weak solenoids associated to these actions.
  
  \vfill   

}

\frame % 1
{

 {\bf Conclusions:}
 
\bigskip

$\bullet$~ The construction of weak solenoids whose monodromy action has interesting properties, such as non-trivial Cantor discriminant group and possibly also wild, is closely related to constructions in geometric group theory and number theory.

\medskip

$\bullet$~ The invariants of weak solenoids obtained using foliation index theorems and Lefschetz trace formulae, are as good a place as any, to go hunting for number theory formulae...
 
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}


  \section{Refs}

    
\frame % 1
{
  \frametitle{References}
 {\scriptsize     
 
 
{A.~Clark and S.~Hurder},
 {\it Homogeneous matchbox manifolds},
 {\bf Transactions A.M.S.},   365:3151--3191, 2013.
 
 {N.~Boston and R.~Jones},
 {\it Arboreal {G}alois representations},
 {\bf Geom. Ded.},  124, 2007.

{J.~Dyer, S.~Hurder and O.~Lukina},
 {\it Molino theory for matchbox manifolds},
 {\bf Pacific Journal 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 Transactions A.M.S.},  to appear, 2018, arXiv:1702.03032.

{S.~Hurder and O.~Lukina},
 {\it Orbit equivalence and   classification of weak solenoids},
 {preprint},   arXiv:1803.02098. 

{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},
 {preprint},   arXiv:1801.01440.

  
{R.W.K. Odoni},
  {\it The {G}alois theory of iterates and composites of polynomials},
  {\bf Proc. London Math. Soc. (3)}, 51:385-414, 1985.


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

  
}
}

 \section{HBD}

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