% New trends on Foliated and Stratified Spaces: 
% Topology, Geometry and Analysis. 
% Commemorating the 70th birthday of Xos�   Masa

% Place: Santiago do Compostela, Spain, France

% 18-19 June 2018

% http://fxc-ow.usc.es/en/congresos/folstratsps/index.html
% talk on June 18, 2018

% slides for talk "A new perspective on Riemannian foliations"
 
 
\documentclass{beamer}

\usepackage{amsfonts}

\usepackage{graphicx}

\usepackage{pgf,pgfarrows,pgfnodes}
\usepackage{wrapfig}

\input xy
\xyoption {all}



% \usetheme{default}
% \usetheme{Singapore}
% \usetheme{CambridgeUS}
% \usetheme{Boadilla}
% \usetheme{Antibes}
% \usetheme{Berkeley}
% \usetheme{Berlin}
% \usetheme{Copenhagen}
% \usetheme{Darmstadt}
% \usetheme{Dresden}
% \usetheme{Frankfurt}
% \usetheme{Goettingen}
% \usetheme{Hannover}
% \usetheme{Ilmenau}
% \usetheme{JuanLesPins}
% \usetheme{Luebeck}
% \usetheme{Madrid}
% \usetheme{Malmoe}
% \usetheme{Marburg}
 \usetheme{Montpellier}
% \usetheme{PaloAlto}
% \usetheme{Pittsburgh}
% \usetheme{Rochester}
% \usetheme{Szeged}
% \usetheme{Warsaw}
 
% \usecolortheme{crane}
 % \beamertemplatesolidbackgroundcolor{craneorange!25}
 
 \parskip=4pt
 

% math notation shortcuts

\newcommand{\ds}{\displaystyle}
\newcommand{\sop}{{\bf Proof:} } % replaced by command \proof
\newcommand{\eop}{\;\;\; \Box} % replaced by command \endproof - still used inside theorem environment

%   bolds
\newcommand{\bZ}{{\bf Z}}
\newcommand{\bC}{{\bf C}}

\newcommand{\bK}{{\bf K}}
\newcommand{\F}{{\mathcal F}} % our favorite foliation
\newcommand{\e}{{\epsilon}} % our favorite constant

\newcommand{\wmW}{\widehat{\mW}}

\newcommand{\cGF}{\cG_{\F}} % denotes the pseudogroup of F
\newcommand{\cGM}{\cG_{\fM}} %
\newcommand{\cGN}{\cG_{\fN}} %
\newcommand{\cGX}{\cG_{\fX}} %
\newcommand{\cGY}{\cG_{\fY}} %
\newcommand{\Dom}{{\rm Dom}} %
\newcommand{\dX}{d_{\fX}} % leafwise metric
\newcommand{\dXp}{d_{\fX}'} % leafwise metric
\newcommand{\dY}{d_{\fY}} % leafwise metric
\newcommand{\diamX}{{\rm diam}_{\fX}} %
\newcommand{\G}{\Gamma}
\newcommand{\ovx}{\overline{x}} % 
\newcommand{\ovy}{\overline{y}} %  
\newcommand{\ovz}{\overline{z}} %  
\newcommand{\oU}{\overline{U}}
\newcommand{\ovgamma}{\overline{\gamma}} %  


% blackboard bolds
\newcommand{\mB}{{\mathbb B}}
\newcommand{\mC}{{\mathbb C}}
\newcommand{\mD}{{\mathbb D}}
\newcommand{\mE}{{\mathbb E}}
\newcommand{\mF}{{\mathbb F}}
\newcommand{\mG}{{\mathbb G}}
\newcommand{\mK}{{\mathbb K}}
\newcommand{\mN}{{\mathbb N}}
\newcommand{\mP}{{\mathbb P}}
\newcommand{\mQ}{{\mathbb Q}}
\newcommand{\mR}{{\mathbb R}}
\newcommand{\mS}{{\mathbb S}}
\newcommand{\mT}{{\mathbb T}}
\newcommand{\mW}{{\mathbb W}}
\newcommand{\mZ}{{\mathbb Z}}



% cal script letters
\newcommand{\cA}{{\mathcal A}}
\newcommand{\cB}{{\mathcal B}}
\newcommand{\cC}{{\mathcal C}}
\newcommand{\cD}{{\mathcal D}}
\newcommand{\cE}{{\mathcal E}}
\newcommand{\cG}{{\mathcal G}}
\newcommand{\cH}{{\mathcal H}}
\newcommand{\cI}{{\mathcal I}}
\newcommand{\cJ}{{\mathcal J}}
\newcommand{\cK}{{\mathcal K}}
\newcommand{\cL}{{\mathcal L}}
\newcommand{\cM}{{\mathcal M}}
\newcommand{\cN}{{\mathcal N}}
\newcommand{\cO}{{\mathcal O}}
\newcommand{\cP}{{\mathcal P}}
\newcommand{\cQ}{{\mathcal Q}}
\newcommand{\cR}{{\mathcal R}}
\newcommand{\cS}{{\mathcal S}}
\newcommand{\cT}{{\mathcal T}}
\newcommand{\cU}{{\mathcal U}}
\newcommand{\cV}{{\mathcal V}}
\newcommand{\cW}{{\mathcal W}}
\newcommand{\cX}{{\mathcal X}}
\newcommand{\cY}{{\mathcal Y}}
\newcommand{\cZ}{{\mathcal Z}}




% fraktur letters

\newcommand{\fA}{{\mathfrak{A}}}
\newcommand{\fB}{{\mathfrak{B}}}
\newcommand{\fC}{{\mathfrak{C}}}
\newcommand{\fD}{{\mathfrak{D}}}
\newcommand{\fG}{\mathfrak{G}}
\newcommand{\fH}{\mathfrak{H}}
\newcommand{\fK}{{\mathfrak{K}}}
\newcommand{\fL}{{\mathfrak{L}}}
\newcommand{\fM}{{\mathfrak{M}}}
\newcommand{\fN}{{\mathfrak{N}}}
\newcommand{\fO}{{\mathfrak{O}}}
\newcommand{\fP}{{\mathfrak{P}}}
\newcommand{\fQ}{{\mathfrak{Q}}}
\newcommand{\fS}{{\mathfrak{S}}}
\newcommand{\fT}{{\mathfrak{T}}}
\newcommand{\fU}{{\mathfrak{U}}}
\newcommand{\fV}{{\mathfrak{V}}}
\newcommand{\fX}{{\mathfrak{X}}}
\newcommand{\fY}{{\mathfrak{Y}}}
\newcommand{\fZ}{{\mathfrak{Z}}}
\newcommand{\fW}{{\mathfrak{W}}}

% greek letters

\newcommand{\vp}{{\varphi}}
\newcommand{\whvp}{\widehat{\varphi}}
\newcommand{\whPhi}{\widehat{\Phi}}
\newcommand{\whe}{\widehat{e}}
\newcommand{\whg}{\widehat{g}}
\newcommand{\whh}{\widehat{h}}
\newcommand{\whU}{\widehat{U}}
\newcommand{\whV}{\widehat{V}}
\newcommand{\whG}{\widehat{G}}
\newcommand{\whH}{\widehat{H}}

\newcommand{\wtSigma}{\widetilde{\Sigma}}

\newcommand{\bRt}{{\bf R}_{0}}
\newcommand{\whcG}{\widehat{\cG}}
 \newcommand{\psg}{{\rm pseudo}{\star}{\rm group}}

  \newcommand{\Aut}{{\bf Aut}}
  \newcommand{\Homeo}{{\bf Homeo}}
\newcommand{\wtM}{\widetilde{M}} %  
\newcommand{\mH}{{\mathbb H}}

\newcommand{\CO}{{\rm CO}} 
\newcommand{\diam}{{\rm diam}} 


\date[June 14, 2018]


  
\title[A New Perspective]{A new perspective on Riemannian foliations}


\author{Steve Hurder\\     \footnotesize {University of Illinois at Chicago}\\....    \\{\footnotesize {\it New trends on Foliated and Stratified Spaces:\\ Topology, Geometry and Analysis}\\ {\scriptsize Commemorating the 70th birthday of Xos\'e  Masa} \\   {\scriptsize Santiago do Compostela-- June 18-19, 2018}}}

 
 
\begin{document}

\frame{\titlepage} % # 1


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

\bigskip
 
 \begin{quotation}
\emph{In the beginning...}
\end{quotation}
\bigskip
 
 Bruce  Reinhart,  ``\emph{Foliated manifolds with bundle-like metrics}'',  
 {\bf Ann. of Math},   69:119--132, 1959 
 
 \bigskip
 
\begin{quote}
 In addition to the foliation, there is postulated the existence of a ``bundle-like''' Riemannian metric, which locally is very similar to a product metric. Such a metric always exists for the special case of a fibre space. 
\end{quote}
 
 \vfill   

}

  
  

  
 
  \frame % 1
{
  \frametitle{}

 
Edmond Fedida,  ``\emph{Sur les feuilletages de Lie}'',  
{\bf C. R. Acad. Sci. Paris S\'er. A-B},  272:A999--A1001, 1971. 

\medskip

 Lawrence Conlon, ``\emph{Transversally parallelizable foliations of codimension two}'',  
{\bf Trans. Amer. Math. Soc.},  194:79--102, 1974. 

\bigskip

\begin{quote}
studied  Riemannian foliations which are transversally Lie, with a dense leaf
\end{quote}

  
 \vfill   

}

 
 
  \frame % 1
{
  \frametitle{}

 
Pierre Molino,  
\emph{Feuilletages transversalement parall\`elisables et feuilletages de Lie. Applications}'', 
{\bf C. R. Acad. Sci. Paris S\'er. A-B},  282:A99--A101, 1976. 
 
 \bigskip
 
 \begin{quote}
Si $\F$ admet un parall\'elisme au sens de L. Conlon, on obtient le th\'eor\`eme suivant: les adh\'erences des feuilles de $\F$ d\'efinissent une fibration $\pi \colon V \to W$ telle que $\F$ induit sur les fibres de $\pi$  des feuilletages de Lie au sens de E. Fedida.
\end{quote}

\bigskip

This result includes the fundamental observation: the closures of the leaves of $\F$ are minimal sets of the foliation, and these closures form a continuous decomposition of the total space of the foliation.
 \vfill   

}

 
 
  \frame % 1
{
  \frametitle{}

Pierre  Molino,  
{\bf Feuilletages riemanniens},  Universit\'e des Sciences et Techniques du Languedoc, Institut de Math\'ematiques, Montpellier, 1983. 

\bigskip

\begin{quote}
The principal technique in this study is to lift the foliation to a foliation of the same dimension on the orthonormal frame bundle associated to the normal bundle of the original foliation. The ``lifted foliation''' is invariant under the action of the orthogonal group on the frame bundle and is transversely parallelizable, i.e., its normal bundle is parallelizable. The structure of the ``lifted foliation'' is studied as a special case of parallelizable foliations, and these results pushed down to the original foliation using the invariance under the orthogonal group. 
\end{quote}
 \vfill   

}

 
 
  \frame % 1
{
  \frametitle{}

These lecture notes, and especially their 1988 translation from   French by Grant Cairns, introduced the study of Riemannian foliations to a broad audience. This explosion of interest is evidenced by the Appendices of the English translation:

$\bullet$ ~ A. \emph{Variations on Riemannian flows}, by Y. Carri\`ere; 

$\bullet$ ~ B. \emph{Basic cohomology and tautness of Riemannian foliations}, by V. Sergiescu; 

$\bullet$ ~ C. \emph{The duality between Riemannian foliations and geodesible foliations}, by G. Cairns; 

$\bullet$ ~ D. \emph{Riemannian foliations and pseudogroups of isometries}, by E. Salem; 

$\bullet$ ~  E. \emph{Riemannian foliations: examples and problems}, by \'E. Ghys. 


 \vfill   

}

  

 \frame % 1
{
  \frametitle{}


The  work of Xos\'e Masa solved  a fundamental problem  combining the topics of Appendices B. and C.

Xos\'e Masa, 
``\emph{Duality and minimality in Riemannian foliations}'',  
{\bf Comment. Math. Helv.},  67:17--27, 1992.

\bigskip

\begin{quote}
In this work, we prove that a Riemannian foliation $\F$ defined on a smooth closed manifold $M$ is minimal, in the sense that there exists a Riemannian metric on $M$ for which all the leaves are minimal submanifolds, if and only if $\F$ is unimodular, that is, the basic cohomology of $\F$ in maximal dimension is nonzero. 
\end{quote}

  \vfill   

}
  

    
\frame
{
  \frametitle{}


This paper by Xos\'e Masa, and the related work by his student, 

Jes\'us    \'Alvarez L\'opez,  ``\emph{The basic component of the mean curvature of Riemannian foliations}'', 
{\bf Ann. Global Anal. Geom.}, 10:179--194, 1992,  

led to an explosion of further works studying the analytic and geometric properties of Riemannian foliations.

\bigskip

\begin{quote}
The study of Riemannian foliations is   reaching mid-life. Perhaps, it is time for a \emph{Little Red Sports Car}...
\end{quote}

In this talk, we consider a new model for further studies of   ``Riemannian foliations''. 
It is sporty, and very algebraic.

  \vfill   

}
  
\section{Equicontinuous foliated spaces}


   
\frame
{
  \frametitle{}
Motivated by a question posed by Ghys in Appendix E. to Molino's book, Alberto Candel and Jes\'us \'Alvarez L\'opez began the study of a topological version of Riemannian foliations, in their work 

``\emph{Equicontinuous foliated spaces}'', 
{\bf Math. Z.},  263:725--774, 2009. 

 The starting point   is the fact that a Riemannian foliation is a foliation whose holonomy pseudogroup is a pseudogroup of local isometries of a Riemannian manifold. More generally, for equicontinuous pseudogroups, one has the notion of equicontinuous foliated spaces. This leads to a version of the Molino structure theory, in the work
 
 \medskip
 
 \'Alvarez L\'opez and Moreira Galicia, 
``\emph{Topological Molino's theory}'', 
{\bf Pacific J. Math.},  280:257--314, 2016. 
 
  \vfill   

}
  

      
\frame
{
  \frametitle{}


 We will consider      foliated spaces which are   transversally totally disconnected.
 
  
  
 \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}, [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 % 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 $\F$, so that 
each leaf of $\F$ is a complete   Riemannian manifold with bounded geometry.

\bigskip

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.

\medskip

{\bf Basic question:} What are these spaces, and their properties?

     \vfill   

}



      
\frame
{
  \frametitle{}

The lamination analog of   a Riemannian foliation   is a matchbox manifold $\fM$ whose holonomy pseudogroup $\cG$ is generated   by   minimal equicontinuous   actions on a Cantor space $\fX$:


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

Here, $\cG$ denotes either the $\psg$ of local actions on the Cantor set $\fX$, or is an actual group action on $\fX$.

      \vfill   

}



      
\frame
{
  \frametitle{}



The simplest examples are given by the     \emph{Vietoris solenoids}.

\medskip

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

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

\bigskip

  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 $\prod_{i \geq 0} \mathbb{S}^1$ .

There is a projection map $\Pi \colon \Sigma_{\bf P} \to \mS^1$, $\Pi(y_0, y_1, y_2, \ldots) = y_0$.
 
 \vfill   

}
  

    
\frame
{
  \frametitle{}

The fibre $\fX_b = \Pi^{-1}(b) = \{(b,y_1,y_2, \ldots)\} \subset \Sigma_{\bf P}$ is a Cantor set, for each $b \in \mS^1$.
 
 
\medskip

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. 

\medskip
 
 {\bf Theorem (Bing 1961, McCord 1965):} The homeomorphism class of a Vietoris solenoid is completely determined by the tail equivalence class of the prime divisors of the degrees of the coverings defining it.
\vfill
 }

 
  
 
  
 \frame % 1
{
  \frametitle{}

  McCord  began  the study of higher-dimensional generalizations of the   Vietoris solenoid in 1965. 
 

Let $M_0$ be a connected closed manifold, and  let $\cP \equiv \{f^i_{i-1}: M_i \to M_{i-1} \mid i \geq 1\}$ be a sequence of finite-to-one proper covering maps. $\cP$ is  called a \emph{presentation}. Then
$$
 \fM_{\infty} = \lim_{\longleftarrow} \{ f^i_{i-1}: M_i \to M_{i-1}  \mid i \geq 1 \}
$$
is a compact connected metrizable space, called a \emph{(weak) solenoid}.

There is a fibration map $\Pi_0 \colon \fM_{\infty} \to M_0$, and for $b \in M_0$ the fiber $\fX_b = \Pi_0^{-1}(b)$ is a Cantor space. The monodromy of the fibration yields a group action $\vp \colon G \times \fX_b \to \fX_b$ where $G = \pi_1(M_0 , b)$.


    
 {\bf Theorem [McCord, 1965].}   A solenoid is a \emph{matchbox manifold}, or \emph{generalized lamination} (Ghys), or \emph{solenoidal manifold} (Sullivan). 

     
 \vfill
 }

  \section{Fedida}

 
\frame
{
  \frametitle{}

A topological space $Z$ is homogeneous if for any pair of points $x,y \in Z$, there is a homeomorphism $h \colon Z \to Z$ with $h(x) = y$.

{\bf Proposition:} 
A transversally parallelizable Riemannnian foliation of a compact connected manifold is homogeneous.

A homeomorphism $h \colon \fM_1 \to \fM_2$ of matchbox manifolds must map the leaves of $\F_1$ to the leaves of $\F_2$ since the leaves consist of the path components of the space.
Thus, the following result can be considered as a matchbox version of Fedida's Theorem:

\medskip

{\bf Theorem (Clark-H, 2013):} Let $\fM$ be an equicontinuous matchbox manifold. Then $\fM$ is homeomorphic to a weak solenoid $\fM_{\infty}$ for some presentation $\cP$. Moreover, if   $\fM$ is a homogeneous space, then the fiber $\fX_b \subset \fM_{\infty}$ is a profinite group.
 
This is proved in the paper  ``\emph{Homogeneous matchbox manifolds}'', {\bf Transactions A.M.S.}, 365:3151-3191,  2013. 

\medskip


 
 \vfill   

}
     
 
   
   
\frame
{
  \frametitle{}

  However, if $\fM$ is not homogeneous, then the fiber $\fX_b$ is not a profinite group, but is a quotient of a profinite group by a non-trivial closed subgroup of some profinite group.
  It is the incredible variety of closed Cantor subgroups of profinite groups that is the source of troubles.
  
  \medskip
    Topological Molino Theory deviates from the smooth theory in this non-homogeneous case, as there are issues with defining the paralellizable frame bundle associated to the foliation, which is a key step introduced by Molino for his general structure theory.

\medskip
    
    To understand this case, we digress into a discussion of the theory of equicontinuous minimal Cantor actions.
 
  

  

 
 \vfill   

}


   
   
 \section{Actions}




\frame % 2
{
  \frametitle{}
 
    
   We recall some   classical notions from Auslander, {\bf Minimal flows and their extensions}, 1988. $G$ denotes a countable group.
  
\medskip

We say that $U \subset \fX$  is \emph{adapted} to the action $\vp \colon G \times \fX \to \fX$  on a Cantor space $\fX$, if $U$ is a   non-empty clopen subset, and for any $g \in G$, 
$\Phi(g)(U) \cap U \ne \emptyset$ implies that $\Phi(g)(U) = U$. 
That is, the translates of $U$ form a  partition of the Cantor set $\fX$.
It follows that
$$
G_U = \left\{g \in G \mid \vp(g)(U) \cap U \ne \emptyset  \right\}  
$$
is a subgroup of finite index in $G$, called the \emph{stabilizer} of $U$. 

 
  

 \vfill   

}


   

\frame % 1
{
  \frametitle{}

  For a Cantor space $\fX$, let $\CO(\fX)$ denote the collection  of all clopen (compact open) subsets of $\fX$.
Note that for $\phi \in \Homeo(\fX)$ and    $U \in \CO(\fX)$, the image $\phi(U) \in \CO(\fX)$.  

\medskip

{\bf Proposition (Glasner and Weiss, 1995):}
A   minimal Cantor action $\vp \colon G \times \fX \to \fX$  is  equicontinuous  if and only if, for the induced action $\Phi_* \colon G \times \CO(\fX) \to \CO(\fX)$, the $G$-orbit of every $U \in \CO(\fX)$ is finite.
 
 \medskip
 
 {\bf Corollary:}  Let $\vp \colon G \times \fX \to \fX$ be   an equicontinuous     Cantor action. Then for all $x \in \fX$ and all $\delta > 0$, there exists an adapted clopen set $U$ with $x \in U \subset B_{\fX}(x,\delta)$.
  

 

 \vfill   

}


   

\frame % 1
{
  \frametitle{}

  
  
 {\bf Definition:} Let $\vp_i \colon G_i \times \fX_i \to \fX_i$ be minimal equicontinuous Cantor actions, for $i=1,2$.
 Say that $\vp_1$ is  \underline{return equivalent} to $\vp_2$ if there exist 
 
  $\bullet$  adapted clopen subsets $U_i \subset \fX_i$ for $i=1,2$
  
   $\bullet$ a homeomorphism  $h \colon U_1 \to U_2$ 
   
     
 such that $h$ induces an isomorphism   $\alpha_h \colon G_1|U_1 \to G_2|U_2$ of the restricted groups, where $G_i|U_i \subset \Homeo(U_i)$.
  
  \medskip
  
{\bf Remark:}   When   $U_i = \fX_i$ for $i=1,2$, and the actions are effective, this reduces to the notion of topological conjugacy of the actions,  where $\alpha_h \colon G_1 \to G_2$ intertwines the actions.
  



 
 \vfill   

}


   

\frame % 1
{
  \frametitle{}

 
{\bf Theorem (Clark, H, Lukina 2017):}
Let $\fM_1$ and $\fM_2$ be weak solenoids. Suppose that there exists a homeomorphism $h \colon \fM_1 \to \fM_2$, then the fiber monodromy  actions associated to $\fM_1$ and $\fM_2$  are return equivalent.
 

 \medskip
 
 In Molino Theory, if two Riemannian foliations are diffeomorphic, then their Molino structure groups are isomorphic.
 
 \medskip
 
 We next consider the structure theory for equicontinuous minimal Cantor actions, and how they behave under return equivalence.
 
 \medskip
 
We introduce the group chain (or odometer) model for an equicontinuous minimal Cantor action, and the related tree model for an action.
  
     \vfill   

}


   \section{Models}


  

 \frame % 1
{
  \frametitle{}

  Let $\vp \colon G \times \fX \to \fX$ be a minimal equicontinuous Cantor action.
   
 For a choice of basepoint $x \in \fX$ and scale $\e > 0$, there exists an adapted clopen set $U \in \CO(\fX)$ with $x \in U$ and $\diam(U) < \e$.      Iterating this construction, for a given basepoint $x$, one can always construct the following:
 
 \bigskip
 
{\bf Definition:} Let  $\vp \colon G \times \fX \to \fX$  be a minimal equicontinuous  action  on a Cantor space $\fX$.
A properly descending chain of clopen sets $\cU = \{U_{\ell} \subset \fX  \mid \ell \geq 1\}$ is said to be an \emph{adapted neighborhood basis} at $x \in \fX$ for the action $\Phi$  if
    $x \in U_{\ell +1} \subset U_{\ell}$ for all $ \ell \geq 1$ with     $\cap  \ U_{\ell} = \{x\}$, and  each $U_{\ell}$ is adapted to the action $\Phi$.


 \vfill   

}
 
 
  
 \frame % 1
{
  \frametitle{}

 

 For such a collection, setting $G_{\ell} = G_{U_{\ell}}$     we obtain a descending chain of finite index subgroups 
 $$\ds \cG_{\cU} = \{G = G_0 \supset G_1 \supset G_2 \supset \cdots \} \ .$$
  
Set $X_{\ell} = G/G_{\ell}$ and note that  $G$ acts transitively on the left on   $X_{\ell}$.    
The inclusion $G_{\ell +1} \subset G_{\ell}$ induces a natural $G$-invariant quotient map $p_{\ell +1} \colon X_{\ell +1} \to X_{\ell}$.
 Introduce the inverse limit 
$$
\fX_{\infty} \equiv \varprojlim \ \{p_{\ell +1} \colon X_{\ell +1} \to X_{\ell} \mid \ell > 0\} 
$$ 
which is a Cantor space with the Tychonoff topology, and the action on the factors $X_{\ell}$ induces   a minimal  equicontinuous action $\Phi_x \colon G \times \fX_{\infty} \to \fX_{\infty}$. 
 

 \vfill   

}
 
  
 \frame % 1
{
  \frametitle{}

 The action $\Phi_x \colon G \times \fX_{\infty} \to \fX_{\infty}$  is called the \emph{generalized odometer} model, or also called a \emph{subodometer}, by Cortez \& Petite in their work ``\emph{$G$-odometers and their almost one-to-one extensions}'',  2008.
 
\bigskip

We give some remarks on this construction.

\smallskip

$\bullet$ ~  Each $X_i = G/G_i$ is a finite set with a left action of $G$. It is a group if $G_i$ is normal in $G$, and then the Cantor space $\fX_{\infty}$ is a profinite group. 

\smallskip

 
$\bullet$ ~  The intersection $\ds K(\cG_{\cU}) = \bigcap_{\ell \geq 0} ~ G_{\ell}$ is called the kernel of $\cG_{\cU}$.   

 \smallskip

$\bullet$ ~ For $g \in K(\cG_{\cU})$, the left action of $g$ on $X_{\ell}$ fixes the coset $e_{\ell} \in X_{\ell}$ and hence fixes the limiting point $e_{\infty} \in \fX_{\infty}$.

 

 \vfill   

}

  \frame % 1
{
  \frametitle{}

We next recall the tree model for the action $\vp \colon G \times \fX \to \fX$.  

 
\bigskip

First, again choose an adapted neighborhood basis at $x \in \fX$ for the action, 
$\ds \cU = \{U_{i} \subset \fX  \mid i \geq 1\}$.

Note that by assumption we have $\bigcap U_i = \{x\}$.
  
  \bigskip
 Next, associate a vertex $v_{i,g}$ at level $i$  to each $g \cdot U_i$.  


 
Join $v_{i,g}$ and $v_{i+1,h}$ by an edge if and only if $h \cdot U_{i+1} \subset g \cdot U_i$.


 
 \vfill   

}

  \frame % 1
{
  \frametitle{}

  
A sequence of vertices $(v_{i,g_i})_{i \geq 0}$ is a path in the space $\mathcal{P}_T$ of paths in $T$, and $\fX \cong \cP_T$.

 
The subgroup $G_i$ of elements which stabilize $U_i$ has finite index in $G$, and there is a group chain of stabilizers $\cG_{\cU} \equiv \{G_i\}_{i \geq 0}$ associated to the action.

  \pgfputat{\pgfxy(1.5, -2.2)}{\pgfbox[left,base]{\pgfimage[width=6cm]{D-ary-tree-2.png}}} 
 
\vfill   

}

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

   

We next introduce the Ellis group associated to an action.
  
 
  An action $\vp \colon G \times \fX \to \fX$ induces a representation $\Phi \colon G \to {\Homeo}(\fX)$ with image group 
  $$H_{\Phi} = \Phi(G) \subset   {\Homeo}(\fX)$$

  


 {\bf Definition:} The  closure $E(\Phi)$ of    $H_{\Phi}$,  
 in the topology of pointwise convergence on maps, is called the   \emph{Ellis (enveloping) semigroup}.
 
   \smallskip
   
 {\bf Proposition (Ellis, 1969):} Let $\vp$ be an equicontinuous    Cantor action. Then $E(\Phi) = \overline{H_{\Phi}} = $     closure of $H_{\Phi}$   
   in the \emph{uniform topology on maps}. In particular, $\overline{H_{\Phi}}$ is a profinite group.


\smallskip

  
 For $x \in \fX$ let $\overline{H_{\Phi}}_x = \{ h \in \overline{H_{\Phi}} \mid h(x) = x \}$ be its isotropy group. 
 
  
 
 
   
  
\vfill   

}

  \frame % 1
{
  \frametitle{}

 {\bf Lemma:} The left action of $\overline{H_{\Phi}}$ on $\fX$ is transitive,  hence  $\fX  \cong \overline{H_{\Phi}}/\overline{H_{\Phi}}_x$ and the closed subgroup $\overline{H_{\Phi}}_x \subset \overline{H_{\Phi}}$   is independent of the choice of basepoint $x$, up to topological isomorphism.

\medskip

We   give a representation for     $\overline{H_{\Phi}}_x$   in terms of the odometer model   for the action.

  \smallskip
The \underline{normal core} $N$ of a subgroup $H \subset G$ is the largest subgroup $N \subset H$ which is   normal in $G$.
   
  
 
 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
 $$G_{\infty} \equiv  \lim_{\longleftarrow} ~  \{q_{i} \colon  G/C_{i +1} \to G/C_{i} \mid i > 0\}~ .$$

  
\vfill   

}

  \frame % 1
{
  \frametitle{}
  
  Each group $G/C_{i}$ acts on the finite set $X_i  = G/G_{i}$, so there is an induced action   $\whPhi_{\infty} \colon G_{\infty} \to {\Homeo}(\fX_{\infty}) \cong {\Homeo}(\fX)$.
 
 \medskip
  
   
  {\bf Theorem (Dyer-H-Lukina, 2016).}    $\overline{H_{\Phi}} \cong \whPhi_{\infty}(G_{\infty})$, 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 \overline{H_{\Phi}}_x ~ .
\end{equation}
 
 The inverse limit group $\cD_{\infty}$ is called the \emph{discriminant group} for the action. 
  Its \emph{non-triviality} is the obstruction to the existence of a \emph{transitive} right  action on $\fX$ that commutes with the left action $\vp$.

\bigskip

 
 \vfill   

}



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

The   approach to the study of weak solenoids via their monodromy   Cantor actions obtained from group chains was initiated in the work of [Fokkink \& Oversteegen, 2002].

\medskip
 
 Let $\Pi_0 \colon \fM_{\infty} \to M_0$ 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:} 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
 
 Then  $ G/G_i = \mathbb{Z}/p_1 \cdots p_i \mathbb{Z}$.

\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 $\cD_{\infty}$ of the Vietoris solenoid is trivial.

 \vfill   

}

 



 \frame % 1
{
  \frametitle{}

\bigskip

 {\bf Example 2:}  Here is a more interesting example,  with $\cD_{\infty}$ 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 $i_\infty: \mathbb{T}_\infty \to K_\infty$.

 
\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 $\mathcal{D}_{\infty}  \cong \mZ/2\mZ$.


\vfill   

}

  
 
 

\section{Wild Actions}


 \frame % 1
{
  \frametitle{}

 The discriminant need not be an invariant of return equivalence for an equicontinuous    Cantor action!
 We use the tree model for the action to analyze this, which leads to the notion of a \underline{wild action}.

Recall that 
$\ds \cU = \{U_{\ell} \subset \fX  \mid \ell \geq 1\}$
is  an adapted neighborhood basis at $x \in \fX$ for the action,   and $\cP_T$ denotes the space of infinite paths starting at the root point corresponding to $\fX$.
 
Then we have a minimal action $\Phi \colon G \times \cP_T \to \cP_T$.

  \medskip
The group $G_i$ stabilizes a branch of a tree, i.e fixes a vertex at level $i$.
Then by  minimality of the action,  the set of vertices at level $i$ is identified with $G/G_i$.
There is a homeomorphism 
  $${\displaystyle \phi: \mathcal{P}_T \to \fX_\infty = \lim_{\longleftarrow}\{G/G_i \to G/G_{i-1}\}},$$ 
equivariant with respect to the actions of $G$ on $\mathcal{P}_T$ and $\fX_\infty$.
 
    
}

 

 

 \frame % 1
{
  \frametitle{}

 The core subgroup $C_i = \bigcap_{g \in G} gG_ig^{-1} \subset G$  fixes every vertex at level $i$, and the quotient group $G/C_i$   acts transitively on the  set of vertices at level $i$, which correspond to the set $G/G_i$. Then 
$$ C_{\infty} = \lim_{\longleftarrow}\{G/C_i \to G/C_{i-1}\}$$
  is a profinite group acting transitively on the path space $\cP_T$.  

   We use this model to consider  the discriminant groups of the     action $\vp$ restricted to an adapted clopen subset $U \subset \fX$.
   
     
  }

 

 

 \frame % 1
{
  \frametitle{}

 
The set of paths through vertex $v_k$ at level $k$ is a clopen set $U_k \subset \mathcal{P}_T$. Assume that $x \in U_k$.

     
  
 The restricted action on $U_k$ is given by $\Phi_k \colon  G_k \to Homeo(U_k)$.
\pgfputat{\pgfxy(-4.6, -2.8)}{\pgfbox[left,base]{\pgfimage[width=6cm]{D-ary-tree-branch.png}}}     
 
 \bigskip
  
 For each $k \geq i$, the normal core of $G_i$ in $G_k$ is
   $$C_{k,i} = \bigcap_{g \in G_k} gG_i g.  \hspace{6cm} .$$
 Observe that $C_{k,i} \supset C_i$ as the action of $C_i$ fixes all vertices at level $i$, while $C_{k,i}$ fixes just those vertices at level $i$ in the branches of the tree through the vertex $v_k$.
 
  The isotropy group of the action of $\overline{\Phi(G_k)}$ at $x$ is   represented  by 
    $$\cD_{x,k} = \lim_{\longleftarrow}\{G_i/C_{k,i} \to G_{i-1}/C_{k,i-1} \mid i \geq k \}$$
which is   the \emph{discriminant group}  of the action $\Phi_k \colon G_k \times U_k \to U_k$.

    
  
}


 \frame % 1
{
  \frametitle{}

Note that there are   coset inclusions $G_i/C_i \to G_i/ C_{k,i}$.

\bigskip

{\bf Theorem  (Dyer-H-Lukina, 2017)}
Let $\vp \colon G \times \fX \to \fX$ be an equicontinuous minimal Cantor   action with group chain $\{G_i\}_{i \geq 0}$ associated to a basepoint $x \in \fX$. 
Then for any $k>j \geq 0$ there is a well-defined surjective homomorphism
  $$\Lambda_{k,j}: \mathcal{D}_{x,j} \to \mathcal{D}_{x,k}$$
of discriminant groups.  
 
 
\vfill
}



 \frame % 1
{
  \frametitle{}


{\bf Definition:} 
The action $\vp$ is said to be \emph{stable}, if there exists $j_0$, such that for all $k>j \geq j_0$ the homomorphism $\Lambda_{k,j}$ is an isomorphism.
If no such   $j_0$  exist, then the action is said to be \emph{wild}.

\bigskip


{\bf Theorem (Dyer-H-Lukina 2017):}
If the global monodromy action of a weak solenoid $\fM_{\infty}$ is stable, then there is a well-defined (homogeneous) Molino space for $\fM_{\infty}$.


This is proved in the paper

Dyer, Hurder, Lukina, ``\emph{Molino theory for matchbox manifolds}''', {\bf Pacific Journal Math.},  289:91-151, 2017.


\vfill
}

\section{Asymptotic discriminant invariant}

 \frame % 1
{
  \frametitle{}

When an action is not stable, so it is wild, then we define a new invariant of the action as follows:

{\bf Definition (H-Lukina, 2017):}
The \emph{asymptotic discriminant} of the action $(\mathfrak{X},G,\Phi)$ is the equivalence class of the chain of surjective group homomorphisms
   $$\mathcal{D}_{x,0} \to \mathcal{D}_{x,1} \to \mathcal{D}_{x,2} \to \cdots$$
with respect to the tail equivalence relation.

The notion of ``tail equivalence'' is precisely defined in the work with  Lukina, 
 {\it Wild solenoids},
 {\bf Transactions A.M.S.},  2018.


\vfill
}

   
   \frame % 1
{
  \frametitle{ }



In our work we also show the following key property:

{\bf Theorem  (H-Lukina, 2017):}
The asymptotic discriminant of an equicontinuous minimal Cantor   action $\vp \colon G \times \fX \to \fX$  is invariant under the return equivalence of actions.
In particular, the property of being stable or wild is an invariant of return equivalence.

\bigskip

{\bf Corollary:} The asymptotic discriminant of the monodromy of a weak solenoid is an invariant of its homeomorphism type.

\bigskip

Our work constructs infinite families of Cantor actions which have distinct asymptotic discriminant invariants. This is completely unique to the topological Molino theory.
\vfill
}

 
 
  

  \section{Quasi-analytic actions}
 
   
 \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.



\vfill   

}


 \frame % 1
{
  \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 Proposition:} Suppose that $\vp \colon G \colon \fX \to \fX$ is the restriction of a $C^{\omega}$ action on $\mD^k$ for some $k \geq 1$. Then the action of $\vp$ is LQA.
   
 \bigskip
 
      
\vfill   

}


 \frame % 1
{
  \frametitle{ }

   
  {\bf Theorem  (H-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 $\fX_{\infty}$ satisfies the LQA property. 
  
    \bigskip
    
    These notions yield  a ``non-realizable'' criteria:
    
      {\bf Corollary.}
  If  an equicontinuous  minimal Cantor action $\vp \colon G  \times \fX \to \fX$   is not LQA, then any weak solenoid whose monodromy action is return equivalent to this action cannot be realized as the minimal set for a $C^{\omega}$-foliation.
  
  \bigskip
  
  {\bf Question:} Is there a version of this result for $C^2$-foliations?
  
\vfill   

}

 
   \frame % 1
{
  \frametitle{ }


 
{\bf Corollary:} A weak solenoid whose monodromy action is not LQA  admits an infinitely increasing chain of closed groups in the fundamental group $G_0 = \pi_1(M_0 , b_0)$ of the base manifold $M_0$. 

\medskip

{\bf Proposition:} Let $\fM_{\infty}$ be a weak solenoid whose base manifold $M_0$ has nilpotent fundamental group $G_0$. Then the monodromy action of the solenoid is stable.

\bigskip

{\bf Problem:} It is unknown if an equicontinuous minimal action of a finitely generated torsion free nilpotent group $G$ must be stable.


  \vfill   

}



 \frame % 1
{
  \frametitle{ }

There is a nice ``geometric proof'' that an LQA Cantor action $\vp \colon G \times \fX \to \fX$ is stable  using   the tree model for the action.
 
\medskip
Let $U_k$ be the clopen set defined by choosing a vertex $v_k$ of the tree model.  Consider the restricted action of $G_k$ on $U_k \subset \fX$ with group chain $\{G_i\}_{i \geq k}$. 

\medskip
The elements in $C_{k,i}\subset G_i$ stabilize all vertices at level $i$ in a branch of $T$,  while the elements in $C_i \subset C_{k,i}$ stabilize all vertices at level $i$. Then let 
$${\displaystyle S_{k} = \lim_{\longleftarrow}\{C_{k,i}/C_i \to C_{k,i-1}/C_i\}}  \cong {\rm ker}\{\mathcal{D}_{x} \to \mathcal{D}_{x,k}\}.$$
Suppose that  $h  \in S_k$, with $h \ne id$, then $h$ acts trivially 
on $U_k$, but acts non-trivially on $\fX$. If the action $\vp$ is not LQA, then  such an $h$ exists for clopen sets $U_k$ with arbitrarily small diameter, and hence the action is not stable.

\vfill   

}



 \frame % 1
{
  \frametitle{ }


 Finally, we note that the LQA property for a group action  $\vp \colon G  \times \fX \to \fX$ can be related to    properties of the germinal groupoid  $\cG(\fX, G, \vp)$ associated to the action.  
   
   \bigskip
   
 Recall that for $g_1, g_2 \in G$, we say that $\vp(g_1)$ and $\vp(g_2)$ are \emph{germinally equivalent} at $x \in \fX$ if $\vp(g_1) (x) = \vp(g_2)(x)$, and there exists an open neighborhood $x \in U \subset \fX$ such that the restrictions agree, $\vp(g_1)|U = \vp(g_2)|U$. We then write $\vp(g_1) \sim_x \vp(g_2)$. 
 
 \medskip
 
 For $g \in G$, denote the equivalence class of  $\vp(g)$ at $x$ by $[g]_x$.
 The collection of germs $\cG(\fX, G, \vp) = \{ [g]_x \mid g \in G ~ , ~ x \in \fX\}$ is given the sheaf topology, and   forms an \emph{\'etale groupoid} modeled on $\fX$. 
 

     
\vfill   

}


 \frame % 1
{
  \frametitle{ }


 \medskip
 
{\bf Theorem  (H-Lukina, 2017).}
If an action $\vp \colon G  \times \fX \to \fX$  is   locally quasi-analytic, then  $\cG(\fX, G, \vp)$ is    Hausdorff.     

 \medskip
   
   The Hausdorff property for a germinal groupoid  $\cG(\fX, G, \vp)$    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. 

\bigskip

{\bf Problem:} Find relations between the wild property for a group action, and the algebraic and topological invariants for the   $C^*$-algebra associated to the action.

Some results on this problem are given in the work by Rui Excel,   ``\emph{Non-{H}ausdorff \'etale groupoids}'', 
 {\bf Proc. A.M.S.},   2011.

 

\vfill   

}

 
 

\section{Examples}


\frame % 1
{
  \frametitle{}

 
 
 {\bf Example 3:}    [Schori, 1966] gave the 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=0.6\textwidth]{SchoriConstr1}
\end{center}
 {\bf Proposition (Dyer-H-Lukina, 2017).} The monodromy action of  $G = \pi_1(\Sigma_2,b_0)$ on the    fiber of the solenoid over $\Sigma_2$  is not LQA, and in particular is wild.
  
   
   
     
 
\vfill   

}
    
 \frame % 1
{
  \frametitle{ }

   
 
 {\bf Example 4:} 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
 $$ \prod_{p \in \cP'} \ {\bf SL}_N(\widehat{\mZ}_{p}) \subset \whG \subset \prod_{p \in \cP} \ {\bf SL}_N(\widehat{\mZ}_p)$$
 
 
\vfill   

}

 \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. 
  
   
 
 \vfill   

}

 \frame % 1
{
  \frametitle{ }

   
 
 {\bf Example 5:}   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.
  
    
   
       
\vfill   

}

 
    
\frame % 1
{

The following discussion concerns  results of [Lukina, 2018].
 
 \pgfputat{\pgfxy(6, -4)}{\pgfbox[left,base]{\pgfimage[width=4.8cm]{D-ary-tree-2.png}}} 

\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.

 
 
\vfill   

}

\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 $f$ and on $\alpha$.

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 $K = \mathbb{Q}$, $\alpha = 2$, $f(x) = x^2-x+1$, 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  not LQA, and in particular is wild.
 

 \bigskip
 
{\bf Remark:} The proof of this result  is geometric, it uses the fact that the action of $Aut(T_d)$ is not locally quasi-analytic.

 \bigskip
 
{\bf Remark:} There are many techniques, in the literature and developing, for calculating   arboreal representations. 

\vfill
}

  \section{Some open problems}  
 
\frame % 1
{

 {\bf Into the future...}
 
  \medskip
  
 
 
{\bf Problem:} Let $\vp \colon G  \times \fX \to \fX$ be a minimal equicontinuous action, where $G$ is a finitely generated, torsion free nilpotent group. Show that the action is tame.
 \medskip


{\bf Problem:} Characterize the algebraic number fields and polynomials whose arboreal representations are wild.

 \medskip


{\bf Problem:} Develop a Molino Theory for equicontinuous foliated spaces which are not minimal.

That is, adapt the Topological Molino Theory of \'Alvarez L\'opez and Moreira Galicia,  to the case where the actions are wild. 

There are   hints at what such a theory may be like in the work on iterated monodromy groups associated to the absolute Galois groups of function fields.

 
  \vfill   

}


  \section{Refs}

    
\frame % 1
{
  \frametitle{References}
 {\scriptsize     
 
   
% {N.~Boston and R.~Jones},
% {\it Arboreal {G}alois representations},
% {\bf Geom. Ded.},  124, 2007.

{ A.~Candel and J.~ \'Alvarez L\'opez}, 
{\it Equicontinuous foliated spaces}, 
{\bf Math. Z.},  263:725--774, 2009. 

  J.~ \'Alvarez L\'opez and M.~Moreira Galicia, 
{\it Topological Molino's theory}, 
{\bf Pacific J. Math.},  280:257--314, 2016. 

{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.

  

{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.

  
}
}

  
\frame % 1
{


\bigskip

 
 


\begin{center}

 \includegraphics[width=0.5\textwidth]{birthday3} 

 
 \end{center}
 

\vfill   

}

 

 

 
  


\end{document}

