% Seminar - UNAM - Mexico City
% Thursday, April 21, 2016
% slides for talk "Homeomorphisms of solenoids"
 
 
\documentclass{beamer}

\usepackage{amsfonts}

\usepackage{pgf,pgfarrows,pgfnodes}
\usepackage{xy}
\usepackage{amsthm}
\usepackage{graphicx}
\usepackage{tikz}
\usepackage{xcolor}
\usepackage{wrapfig}

 \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{\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}} %  


% 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}}}

% greek letters

\newcommand{\vp}{{\varphi}}

\newcommand{\bRt}{{\bf R}_{0}}

 
 
\date[April 19, 2016]


  
\title[Weak solenoids]{Homeomorphisms of solenoids}


\author{Steve Hurder}

\institute[UIC] {University of Illinois at Chicago\\www.math.uic.edu/$\sim$hurder}

\date{April 21, 2016}

\begin{document}

\frame{\titlepage} % # 1


 
  \section{Introduction}
 

 \frame % 1
{
  \frametitle{Dynamics of flows}
 
  Consider a non-singular $C^1$-vector field $\vec{V}$ on a compact $n$-manifold $M$, and its  flow $\vp_t \colon M \to M$ defined for all values of $t$.


\medskip

 {\bf Definition:} A point $x \in M$ is \emph{non-wandering} if its future and past orbits return infinitely often to a neighborhood of the point.  
 
$\Omega \subset M$ denotes the set of all non-wandering points for the system.


\medskip

A closed subset $\fM \subset M$ is \emph{minimal}  if it is flow invariant, and the orbit of every point in $\fM$ is dense in $\fM$. 

 

\vfill   

}


 \frame % 1
{
  \frametitle{}
 
  

\medskip

{\bf Main Problems:} 

$\bullet$~ Describe the topology of the non-wandering set $\Omega$.

$\bullet$~ Describe the dynamics of the flow on  $\Omega$.

$\bullet$~ Describe the minimal sets of the flow, and their dynamics.

 \medskip
 
  This program has been carried out for $C^1$-flows that satisfy Smale's \emph{Axiom A} property, for example, and for \emph{Morse-Smale} flows. Otherwise, simple questions such as what is the topology of the minimal sets for a flow are intractable.

Alternate approach: ask whether every flow is $C^1$-close to one of these standard types, as in the Palis Conjectures.


\vfill   

}


 \frame % 1
{
  \frametitle{Example: a hyperbolic attractor}
 
\vspace{-0.8in}  
A \emph{Smale solenoid} gives an example of a hyperbolic attractor:
\pgfputat{\pgfxy(-11, -3.5)}{\pgfbox[left,base]{\pgfimage[width=4.0cm]{Smale_Solenoid}}}
\pgfputat{\pgfxy(-0.5, -7)}{\pgfbox[left,base]{\pgfimage[width=2.5cm]{Section.jpg}}}

 

Let $M = \mathbb{S}^1 \times \mathbb{D}^2$. 
The map $g: \mathbb{S}^1 \times \mathbb{D}^2\to \mathbb{S}^1 \times \mathbb{D}^2$ is defined by 
\pgfputat{\pgfxy(-7, -0.7)}{\pgfbox[left,base]{$g(\theta, (y_1,y_2)) = \left( 2 \theta, \frac{1}{10} y_1 + \frac{1}{2} \cos \theta, \frac{1}{10}y_2 + \frac{1}{2} \sin \theta\right)$.}}
\pgfputat{\pgfxy(-7.1, -1.5)}{\pgfbox[left,base]{Let $C = \{0\} \times \mathbb{D}^2$ be a section.}}
\pgfputat{\pgfxy(-7.2, -2.3)}{\pgfbox[left,base]{Then $g^n(M) \cap C$ is the union of $2^n$ disjoint disks.}} 
\pgfputat{\pgfxy(0, -2.9)}{\pgfbox[left,base]{$\Lambda = \bigcap_{n \geq 1} g^n(M)$ is a hyperbolic attractor (called the \emph{Smale} attractor):}} 
\pgfputat{\pgfxy(-0.2, -3.5)}{\pgfbox[left,base]{There are expanding ($\mathbb{S}^1$) and contracting ($\mathbb{D}^2$) directions.}} 
\pgfputat{\pgfxy(-0.3, -4.5)}{\pgfbox[left,base]{$C \cap \Lambda$ is a Cantor set, and $(C \cap \Lambda,g)$ is a dynamical system}} 
\pgfputat{\pgfxy(-0.35, -5.1)}{\pgfbox[left,base]{on a totally disconnected set $C \cap \Lambda$.}}


\vfill   

}




 \frame % 1
{
  \frametitle{Dynamics of foliations}
 
  Consider an integrable distribution ${\it F} \subset TM$ on a compact $n$-manifold $M$, and the foliation $\F$  whose leaves consist of the connected submanifolds of $M$ which are tangent to  ${\it F}$. 

\medskip

 {\bf Definition:} A point $x \in M$ is \emph{non-wandering} if the leaf $L_x$ through $x$ returns infinitely often to a neighborhood of the point.  
 
$\Omega \subset M$ denotes the set of all non-wandering points for the system.

The complement $M - \Omega$ consists of the \emph{proper} leaves of $\F$.

\medskip

A closed subset $\fM \subset M$ is \emph{minimal}  if it is a union of leaves, and   each leaf $L \subset \fM$ is   dense in $\fM$. 

For example, a compact leaf is minimal.

 


\vfill   

}


 \frame % 1
{
  \frametitle{}
 
 
{\bf Main Problems:} 

$\bullet$~ Describe the topology of the non-wandering set $\Omega$.

$\bullet$~ Describe the dynamics of $\F$.

$\bullet$~ Describe the minimal sets $\fM$ of    $\F$ - what is their topology?

$\bullet$~ Describe the dynamics of $\F | \fM$.

\medskip

This program has been investigated in the case of $C^2$-foliations of codimension-one, in particular    by Richard Sacksteder,  John Cantwell \& Larry Conlon, and  Gilbert Hector.

\bigskip

 {\bf Question:} What hope is there to obtain meaningful results for $C^r$-foliations with codimension $q > 1$, and $r \geq 1$?

\medskip
 
$\bullet$ ~ {S.~Hurder}, {\it Lectures on Foliation Dynamics},\\
  {\bf Foliations: Dynamics, Geometry and Topology}, \\  
  Advanced Courses in Mathematics CRM Barcelona, Springer,   2014.

\vfill   

}

 \frame % 1
{
  \frametitle{}
 
  
Look at special cases:


$\bullet$ ~ Minimal sets with positive entropy

$\bullet$ ~ Minimal sets with equicontinuous or distal dynamics

$\bullet$ ~ Exceptional minimal sets

\medskip

A minimal set $\fM$ is \emph{exceptional} if its intersections with open transversals $\cT \subset M$ to $\F$ have closures which are Cantor sets.


The dynamics of $\F | \fM$ is \emph{equicontinuous} if for each transversal $\cT$ to $\F$ has a metric such that the induced return map of the foliation to the set $\cT \cap \fM$ is equicontinuous. 

\medskip

This is analogous to saying that $\F | \fM$ is a topological Riemannian foliation, except that when $\fM$ is exceptional, there is no normal bundle to $\F | \fM$.

\medskip 
{\bf Question:} Can we classify the equicontinuous foliations on exceptional minimal sets?



\vfill   

}


 
 
\section{MM}
 \frame % 1
{
  \frametitle{Matchbox manifolds}


 A compact connected metrizable space $\mathfrak{M}$ is a $n$-dimensional \emph{matchbox manifold} if it admits an atlas $\{U_i\}_{1 \leq i \leq \nu}$, where $U_i$ is an open set equipped with a homeomorphism
  $$\varphi_i: \overline{U}_i \to [-1,1]^n \times X_i,$$
and $U_i = \varphi_i^{-1}((-1,1)^n \times X_i$, where $X_i$ is totally disconnected.
 

\bigskip
The term a \emph{matchbox manifold} is due to {\bf Aarts} \\ {\bf and Martens 1988}, who studied $1$-dimensional \\ matchbox manifolds (for $n=1$). \pgfputat{\pgfxy(2.5,-1.)}{\pgfbox[left,base]{\pgfimage[width=2.5cm]{blue_matches}}} 

\vfill   

}

 \frame % 1
{
  \frametitle{}
 
 {\bf Theorem:} [Clark  \& Hurder, 2012]
Let  $\fM$ be an equicontinuous  matchbox manifold. Then:

$\bullet$ ~ There exists an open transversal $\cT$ such that the induced return map on $\cT \cap \fM$ is given by an equicontinuous action of a finitely generated group $\G$ on a Cantor set $X$.

$\bullet$ ~ $\fM$ is foliated homeomorphic to a weak solenoid $\cS$ which admits a Cantor fibration $\pi \colon \cS \to M_0$ over a compact manifold $M_0$ with fiber $X$.

\bigskip
 {\bf Theorem:} [Clark, Hurder \& Lukina, 2013] 
 An   equicontinuous $\fM$ is classified (up to foliated homeomorphism) by the induced holonomy action $\vp \colon \G \times X \to X$ on a transversal Cantor set $X$.
 
 \medskip
 
 {\bf Problem:}
 Understand the group of self-homeomorphisms of an $n$-dimensional matchbox manifold, 
 and the maps between matchbox manifolds.
 

\vfill   

}

\section{Solenoids}


 \frame % 1
{
  \frametitle{Solenoids}
 

Let $M_0$ be a connected closed manifold, and, recursively, let $f^i_{i-1}: M_i \to M_{i-1}$ be a finite-to-one covering map. Then
$$M_\infty = \lim_{\longleftarrow} \{ f^i_{i-1}: M_i \to M_{i-1}\} = \{ (y_0, y_1, y_2, \ldots) \mid f^i_{i-1}(y_i) = y_{i-1}\}$$
is a compact connected metrizable space called a \emph{solenoid}.

 
Let $x _i \in M_i$, then the cylindrical clopen set (the fiber)
    $$F_i = \{(f^i_0(x_i),  \ldots , x_i, y_{i+1}, \ldots) \in M_\infty \} \textrm{ is a Cantor set }.$$
 
 {\bf Theorem (McCord 1966):}   A solenoid is a matchbox manifold. 
    
 

\vfill   

}




 \frame % 1
{
  \frametitle{}
 
 {\bf Example:} Let $M_i = \mathbb{T}^n$, and let $f^i_{i-1}: \mathbb{T}^n \to \mathbb{T}^n$ be a proper covering. The inverse limit $\mathbb{T}_\infty$ is an abelian topological group.

\medskip

{\bf Theorem: [Bowen \& Franks 1976, Gambaudo \& Tresser 1990]} If $\cS$ is a solenoid with base $\mS^1$, and the degrees of the covering maps tend to $\infty$ sufficiently fast, then $\cS$ is homeomorphic to the minimal set of a smooth flow on a $3$-manifold.

\bigskip

{\bf Theorem: [Clark \& Hurder 2010]} If $\cS$ is a solenoid with base $\mT^n$, and the degrees  of the covering maps tend to $\infty$ sufficiently fast, and are not too wild as matrices, then $\cS$ is homeomorphic to the minimal set of a smooth foliation.

\bigskip

For other classes of solenoids, it remains an open problem, whether they can be realized as minimal sets of $C^r$-foliations for $r \geq 1$.

{\bf Problem:} Study the properties of solenoids as equicontinuous dynamical systems.

\vfill   

}


 \frame % 1
{
  \frametitle{}
 
 Let $\pi \colon \cS \to M_0$ be a weak solenoid defined by the system of maps $\{f_0^i \colon M_i \to M_0 \mid i > 0\}$. Choose a basepoint $x_0 \in M_0$, and   basepoints $x \in M_i$ such that $f_0^i(x_i) = x_0$ then let 
 $$x = \lim x_i \in X = \pi^{-1}(x_0) \subset \cS \ . $$
 Define $G = G_0 = \pi_1(M_0, x_0)$, and let $G_i \subset G$ be the subgroup defined by 
 $\ds G_ i = {\rm Image} \{ (f_0^i)_{\#} \colon \pi_1(M_i , x_i) \to \pi_1(M_0 , x_0)\}$. 
 
 The collection $\{G_i\}_{i \geq 0}$ forms a group chain in $G$.

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 of $\cS$ is identified with 
$$X \cong \varprojlim ~ \left\{  X_i \to X_{i-1} \right\}  = \varprojlim ~ \left\{  G/G_i \to G/G_{i-1} \right\}  = G_{\infty}$$
The  left $G$-action   $\Phi \colon G \to {\it Aut}(X)$  is minimal and equicontinuous. 
We say that $(X, G)$ is an \emph{equicontinuous minimal Cantor action}.
\vfill   

}


 \frame % 1
{
  \frametitle{Homogeneous solenoids}
 
A topological space $\mathfrak{M}$ is \emph{homogeneous} if for every $x,y \in \mathfrak{M}$ there is a homeomorphism $h: \mathfrak{M} \to \mathfrak{M}$ such that $h(x) = y$.
 
 \bigskip
 {\bf McCord 1965:} If $G_i$ is a normal subgroup of $G$ for all $i \geq 0$, then $G/G_i$ is a group, and $G_\infty$ is a profinite group. Then $M_\infty$ is homogeneous.
 
 \medskip
 {\bf Example:}   Let $M_i = \mathbb{T}^n$, and let $f^i_{i-1}: \mathbb{T}^n \to \mathbb{T}^n$ be a proper covering. Then the inverse limit $\mathbb{T}_\infty$ is a homogeneous matchbox manifold. 
 
   \bigskip
 Are there solenoids which are non-homogeneous?

\vfill   

}


 \frame % 1
{
  \frametitle{Non-homogeneous solenoids}
 

{\bf Schori 1966:} Not all solenoids are homogeneous. 
 
 \pgfputat{\pgfxy(1, -3.5)}{\pgfbox[left,base]{\pgfimage[width=8cm]{SchoriConstr1.pdf}}} 
 
 \vspace{3.5cm}
$M_\infty$ is the inverse limit of $3$-to-$1$ coverings of the genus $2$ surface $\Sigma_2$.
 Every leaf in $M_\infty$ is a non-compact surface of infinite genus.
  Schori found a closed loop in $\Sigma_2$, which, depending on the point in the fiber of $M_\infty \to \Sigma_2$, lifts either to a closed loop, or to a non-closed curve. So $M_{\infty}$ is not homogeneous. 
 
 
\vfill   

}



 \frame % 1
{
  \frametitle{}
 
{\bf Rogers \& Tollefson 1971:} There exists a closed manifold $M_0$ and a group chain $\{H_i\}_{i \geq 0}$ with $H_0=\pi_1(M_0,x_0)$   such that $H_i$ is not a normal subgroup of $H_0$, but the associated solenoid $M_\infty$ is a homogeneous space.

 \medskip
  That is, it is possible to obtain a homogeneous solenoid as an inverse limit of non-regular coverings.
  
  \medskip
 {\bf Question:}  How one can determine if a solenoid is homogeneous?

\vfill   

}

\section{Chains}

 \frame % 1
{
  \frametitle{Equivalent group chains}
 
 Rogers and Tollefson suggested to use the following notion to study the problem of homogeneity.
 
 {\bf Definition:} (Equivalent group chains)
 Group chains $\{G_i\}_{i \geq 0}$ and $\{H_i\}_{i \geq 0}$ are \emph{equivalent} if, possibly for a subsequence, one has 
 $$G_0 = H_0 \supset G_{1} \supset H_{1} \supset G_{2} \supset H_{2} \supset  \cdots $$
 
 
 \medskip

 Dynamically, $\{G_i\}_{i \geq 0}$ and $\{H_i\}_{i \geq 0}$ are equivalent if and only if there is a conjugacy $h \colon G_\infty \to H_\infty$ with $h(eG_i) = (eH_i)$.
 
 \medskip
 If $\{G_i\}_{i \geq 0}$ and $\{H_i\}_{i \geq 0}$ are not equivalent, there may still exist a conjugacy $h \colon  G_\infty \to H_\infty$, but it cannot preserve the basepoint.


\vfill   

}



 \frame % 1
{
  \frametitle{A criterion for homogeneity}
 
A group chain $\{H_i\}_{i \geq 0}$ is \emph{weakly normal} if it is equivalent to a group chain $\{G_i\}_{i \geq 0}$ such that there is $i_0 \geq 0$ such that $i \geq i_0$ 
  $$G_i \subset G_{i_0} \subset N(G_i), ~ N(G_i) \textrm{ is the normalizer of } G_i \textrm{ in }  G_0.$$
  
 {\bf Theorem: (Fokkink and Oversteegen 2002)}
A solenoid $\mathcal{S}$ is homogeneous if and only if its associated group chain $\{H_i\}_{i \geq 0}$ is \emph{weakly normal}. 
 

\medskip
Topologically, this means that, possibly restricting to clopen subset of the fiber, one can represent the solenoid as the inverse limit of a sequence of regular coverings of a closed manifold.

\vfill   

}



 \frame % 1
{
  \frametitle{Ellis semigroup}
 

Let $\Phi \colon G \to {\it Homeo}(X) : g \mapsto \phi_g $  be a group action.

\medskip
Let $\phi(G) = \{ \phi_g \mid g \in G\} \subset {\it Maps}(X,X)$.

 {\bf Theorem: (Ellis 1969)}
The closure $\mathfrak{G} = \overline{\phi(G)} \subset Maps(X,X)$ in the topology of \emph{pointwise convergence} on maps has a structure as a semigroup, called the \emph{enveloping (Ellis) semigroup}.
 

{\bf Ellis 1969}, see also {\bf Auslander 1988}: If the action $(X,G)$ is \emph{equicontinuous}, then the Ellis semigroup $\mathfrak{G}$ is a group. 

\medskip
{\bf Note:} The action $(G_\infty,G)$ is equicontinuous, so its Ellis semigroup is a group.
Denote by 
$$\mathfrak{G}_x = \{ h \in \mathfrak{G} \mid h(x) = x\}$$
the isotropy subgroup of the $\mathfrak{G}$-action at $x \in X$.
 


\vfill   

}



 \frame % 1
{
  \frametitle{A representation of the Ellis (semi)group}
 
 {\bf Remark:} In general, it is quite difficult to compute the Ellis semigroup of an action  $(X, G)$.
Let $\{G_i\}_{i \geq 0}$ be a group chain in G such that  $(X, G)$ is conjugate to $(G_\infty,G)$.

\medskip
Let $C_i = \bigcap_{g \in G} gG_ig^{-1}$, then $C_i$ is a normal subgroup of $G$, and 
  $$C_\infty = \lim_{\longleftarrow}\{G/C_i \to G/C_{i-1}\} \textrm{ is a profinite group. }$$
  
 {\bf Theorem: [Dyer, Hurder, Lukina 2015]}
The profinite group $C_\infty$ is isomorphic to the Ellis group $\mathfrak{G}$ of the action $(X,G)$. Also, if $\mathfrak{G}_x$ is the isotropy group of the $\mathfrak{G}$-action on $X$, then 
  $$\mathfrak{G}_x \cong \lim_{\longleftarrow}\{G_i/C_i \to G_{i-1}/C_{i-1}\} \equiv \cD_x.$$
 
$\cD_x$ is called the \emph{discriminant group} of the action. 



\vfill   

}



 \frame % 1
{
  \frametitle{A criterion for homogeneity}
 
Since $\mathfrak{G}_x$ is a profinite group, it is either finite, or a Cantor group.

 {\bf Proposition: [Dyer, Hurder, Lukina 2015]}
The isotropy group $\mathfrak{G}_x$ of the Ellis group action on $X$ is a normal subgroup of $\mathfrak{G}_x$ if and only if $\mathfrak{G}_x$ is trivial.
 
\medskip

Therefore, if the isotropy group $\mathfrak{G}_x$ is non-trivial, then $X \cong \mathfrak{G}/\mathfrak{G}_x$ does not admit group structure.

\medskip

 {\bf Corollary:}
Let $\mathfrak{M}$ be a solenoid with associated group chain $\{H_i\}_{i \geq 0}$. Then $\mathfrak{M}$ is homogeneous if and only if the isotropy group $\mathfrak{G}_x$ of the Ellis group action on $X$ is trivial.
 




\vfill   

}



 \frame % 1
{
  \frametitle{Advantages of the Ellis group approach}
 
Computing the isotropy group $\mathfrak{G}_x$ has the following advantages as opposed to working with equivalence classes of group chains:

\begin{enumerate}
\item Unlike the normality properties of group chains, finiteness of the discriminant group $\cD_x$ does not depend on the choice of a group chain.
\item In many examples, one can explicitly compute the quotients $G_i/C_i$ and the discriminant group $\cD_x$, while it may be quite difficult to show that there is no chain of normal subgroups equivalent to a given group chain $\{G_i\}_{i \geq 0}$.
\end{enumerate}



\vfill   

}



 \frame % 1
{
  \frametitle{Virtually homogeneous solenoids}
 
We now have a criterion for a solenoid to be homogeneous, and we know that there are solenoids which are non-homogeneous.

\medskip
{\bf Question:} Are there solenoids which are `less' non-homogeneous than other ones?
That is, is there any way to quantify the degree of non-homogeneity?

\medskip
We introduce the notion of a   \emph{virtually homogeneous solenoid}:

 {\bf Definition:}
A solenoid $\mathfrak{M} \to M_0$ is \emph{virtually homogeneous}, if there exists a finite-to-one covering $p \colon  N_0 \to M_0$ such that the pullback solenoid $\mathfrak{N} = p^* \mathfrak{M}$ is homogeneous.
 





\vfill   

}



 \frame % 1
{
  \frametitle{A criterion for virtual  homogeneity}
 
Recall that $\mathfrak{G} = \lim_{\longleftarrow} \{G/C_i \to G/C_{i-1}\}$ is the Ellis group of the action, and  $\cD_x$ is the discriminant group at $x$.

\medskip
If  $\cD_x$  is non-trivial, it is either a finite group or a Cantor group.

\medskip

 {\bf Theorem: (Dyer, Hurder, Lukina 2015)}
 If $\cD_x$ is a finite group, then a solenoid $\mathfrak{M}$ with associated group chain $\{H_i\}_{i \geq 0}$ is virtually homogeneous.

\medskip


\emph{Proof.} One can show that $\cD_x$ is finite if and only if $\{H_i\}_{i \geq 0}$ is equivalent to a chain $\{G_i\}_{i \geq 0}$ such that
 $H_i \cap C_1 = C_i$. Then use Galois theory to construct a covering of the base manifold.

\vfill   

}

\section{Examples}

 \frame % 1
{
  \frametitle{The discrete Heisenberg group}
 
Let $\mathcal{H} = (\mathbb{R}^3,*)$ be the Heisenberg group, and $\mathcal{H} = (\mathbb{Z}^3,*)$ be the discrete Heisenberg group, with
  $$(x,y,z)*(x',y',z') = (x+x', y+ y', z + z' + xy').$$
Then $M_0 = \mathbb{H}/\mathcal{H}$ is a closed $3$-manifold, and $\pi_1(M_0,0) = \mathcal{H}$. 

\medskip
Group chains in the Heisenberg group, giving homogeneous actions, were classified by {\bf Lightwood, Sahin and Ugarcovici 2014}.

\medskip
We are interested in group chains which give non-homogeneous actions.


\vfill   

}





 \frame % 1
{
  \frametitle{A virtually homogeneous solenoid}
 
 Let $p,q$ be distinct primes, and let $M_n = \left( \begin{array}{rr} qp^n & pq^n \\ p^{n+1} & q^{n+1} \end{array}\right)$. Then
  $\{G_n\} =\{ M_n \mathbb{Z}^2 \times p\mathbb{Z}\}$ defines a nested group chain. Let $G_\infty$ be the corresponding inverse limit space. Let $M_n = \mathbb{H}/G_n$.

 {\bf Proposition:}
The solenoid defined by the group action $(G_\infty, \mathbb{H})$ is virtually homogeneous, but not weakly normal.
 

\emph{Proof.} Let $L_n = \left( \begin{array}{rr} qp^n & p^2q^n \\ p^{n+1} & pq^{n+1} \end{array}\right)$. Then $C_n = L_n \mathbb{Z}^2 \times p \mathbb{Z}$ is a normal subgroup of index $p$ in $G_n$. Since $p$ is a prime, $C_n$ is a maximal normal subgroup of $G_n$. Then for $n \geq 1$ we have $|G_n/C_n| = p$, and it follows that $\cD_x$ is nontrivial and finite. 



\vfill   

}



 




 \frame % 1
{
  \frametitle{}
 

{\scriptsize

Based on the works

\begin{itemize}
\item {\it J. Dyer, S. Hurder and O. Lukina}, The discriminant invariant of Cantor group actions, {arXiv: 1509.06227}.
\item {\it J. Dyer, S. Hurder and O. Lukina}, Growth and homogeneity of matchbox manifolds, {arXiv: 1602.00784}.
\end{itemize}

Other works on equicontinuous and non-equicontinuous matchbox manifolds

\begin{itemize}
\item {\it A. Clark, R. Fokkink and O.Lukina}, The Schreier continuum and ends, Houston J. Math., 40:569-599, 2014.
\item {\it A. Clark and S. Hurder}, Embedding solenoids in foliations, Topology Appl., 158:1249--1270, 2011..  
\item {\it A. Clark and S. Hurder}, Homogeneous matchbox manifolds, Trans. AMS, 365:3151--3191, 2013.  
\item {\it A. Clark, S. Hurder and O. Lukina}, Classifying matchbox manifolds, arXiv: 1311.0226.
\item {\it A. Clark, S. Hurder and O. Lukina}, Shape of matchbox manifolds, Indagationes Mathematicae, 25:669--712, 2014.
\item {\it A. Clark, S. Hurder and O. Lukina}, Voronoi tessellations of matchbox manifolds, Top. Proc. 41:167--259, 2013.
\item {\it O. Lukina}, Hierarchy of graph matchbox manifolds, Topology Appl., 159:3461--3485, 2012.
\item {\it O. Lukina}, Hausdorff dimension of matchbox manifolds, arXiv: 1407.0693.
\end{itemize}
}


\vfill   

}



  
\frame % 1
{
  \frametitle{ }
\begin{center}
\Large{!`Gracias por su atenci\'on!}
\end{center} 

\vfill   

 }
  
  
\end{document}

