% Automatic Sequences, Aperiodic Order, Number Theory
% 28 - 30 October 2015
% Technical University Delft
% Conference venue TU Delft EWI
% slides for talk "Classification problems for laminations"
% slides completed October 26, 2015

 

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\date[October 29, 2015]


  
\title[Classification problems]{Classification problems for laminations}


\author{Steve Hurder}

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



\begin{document}

\frame{\titlepage} % # 1


 
  \section{Introduction}
 
 

\frame % 1
{
  \frametitle{ }
  
 A \emph{generalized lamination} is a compact connected metric space $\fM$, which admits a covering by open sets, each  homeomorphic to a product of a totally disconnected space with an open subset of euclidean space $\mR^n$. The path components of $\fM$ define the leaves of dimension $n$ and form a foliation $\F$ of $\fM$.  
 
 $\bullet$~ 
 A lamination is a hybrid of a compact manifold and a profinite group or groupoid  structure.
 
  $\bullet$~  We call these \emph{matchbox manifolds} in our works, following usage introduced by Aarts \& Oversteegen.
 
 
 $\bullet$~ These arise in the study of aperiodic tilings, minimal sets for dynamical systems,   the study of complex rational maps, \ldots .  
   
 \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 change of coordinates along intersection is smooth diffeomorphism; 

{\bf +}~ some other technicalities.

 
 \bigskip
 
  The path connected components of $\fM$ are the leaves of the foliation  $\F$. 

  

 
 \vfill   

}


 
  \frame % 1
{
  \frametitle{}
 
 
The   charts $\vp_{i} \colon U_i \to [-1,1]^n \times \fX_i$ on the open covering satisfy a compatibility condition on their overlaps.

\bigskip

\begin{center}
\includegraphics[width=0.6\textwidth]{pix/chart.pdf}
\end{center}
 
 
   \vfill   

}

\frame % 1
{
  \frametitle{ }
  
  {\bf Problem:} Given matchbox manifolds $\fM_1$ and $\fM_2$ of the same leaf dimension $n \geq 1$,  find invariants which are sufficient to imply that the spaces are homeomorphic.
 
 \medskip
 
 ``Manifold invariants'' + ``algebraic invariants'' = ``Classification'' 
 
  \bigskip
  
    {\bf Problem:} Given a matchbox manifold $\fM$, what are the properties of the group of self-homeomorphisms ${\bf Homeo}(\fM)$. 

\bigskip
    
  $\bullet$~  $\fM$ is  \emph{homogeneous}  if  ${\bf Homeo}(\fM)$ acts transitively on $\fM$.  

\medskip
   
     {\bf Problem:} [Bing]  Characterize the homogeneous laminations.
     
  
 \vfill   

}


\frame % 1
{
  \frametitle{ }
  
  
{\bf Theorem:} [Hagopian (1977), Aarts, Hagopian \& Oversteegen (1991)]  The homogeneous $1$-dimensional matchbox manifolds are precisely the Vietoris solenoids.

 \bigskip

 {\bf Theorem:} [Fokkink (1991), Aarts and Oversteegen (1995)] \\ Two orientable,  minimal, $1$--dimensional matchbox manifolds are homeomorphic if and only if they are return equivalent.

\bigskip
 
{\bf Theorem:} [Clark \& Hurder (2013)]  The homogeneous $n$-dimensional matchbox manifolds are precisely the   regular solenoids.


 

 \vfill   

}

 \section{Solenoids}
 
 
\frame % 1
{
  \frametitle{}
  
  
  {\bf Classic example:} Vietoris solenoid (1927), defined by tower of coverings: 
  $$\cP ~ \equiv ~ \cdots \longrightarrow \mS^1  \stackrel{p_{\ell +1}}{\longrightarrow} \mS^1    \stackrel{p_{\ell}}{\longrightarrow} \cdots  \stackrel{p_{2}}{\longrightarrow}\mS^1   \stackrel{p_{1}}{\longrightarrow} \mS^1$$
where each $p_{\ell}$ is a covering map of   degree $n_{\ell} > 1$.  

$\cP$ is called a presentation.   Set     $n_\cP = \{n_1, n_2, n_3, \ldots \}$. Can assume that each $n_i > 1$ is prime, otherwise randomly chosen.


 

 
$$\cS_{\cP} \equiv \lim_{\longleftarrow} ~ \{ p_{\ell +1} \colon \mS^1 \to \mS^1\} ~ \subset \prod_{\ell \geq 0} ~ \mS^1  $$
 $\cS_{\cP}$ is given  the (relative) product topology.
   
   \medskip
   
     {\bf Proposition:} The space  $\cS_{\cP}$ is a matchbox manifold, for which every leaf is diffeomorphic to $\mR$, and dense in $\cS_{\cP}$.

  
  
   
\vfill   

}

  
 \frame % 1
{
  \frametitle{}
 
   
 \bigskip
 
   
 
  \begin{center}
  Van Dantzig - Vietoris solenoid \\
  The case where $n_i = 2$ for all $i \geq 1$
\bigskip


\includegraphics[width=0.8\textwidth]{pix/Smale-Williams_Solenoid_Large.png}
\end{center}
   \vfill   

}

\frame % 1
{
  \frametitle{}
  
 
   {\bf Proposition:} The homeomorphism type of $\cS_{\cP}$ depends only on the set of integers $n_\cP$.

 \medskip

More is true. Let $\cP$ and $\cQ$ be presentations, and let $P$ be the infinite set of prime factors of the integers in the set $n_\cP$, included with multiplicity, and $Q$ the same of $n_\cQ$.

\medskip

 {\bf Theorem:} [Bing (1960), McCord (1965), Aarts \& Fokkink (1991)] The solenoids $\cS_{\cP}$ and $\cS_{\cQ}$ are homeomorphic, if and only if there is a \emph{bijection} between a cofinite subset of $P=(p_1,p_2,\ldots)$ with a cofinite subset of $Q = (q_1,q_2,\ldots)$. 
 
 
 \bigskip
 
 {\bf Conclusion:} no matter how chaotic the choice of the sequence of primes, the classification reduces to an analytic problem in the sense of descriptive set theory. 



 \vfill   

}


\frame % 1
{
  \frametitle{}
 
A weak solenoid $\cS_{\cP}$ is obtained by considering a closed connected $n$-manifold $M$, and a   
 \emph{presentation}   
 $$\cP = \{ p_{\ell+1} \colon M_{\ell+1} \to M_{\ell} \mid \ell \geq 0\}$$ 
 which is a collection of maps satisfying:
 
 $\bullet$~    $M_{\ell}$ is a connected compact   manifold of dimension $n$;
 
 $\bullet$~  each  \emph{bonding map}  $p_{\ell +1}$  is a proper  covering map. Then set
 
  
    $$\cS_{\cP} \equiv \lim_{\longleftarrow} ~ \{ p_{\ell +1} \colon M_{\ell +1} \to M_{\ell}\} ~ \subset \prod_{\ell \geq 0} ~ M_{\ell} $$

 $\bullet$~ $\cS_{\cP}$ is given  the restriction of the product topology.
 
 
{\bf Proposition:}[McCord (1965)] A weak solenoid is a matchbox manifold.

\vfill   

}

 
\frame % 1
{
  \frametitle{}

 {\bf Examples:} {\it  $2$-dimensional matchbox manifolds.}

\medskip
 
 $M_0$ is the $2$-torus, $\mT^2 = \mR^2 / \mZ^2$.
 
 Suppose that   $A \in GL(2, \mZ) \subset GL(2,\mR)$ is a  $2 \times 2$ invertible integer matrix, then $\Gamma_A = A \cdot \mZ^2$ is a subgroup of finite index in $\Gamma_0 = \mZ^2$. 
 
 Then there is an induced proper covering map $\phi_A \colon \mT^2 \to \mT^2$, where the degree of the covering is the index of $\Gamma_A $ in $\Gamma_0$, which equals the determinant $\det(A) \in \mZ$. 
 
 Given an infinite collection $\cA \equiv \{A_{\ell} \in GL(2, \mZ) \mid \ell =1,2, \ldots\}$ set $p_{\ell} = \phi_{A_{\ell}}$ and we obtain a presentation
 
  $$\cP_{\cA} = \{p_{\ell} \colon \mT^2 \to \mT^2 \mid \ell = 1,2, \ldots\}$$
  
  which defines a solenoid  $\cS({\cA})$.
  
  \bigskip
  
   {\bf Fact:} The solenoids of the form  $\cS({\cA})$ are not classifiable.
\vfill   

}


\frame % 1
{
  \frametitle{}

 {\bf Examples:} {\it  $2$-dimensional matchbox manifolds.}

\medskip
 
Let  $M_0 = \Sigma_g$ be a Riemann surface of genus $g \geq 2$, pick a basepoint $x_0 \in M_0$ and let $\Gamma_0 = \pi_1(M_0, x_0)$.

  $\Gamma_0$   is residually finite, and each subgroup $\Gamma \subset \Gamma_0$ of finite index defines a proper covering 
$\pi \colon \Sigma_{\Gamma} \to \Sigma_g$.
 Given an infinite descending chain of subgroups with $\Gamma_{i+1}$ finite index in $\Gamma_{i}$, 
 $$\cG \equiv  \Gamma_0 \supset \Gamma_1 \supset \Gamma_2 \supset \cdots $$
 Let $M_{\ell} = \Sigma_{\Gamma_{\ell}}$ and $p_{\ell+1} \colon M_{\ell+1} \to M_{\ell}$ be the induced proper covering maps. Then this defines a presentation $\cP_{\cG}$ and a corresponding solenoid  $\cS({\cG})$.
  
   Note that the intersection $\ds \bigcap_{n\geq 0}~ \Gamma_n$ need not be the trivial group.

  \bigskip
  
   {\bf Fact:} The solenoids of the form  $\cS({\cG})$ are not classifiable.

 \vfill   

}

\frame % 1
{
  \frametitle{}

 {\bf Examples:} {\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.

\medskip

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 lemma. 
 \vfill   

}

\frame % 1
{
  \frametitle{ }
  
  
{\bf Theorem:} [Dyer, Hurder, Lukina (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 $, $\{G_n\}_{n \geq 1} = \{A_n \mZ^2 \times p^n \mZ\}_{n \geq 1}$.
 Then the weak solenoid $\cS(\{A_n\})$ 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 the leaves of the foliation $\F$ on $\cS(\{A_n\})$ are all isometric to the real Heisenberg group $\mH$.

\medskip

By a result of Clark, Hurder \& Lukina (2013), the  classification problem for the Heisenberg solenoids reduces to an algebraic problem, as we discuss next.

 \vfill   

}

 \section{Group chains}


  \frame % 1
{
  \frametitle{}

{\bf Definition:}   Let $G$ be a finitely generated group. A \emph{group chain} $\cG = \{G_i \mid i \geq 0\}$, with $G_0=G$, is a properly descending chain of subgroups of $G$, such that $|G:G_i| < \infty$ for every $i \geq 0$.

  
\medskip

{\bf Definition:} [Rogers \& Tollefson (1971)]  Let $G$ be a finitely generated group, and $\{G_i\}_{i \geq 0}$ and $\{H_i\}_{i \geq 0}$ be group chains with $G_0=H_0=G$. We say they  are \emph{equivalent}, if and only if, there is a group chain $\{K_i\}_{i \geq 0}$ and infinite subsequences $\{G_{i_k}\}_{k \geq 0}$ and $\{H_{j_k}\}_{k \geq 0}$ such that $K_{2k} = G_{i_k}$ and $K_{2k+1} = H_{j_k}$ for $k \geq 0$.

\medskip
 
 $\{G_i\}_{i \geq 0}$ and $\{H_j\}_{j \geq 0}$ are \emph{weakly equivalent} if there exists $i_0 \geq 0$ and $j_0 \geq 0$ such that the   subchains 
$\{G_i\}_{i \geq i_0}$ and $\{H_j\}_{j \geq j_0}$ are equivalent.

 
   \vfill   

}

 
\frame % 1
{
  \frametitle{ }
  
 
 

 {\bf Definition:} [Fokkink \& Oversteegen (2002)] Let $G$ be a finitely generated group, and $\{G_i\}_{i \geq 0}$ and $\{H_i\}_{i \geq 0}$ be group chains with $G_0=H_0=G$.   We say they  are \emph{conjugate equivalent}, if and only if, there exists a collection $(g_i) \in G$, such that the group chains $\{g_iG_ig_i^{-1}\}_{i \geq 0}$ and $\{H_i\}_{i \geq 0}$ are equivalent. Here $g_iG_i = g_jG_i$ for all $i\geq 0$ and all $j \geq i$.



 $\{G_i\}_{i \geq 0}$ and $\{H_j\}_{j \geq 0}$ are \emph{weakly conjugate equivalent} if there exists $i_0 \geq 0$ and $j_0 \geq 0$ such that the   subchains 
$\{G_i\}_{i \geq i_0}$ and $\{H_j\}_{j \geq j_0}$ are conjugate equivalent.

 \bigskip
 
$\bullet$~   $G$   abelian $\Longrightarrow$   conjugate equivalence $\equiv$ equivalence.

\medskip

  \vfill   

}


 
 \frame % 1
{
  \frametitle{}
 
   Let $G$ be a finitely generated group.   Denote by $\fG$ the collection of \emph{all possible} nested proper chains of subgroups of finite index. 

\medskip

{\bf Proposition:} (Weak) \emph{equivalence} and \emph{conjugate equivalence} of group chains form equivalence relations on $\fG$.

\medskip

{\bf Problem:} Let $G$ be a finitely generated group. Determine the (weak) equivalence and conjugate equivalence classes in $\fG$.

\medskip

  
   \vfill   

}

 \section{Dynamics}
 
 \frame % 1
{
  \frametitle{}
 
 Let $G$ be a finitely generated group, and  $\cG \in \fG$ a group chain.  
  
    $$X = G_\infty = \lim_{\longleftarrow}\, \{G/G_i \to G/G_{i-1}\} $$
is a Cantor set, and there is a minimal action $\phi \colon G \to {\bf Homeo}(X)$ given by left multiplication on each factor.

This is a \emph{Cantor minimal system}, and is equicontinuous.

 \medskip
 
Equicontinuous $\Leftrightarrow$ orbits of pairs of points stay uniformly close.

Expansive $\Leftrightarrow$ orbits of pairs of points spread uniformly apart.


{\bf Theorem:} [Auslander, Glasner \& Weiss (2007)] A Cantor minimal system $(X,G,\phi)$ is either equicontinuous or expansive. That is, there are no distal Cantor minimal systems.
 
 
   \vfill   

}


 \frame % 1
{
  \frametitle{}
 
 {\bf Definition:} Let $(X_1,G_1,\phi_1)$ and $(X_2,G_2,\phi_2)$ be a Cantor minimal systems. A homeomorphism $h \colon X_1 \to X_2$ is
 
 $\bullet$~ a \emph{conjugacy} if \\
 $\ds  h(\phi_1(g))(x) = \phi_2(g)(h(x))$   ,   $\forall$   $g \in G_1$, $x \in X_1$
 
 $\bullet$~ an \emph{orbit equivalence} if $\ds h\left(\{\phi_1(g)(x) \mid g \in G_1\}\right) = \{\phi_2(g)(h(x)) \mid g \in G_2\}$   ,   $\forall$ $x \in X_1$.
 
  \medskip
 
 The classification of Cantor minimal systems is a major industry, especially for the case where $G = \mZ^n$ and the action is expansive. 
 
 Giordano, Matui, Putnam, Skau in a series of papers classified these actions \emph{up to orbit equivalence}.
 
  
    \vfill   

}


 \frame % 1
{
  \frametitle{}
 
   {\bf Definition:} ${Aut}(X,G,\phi)$ is the group of automorphisms of the Cantor minimal system $(X,G,\phi)$.
\bigskip

 {\bf Definition:} A Cantor minimal system  $(X,G,\phi)$   is:
 
 $\bullet$~ \emph{regular}  if the action of ${Aut}(X,G,\phi)$ on $X$ is transitive;\\
 $\bullet$~  \emph{weakly regular} if the action of ${Aut}(X,G,\phi)$ decomposes $X$ into a finite collection of  orbits;\\
 $\bullet$~ \emph{irregular}  if the action of ${Aut}(X,G,\phi)$ decomposes $X$ into an infinite collection of  orbits.
  
  
   \vfill   

}

  
 \section{Discriminant}
 
 \frame % 1
{
  \frametitle{}
 
  Let    $(X,G,\phi)$  be a Cantor minimal system. Each $g \in G$ defines an element  $\whphi(g) \in X^X$, and let 
  $$\whG = \{ \whphi(g) \mid g \in G\} \subset X^X$$
  
  Endow the space $\whG$ with     the topology of pointwise convergence. 
  
     The closure $E(X,G,\phi)$ is called the \emph{enveloping Ellis semi-group}. There is a natural embedding $G \subset E(X,G,\phi)$. 
 \medskip
 
    For an expansive Cantor minimal system,  $E(X,G,\phi)$ is a monster. Though for $G$ amenable, it has been studied extensively.
    
    \medskip
    
 {\bf Theorem:} [Ellis (1960)]    If $(X,G,\phi)$ is equicontinuous, then the topology on $E(X,G,\phi)$ is separable, and $E(X,G,\phi)$ is a Cantor group which acts transitively on $X$. 
 
 Identify the action of $G$ on $X$ with the action of $G$ on $E(X,G,\phi)/\cH_x$ where $\cH_x$ is the isotropy subgroup at $x$.
 
 
  
   \vfill   

}


 \frame % 1
{
  \frametitle{}
 
  Let $G$ be a finitely generated group, and  $\cG \in \fG$ a group chain.  
    $$X = G_\infty = \lim_{\longleftarrow}\, \{G/G_i \to G/G_{i-1}\} $$
 Define $\ds C_i = \bigcap_{g \in G} gG_ig^{-1}$ which is  the maximal normal subgroup of $G_i$ called the \emph{core} of $G_i$ in $G$.
 
 $G_i$ finite index in $G$ $\Longrightarrow$ $C_i$ has finite index in $G$.
 
The   normal subgroups $\{C_i\}_{i \geq 0}$ form a nested chain, and the inverse limit space  is a Cantor group, 
$$C_\infty(\phi) = \lim_{\longleftarrow}\{G/C_i \to G/C_{i-1}\}$$
  
 {\bf Proposition:} There is a natural topological isomorphism $\ds C_\infty \cong E(X,G,\phi)$.
   \vfill   

}


 \frame % 1
{
  \frametitle{}

For each $x \in X$, there is a natural map $\ds q_x: C_\infty \to G_\infty $
which is the quotient of $C_\infty$ by the action of the subgroup 
$$\cD_x = \lim_{\longleftarrow}\, \{G_i/C_i \to G_{i-1}/C_{i-1}\} .$$

$\cD_x$ is called  the \emph{discriminant group} of $(X,G,\Phi)$ at $x$.

 {\bf Proposition:} There is a natural topological isomorphism $\ds \cD_x \cong \cH_x$.

 Thus, for an equicontinuous Cantor minimal system $(X,G,\phi)$ defined by the group chain $\cG \in \fG$, the abstract group $\cH_x$ can be calculated in terms of the  quotient group chain $\{G_i/C_i \mid i \geq 0\}$.
 
   \vfill   

}



 \frame % 1
{
  \frametitle{}

 {\bf Theorem:}  [Dyer, Hurder \& Lukina (2015)] Let $(X,G,\Phi)$ be an equicontinuous Cantor minimal system defined by a group chain $\{G_i\}_{i \geq 0}$ at $x$. 
Then 


$\bullet$~ The action $(X,G,\Phi)$ is regular if and only if $\cD(\phi)_x$ is trivial for some $x \in X$.\\
$\bullet$~ The action $(X,G,\Phi)$ is weakly regular if    $\cD(\phi)_x$ is finite for some $x \in X$.  
 
 \bigskip
 
  {\bf Corollary:} If the equicontinuous Cantor minimal system $(X,G,\Phi)$ defined by the group chain $\{G_i\}_{i \geq 0}$ is irregular, then $\cD(\phi)_x$ is a Cantor group for all $x \in X$. 

   
   \vfill   

}


 \frame % 1
{
  \frametitle{}

{\bf Theorem:}  [Dyer  (2015)] The discriminant group $\cD(\phi)_x$  for the Cantor minimal system defined by the 
   group chain $G_0 =\cH $, $\{G_n\}_{n \geq 1} = \{A_n \mZ^2 \times p^n \mZ\}_{n \geq 1}$ in the discrete Heisenberg group $\cH$ is a Cantor group.
  
 \medskip
 
 Thus, when we pass from the abelian case, where $G = \mZ^n$, to the next most complicated groups, the Heisenberg groups, the structure of    ${Aut}(X,G,\phi)$ becomes non-trivial. 
 
 Correspondingly, the study of the conjugacy problems for weak solenoids defined by such chains is much more delicate.
 
     \vfill   

}

 \frame % 1
{
  \frametitle{}

   
  {\bf Problem:}   Explore the relations between the dynamics of an equicontinuous Cantor minimal system $(X,G,\Phi)$ and its discriminant groups $\cD(\phi)_x$ in the cases where this is Cantor group.
  
   \bigskip
  
  {\bf Problem:}   Explore the relations between the dynamics of an expansive Cantor minimal system $(X,G,\Phi)$ and the Ellis enveloping semi-group $E(X,G,\Phi)$.
  
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 {\it  Thank you for your attention!}
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{
  \frametitle{References}
 {\footnotesize     
 
 {J.M.~Aarts and R.J.~Fokkink},
  {\it The classification of solenoids}, {\bf Proc. Amer. Math. Soc.}, 111: 1161-1163, 1991.

{J.M.~Aarts and L.~Oversteegen},
  {\it Matchbox manifolds}, 
  {Lecture Notes in Pure and Appl. Math., Vol. 170},
  {Dekker, New York}, 1995, pages 3--14..
     
{A.~Clark, S.~Hurder and O.~Lukina},
 {\it Classifying matchbox manifolds}, {arXiv:1311.0226}.
 
{J.~Dyer},
 {\it Dynamics of equicontinuous group actions on {C}antor sets},
{\bf Thesis}, University of Illinois at Chicago, 2015.

{J.~Dyer, S.~Hurder and O.~Lukina},
 {\it The discriminant invariant of Cantor group actions },
{arXiv:1509.06227}.

{R.~Fokkink and L.~Oversteegen},
  {\it Homogeneous weak solenoids},
  {\bf Trans. Amer. Math. Soc.}, 354: 3743--3755, 2002.

{E.~Glasner},
  {\it Enveloping semigroups in topological dynamics},
  {\bf Topology Appl.}, 154: 2344--2363, 2007.

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


 {R.~Schori},
  {\it Inverse limits and homogeneity},
  {\bf Trans. Amer. Math. Soc.}, 124: 533--539, 1966.

   

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