% Tiling and Recurrence
% Dates: 4-8 December 2017
% Place: CIRM (Marseille Luminy, France)

% talk on December 4, 2017

% slides for talk "Wild solenoids and tilings"
 
 
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\date[December 4, 2017]


  
\title[Wild tilings]{Wild solenoids and tilings}


\author{Steve Hurder\\ ... \\{\footnotesize {\it Tiling and Recurrence} - CIRM Luminy\\December 4, 2017}}

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

 
\begin{document}

\frame{\titlepage} % # 1


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

% \begin{center}
% $\star$ Search for ``hidden symmetries'' in Cantor actions $\star$
% \end{center}

 

{\bf Set-up:} 

$\bullet$   $G$   a   group (or $\psg$ $\equiv$ a pseudogroup without the disjoint union property for maps),

$\bullet$ $\fX$ a Cantor space, 

$\bullet$  $\vp \colon G \times \fX \to \fX$ a minimal action.

 
\medskip

{\bf Program:}  

Define and calculate invariants which distinguish such actions up to an appropriate notion of equivalence.    
 
  \medskip
 
 {\bf Applications:} 
 
 $\bullet$ Classification of weak solenoids up to homeomorphism
 
 $\bullet$ Classification of tiling spaces
 
 $\bullet$ Invariants of iterated polynomial equations
 
\vfill   

}


 
 \frame % 1
{
  \frametitle{ }

 
 {\bf Definition:} Let $\vp_i \colon G_i \times \fX_i$ be minimal Cantor actions, for $i=1,2$.
 Say that $\vp_1$ is  \underline{return equivalent} to $\vp_2$ if there exist 
 
  $\bullet$  clopen subsets $U_i \subset \fX_i$ for $i=1,2$
  
   $\bullet$ 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 $\psg$s
  $$G_i|U_i = \{g_U \colon U \to V \mid g \in G_i ~ , ~ U \subset Dom(g) ~, ~ V = g(U) \subset U_i\}$$
 
  \medskip
  
{\bf Remark:}   When $G_i$ is a group   and $U_i = \fX_i$ for $i=1,2$,  this reduces to the notion of topological conjugacy of the actions, allowing an isomorphism $\alpha_h \colon G_1 \to G_2$ intertwining the actions.
  
  
 
\vfill   

}

\section{LQA}
 
 \frame % 1
{
  \frametitle{}
{\bf ``Analytic'' Cantor actions:} 

\bigskip

  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} (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.

\medskip
 
$\bullet$  A   homeomorphism $h \colon U \to V$ is \underline{locally quasi-analytic} (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 $\psg$ action $\vp \colon G \colon \fX \to \fX$ is LQA if  for each $x \in \fX$, there exists a clopen neighborhood $x \in U$, such that each map in the restricted $\psg$ $G|U$   is QA.



\vfill   

}


 \frame % 1
{
  \frametitle{ }

{\bf Remarks:} 

$\bullet$ A free action $G \times \fX \to \fX$ is  QA.

 $\bullet$ A minimal action of $G = \mZ^n$ is always QA.
 
 $\bullet$ The pointed automorphism group of an equi-valent  tree (acting on the Cantor set of ends) is not LQA. Such actions are also   not finitely generated.
 
 \bigskip
 
 {\bf Proposition:} The LQA property is an invariant of return equivalence for minimal $\psg$ actions.

\vfill   

}


 \frame % 1
{
  \frametitle{ }

 Let $\vp \colon G \colon \fX \to \fX$ be a $\psg$ action. A point $x \in \fX$ is said to be a  \emph{holonomy point} for the action if for all $g \in G$ with $x \in Dom(g)$, $g(x) = x$, and for all open $x \in U \subset Dom(g)$, the restriction $g|U \colon U \to g(U)$ is not the identity.
 
 \bigskip
 The following result was first shown for foliation $\psg$'s by Epstein, Millett and Tischler in 1977:
 
 {\bf Proposition:} For $\vp \colon G \colon \fX \to \fX$ a $\psg$ action,   then 
 $$Holo(\vp) = \{x \in \fX \mid x ~ {\rm is ~ a ~   holonomy~ point} \}$$
 is meager; in particular, it has empty interior in $\fX$.
\vfill   

}

 
  \section{TOE}

  
 \frame % 1
{
  \frametitle{ }

 
 {\bf Definition:} Let $\vp_i \colon G_i \times \fX_i$ be minimal Cantor actions, for $i=1,2$.
 Say that $\vp_1$ is  \underline{topologically orbit equivalent} (TOE) to $\vp_2$ if there exist a homeomorphism  $h \colon \fX_1 \to \fX_2$ 
 such that for all $x \in \fX_1$
 $$h \left( \{\vp_1(g)(x) \mid g \in G_1\}\right)  = \{\vp_2(g')(h(x)) \mid g' \in G_2\}$$
 
 \bigskip
 
 

{\bf Definition:} A continuous cocycle $\phi \colon G_1 \times \fX_1 \to G_2$ over the action $\vp_1$  is said to be a \underline{rearrangement cocycle}  for a TOE $h \colon \fX_1 \to \fX_2$ between actions $\vp_1$ and $\vp_2$ if we have
\begin{equation}\label{cocycle}
h(\vp_1(g)(x)) = \vp_2(\phi(g, x))(h(x)) \quad {\rm for ~ all} ~ g \in G_1 ~, ~ x \in \fX_1 \ .
\end{equation}


 

     
 
\vfill   

}

 
 \frame % 1
{
  \frametitle{ }

 
 {\bf Lemma:} For $i=1,2$, let  $\vp_i \colon G_i \times \fX_i$ be a free minimal  Cantor action.
 Let     $h \colon \fX_1 \to \fX_2$ be a TOE between $\vp_1$ and $\vp_2$.
 Then there exists a rearrangement cocycle $\phi \colon G_1 \times \fX_1 \to G_2$ for $h$.
 
 {\it Proof.} The formula \eqref{cocycle} defines $\phi$ as the actions are free.
 
 \bigskip
 
 {\bf Proposition:} For $i=1,2$, let $\vp_i \colon G_i \times \fX_i$ be an LQA  minimal  Cantor action.
 Let     $h \colon \fX_1 \to \fX_2$ be a TOE between $\vp_1$ and $\vp_2$.
 Then there exists a rearrangement cocycle $\phi \colon G_1 \times \fX_1 \to G_2$ for $h$.

{\it Proof.} The LQA assumption implies that there exists a dense $\vp_2$-invariant subset   $\fX_2' \subset \fX_2$ on which the action is free. Then $\fX_1' = h^{-1}(\fX_2') \subset \fX_1$ is dense and $\vp_1$-invariant. Use
the formula \eqref{cocycle} to define $\phi$ on $\fX_1'$, then extend to all of $\fX_1$ by continuity.

  \bigskip
 
 We apply these concepts for a special class of actions.
 
\vfill   

}

  
  
 \frame % 1
{
  \frametitle{ }

 {\bf Definition:} A Cantor action $\vp \colon G  \times \fX \to \fX$ is \underline{equicontinuous} if for some metric $\dX$ on $\fX$,   for every $\e > 0$ there exists $\delta > 0$ such that 
 $$\dX(x,y) < \delta ~ \Longrightarrow ~ \dX(\vp(g)(x) , \vp(g)(y)) < \e \quad {\rm for ~ all } ~ g \in G   .$$
   
  
 
 {\bf Theorem:} (folkore) Assume that $G$ is finitely generated. Given $x \in \fX$, then the equicontinuous minimal Cantor action $\vp$ is conjugate to a generalized odometer 
 $(X_{\infty} , G, \Phi)$ associated to a properly descending chain of subgroups of finite index in $G$,
 $$G = G_0 \supset G_1 \supset G_2 \supset \cdots \supset G_{\ell} \supset \cdots $$
 where 
 $\ds X_{\infty} \equiv  \lim_{\longleftarrow} ~  \{p_{\ell} \colon  G/G_{\ell +1} \to G/G_{\ell} \mid \ell > 0\}$
 has the natural  left $G$-action, but need not be a profinite group.    Moreover, the conjugating homeomorphism $\tau \colon X_{\infty} \to \fX$ satisfies $\tau(e_{\infty}) = x$, where $e_{\infty} \in X_{\infty}$ is the   coset of the identity   $e \in G$. The map $\tau$ represents ``coordinates'' for $\fX$ around $x$.
 
\vfill   

}




 \frame % 1
{
  \frametitle{ }

   The following result was proved by Cortez and Medynets:
   
 {\bf Theorem:} Let $\vp_i \colon G_i \times \fX_i \to \fX_i$ be free  equicontinuous minimal  Cantor actions, for $i=1,2$.
 Let     $h \colon \fX_1 \to \fX_2$ be a TOE between $\vp_1$ and $\vp_2$.
 Then  $h$ induces a return equivalence between the actions $\vp_1$ and $\vp_2$.

\bigskip

  In fact, a stronger statement is true:
  
     {\bf Theorem:} Let $\vp_i \colon G_i \times \fX_i \to \fX_i$ be an LQA  equicontinuous  minimal Cantor action, for $i=1,2$.
 Let     $h \colon \fX_1 \to \fX_2$ be a TOE between $\vp_1$ and $\vp_2$.
 Then  $h$ induces a return equivalence between the actions $\vp_1$ and $\vp_2$.

 {\it Proof.} The existence of a continuous rearrangement cocycle is a consequence of the LQA hypothesis. The proof then follows as in Cortez and Medynets [2016] and Clark and Hurder [2013].
\vfill   

}

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

 A Cantor action $\vp \colon G  \times \fX \to \fX$ of a group $G$ defines a homomorphism $\whvp \colon G \to H_{\vp} \subset {\rm Homeo}(\fX)$.

 


 {\bf Definition:} The \underline{Ellis (enveloping) semigroup} $E(\vp) \subset {\rm Homeo}(\fX)$ is the closure of   the group $H_{\vp}$ in the topology of pointwise convergence on maps.
 
 \bigskip
 
 The Ellis semi-group  of an action $\vp$ seems very difficult to calculate in general, except in the following case:
 
 {\bf Proposition:} Let $\vp$ be an equicontinuous    Cantor action. Then $E(\vp)$ is the closure  $\overline{H_{\vp}}$ of $H_{\vp}$   
   in the \emph{uniform topology on maps}. 

\vfill   

}

 
  
 \frame % 1
{
  \frametitle{ }


{\bf Lemma:} The left action of $\overline{H_{\vp}}$ on $\fX$ is transitive if $\vp$  is  minimal.

   {\it Proof.} Let $x \in \fX$, then the orbit $ \overline{H^{\vp}}\cdot x$ is a closed subset of $\fX$ that contains a dense orbit under the action of  $\whvp(G)$. 

 \bigskip
 
 
 Let $\overline{H^{\vp}}_x = \{ h \in \overline{H^{\vp}} \mid h(x) = x \}$ be the isotropy group of $x \in \fX$. 
 
 {\bf Corollary:}  $\fX \cong \overline{H^{\vp}}/\overline{H^{\vp}}_x$.
 
  
 \medskip
 
 Note that  group $\overline{H^{\vp}}_x$ is independent of the choice of basepoint $x \in \fX$, up to topological isomorphism of groups.
 
 \bigskip
 
The \underline{normal core} $N$ of a subgroup $H \subset G$ is the largest subgroup $N \subset H$ which is   normal in $G$.
   
   {\bf Proposition:} The normal core of    $\overline{H^{\vp}}_x$ in $\overline{H^{\vp}}$ is trivial. 
 
 {\it Proof.} The action of $\overline{H^{\vp}}$ on $\fX$   is effective,   the  normal core of    $\overline{H^{\vp}}_x$ acts trivially on  
 the quotient space $\fX \cong \overline{H^{\vp}}/\overline{H^{\vp}}_x$, hence it 
 is trivial.

\vfill   

}

 
  
 \frame % 1
{
  \frametitle{ }

  Let $G$ be finitely generated,   $\vp \colon G  \times \fX \to \fX$   an equicontinuous  minimal Cantor action, and $x \in \fX$ a choice of basepoint. 
  
  Then the action is conjugate to   a generalized odometer 
 $(X_{\infty} , G, \Phi)$ associated to a properly descending chain of subgroups $\{ G_{\ell} \mid \ell \geq 0\}$ of finite index in $G$, where  the conjugating homeomorphism $\tau \colon X_{\infty} \to \fX$ satisfies $\tau(e_{\infty}) = x$.  

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

The left action of $G$ on $G_{\infty}$ yields a minimal Cantor action $(G_{\infty}, G, \Phi_{\infty})$  and homomorphism $\whPhi_{\infty} \colon G_{\infty} \to {\rm Homeo}(\fX)$.
 
  
     
 
\vfill   

}

  
 \frame % 1
{
  \frametitle{ }

  {\bf Theorem:}  Let $\vp \colon G  \times \fX \to \fX$ be an equicontinuous  minimal Cantor action. Then $\overline{H^{\vp}} = \whPhi_{\infty}(G_{\infty})$.

  \medskip
  
     
  {\bf Theorem:} There is a natural isomorphism 
 \begin{equation}\label{eq-discriminant}
\overline{H^{\vp}}_x \cong  \cD_x \equiv \lim_{\longleftarrow} ~  \{\pi_{\ell} \colon  G_{\ell+1}/C_{\ell +1 } \to G_{\ell}/C_{\ell} \mid \ell \geq 0\}~ .
\end{equation}
 The profinite group $\cD_x$ defined by the right-hand-side of \eqref{eq-discriminant} is called the \underline{discriminant} of the group chain   $\{ G_{\ell} \mid \ell \geq 0\}$ at $x$.

\medskip

 {\bf Corollary:} The isomorphism class of the discriminant group $\cD_x$   is independent of the choice of basepoint $x \in \fX$ and the group chain $\{ G_{\ell} \mid \ell \geq 0\}$  used to define the odometer model of the action $\vp$. 
 
 \medskip
 
 {\bf Problem:} Show that  the  isomorphism class of the discriminant group    an invariant of return equivalence.

\vfill   

}



  \section{Wild}

  
 \frame % 1
{
  \frametitle{ }
A minimal Cantor action is wild if there exist ``hidden symmetries'' in the large scale structure of the associated colored tiling. 

\bigskip

 Let   $(\fX, G, \vp)$ be conjugate to the   odometer 
  $(X_{\infty} , G, \Phi)$ associated to a   properly descending chain of subgroups $\{ G_{\ell} \mid \ell \geq 0\}$ of finite index in $G$, with basepoint $x$.
 Corresponding to each subgroup $G_{\ell}$ for $\ell \geq 0$, there is a clopen neighborhood $U_{\ell}$ of $x$ with $x \in U_{\ell +1} \subset U_{\ell}$ and 
 $G_{\ell} = G | U_{\ell}$. 
 
 Introduce the nested clopen neighborhood subgroups of $\whe \in G_{\infty}$
 $$\whU_{\ell}   \equiv  \lim_{\longleftarrow} ~  \{\pi_{\ell} \colon  G_{\ell}/C_{\ell +k+1} \to G_{\ell}/C_{\ell + k} \mid k > 0\} \subset  G_{\infty} \, .$$
   Note that $\cD_x \subset   \whU_{\ell}$ and  $U_{\ell} \cong \whU_{\ell}/\cD_x$  for all $\ell \geq 0$. 
    
 
 
     
 
\vfill   

}

 
 \frame % 1
{
  \frametitle{ }
 
   The isotropy action of $\cD_x$ on $U_{\ell}$ is induced by the 
adjoint action: 

for $\whh \in \cD_x$ and $y \in U_{\ell}$, let $y = \whg \,  \cD_x$ for $\whg \in \whU_{\ell}$, then
$$\whh \cdot y = Ad(\whh)(\whg) \,   \cD_x = \whh \  \whg \  \whh^{-1} \,   \cD_x \, .$$


The ``trivial core'' property of $\cD_x$ implies that the action of $\whh \in \cD_x$ on $X_{\infty} \cong G_{\infty}/\cD_x$ is non-trivial for all non-trivial $\whh$.

 \medskip
 
 {\bf Definition:} An equicontinuous  minimal Cantor action $\vp$ is \underline{wild} if for all $\ell > 0$ there exists  non-trivial $\whh \in \cD_x$ and $\ell' > \ell > 0$ such that the left isotropy action of $\whh$   on   $U_{\ell'}$ is trivial.

\medskip

  {\bf Definition:} A  minimal Cantor action $\vp$ is \underline{wild} if its maximal equicontinuous factor is wild.

\vfill   

}


 \frame % 1
{
  \frametitle{ }



  
 
 
  An action which is not wild is said to be \underline{stable}. 
  
  \medskip
  
  {\bf Theorem:}  Let $\vp \colon G_i \times \fX \to \fX$ be a stable  equicontinuous  minimal Cantor action. Then for all suitably small clopen subsets $U \subset \fX$ the discriminant group of the restricted action $G|U$ is independent (up to isomorphism) of the choice of $U$.
  
  The limiting isomorphism class of the discriminant for a stable action is called the \underline{stable discriminant} of the action.
  
  \medskip
  

   {\bf Corollary:} Let $\vp \colon G_i \times \fX \to \fX$ be a stable  equicontinuous  minimal Cantor action.
   Then  the  isomorphism class of the discriminant group    an invariant of return equivalence. 
   
   That is, any   minimal action which is return equivalent to $\vp$ is equicontinuous and stable, and their   stable discriminant groups are isomorphic. 

     
 
\vfill   

}


 \frame % 1
{
  \frametitle{ }

 
 
    
  {\bf Theorem:}  (Hurder \& Lukina [2017])
  Let $\vp \colon G  \times \fX \to \fX$ be an equicontinuous  minimal Cantor action, where $G$ is finitely generated. Then the action $\vp$ is stable if and only if the action of $G_{\infty}$ on $X_{\infty}$ satisfies the LQA property. 
  
  \medskip
  
 {\it Proof.} Suppose the action of $G_{\infty}$ on $X_{\infty}$ satisfies the LQA property. Choose a metric $\dX$ on $\fX$ and choose group chain model for the action. Then there exists $\e > 0$ such that 
 if $U \subset \fX$ is a clopen subset with ${\rm diam}(U) < \e$ then the restricted $\psg$ $G|U$ is $QA$. Let $\ell > 0$ be sufficiently large so that ${\rm diam}(U_{\ell}) < \e$.
  Then for all non-trivial   $\whh \in \cD_x$ and $\ell' > \ell > 0$   the isotropy action of $\whh$ is non-trivial on  $U_{\ell'}$.
  
  The converse follows in a similar fashion.
  
  
 
\vfill   

}




\section{Examples}

 \frame % 1
{
  \frametitle{ }

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

}


 \frame % 1
{
  \frametitle{ }

   
 
 {\bf Example 2:}   Arboreal actions of Galois groups
 
 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.
 
$\bullet$ 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.


$\bullet$     2013 survey paper by   Rafe Jones gives   introduction to this program  from the point of view of arithmetic dynamical systems and number theory.
  
$\bullet$  Lukina [2017] gives concrete examples of polynomials with non-trivial discriminant invariants for their arboreal actions, and also describes some wild arboreal actions.
    
\vfill   

}


 \frame % 1
{
  \frametitle{ }

   
 
 {\bf Example 3:} 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 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 $\cD$ are not return equivalent. 
 
 \bigskip
 
 {\bf Conclusion:} The methods for constructing wild actions are not systematically understood, yet they seem to be an interesting class of minimal Cantor actions worthy of further study.
 
 
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}

    \section{Tilings}

  
 \frame % 1
{
  \frametitle{ }

  Finally, we give a ``geometric'' interpretation of the notion of a wild action,  as the existence of``hidden symmetries'' in the   periodic tilings associated to periodic semi-cocycles over the action. 
  
  \medskip
  
  Let $\vp \colon G  \times \fX \to \fX$ be an equicontinuous  minimal Cantor action, where $G$ is finitely generated.  
    Let  $x \in \fX$, and  $(\fX, G, \vp)$ be conjugate to the   odometer 
  $(X_{\infty} , G, \Phi)$ associated to a   properly descending chain of subgroups $\{ G_{\ell} \mid \ell \geq 0\}$ of finite index in $G$, with basepoint $x$.

\medskip

For $\ell \geq 0$, recall that  $U_{\ell} \subset \fX$ is the clopen subset satisfying
 $$G_{\ell} =   \{g \in G \mid \vp(g)(U_{\ell}) \cap U_{\ell} \ne \emptyset\} \ .$$ 
 In particular, $G_0 = G$ and $U_0 = \fX$. 
 
 
  \vfill   

}

  
  
  
 
 \frame % 1
{
  \frametitle{ }

 We next construct a periodic semi-cocycle for a group acting cocompactly on the Poincar\'e disc $\mD^2$ of constant   curvature $-1$.
 
 Let $\{g_1, \ldots, g_k\} \subset G$ be a generating set. Let $\Sigma_k$ be a closed surface of genus $k$ with a Riemannian  metric of constant negative curvature $-1$. Let  $\pi \colon \wtSigma_k \to \Sigma_k$ be the universal covering of $\Sigma_k$ then $\wtSigma_k$ is isometric to the Poincar\'e disk $\mD^2$, and the fundamental group $\G_k \equiv \pi_1(\Sigma_k, b_0)$ acts on $\mD^2$ on the right by   deck transformations, which are isometries.
 
  
 Let $\tau_k \colon \G_k \to \star_k \mZ$ be a surjective map onto the free group on $k$ generators. Let $\xi_k \colon \star_k \mZ \to G$ be the surjective map defined by the choice of generating set for $G$, then the composition $\sigma_k \equiv  \xi_k \circ \tau_k \colon \G_k \to G$ is a surjection. 
 
 For $\ell \geq 0$ set $\G_{k,\ell} = \sigma_k^{-1}(G_{\ell})$ 
 which has finite index in $\G_k$. 


\vfill   

}


 \frame % 1
{
  \frametitle{ }

   
 Let $M_{k,\ell} \to \Sigma_k$ be the finite covering associated to $\G_{k,\ell}$ so $\pi_1(M, b) \cong \G_{k,\ell}$, which is again covered by $\mD^2 = \wtSigma_k$. Choose a convex fundamental domain $T_{k,\ell} \subset \mD^2$ for this covering. Then $\mD^2$ has a periodic tiling by $T_{k,\ell}$ invariant under the deck transformations by elements of  $\G_{k,\ell}$.
 
  
 
The covering  $M_{k,\ell} \to \Sigma_k$ is the   tiling space for the periodic tiling $\cT_{k,\ell}$ of $\mD^2$ associated to the tile   $T_{k,\ell}$ and the semi-cocycle   
$$f_{k,\ell} \colon \G_k \to X_{\ell} = G/G_{\ell}$$
defined by composing $\sigma_k$ with the quotient map. 

As $\ell \to \infty$, the sequence of coverings $M_{k,\ell} \to \Sigma_k$ converge to the weak solenoid $\cS_{\vp} \to \Sigma_k$ defined by the suspension of  the induced action 
$\vp \circ \sigma_k \colon \G_k  \times \fX \to \fX$: 
$$\cS_{\vp} \equiv  (\mD^2 \times \fX)/ (w \cdot \gamma^{-1}, x) \sim (w, \vp \circ \sigma_k(\gamma)(x) \ . $$
 
  
\vfill   

}


 \frame % 1
{
  \frametitle{ }

The automorphism group of $\cS_{\vp}$ is typically trivial without additional assumptions on $G$, such as $G = \mZ^n$.

Assume that the action  $\vp$  is wild.  Then there exists $\ell' > \ell$, and $\whh \in   \cD_x \subset G_{\infty}$  such that   
 the  left isotropy action of $\whh$ on $U_{\ell} = \whU_{\ell}/\cD_x$ is non-trivial, but the isotropy action on $U_{\ell'} = \whU_{\ell'}/\cD_x$ is trivial.



Let  $\pi \colon M_{k,\ell'} \to M_{k,\ell}$ be the   covering associated to $\G_{k,\ell'} \subset \G_{k,\ell}$ whose fiber is the   coset space $X_{\ell, \ell'} = G_{\ell}/G_{\ell'}$.

   Choose a convex fundamental domain $T_{k,\ell'} \subset \mD^2$ for this second covering, which is a union of contiguous translates of    the tile $T_{k,\ell}$.

\bigskip

  {\bf Claim:} 
  There exists $\gamma \in \G_k$ whose covering translation action on $\mD^2$ permutes the tiling by $T_{k,\ell}$ but preserves the tiling by $T_{k,\ell'}$.
  
 
 
  \vfill   

}

 \frame % 1
{
  \frametitle{ }

{\bf Remark:} The action of $\gamma$ on the tiling $\cT_{k,\ell}$ preserves the super-tile structure given by the tiles in $\cT_{k,\ell'}$.
   This is a ``hidden symmetry'' of the tiling $\cT_{k,\ell}$.

\medskip

Now recall that  $\whh \in   \cD_x \subset G_{\infty}$ is given. 
Definition   \eqref{eq-discriminant} of $\cD_x$ implies there exists $h_{\ell'} \in  G_{\ell'}$ so that    
 $$\whh \ mod \ C_{\ell'} = h_{\ell'} \ mod \ C_{\ell'}$$
Let  $\overline{h_{\ell'}} \in G_{\ell'}/C_{\ell'}$ be its image in the finite group $G_{\ell'}/C_{\ell'}$.
Then by the choice of $\whh$ the left action of  $\overline{h_{\ell'}}$ commutes with the left action of $G_{\ell'}$ on the coset space $X_{\ell, \ell'} = G_{\ell}/G_{\ell'}$.
 
 Let $\gamma \in \G_{k,\ell'}$ be chosen such that $\sigma_k(\gamma) =  h_{\ell'}$.
 
   \bigskip
  
 {\bf Remark:}  A similar analysis can be applied to the non-periodic Toeplitz semi-cocycles over $\G_k$ as constructed by   Krieger [2010] and Cortez and Medynets [2016].
 

  
 
 
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}

 
  \section{Refs}

    
\frame % 1
{
  \frametitle{References}
 {\scriptsize     
 
 
{A.~Clark and S.~Hurder},
 {\it Homogeneous matchbox manifolds},
 {\bf Transactions A.M.S.},   365:3151--3191, 2013.
 
 {M.-I.~Cortez and S.~Petite}, {\it $G$-odometers and their almost one-to-one extensions},
  {\bf J. London Math. Soc.}, 78:1--20, 2008. 

 {M.-I.~Cortez and K.~Medynets}, {\it Orbit equivalence rigidity of equicontinuous systems},
  {\bf J. London Math. Soc.}, 94:545--556, 2016. 

 
 
{J.~Dyer, S.~Hurder and O.~Lukina},
 {\it Molino theory for matchbox manifolds},
 {\bf Pacific Journal Math},  Vol. 289:91-151, 2017.
 

{S.~Hurder and O.~Lukina},
 {\it Wild solenoids},
 {\bf Transactions A.M.S.},  to appear, 2017.

{S.~Hurder and O.~Lukina},
 {\it Orbit equivalence for quasi-analytic Cantor actions},
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{F.~Krieger},
 {\it Toeplitz subshifts and odometers for residually finite groups},
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{A.~Lubotzky},  {\it Torsion in profinite completions of torsion-free groups},
 {\bf Quart. J. Math.} 44:327--332, 1993.


{O.~Lukina},
 {\it Arboreal {C}antor actions},
 {in preparation}, 2017.

{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.

  
}
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