%J oint Mathematics Meetings
% AMS Special Session on Riemannian Foliations and Applications
% January 15-18, 2020 (Wednesday - Saturday)
% Meeting # 1154, Abstract Issue: 

 
% slides for talk "Classification of weak solenoids"
 
 
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\date[January 18, 2020]


  
\title[Wild things]{Classification of weak solenoids}


\author{Steve Hurder, University of Illinois at Chicago\\   {\footnotesize  Joint work with}  \\ Olga Lukina, University of Vienna} 

 
 
\begin{document}
 
  \frame{\titlepage} % # 1

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

\bigskip
 
 A \emph{Riemannian foliation} is one whose holonomy pseudogroup is generated by  local isometries of a Riemannian manifold. 

 

\medskip
 
 Bruce  Reinhart:
    ``\emph{Foliated manifolds with bundle-like metrics}'',  
 {\bf Ann. of Math},   69:119--132, 1959 
 
 \medskip
 
   
Pierre  Molino: {\bf Feuilletages riemanniens},    Montpellier, 1983\\ \& 
{\bf Riemannian Foliations}, Birkhauser, 1988.
 

\bigskip
An \emph{equicontinuous foliated space} is one whose holonomy pseudogroup is equicontinuous $\sim$ isometric.

\bigskip

{\bf Problem:} How much of the theory of Riemannian foliations  in Molino's book   can be extended to equicontinuous foliated spaces?

 \vfill   

}



\frame
{
  \frametitle{}
  
 This question has been studied in the works of Jes\'{u}s \'{A}lvarez L\'{o}pez:
 
 \smallskip
 
$\star$~ (with Alberto Candel),   \\
\quad ``\emph{Equicontinuous foliated spaces}'', \\
\qquad{\bf Math. Z.},  263 (2009), 725--774. 

 
 \smallskip

$\star$~ (with  Manuel Moreira Galicia), \\
\quad``\emph{Topological Molino's theory}'', \\
\qquad {\bf Pacific J. Math.},  280 (2016), 257--314. 
 
  \smallskip


$\star$~ (with  Ram\'{o}n Barral Lij\'{o}), \\
 \quad ``\emph{Molino's description and foliated homogeneity}'',\\
\qquad {\bf   Topology Appl.}, 260 (2019), 148--177. 
  
  \vfill   

}


 \frame % 1
{
  \frametitle{}

 Our interest is the special case where the 
   foliated spaces     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 % 1
{
  \frametitle{}

 
{\bf Theorem [Clark-Hurder, 2013]} Let $\fM$ be an equicontinuous matchbox manifold. Then $\fM$ is homeomorphic to a \emph{weak solenoid}. 

\smallskip

If $\fM$ is homogeneous space, then the weak solenoid is a profinite  group fibration over a compact manifold. 
 
\smallskip

If $\fM$ is not homogeneous, then it is homeomorphic to a quotient of a profinite group fibration by a non-trivial closed subgroup.

 \medskip
 
Alex Clark \& S.H.,   ``\emph{Homogeneous matchbox manifolds}'', {\bf Transactions A.M.S.}, 365 (2013), 3151-3191. 

 
\medskip

 

 \vfill   

}

  

\section{Solenoids}


 \frame % 1
{
  \frametitle{}
{\bf Weak Solenoids}

 \medskip
 $\bullet$ ~ $M$ is compact manifold without boundary
 
 \medskip
 
 $\bullet$ ~ $G = \pi_1(M,x_0)$ is finitely generated group.
 
  
 
 $$M = M_0 \stackrel{p_1}{\longleftarrow} M_1 \stackrel{p_2}{\longleftarrow} M_2 \stackrel{p_3}{\longleftarrow} M_3  \cdots $$
 Choose $x_{\ell} \in M_{\ell}$ with $p_{\ell}(x_{\ell}) = x_{\ell -1}$,   set $G_{\ell} = \pi_1(M_{\ell}, x_{\ell})$
 
Inclusion maps $q_{\ell} \colon G_{\ell} \subset G_{\ell-1}$, descending chain of groups
  $$G = G_0 \stackrel{q_1}{\longleftarrow} G_1 \stackrel{q_2}{\longleftarrow} G_2 \stackrel{q_3}{\longleftarrow} G_3  \cdots $$

Tower of coverings is \emph{normal} if each $G_{\ell} \subset G_0$ is a normal subgroup.

 \medskip
 
  {\bf Example:} Vietoris solenoid is given coverings of $\mS^1$, so is determined by a chain of normal subgroups of $\mZ$.


 \vfill   

}
  
   
 
  

   
 \frame % 1
{
  \frametitle{}

 
Inverse limit space for   a tower of coverings:
\begin{eqnarray*}
 \lefteqn{ M_{\infty} = \varprojlim ~ \{ p_{\ell+1}^{\ell}\colon M_{\ell+1} \to M_{\ell}  \mid \ell \geq 0 \}}  \\
& =&  \{ (y_0, y_1, y_2, \ldots) \mid p_{\ell+1}^{\ell}(y_{\ell+1}) = y_{\ell}  \mid \ell \geq 0\} \\
&  \subset & \prod_{\ell \geq 0} ~ M_{\ell}
\end{eqnarray*}
is a compact connected metrizable space called a \emph{(weak) solenoid}.

For each $\ell > 0$, there is a fibration map $\Pi_{\ell} \colon M_{\infty} \to M_{\ell}$.
\medskip

For fixed $x_{\ell} \in M_{\ell}$ the fiber $\fX_{\ell} = \Pi_{\ell}^{-1}(x_{\ell}) \subset \fX_0$ is a Cantor space.

\medskip

$\bullet$~ The path connected components of $M_{\infty}$ are manifolds, 

$\bullet$~ leaves are non-compact covering spaces of $M_0$, 

$\bullet$~  $M_{\infty}$ is a   foliated space with Cantor transversals.
     

 \vfill   

}
 
  
  
    
  
 \frame % 1
{
  \frametitle{}

  The monodromy action on the fiber,  $\Phi \colon G_0 \times \fX_0 \to \fX_0$
  
Fundamental group $G_0 = \pi_1(M_0,x_0)$ acts on the fiber $\fX_0$ via lifts of paths in $M_0$ to the leaves of $\F_{\fM}$.
 
 This action is 
 
 $\bullet$ ~ \emph{minimal} =  the orbit of each point is dense in $\fX_0$.

 $\bullet$ ~  \emph{equicontinuous}:  for every $\e > 0$ there exists $\delta > 0$ such that 
 $$\dX(x,y) < \delta ~ \Longrightarrow ~ \dX(\vp(g)(x) , \vp(g)(y)) < \e \quad {\rm for ~ all } ~ g \in G   .$$
   
  Cantor action $(\fX, G, \Phi)$ $\equiv$    minimal  \& equicontinuous 
  \medskip

Conclusion of works with   Clark \&   Lukina can be summarized:
 \begin{center}
 \emph{Analyze/Classify  weak solenoids} $\Leftrightarrow$    \emph{Analyze/Classify   Cantor actions}
  \end{center}
 
   \vfill   

}

 
 \section{Dynamics}


  \frame % 1
{
  \frametitle{ }


   \underline{Profinite model} for Cantor action $(\fX, G, \Phi)$. 

\medskip

 {\bf Definition:}  $\oG = \overline{H_{\Phi}} =$       closure of $H_{\Phi} = \Phi(G) \subset \Homeo(\fX)$   \\
   in the \emph{uniform topology on maps}.  $\oG$ is profinite group.


\medskip

  
 For $x \in \fX$,  $\cD_x = \{ \whh \in \overline{H_{\Phi}} \mid \whh \cdot x = x \}$   ~  (\emph{isotropy group})
 
  \medskip
 
 {\bf Lemma:}   Left action of $\oG$ on $\fX$ is transitive.
  Hence 
  
  $\bullet$ ~  $\fX \cong \oG/\cD_x$

  $\bullet$ ~
  $\cD_x$     independent of the choice of basepoint $x$. 

\medskip

 
The  \underline{normal core} of $G_{\ell}$ is 
     $  C_{\ell} =  \bigcap_{g \in G} gG_{\ell} g^{-1}  \subset G_{\ell}$

 \medskip
    
  {\bf Theorem [Dyer-Hurder-Lukina, 2016]}  
   \begin{equation*} 
  \cD_x \cong \varprojlim ~ \{\pi_{\ell+1} \colon  G_{\ell+1}/C_{\ell +1 } \to G_{\ell}/C_{\ell} \mid \ell \geq 0\} ~ .
\end{equation*}

\vfill   

}

  
 \frame % 1
{
  \frametitle{}

 \begin{enumerate}
\item $\cD_x$ is trivial for   Cantor action $(\fX, G, \Phi)$   with $G$ abelian.
\smallskip
\item $\cD_x$ can be a Cantor group for a Cantor action $(\fX, G, \Phi)$ when  $G$ is 3-dimensional Heisenberg group.
 \smallskip
\item  Every \underline{finite group} and every \underline{separable profinite group} can be realized as  $\cD_x$ for a Cantor action by   a torsion-free, finite index subgroup of  $ {\bf SL}(n,\mZ)$, $n \geq 3$.
 \smallskip
\item  $\cD_x$ can be wide-ranging for arboreal representations of absolute Galois groups of number fields and function fields.
 \smallskip
\item Every Cantor action by a finitely generated group $G$ can be realized by a tower of finite coverings of a closed   surface.
\end{enumerate}

 \medskip
 
 {\bf Problem:} Can one ``hear'' $\cD_x$ in the spectrum of leafwise elliptic operators (e.g. Lapacians) on weak solenoids?
 
  \vfill   

}


  \frame % 1
{
  \frametitle{ }

 \medskip

 The inverse limit of a weak solenoid can begin at any level of the tower of coverings:
 \begin{eqnarray*}
  M_{\infty} & =& \varprojlim ~ \{ p_{\ell+1}^{\ell}\colon M_{\ell+1} \to M_{\ell}  \mid \ell \geq 0 \}   \\
& \cong &  \varprojlim ~ \{ p_{\ell+1}^{\ell}\colon M_{\ell+1} \to M_{\ell}  \mid \ell \geq k \} 
\end{eqnarray*}
 
 \medskip
 
{\bf Conclusion:}  Dynamical invariants for weak solenoids must be unchanged upon passing to restrictions to clopen subsets which are  adapted to the action of the monodromy. 
  
 

$\star$ ~ Study dynamics up to    \emph{return equivalence}.
    \vfill   

}

 

\frame % 2
{
  \frametitle{}
 
For  Cantor action $(\fX, G, \Phi)$,  $U \subset \fX$  is \emph{adapted} to the action $\Phi$  

$\bullet$ ~ $U$ is a   non-empty clopen subset, 
 
$\bullet$ ~ for any $g \in G$, 
$\Phi(g)(U) \cap U \ne \emptyset$ implies that $\Phi(g)(U) = U$. 

Translates of $U$ form a  partition of the Cantor set $\fX$.

The set of ``return  times'' to $U$,
$$
G_U = \left\{g \in G \mid \vp(g)(U) \cap U \ne \emptyset  \right\}  
$$
is a subgroup of finite index in $G$, called the \emph{stabilizer} of $U$. 

 
  

 \vfill   

}


 

\frame % 1
{
  \frametitle{}

  
  
 {\bf Definition:} Let $\Phi_i \colon G_i \times \fX_i \to \fX_i$ be  Cantor actions, for $i=1,2$.

Then  $\Phi_1$ is  \underline{return equivalent} to $\Phi_2$ if there exist 
 
  $\bullet$~   for $i=1,2$ a clopen subset  $U_i \subset \fX_i$ adapted for action $\Phi_i$
  
   $\bullet$~  homeomorphism  $h \colon U_1 \to U_2$ 
   
       $\bullet$ ~  isomorphism   $\alpha_h \colon H_1  \to H_2$ of the  action groups,  induced by $h$,  where $H_i = \Phi_i(G_{U_i})  \subset \Homeo(U_i)$
  
  \medskip
  
{\bf Remark:}   When   $U_i = \fX_i$ for $i=1,2$, and the actions are effective, this reduces to the notion of isomorphism, or just topological conjugacy of the actions,  where $\alpha_h \colon G_1 \to G_2$ intertwines them.
  
  \bigskip

{\bf Problem:} Find return equivalence invariants of Cantor actions.



 
 \vfill   

}



 
 \section{Wild actions} 
   
 
  

 \frame % 1
{
  \frametitle{}

  Let  $(\fX, G, \Phi)$ be  Cantor action.
 Fix basepoint $x \in \fX$ and   $\e > 0$.
 
 There exists an adapted clopen set $U \subset \fX$   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:}    
A properly descending chain of clopen sets $\cU = \{U_{\ell} \subset \fX  \mid \ell \geq 1\}$ is  an \emph{adapted neighborhood basis} at $x \in \fX$ for the action $\Phi$  if


    $\bullet$~   $x \in U_{\ell +1} \subset U_{\ell}$ for all $ \ell \geq 1$ with     $\cap  \ U_{\ell} = \{x\}$, 
    
     $\bullet$~   each $U_{\ell}$ is adapted to the action $\Phi$ , set $G_{\ell} = G_{U_{\ell}}$
     
We obtain a sequence of localized Cantor actions $(U_{\ell}, H_{\ell}, \Psi_{\ell})$:

$H_{\ell} = \Phi_{\ell}(G_{\ell})  \subset \Homeo(U_{\ell})$, 
$ \Psi_{\ell} \colon H_{\ell} \times U_{\ell} \to U_{\ell}$ 

 \vfill   

}
 

 




 \frame % 1
{
  \frametitle{}

A Cantor action $(\fX, G, \Phi)$ is  either \underline{stable} or \underline{wild}.

Depends on whether the ``sheaf'' of local actions is stable, or not.
\medskip

 Let $\cD(\Psi_{\ell}) \subset \Homeo(U_{\ell})$  be discriminant group of $(U_{\ell}, H_{\ell}, \Psi_{\ell})$.
  
  There is surjective homomorphism $\rho_{\ell} \colon \cD_x = \cD(\Psi_0) \to \cD(\Psi_{\ell})$.
  
 
Set $K_{\ell} \equiv \ker \{\rho_{\ell}\}$ for $\ell \geq 1$. Then $K_1 \subset K_2 \subset \cdots$

 
 {\bf Theorem [Hurder-Lukina, 2019] } The isomorphism class of the direct limit group 
 $$\Upsilon(\Phi) = \varinjlim \ \{ K_{\ell} \subset K_{\ell+1} \mid \ell \geq 1\}$$
 is a well-defined conjugacy invariant of a Cantor action  $(\fX, G, \Phi)$.
 


   \vfill   

}

 
 \frame % 1
{
  \frametitle{}

 
 A Cantor action  $(\fX, G, \Phi)$ is:
 
 $\bullet$ ~  \underline{stable} if the  chain $\{K_{\ell} \mid \ell \geq 1\}$  is bounded.
 
That is,  if there exists $\ell_0$ so that $K_{\ell} = K_{\ell+1}$ for $\ell \geq \ell_0$. 

\medskip

 $\bullet$ ~    \underline{wild} if the  chain $\{K_{\ell} \mid \ell \geq 1\}$  is unbounded.  
 
  \bigskip
{\bf Theorem [Hurder-Lukina, 2019]:} The wild property for a Cantor action   is invariant under continuous orbit equivalence.

 \vfill   

}



   

  \frame % 1
{
  \frametitle{}


{\bf Example:} The   examples of group actions on trees generated by automata studied by Nekrashevych, Bartholdi, Grigorchuk \emph{et al}  typically induce wild actions on the boundary of the trees.

\bigskip
 
 {\bf Theorem [Lukina, 2019]:}
Let $p$ and $d$ be distinct odd primes, let $K=\mQ_p$ be the field of $p$-adic numbers. Let
  $ f(x) = (x+p)^d - p$.
Then the action of ${\rm Gal}_\infty(f)$ is stable.
 
\bigskip

 {\bf Theorem [Lukina, 2018]:}
Let $f(x)$ be a quadratic polynomial with critical point $c$.
  If the post-critical set $P_C$ is infinite, then the action of ${\rm Gal}_{\rm geom}(f)$, and so of ${\rm Gal}_{\rm arith}(f)$ is wild.

 \vfill   

}



   

  \frame % 1
{
  \frametitle{}
 
 
There is a geometric interpretation of the stable/wild property.

\medskip
 
{\bf Definition:} 
A topological action $\Phi \colon G \times \fX \to \fX$ is \emph{locally quasi-analytic (LQA)} if there exists $\epsilon>0$ such that for any open set $U \subset \mathfrak{X}$ with $diam (U) < \epsilon$,   and for any open $V \subset U$ and $g_1,g_2 \in G$ if
  $$ \textrm{if } \Phi(g_1)|V = \Phi(g_2)|V \textrm{ then }\Phi(g_1)|U = \Phi(g_2)|U ~ .$$
 

  
Alternatively, the action is locally quasi-analytic if and only if for all $g \in G$ if $\Phi(g)|V = id$, then $\Phi(g)|U = id$, for open sets $V \subset U$.

\medskip

{\bf Theorem  [Hurder-Lukina, 2017]:}
A Cantor action $(\fX, G, \Phi)$ with $G$ finitely generated is   stable,  if and only if the pro-finite action     $\whPhi  \colon  \oG \times \fX \to \fX$ is locally quasi-analytic.
 

\vfill   

}

 
 

   
 \frame % 1
{
  \frametitle{}
 
 Here are two further results:
 \medskip
 
 
 {\bf Theorem [Hurder-Lukina, 2018]:}  Let  $\Phi \colon G \times \fX \to \fX$ be a Cantor action with $G$ a finitely-generated \emph{nilpotent} group. Then the action is stable. Moreover, any Cantor action which is continuously orbit equivalent must be return equivalent. 
 
\bigskip

{\bf Theorem [Hurder-Lukina, 2018]:} There exists  uncountably many wild actions  of torsion-free finite index subgroups of ${\bf SL}(n,\mZ)$ with distinct pro-isomorphism classes of direct limit groups  $\Upsilon(\Phi)$.

 \bigskip
 
 {\bf Problem:} The wild property lurks in the spectrum of the leafwise laplacians for weak solenoids. 
\emph{Find it}.

 

    
  
  
\vfill   

}
  

   
 
    
\frame % 1
{
  \frametitle{References}
 {\scriptsize    
 
 {A.~Clark and S.~Hurder},
 {\it Homogeneous matchbox manifolds},
 {\bf Trans. A.M.S.},   365 (2013), 3151--3191.
 
  {J.~Dyer, S.~Hurder and O.~Lukina},
 {\it Molino theory for matchbox manifolds},
 {\bf Pac. Jour. Math.},  289 (2017), 91-151.
 
  {S.~Hurder and O.~Lukina},
 {\it Wild solenoids},
 {\bf Trans. A.M.S.},  371 (2019), 4493--4533; {arXiv:1702.03032}
 
 {A.~Clark, S.~Hurder and O.~Lukina},
 {\it Classifying matchbox manifolds},
 {\bf Geometry \& Topology}, 23 (2019),   1--27.


{S.~Hurder and O.~Lukina},
 {\it Orbit equivalence and classification of weak solenoids},
 {\bf Indiana Univ. Math. J.}, 2020; {arXiv:1803.02098}.

{A.~Clark, S.~Hurder and O.~Lukina},
 {\it Pro-groups and a generalization of a theorem of Bing},
 {\bf Topology and Its Applications}, 2020; {arXiv:1811.00288}.

 {S.~Hurder and O.~Lukina},
 {\it Limit group invariants for non-free Cantor actions},
 {\bf submitted}; {arXiv:1904.11072}.

 }  

 
\bigskip

 \begin{center}
 
 {\large 
 Thank you for your attention!
 }
 \end{center}
 


}

 \section{HBD}
 
 

 
  


\end{document}

