% Tokyo Zoom Seminar
% June 23, 2020 
 
 
% slides for talk "Cantor dynamics of renormalizable groups"
 
 
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\date[June 23, 2020]


  
\title{Cantor dynamics of renormalizable groups }


\author{Steve Hurder, UIC\\     Joint work with   \\ Olga Lukina, University of Vienna \\ Wouter Van Limbeek, UIC\\ .\\.\\
 http://www.math.uic.edu/$\sim$hurder/talks/Tokyo20200623.pdf} 
 
 
\begin{document}
 
   \frame{\titlepage} % # 1

 
       \frame % 1
{
  \frametitle{}
{\bf The Hirsch foliation} is a  codimension-one foliation $\F$ of a compact 3-manifold $N$.

 \bigskip
With appropriate choices in its construction, the foliation $\F$ is real analytic with an exceptional minimal set.

\bigskip

$\bullet$~ \underline{Morris Hirsch}, \emph{A stable analytic foliation with only exceptional minimal sets}, in {\bf Dynamical Systems,
Warwick, 1974}, Lect. Notes in Math. vol. 468,  1975, 9--10.

\bigskip

  \vfill   

}
 
 
       \frame % 1
{
  \frametitle{}

{\bf Construction of the Hirsch foliation:}

\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.4\textwidth]{hirschplug.eps}
\label{figure-originalhirsch}
\end{center}
\end{figure}

Get a foliated 3-manifold by gluing outer torus $\mT^2$ to the inner torus $\mT^2$ preserving foliation by circles, so is determined by a covering  map $\phi \colon \mS^1 \to \mS^1$ of degree $d > 1$.
 
  Choose the gluing map $\phi \colon \mS^1 \to \mS^1$ carefully, to obtain a  foliation which is analytic and has an exceptional minimal set, whose pseudogroup dynamics is determined by the map $\phi$.

 \vfill   

}
 
   

       \frame % 1
{
  \frametitle{}
{\bf Properties of the Hirsch foliation:}

\bigskip

The generic  leaves of $\F$    are ``tree-like'' surfaces.


\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.4\textwidth]{hirschq1.eps}
\label{figure-typicalleaf}
\end{center}
\end{figure}

There are also a finite number of leaves with ``loops'', corresponding to fixed points for the map $\phi \colon \mS^1 \to \mS^1$.
  \vfill   

}
 
 
       \frame % 1
{
  \frametitle{}

The geometry, dynamics and topology of ``Hirsch-like'' foliations have been well-studied, for example in:

   \bigskip
  
$\bullet$~     \underline{Alberto Pinto} \& \underline{Dennis Sullivan}, 
{\it The circle and the solenoid}, 
{\bf Discrete Contin. Dyn. Syst.},   vol. 16,  2006, 463--504.


 
  \bigskip
 
 
$\bullet$~     \underline{Bin Yu}, {\it Affine {H}irsch foliations on 3-manifolds}, 
{\bf Algebr. Geom. Topol.},   vol. 17,  2017, 1743--1770.

 \bigskip
 
 
$\bullet$~  \underline{S\'ebastien Alvarez} \& \underline{Pablo Lessa}, {\it The {T}eichm\"{u}ller space of the {H}irsch foliation}, 
{\bf Ann. Inst. Fourier},   vol. 68,  2018, 1--51.

        
   \vfill   

}
 
 
       \frame % 1
{
  \frametitle{}


{\bf Definition:} A closed connected manifold $M$ is said to be \\
\underline{non co-Hopfian} if it admits a proper self-covering map $\phi \colon M \to M$.


\medskip

$\bullet$~ The circle  $\mS^1$ is non co-Hopfian.

\medskip

$\bullet$~ The $n$-torus $\mT^n = \mS^1 \times \cdots \times \mS^1$ is non co-Hopfian.

\medskip

$\bullet$~  $N$   closed connected manifold, then  $\mS^1 \times N$ is non co-Hopfian.

\medskip
$\bullet$~ The  nilmanifold $N = \cH/\G$ where $\G \subset \cH$ is the integer lattice in the 3-dimensional Heisenberg Lie group $\cH$ admits many inequivalent proper self-covering maps.

\bigskip

There are many other constructions of non co-Hopfian manifolds.

 \medskip
 
Non co-Hopfian manifolds  have    applications in dynamical systems, foliation theory, and spectral theory.

 \vfill   

}
 


       \frame % 1
{
  \frametitle{}

Constructions of generalized Hirsch foliations were given  in

$\bullet$~ \underline{Bis,  Hurder    \& Shive}, 
\emph{Hirsch foliations in codimension greater than one}.  {\bf Foliations 2005}, World Sci. Publ., 2006, 71--108.

\bigskip

{\bf Theorem:} Associated to a proper self-covering map $\phi \colon M \to M$, there is a generalized Hirsch foliation $\F$ on a  closed  manifold $N$,  with codimension-q equal to the dimension of $M$.


  


 \vfill   

}
 


       \frame % 1
{
  \frametitle{}
 \medskip

  
 \bigskip
 
  {\bf Question 1:}
 Which closed manifolds are non co-Hopfian?
 
  
\medskip
  {\bf Question 2:} What are the dynamical properties of Hirsch foliations?
  
  \medskip
   
{\bf Question  3:} What are the properties of minimal sets for Hirsch foliations?
 

 \medskip
 The answer to Questions 2 and 3 depend on the choice of the proper covering map $\phi \colon M \to M$ of course.
 
 \bigskip
 
 
We address  Question 1,   using ideas from algebra and dynamics. 
 \vfill   

}
 


       \frame % 1
{
  \frametitle{}
{\bf The Smale solenoid}

For $m > 1$, let  $\phi_m \colon \mS^1 \to \mS^1$, given by $\phi_m(e^{\iota \theta}) = e^{\iota m \theta}$.

$\phi_m$ is a proper self-covering map of the circle of degree $m$.

Iterate the map   $\phi_m$ repeatedly to obtain the Smale solenoid:
$$\cS_m ~ \equiv ~ \varprojlim \ \{  \mS^1 \stackrel{\phi_m}{\longleftarrow} \mS^1  \stackrel{\phi_m}{\longleftarrow} \mS^1  \stackrel{\phi_m}{\longleftarrow}  \cdots \} \subset \prod_{\ell \geq 0} \ \mS^1 \ .$$


   
 \hspace{60pt} \includegraphics[width=0.5\textwidth]{Smale-Williams_Solenoid_Large} 

  
 \vfill   

}
 

       \frame % 1
{
  \frametitle{}
  
 \medskip
  
  
Associated to a proper self-map $\phi \colon M \to M$ we can form a generalized solenoid 

$$\cS_{\phi} ~ \equiv ~ \varprojlim \ \{  M \stackrel{\phi}{\longleftarrow} M  \stackrel{\phi}{\longleftarrow} M  \stackrel{\phi}{\longleftarrow}  \cdots \} \subset \prod_{\ell \geq 0} \ M \ .$$



These are a special class of the \underline{weak solenoids} introduced by Chris McCord in 1966.

 
\medskip
{\bf Problem:} Characterize the   properties of the weak solenoids $\cS_{\phi}$, and the dynamics of the induced shift map $\sigma_{\phi} \colon  \cS_{\phi} \to  \cS_{\phi}$. 

 \vfill   

}
   
  

  \frame % 1
{
  \frametitle{}
{\bf Group chains}

For the Smale solenoid, given the tower of maps
$$\cS_m ~ \equiv ~ \varprojlim \ \{  \mS^1 \stackrel{\phi_m}{\longleftarrow} \mS^1  \stackrel{\phi_m}{\longleftarrow} \mS^1  \stackrel{\phi_m}{\longleftarrow}  \cdots \} \subset \prod_{\ell \geq 0} \ \mS^1 \ ,$$
let $x_0 \in \mS^1$ be the identity element, then $\pi_1(\mS^1, x_0) \cong \mZ$.  

\medskip

We get a chain of subgroups of finite index
$$ \cG_m = \{ \mZ \supset m \cdot \mZ \supset m^2 \cdot \mZ \supset \cdots \}$$
 
 
 \vfill   

}
   
  

  \frame % 1
{
  \frametitle{}

Next, do this for a non co-Hopfian manifold $M$ of dimension $n > 1$.

\medskip

Let   $\phi \colon M \to M$ be a proper self-covering.

\medskip

Choose a basepoint $x_1 \in M$ and set $x_0 = \phi(x_1)$. Then we have
$$\phi_* \colon \pi_1(M, x_1) \to \pi_1(M,x_0)  \equiv \G_0$$

\medskip

Choose an isomorphism $\pi_1(M, x_1) \cong \pi_1(M,x_0)$. 

 $\star$~    $\phi_*$ induces a self-embedding  $\vp \colon \G_0 \to \G_0$.  

 
 
 
$\star$~ $\G_0$ is finitely generated.

 

$\star$~ $\vp(\G_0) \subset \G_0$ is proper subgroup with  finite index.

$\star$~ Group chain $\cG_{\vp} = \{ \G_0 \supset \G_1 = \vp(\G_0) \supset \G_2 = \vp(\G_1) \supset \cdots\}$.

\medskip
 

 \vfill   

}


  \frame % 1
{
  \frametitle{}

A finite index inclusion $\vp \colon \mZ^n \to \mZ^n$ is called a \underline{renormalization} of the lattice $\mZ^n \subset \mR^n$ in the percolation \& physics literature. 

\medskip
 
{\bf Definition:} Let $\G$ be a finitely generated group, then an inclusion $\vp \colon \G \to \G$ with finite index image is called a \underline{renormalization} of $\G$. 

\bigskip

 $\G$ is said to be \underline{renormalizable} if it admits a renormalization.

\medskip

$\G$ is also called a \underline{finitely non-co-Hopfian} group.

\medskip
 
{\bf Fact:} $M$ is non co-Hopfian $\Leftrightarrow$ $\pi_1(M,x)$ is renormalizable.

\medskip

{\bf Questions:} 

{\bf 1.} What finitely-generated groups are renormalizable?

{\bf 2.} What are the invariants of renormalization maps?
 
 
  \vfill   

}

 

  \frame % 1
{
  \frametitle{}
{\bf Irreducibility:}
\medskip

Let $\vp \colon \G \to \G$ be a renormalization. Recursively define a descending chain of subgroups 
$\G_{\ell +1} = \vp(\G_{\ell})$ for $\ell \geq 0$, so 
$$\G \equiv \G_0 \supset \G_1   \supset \G_2   \supset \cdots $$

 

Let $\cG_{\vp} = \{\G_{\ell} \equiv \vp^{\ell}(\G) \mid \ell \geq 0\}$ be the descending chain of subgroups of finite index associated to $\vp$, then  
$$K(\vp) = \bigcap_{\ell \geq 0} \ \G_{\ell}$$
  is called the \underline{kernel} of the chain.

\medskip

 {\bf Definition:} A renormalization $\vp \colon \G \to \G$ is said to be \underline{irreducible} if $K(\vp)$ is the trivial group, and \underline{almost irreducible} if $K(\vp)$ is finite.



 \vfill   

}

  
    
\frame % 1
{


\bigskip

 
\medskip

 

{\bf Definition:}  $\G$ is said to be \underline{strongly scale-invariant} if there is an almost irreducible renormalization for $\G$.

\medskip


This terminology was introduced in

$\bullet$~ {\it Nekrashevych and Pete}, {\bf Scale-invariant groups}, {Groups Geom. Dyn.}, 2011 
 

\bigskip



{\bf Question:}   Is  a strongly scale-invariant group $\G$   virtually nilpotent?

 \medskip
 
 This question is inspired by a celebrated result of Gromov
.
  \vfill   

}

   
      
  \frame % 1
{
  \frametitle{}
{\bf Example: Expanding manifolds} 

Let $M$ be a closed Riemannian manifold. A smooth map $f \colon M \to M$ is \underline{expanding} if there exists some $\lambda > 1$ such that 
$$\| df(\vec{v}) \| \geq \lambda \|\vec{v}\| \quad {\rm for ~ all} ~ x \in M ~ {\rm and} ~ \vec{V} \in T_xM$$
The map $f$ must be a proper covering.

\medskip

{\bf Theorem:} [Franks 1968] If $M$ admits an expanding map, then $\G_0 = \pi_1(M, x_0)$ has polynomial growth rate.

 \medskip
 
{\bf Theorem:} [Gromov 1979] If $\G$ is a finitely generated group with polynomial growth rate, then $\G$ admits a nilpotent subgroup $\Lambda \subset \G$ with finite index. i.e., $\G$ is virtually nilpotent.

 \vfill   

}

    
    
\frame % 1
{

Our work is motivated by a result in
    
     $\bullet$~ \underline{ Van Limbeek}, {\it Towers of regular self-covers and linear endomorphisms of tori}, {\bf Geom. Topol.}, 2018.

 \medskip
 
  {\bf Theorem:} Let $\G$ be a strongly scale-invariant group, with a renormalization $\vp \colon \G \to \G$ such that  $\G_{\ell} = \vp^{\ell}(\G)$ is \underline{normal} in $\G$.
    Then $\G/K(\vp)$ is abelian.
 
 
\medskip

 \bigskip
 {\bf Question:} Is there a weaker assumption than normality for the subgroups $\G_{\ell}$ that yields a solution to the nilpotent question?
 
 \medskip
 
We approach this using ideas from Cantor dynamical systems,
 
   
$\bullet$~  \underline{Hurder, Lukina \& Van Limbeek}, {\it Cantor dynamics of renormalizable groups},  {\bf arxiv:2002.01565}
 
\vfill   

}

 
   \frame % 1
{
  \frametitle{}
{\bf Construction of   Cantor actions}

\medskip

Consider again the Smale solenoid. Fix the integer $m > 1$, so we have an embedding $\vp \colon \mZ \to \mZ$, given by $\vp(k) = m \cdot k$.

\medskip
Then $\G_{\ell} = m^{\ell} \cdot \mZ \subset \mZ$.

 
  Pass to quotient groups and form the inverse limit space  
$$\fX \equiv ~ \varprojlim \ \{ 0 = \mZ/\mZ   \stackrel{m_*}{\longleftarrow} \mZ/m\mZ  \stackrel{m_*}{\longleftarrow} \mZ/m^2\mZ  \stackrel{m_*}{\longleftarrow}  \cdots \}  \ $$
The inverse limit $\fX$ is a Cantor group,     the $m$-adic integers $\widehat{\mZ}_m$.

\medskip

The group $\G = \mZ$ acts by addition on each quotient group $\mZ/m^{\ell} \mZ$.
 
 \medskip
Get dynamical system $\mZ \times \fX \to \fX$ which is $m$-adic odometer.
\vfill   

}

 
   \frame % 1
{
  \frametitle{}



\medskip
Let $\G$ be a finitely generated group.

Let $\cG = \{\G_{\ell} \mid \ell \geq 0\}$ be a group chain, where $\G_0 = \G$ and $\G_{\ell+1} \subset \G_{\ell}$ is a proper subgroup of finite index.

 
\smallskip

   $X_{\ell} = \G/\G_{\ell}$ is a finite set with transitive left $\G$-action.

\smallskip

Inclusion $\G_{\ell +1} \subset \G_{\ell}$ induces a surjection $p_{\ell+1} \colon X_{\ell+1} \to X_{\ell}$. Define
$$\fX ~ \equiv ~ \varprojlim \ \{p_{\ell +1} \colon X_{\ell +1} \to X_{\ell} \mid \ell \geq 0\} \subset \prod_{\ell \geq 0} \ X_{\ell} \ .$$
The product of finite sets is given the Tychonoff topology - cylinder sets generate the topology.

Then $\fX$ is a closed subset, so is a Cantor space with left $\G$-action. 

 
\smallskip
Obtain minimal $\G$-action   $\Phi  \colon \G \times \fX \to \fX$.

Called a \underline{subodometer} by Cortez and Petite.

 

 \vfill   

}

  \frame % 1
{
  \frametitle{}

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

For the ultrametric  metric on $\fX$, the action $\Phi$ is isometric: 

$\bullet$~ $(\fX, \G, \Phi)$ is an \underline{equicontinuous Cantor action}.



\bigskip
{\bf Remark:} A smooth equicontinuous   action on a manifold is analogous to an isometric action.


\medskip
{\bf Remark:} A minimal equicontinuous Cantor action can also be viewed as a group action  on a rooted tree. 


 
 \vfill   

}
 
 
   
    \frame % 1
{
  \frametitle{}

   Let $\Phi \colon \G \times \fX \to \fX$ be an equicontinuous Cantor action.

 This defines a homomorphism $\Phi  \colon \G \to \Homeo(\fX)$

\smallskip

   $\whGamma = \overline{\Phi(\G)} \subset \Homeo(\fX)$ is the closure in uniform topology

\smallskip


{\bf Theorem:} [Ellis, 1969] $\Phi$ equicontinuous implies that   $\whGamma$ is a  profinite group, compact and totally disconnected.

\bigskip

  This result of  Ellis   is the analog in topological dynamics for the method used to study Riemannian pseudogroups in:
   
$\bullet$~  \underline{Andr\'{e} Haefliger} and \underline{\'{E}liane Salem}, {\it Pseudogroupes d'holonomie des feuilletages riemanniens sur des   vari\'{e}t\'{e}s compactes {$1$}-connexes},  in {\bf G\'{e}om\'{e}trie diff\'{e}rentielle ({P}aris, 1986)}, 1988.

   

 \vfill   

}
  
   



   \frame % 1
{
  \frametitle{}

\bigskip


{\bf Lemma:} Let $\vp \colon \G \to \G$ be a renormalization for $\G$ with  associated Cantor action $(\fX_{\vp}, \G, \Phi_{\vp})$.
  Then       $\ker(\Phi_{\vp}) \subset K(\vp)$,
where $\Phi_{\vp} \colon \G \to \whGamma_{\vp}$ is the map to the completion.

\bigskip

{\bf Strategy:}  For $K(\vp)$    finite,  find conditions on renormalization $\vp \colon \G \to \G$ which imply that   $\whGamma_{\vp}$ is a virtually nilpotent   group, and  hence $\G$ is virtually nilpotent.

 
 \vfill   

}
 
 
   
    \frame % 1
{
  \frametitle{}
  
 {\bf Lemma:} $\Phi_{\vp}$  induces an equicontinuous action $\whPhi_{\vp} \colon \whGamma \times \fX_{\vp} \to \fX_{\vp}$.
 
  For a   sequence  
 $\whgamma = \{ \Phi_{\vp}(\gamma_i) \in \Homeo(\fX) \mid i > 0\} \in \whGamma$ which converges in the uniform topology of maps, 
 given $x \in \fX_{\vp}$ set $\whgamma \cdot x = \lim \  \Phi_{\vp}(\gamma_i)(x)$.
 
 \medskip
 
 {\bf Lemma:}   $\Phi_{\vp}$ minimal implies that  $\whGamma_{\vp}$ acts transitively       on $\fX_{\vp}$.
 \medskip
 
  For  $x \in \fX_{\vp}$, 
define the isotropy subgroup 
$$\cD_x = \{\whgamma \in \whGamma_{\vp} \mid \whPhi_{\vp}(\whgamma)(x) = x\}$$  

\medskip

Isomorphism class of $\cD_x$ is independent of choice of $x$ and invariant of isomorphism of actions.

  
 \vfill   

}
  
   



   \frame % 1
{
  \frametitle{}

 

 \smallskip

{\bf Proposition:} $\fX_{\vp}$ is a homogeneous space for $\whGamma_{\vp}$. That is,
 $$\fX_{\vp} \cong \whGamma_{\vp}/\cD_x \quad {\rm as~left} ~ \G-{\rm spaces}$$

 
 \bigskip
  
 {\bf Remark:} If $\G$ is abelian group, then $\cD_x$ is trivial, so $\fX_{\vp}$ is a profinite group and   $\G$ acts on $\fX_{\vp}$  by group multiplication.
 
 \medskip
 
  Say that $(\fX_{\vp}, \G, \Phi_{\vp})$ is a \underline{generalized odometer}.

 

 
  

 \vfill   

}

      
\frame % 1
{
  \frametitle{}
 
 
 {\bf Strategy:} We obtain invariants of the self-embedding $\vp$ by studying the dynamics of the \underline{adjoint action} of $\cD_x$ on $\whGamma_{\vp}$.  

\bigskip
 
First, there is a canonical basepoint for  $(\fX_{\vp}, \G, \Phi_{\vp})$:
   
   \medskip
   
  {\bf Proposition:} There is a rescaling $\lambda_{\vp} \colon X_{\vp} \to X_{\vp}$ whose image $U_1 = \lambda_{\vp}(X_{\vp})$ is a clopen subset of $X_{\vp}$.
Moreover, the action $(X_{\vp}, \G, \Phi_{\vp})$ is conjugate to the restricted action $(U_1, \G_{U_1}, \Phi_{U_1})$.

\medskip

{\it Idea of   proof:}   $\vp$ induces a map of quotients $\ovp \colon \G/\G_{\ell} \to \G_1/\G_{\ell+1}$. This induces the shift map $\lambda_{\vp} \colon X_{\vp} \to U_1 \subset X_{\vp}$.


\bigskip

{\bf Definition:}     $\cD_{\vp} \subset \Homeo(X_{\vp})$ is the isotropy subgroup at the unique fixed-point $x_{\vp}$ of the contraction map $\lambda_{\vp}$.

  \vfill   

}
   

    
    \frame % 1
{
  \frametitle{}

 
The study of invariants for the adjoint action of  $\cD_{\vp}$ on $\whGamma_{\vp}$ leads into analyzing the regularity properties of    Cantor actions. 
\medskip

Let  $\Phi \colon \G \times \fX \to \fX$ be a Cantor action of a countable group $\G$.
 

The action is:

$\star$~     \underline{effective}, or \underline{faithful},  if   $\Phi \colon \G \to \Homeo(\fX)$ has  trivial kernel.
   
$\star$~   \underline{free} if for all $x \in \fX$ and $g \in \G$,   $g \cdot x = x$ implies that $g = e$


$\star$~ \underline {isotropy group} of $x \in \fX$ is  $\G_x = \{ g \in \G \mid g \cdot x = x\}$

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$\star$~  $\Fix(g) = \{x \in \fX \mid g \cdot x = x \}$, and   \underline{isotropy set}
$$ \Iso(\Phi) = \{ x \in \fX \mid \exists ~ g \in \G ~ , ~ g \ne id ~, ~g \cdot x = x    \} = \bigcup_{e \ne g \in \G} \ \Fix(g) $$
 
 

   

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Cantor action $\Phi \colon \G \times \fX \to \fX$  is \underline{topologically free}  if  $\Iso(\Phi) $ is meager in $\fX$ $\Longrightarrow$  $\Iso(\Phi)$ has empty interior. 
\medskip

For $\G$ a countable group, this is a natural hypothesis to impose.

\medskip

However, for a Cantor action $\Phi \colon H \times \fX \to \fX$ where $H$ is not countable, we introduce another definition of regularity.

\bigskip 
 First, recall the topology of Cantor space $\fX$ is generated by   clopen subsets: U is   closed and open.
A non-empty clopen   $U \subset \fX$ is \underline{adapted} if the return times to $U$ form a subgroup:
$$\G_U = \{ g \in \G \mid \Phi(g)(U) = U  \} \subset \G$$


{\bf Lemma:} For   $x \in \fX$ and open $x \in V$, there is adapted $U$ with $x \in U \subset V$.

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  {\bf Definition:}    An action $\Phi \colon H \times \fX \to \fX$, where 
    
 $\bullet$~    $H$ is a topological group and  
 
 $\bullet$~  $\fX$ is a Cantor space   
 
 is  \underline{quasi-analytic}  if for each clopen set $U \subset \fX$,  $g \in H$ 
 \smallskip
 
 
  $\bullet$~ if   $\Phi(g)(U) = U$ and the restriction $\Phi(g) | U$ is the identity map on $U$, 
  then $\Phi(g)$ acts as the identity on all of $\fX$.  

\medskip
For $H$ a countable group, this is equivalent to topologically free.


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{\bf Profinite Actions:}
\medskip

 $\star$~  $\vp \colon \G \to \G$ is a renormalization for $\G$
  
 $\star$~   associated Cantor action $(\fX_{\vp}, \G, \Phi_{\vp})$
  
 $\star$~   induced profinite action $\whPhi_{\vp} \colon \whGamma_{\vp} \times X_{\vp} \to X_{\vp}$ 
  
 \bigskip


 Here are our   key   results:
 
 \bigskip
 
 {\bf Theorem 1:}  The  action  $\whPhi_{\vp}$ is quasi-analytic.

\medskip
 {\bf Corollary 1:}  Let $\whgamma \in \whGamma_{\vp}$.
   The homeomorphism $\whPhi_{\vp} (\whgamma) \colon \fX_{\vp} \to \fX_{\vp}$ is uniquely determined by its restriction to an adapted subset of $\fX$.


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\medskip


{\bf Theorem 2:} A renormalization map $\vp$ induces a contraction map on the closure,
$\whvp \colon \whGamma_{\vp} \to \whGamma_{\vp}$ with open image.

   \bigskip
  
 The proof of Theorem~2 looks   ``obvious'', except that it isn't.  
 
 Here is the issue: 
 
The renormalization  $\vp$   induces a   map
 $\whvp \colon \whGamma_{\vp} \to \Homeo(U_1)$. 
 
We need to show that the maps in the image of $\whvp$ have unique extensions to $\Homeo(X_{\vp})$.

This is exactly what Theorem~1 says is true.



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{\bf Theorem 3:} $\ds \cD_{\vp} =   \bigcap_{n > 0} \ \whvp^n(\whGamma_{\vp}) \subset \whGamma_{\vp}$
 
 \medskip
 
 This connects the discriminant invariant  for a Cantor action, with an invariant  for a contraction profinite group.

\bigskip

The proof of Theorem~3 follows almost directly from the algebraic definition for $\cD_{\vp}$ developed in

$\bullet$~  \underline{Jessica Dyer}, {\it Dynamics of Equicontinuous Group Actions on Cantor Sets}, {\bf Thesis UIC}, 2015.


 
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  Theorems~2 and 3 are applied to show that $\whGamma_{\vp}$ is virtually nilpotent.

\medskip
There is an extensive literature on the structure of profinite groups with an open contraction mapping,  in particular by:

$\star$~   Baumgartner,  Caprace,  Reid,  Wesolek,  Willis, Wilson

\bigskip
 
The following result is based on results of 
 
 Udo Baumgartner \& George Willis, and Colin Reid:




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 \medskip

 {\bf Theorem:}  
 Let $\whvp \colon \whGamma_{\vp} \to \whGamma_{\vp}$ be a  contraction map with open image.  Then there is an isomorphism with a semi-direct product
  $$\whGamma_{\vp} \cong \cN_{\vp} \rtimes \cD_{\vp}$$
\begin{eqnarray*}
\cN_{\vp} & = &  \{ \whg \in \whGamma_{\vp} \mid \lim_{\ell \to \infty} \ \whvp^{\ell}(\whg) = \whe \} \\
\cD_{\vp} & = &  \bigcap_{n > 0} \ \whvp^n(\whGamma_{\vp}) \subset \whGamma_{\vp}
\end{eqnarray*}
Moreover,  the contraction factor $\cN_{\vp}$ is pro-nilpotent.

 \medskip
 

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 We use this structure theorem for contraction maps to show:
 \medskip
 
 {\bf Theorem [HLvL2020]:}
  Let $\vp$ be a renormalization of the finitely generated group $\G$. Suppose that  
$$K(\vp) = \bigcap_{\ell > 0} \vp^{\ell}(\G)  \subset \G \quad , \quad  \cD_{\vp} =   \bigcap_{n > 0} \ \whvp_0^n(\whGamma_{\vp}) \subset \whGamma_{\vp}$$
are both finite groups,  then 

$\bullet$~ $\G$ is  virtually nilpotent, 

$\bullet$~ If both groups  are trivial, then $\G$ is nilpotent.


\bigskip

{\bf Remark:} The normality assumption in Van Limbeek's Theorem is replaced by the assumption that $\cD_{\vp}$ is a finite group.

  
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{\bf Next Steps:}

\medskip


$\star$~ Let $\vp$ be an irreducible renormalization of a finitely generated group $\G$. Show that $\cD_{\vp}$ is  nilpotent, and thus   $\G$ is virtually nilpotent.
\medskip

This is true in all examples calculated.   Need better understanding of closed subgroups of  profinite groups to complete  the proof.

 \bigskip
 
  $\star$~ Develop general ``formula'' for calculating the discriminant invariant $\cD_{\vp}$ for renormalization map $\vp$
  
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{\bf Dynamics:}

\medskip


A  proper self-covering proper   $\phi \colon M \to M$ is called an \underline{endomorphism} in the dynamical systems literature.

\smallskip

$\bullet$~  {\it Michael Shub}, {\bf Endomorphisms of compact differentiable manifolds}, Amer. J. Math., Vol. 91, 1969.

\bigskip

When $\phi$ is an expanding map on $M$, the induced dynamics on the weak solenoid $\cS_{\phi}$ is of hyperbolic type, and   well-studied.
 
  
  
  Let  $\vp = \phi_* \colon \G \to \G$ be the renormalization associated to   $\phi \colon M \to M$ which is not assumed to be expansive. 
  
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{\bf Question 1:} What can be said about the dynamics on the minimal sets of the induced action on $\cS_{\phi}$?
 
 \bigskip
 
{\bf Question 2:}  Suppose the discriminant   $\cD_{\vp}$ is a Cantor group. How does this influence the dynamics of the shift map on $\cS_{\phi}$?
 
  \bigskip
 
{\bf Problem 3:}  Let $\F_{\phi}$ be the Hirsch foliation associated to a proper self-covering $\phi \colon M \to M$. Show   the discriminant group $\cD_{\vp}$ is a Morita equivalence  invariant     of the holonomy pseudogroup of $\F_{\phi}$.


\medskip
 The discriminant group $\cD_{\phi}$ is an example of the phenomenon of shape dynamics discussed in Section 6 of:
 
 $\bullet$~ \underline{Hurder}, {\it Lectures on Foliation Dynamics: Barcelona 2010}, 
 in {\bf Foliations: Dynamics, Geometry and Topology}, Advanced Courses in Mathematics CRM Barcelona, 2014.
 
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 {\scriptsize    
 

{U.~Baumgartner and G.~Willis}, {\it Contraction groups and scales of automorphisms of totally disconnected locally compact groups}, {\bf  Israel J. Math.}, 2004.
 

 {J.~Dyer, S.~Hurder and O.~Lukina},
 {\it The discriminant invariant of Cantor group actions},
 {\bf Topology and Its Applications},  2017.
 
  
 {J.~Dyer, S.~Hurder and O.~Lukina},
 {\it Molino theory for matchbox manifolds},
 {\bf Pac. Jour. Math.}, 2016.
 
 {M.~Gromov},  {\it Groups of polynomial growth and expanding maps}, 
 {\bf Inst. Hautes \'{E}tudes Sci. Publ. Math.},   1981.


   {S.~Hurder and O.~Lukina},
 {\it Wild solenoids},
 {\bf Trans. A.M.S.}, 2019.
 
 
 {S.~Hurder and O.~Lukina},
 {\it Orbit equivalence and classification of weak solenoids},
 {\bf Indiana Univ. Math. J.}, to appear; {arXiv:1803.02098}.


 {S.~Hurder, O.~Lukina and W.~Van Limbeek},
 {\it Cantor dynamics of renormalizable groups},
 {\bf submitted}; {arXiv:2002.01565}.

 {V.~Nekrashevych and G.~Pete},
 {\it Scale-invariant groups},
 {\bf Groups Geom. Dyn.}, 2011.

{C.~Reid},
  {\it Endomorphisms of profinite groups},
  {\bf Groups Geom. Dyn.},   2014.   



{M.~Shub}, 
  {\it Expanding maps}, 
  In {\bf Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968)},  
  {Amer. Math. Soc.}, 1970.


 {W.~Van Limbeek},
 {\it Towers of regular self-covers and linear endomorphisms of tori},
 {\bf Geom. Topol.}, 2018.   


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