% Vienna Zoom Seminar
% June 4, 2020 
 
 
% slides for talk "Cantor dynamics of renormalizable groups"
 
 
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\date[June 4, 2020, 2020]


  
\title{Cantor dynamics of renormalizable groups}


\author{Steve Hurder, UIC\\     Joint work with   \\ Olga Lukina, University of Vienna \\ Wouter Van Limbeek, UIC} 

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

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

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

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

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


   
 \hspace{60pt} \includegraphics[width=0.6\textwidth]{Smale-Williams_Solenoid_Large} 

  
 \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.


\medskip

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

The $n$-torus $\mT^n = \mS^1 \times \cdots \times \mS^1$ is non co-Hopfian.
 
 \medskip
  
   $N$   closed connected manifold, then  $\mS^1 \times N$ is non co-Hopfian.

 \bigskip

Associated to a proper self-map $\Pi \colon M \to M$ we can form a generalized solenoid $\cS_{\Pi}$ as before.
These are a special class of the \underline{weak solenoids} introduced by Chris McCord in 1966.

More generally, non co-Hopfian manifolds  have    applications in dynamical systems, foliation theory, and spectral theory.

\medskip
{\bf Problem:} Characterize the   non co-Hopfian manifolds. 

 \vfill   

}
   
  

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

For the Smale solenoid, given the tower of maps
$$\cS_m ~ \equiv ~ \varprojlim \ \{  \mS^1 \stackrel{\Pi_m}{\longleftarrow} \mS^1  \stackrel{\Pi_m}{\longleftarrow} \mS^1  \stackrel{\Pi_m}{\longleftarrow}  \cdots \} \subset \prod_{\ell \geq 0} \ \mS^1 \ ,$$
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 of dimension $n > 1$.

\medskip

Let $M$ be non co-Hopfian and $\Pi \colon M \to M$ a proper self-covering.

\medskip

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

\medskip

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

 $\star$~    $\Pi_*$ 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 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

 

In the terminology of Benjamini, and Nekrashevych and Pete,  

$\star$~   {\it Scale-invariant groups}, {\bf Groups Geom. Dyn.}, 2011 
\medskip


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

\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 was motivated by the following result from

 $\bullet$~{\it Van Limbeek}, {\bf Towers of regular self-covers and linear endomorphisms of tori}, {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.
    
 \bigskip
 {\bf Question:} Is there a weaker assumption than normality for the subgroups $\G_{\ell}$ that yields a solution to the nilpotent question?
 
 \medskip
 
Our approach uses ideas from Cantor dynamical systems,
 
   
$\bullet$~ {\it Hurder, Lukina \& Van Limbeek}, {\bf Cantor dynamics of renormalizable groups},  {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  ultrametric  metric on $\fX$, the action $\Phi$ is isometric: 

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



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

\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 the action of $\G$ on it is by ``addition''.
 
  Then $(\fX_{\vp}, \G, \Phi_{\vp})$ is a \underline{generalized odometer}.

\bigskip

 

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

 \vfill   

}
  
   
  
    
\frame % 1
{
  \frametitle{}
  
We first observe 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

$\bullet$~  Let  $\cD_{\vp} \subset \Homeo(X_{\vp})$ be 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  $\G_x = \{ g \in \G \mid g \cdot x = x\}$

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

   

 \vfill   

}
  
   

   \frame % 1
{
  \frametitle{}
      
 
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 $x \in U \subset V$.

 \vfill   

}

    \frame % 1
{
  \frametitle{}
  
  {\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.

\medskip

 Here is our key technical result:
 
 {\bf Theorem 1:}  The  action $\whPhi_{\vp} \colon \whGamma_{\vp} \times X_{\vp} \to X_{\vp}$  is quasi-analytic.

\smallskip
 {\bf Corollary:}  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 $\fX$.


 \vfill   

}
  

   
  
 
    
  
 \frame % 1
{
  \frametitle{}

 
Return now to study of $\whGamma_{\vp}$ associated to a renormalization $\vp \colon \G \to \G$.   
Here is our two main (technical) results:

\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

{\bf Theorem 3:} $\ds \cD_{\vp} =   \bigcap_{n > 0} \ \whvp^n(\whGamma_{\vp}) \subset \whGamma_{\vp}$
 
 This connects discriminant invariants for Cantor actions, with invariants for contraction profinite groups.

\medskip

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

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


 
  \vfill   

}


   \frame % 1
{
  \frametitle{}
      
   
  \bigskip
  
 The proof of Theorem~2 looks equally ``obvious'', except that it isn't.  
 Here is the issue: 
 
The renormalization  $\vp$ naturally 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.

\bigskip

  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


 \vfill   

}
 
  
 
 
 

  
    
 
 

  
    
 
    
\frame % 1
{
  \frametitle{}

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

\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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  \frametitle{}


{\bf Basic Problem:}   Let $\vp$ be an irreducible renormalization of the finitely generated group $\G$. Show that $\cD_{\vp}$ is  nilpotent.
\bigskip

This is true in all examples calculated.   Need better understanding of closed subgroups of  profinite groups to prove it in general.

 \bigskip
 
 One expects further properties of renormalizable groups can be obtained from applying results on contraction groups and scales of automorphisms of totally disconnected locally compact groups to the contraction map 
$\whvp \colon \whGamma_{\vp} \to \whGamma_{\vp}$ induced by a renormalization.

 


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  \frametitle{}
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}

 
 
  


\end{document}

