% GTDS Seminar
% March 9, 2020 
 
 
% slides for talk "Cantor dynamics of renormalizable groups"
 
 
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\date[March 9, 2020]


  
\title{Cantor dynamics of renormalizable groups}


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

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

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

$\bullet$~ A compact closed connected manifold $M$ is \underline{non-co-Hopfian}\\ if it admits a \emph{proper} self-covering map
$\Pi \colon M \to M$.

\bigskip

 $\mS^1$ is non-co-Hopfian. Pick an integer $m > 1$ then $\Pi_m \colon \mS^1 \to \mS^1$.
 
 \medskip

Let $N$ be a compact closed connected manifold, then $\mS^1 \times N$ is non-co-Hopfian.

 \medskip

Formulate a condition of ``irreducibility''.

 \vfill   

}


  \frame % 1
{
  \frametitle{}

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
$$\Pi_* \colon \pi_1(M, x_1) \to \pi_1(M,x_0)$$

\medskip

Since $\pi_1(M, x_1) \cong \pi_1(M,x_0)$, the map $\Pi_*$ defines a self-embedding 
$\vp \colon \G_0 \to \G_0$, where $\G_0 \equiv \pi_1(M,x_0)$.  Note that:

\bigskip

 

\medskip
$\bullet$~ $\G_0$ is finitely generated.

 

$\bullet$~ $\vp(\G_0) \subset \G_0$ has finite index.

\medskip
 

 \vfill   

}


  \frame % 1
{
  \frametitle{}


 
{\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 $\G$. 

\medskip

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

\medskip

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

\medskip

For $\G = \mZ^n$ 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.

\bigskip

{\bf Question:} What groups are renormalizable?

 
 \vfill   

}


  \frame % 1
{
  \frametitle{}

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

\medskip

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.

\bigskip

{\bf Problem:} Let $\G$ be a renormalizable group with group chain $\cG_{\vp}$ whose kernel $K(\vp)$ is a finite group. What can be said about $\G$?



 \vfill   

}

 
 
    
\frame % 1
{

  {\bf Theorem:} [Van Limbeek 2018] Let $\G$ be a renormalizable group , and suppose that each subgroup $\G_{\ell} = \vp^{\ell}(\G)$ is normal in $\G$.
  
  Then $\G/K(\vp)$ is abelian.
    
 \bigskip
 
 $\star$~{\it Van Limbeek}, {\bf Towers of regular self-covers and linear endomorphisms of tori}, {Geom. Topol.}, 2018.

\medskip

The assumption that the images $\vp^{\ell}(\G)$ are all normal is very strong, and not satisfied for many renormalizable groups.
   
\vfill   

}


   \frame % 1
{
  \frametitle{}
  
  
{\bf Example: Heisenberg group}

$$\G  \ = \ \left\{ [x,y,z] = \left[ \begin{array}{ccc} 1 & x & z \\ 0 & 1 & y \\ 0 & 0 & 1 \end{array} \right] \mid x,y,z \in \mZ \right\} $$

  
  
\bigskip
 
 $\bullet$~   $p,q > 1$. Define  $\vp[x,y,z] = [px,qy,pqz]$

$\bullet$~   $\G_{\ell}  =  \left\{  [p^{\ell} x, q^{\ell} y, p^{\ell}q^{\ell} z] \mid  x,y,z \in \mZ  \right\}$

 $\bullet$~  $\G_{\ell}$ is not normal in $\G$.

 $\bullet$~  $K(\vp)$ is trivial group.


 \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
{\small

{\it Idea of   proof:} For some $\lambda_{min} > 1$, the $\ell$-th power of the expanding map expands the inner radius of disks in the universal covering $\widetilde{M}$ at a rate bounded below by  $\lambda_{min}^\ell$. 
For some $\lambda_{max} \geq \lambda_{min}$, it expands their volumes at a rate bounded above by $\lambda_{max}^{n \ell}$. Thus the growth rate is bounded by a polynomial of degree $n = \dim M$.
}
 \vfill   

}


  \frame % 1
{
  \frametitle{}


{\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.

\bigskip
 
{\small  
The idea of the proof of this celebrated result is not so simple. Gromov shows that there is a representation of $\G$ into a linear group, and then invokes the ``Tits alternative'' for linear groups.}

\bigskip

Let $f \colon M \to M$ be an expanding map. Choose $x_0 \in M$ and set $\G_0 = \pi_1(M, x_0)$. 
Then   $\vp = f_* \colon \G_0 \to \G_0$ is a renormalization of $\G_0$ with $K(\vp)$ the trivial group.

 


 \vfill   

}


  \frame % 1
{
  \frametitle{}

 
Gromov's Theorem motivates the following:

\medskip

{\bf Conjecture:}   $\G$ a finitely generated group with a renormalization $\vp$ such that $K(\vp)$ is trivial. Then $\G_0$ is virtually nilpotent.

\medskip

In the work of Nekrashevych \& Pete [2011] they called a chain 
$\cG = \{\G_0 \supset \G_1 \supset \G_2 \supset \cdots\}$ with $K(\vp)$ finite a \underline{strong scale}, and asked if this implies that $\G_0$ must be virtually nilpotent?

\medskip
We give a positive solution to a modified version of this conjecture:

$\star$~ {\it Hurder, Lukina \& Van Limbeek}, {\bf Cantor dynamics of renormalizable groups},  {arxiv:2002.01565}

 \medskip
 
{\small 
 Idea is to replace the metric space rescaling methods of Gromov, with induced actions on Cantor spaces, and then use the structure theory for profinite groups to again reduce the problem to linear algebra.	
 }
	 \vfill   

}

\section{Cantor Actions}

  \frame % 1
{
  \frametitle{}
{\bf Some generalities:}

\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

For each $\ell > 0$,  $X_{\ell} = \G/\G_{\ell}$ is a finite set with left $\G$-action.

\smallskip

Inclusion $\G_{\ell +1} \subset \G_{\ell}$ induces a surjection $p_{\ell+1} \colon X_{\ell+1} \to X_{\ell}$ so
$$\fX ~ \equiv ~ \varprojlim \ \{p_{\ell +1} \colon X_{\ell +1} \to X_{\ell} \mid \ell \geq 0\} \subset \prod_{\ell \geq 0} \ X_{\ell}$$
is a Cantor space with left $\G$-action. 

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

 
  

 \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 
 $$\dX(x,y) < \delta ~ \Longrightarrow ~ \dX(\Phi(g)(x) , \Phi(g)(y)) < \e \quad {\rm for ~ all } ~ g \in \G   .$$
   
\smallskip

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

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

\bigskip
Our study of these actions began with the work

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


\medskip

 
 \vfill   

}

   \frame % 1
{
  \frametitle{}

 Three approaches to the subject:
 
 \medskip
 
$\bullet$~ Equicontinuous Cantor actions

\qquad Dynamical systems approach

\medskip

$\bullet$~ Group actions on rooted trees

\qquad Geometric group theory approach

\medskip

$\bullet$~ Clopen subset chains for profinite groups

\qquad Representations into finite groups

\medskip

 \vfill   

}

   \frame % 1
{
  \frametitle{}

Topology of Cantor space $\fX$ is generated by   clopen subsets: \\U is   closed and open.
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 Proposition:}
A Cantor action $\Phi \colon \fX \to\fX$ is equicontinuous if and only if all orbits are finite for  its action on the set 
$$\cCO = \{ U \subset \fX \mid U ~ {\rm is ~ clopen} \}$$
{\bf Corollary:} Given   $x \in \fX$ and open $x \in V$, there is adapted $U$ with $x \in U \subset V$.
  
  $\bullet$~ Action of $\G$ on coset space $X_U = \G/\G_U$ is a finite approximation to action $\Phi$, so gives a linear model.

 \vfill   

}

   
    \frame % 1
{
  \frametitle{}

$\bullet$~  Equicontinuous Cantor action $\Phi \colon \G \times \fX \to \fX$,  same  as homomorphism $\Phi  \colon \G \to \Homeo(\fX)$

\smallskip

$\bullet$~  $\whGamma = \overline{\Phi(\G)} \subset \Homeo(\fX)$ - closure in uniform topology

\smallskip


$\bullet$~  $\whGamma$ is separable profinite group, compact and totally disconnected

\smallskip


$\bullet$~ $\whPhi \colon \whGamma \times \fX \to \fX$ is transitive equicontinuous action

\smallskip


$\bullet$~ For $x \in \fX$, define $\cD_x = \{\whg \in \whGamma \mid \whPhi(\whg)(x) = x\}$

\smallskip


$\bullet$~ $\cD_x$ is called the \underline{discriminant} of the action $\Phi$

\smallskip


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

 
 

 \vfill   

}
  
   
   
   
   
 \frame % 1
{
  \frametitle{}

{\bf Dyer formula for $\cD$}
 $$
C_{\ell} ~ = ~ \bigcap_{g \in \G} ~ g \ \G_{\ell} \ g^{-1} ~ \subset \G_{\ell}  .
$$
is  the largest normal subgroup (the \emph{core}) of   $\G_{\ell}$.

 For all $\ell \geq 1$,  $C_{\ell +1} \subset C_{\ell}$ and $C_{\ell}$ has finite index in $\G$.

Quotient group  $Q_{\ell} = \G/C_{\ell}$ with identity element $e_{\ell} \in Q_{\ell}$. 

There are natural quotient maps $q_{\ell+1} \colon Q_{\ell +1} \to Q_{\ell}$.

Form the inverse limit Cantor group
$$
\whGamma_{\infty} = \varprojlim \ \{q_{\ell +1} \colon Q_{\ell +1} \to Q_{\ell} \mid \ell \geq 0\} ~ .
$$

 
 \vfill   

}
  
   
   
   
   
 \frame % 1
{
  \frametitle{}


{\bf Theorem:} [Dyer 2015]  
There is a natural isomorphism $\whtau \colon \whGamma  \to \whGamma_{\infty} $ which identifies the discriminant group $\cD_x$ with the inverse limit  
$$
\cD_x \cong \cD_{\infty}  = \varprojlim \ \{q_{\ell +1} \colon \G_{\ell +1}/C_{\ell+1} \to \G_{\ell}/C_{\ell} \mid \ell \geq 0\} \subset \whGamma_{\infty}~ .
$$
 
\bigskip

For each $\ell > 0$, have subgroup  ~ $D_{\ell} \equiv \G_{\ell}/C_{\ell} \subset  \G/C_{\ell} \equiv Q_{\ell}$ 
\medskip

 $Q_{\ell}$ is a finite approximation to $\whGamma$ 
 
 \medskip
 
 $D_{\ell}$ is the fiber of the map $Q_{\ell} \to X_{\ell} = \G/\G_{\ell}$ 
 
 \medskip
 
 $D_{\ell}$ is \underline{totally not normal}.
 \medskip
 
 Quotient by $D_{\ell}$ breaks the   symmetry of $Q_{\ell}$ 


 \vfill   

}
  
   
   
   
   
 \frame % 1
{
  \frametitle{}
For an equicontinuous Cantor action $\Phi \colon \G \times \fX \to \fX$

 \begin{enumerate}
\item $\cD$ 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  $\cD$ for a Cantor action by   a torsion-free, finite index subgroup of  $ {\bf SL}(n,\mZ)$, $n \geq 3$.
\smallskip

$\star$~ {\it Dyer, Hurder \& Lukina}, {\bf Molino theory for matchbox manifolds}, Pacific J. Math. 2017


 \smallskip
 
\item  $\cD$ can be wide-ranging for arboreal representations of absolute Galois groups of number fields and function fields.

\smallskip

$\star$~{\it Lukina}, {\bf Arboreal Cantor actions}, {Jour. L. M. S.}, 2019

$\star$~{\it Lukina}, {\bf Galois groups and Cantor actions}, {arXiv:1809.08475}

  




 \end{enumerate}

   \vfill   

}


    
\frame % 1
{
  \frametitle{}
  
We apply the above ideas to our situation.    

$\bullet$~ Let   $\vp \colon \G \to \G$ be a renormalization.

$\bullet$~  Let  $\cG_{\vp} = \{\G_{\ell} = \vp^{\ell}(\G) \mid \ell \geq 0\}$ be the associated group chain.

$\bullet$~  Let  $\Phi_{\vp} \colon \G \times X_{\vp} \to X_{\vp}$ be the  associated Cantor action.

$\bullet$~  Let  $\cD_{\vp} \subset \Homeo(X_{\vp})$ be the isotropy at unique fixed-point $x_{\vp}$

   
   \medskip
   
  {\bf Proposition:} There is a rescaling $\lambda_{\vp} \colon X_{\vp} \to X_{\vp}$ whose image $U_1 = \lambda_{\vp}(X_{\vp})$ is 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 $\lambda_{\vp} \colon X_{\vp} \to U_1 \subset X_{\vp}$.

  \vfill   

}
  

    
   \frame % 1
{
  \frametitle{}
  
  
{\bf Heisenberg group actions continued:}

$$\G  \ = \ \left\{ [x,y,z] = \left[ \begin{array}{ccc} 1 & x & z \\ 0 & 1 & y \\ 0 & 0 & 1 \end{array} \right] \mid x,y,z \in \mZ \right\} $$

 {\bf Example 1:}
 
 $\bullet$~  Let $m > 1$. Define  $\vp_1[x,y,z] = [mx, my, m^2z]$

$\bullet$~   $\G_{\ell}  =  \left\{  [m^{\ell} x, m^{\ell} y, m^{2 \ell} z] \mid  x,y,z \in \mZ  \right\} $


 $\bullet$~ $\cD_{\vp_1}$ is the trivial group.

$\bullet$~  $X_{\vp_1}$ is a profinite Heisenberg group, 


$\bullet$~ $\Phi_{\vp_1}$ is left multiplication by $\G$.


\bigskip

  \vfill   

}
    
     
   \frame % 1
{
  \frametitle{}


{\bf Example 2:}
 
 $\bullet$~   $p,q > 1$   distinct primes. Define  $\vp_2[x,y,z] = [px,qy,pqz]$

$\bullet$~   $\G_{\ell}  =  \left\{  [p^{\ell} x, q^{\ell} y, p^{\ell}q^{\ell} z] \mid  x,y,z \in \mZ  \right\} $


 $\bullet$~ $\cD_{\vp_2}\cong \widehat{\mZ}_{q} \times \widehat{\mZ}_{p}$.



$\bullet$~  $\ds X_{\vp_2}  \cong    \{[x,y,z] \mid x \in \widehat{\mZ}_{p} \ , \ y \in \widehat{\mZ}_{q} \ ,  \ z \in \widehat{\mZ}_{pq} \}$.


$\bullet$~ $\Phi_{\vp_2}$ is left multiplication by $\G$.
\bigskip

So for $p,q$ and $p',q'$ distinct pairs of primes, the Cantor actions associated to the corresponding renormalizations are not conjugate. 

They are not even continuously orbit equivalent.
\medskip

{\bf Conclusion:}
Renormalizable groups give rise to a large class of inequivalent minimal equicontinuous Cantor actions.
 
 \vfill   

}
   
\section{Profinite actions}
  

    
  
 \frame % 1
{
  \frametitle{}

We want to apply the Cantor action approach to the question   whether a renormalizable group must be virtually nilpotent. This requires new techniques from the profinite literature.

First, there is a map induced on closures:

 $\bullet$~  $\whPhi_{\vp} \colon \whGamma_{\vp} \times X_{\vp} \to X_{\vp}$ action of closure $\whGamma_{\vp} \subset \Homeo(X_{\vp})$.

\medskip

Here is our main result:


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

\medskip

This has many implications;  there is an extensive literature of the structure of profinite groups with a contraction mapping,   by:

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

 
 

 
  \vfill   

}

    \frame % 1
{
  \frametitle{}

 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 $\Homeo(X_{\vp})$.

\bigskip 
Take detour into regularity properties of   action $\Phi \colon \G \times \fX \to \fX$

$\bullet$~     \underline{effective}, or \underline{faithful},  if   $\Phi_0 \colon \G \to \Homeo(\fX)$ has  trivial kernel.
   
$\bullet$~   \underline{free} if for all $x \in \fX$ and $g \in \G$,   $g \cdot x = x$ implies that $g = e$

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


$\bullet$~  $\Fix(g) = \{x \in \fX \mid g \cdot x = x \}$, 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
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  \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. 

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

\bigskip

 
  
  {\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.
 \vfill   

}
  

  

     
   
  
   
 
     
\frame % 1
{
  \frametitle{}

Let $G$ be the \underline{profinite completion} of $\G$.

$G$ is the inverse limit over the  partially ordered set of   finite-index normal subgroups of $\G$.

 $G$ acts on $X_{\vp}$ via the projection $G \to \whGamma_{\vp}$.

\medskip

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

\medskip

 
{\it Idea of proof.} Suppose  $\whh \in G$ and $\whPhi_{\vp}(\whh)$ fixes  $U \subset X_{\vp}$.

By conjugating $\whh$ we can assume $U_{\ell} \subset U$ for some $\ell > 0$.

 

  \vfill   

}
  
\frame % 1
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  \frametitle{}

By universality, $\vp$ extends to a map $\whvp \colon G \to G$ with open image.

 Get a chain of clopen subgroups $\whV_{\ell} = \whvp(G)$ for $\ell \geq 0$.
 
 Then $\whV_{\ell}$ projects to clopen set $\whU_{\ell} \subset \whGamma_{\vp}$
 
  $\whU_{\ell}$ is stabilizer of   $U_{\ell} = \lambda_{\vp}^{\ell}(X_{\vp})$.
So for all $x \in X_{\vp}$, 
$$\whPhi_{\vp}(\whh) \circ \lambda_{\vp}^{\ell}(x) = \lambda_{\vp}^{\ell}(x)  $$
 
  By some technical wizardry, we use this relation to show that $\whPhi_{\vp}(\whh)$ acts as the identity on all of $X_{\vp}$.
  
  This shows that   the action $\whPhi_{\vp} \colon G \times X_{\vp} \to X_{\vp}$  is quasi-analytic.
 
 {\bf Corollary:}  The action $\whPhi_{\vp} \colon \whGamma_{\vp}  \times X_{\vp} \to X_{\vp}$  is quasi-analytic.
 
 Main result then follows.
 
\vfill   

}


  
 \section{Applications}   
 
    
\frame % 1
{
   \frametitle{}
 
{\bf Summary:}  Given a renormalization  $\vp \colon \G \to \G$, there is 

$\bullet$~ a group chain $\cG_{\vp} = \{\G_{\ell} = \vp^{\ell}(\G) \mid \ell \geq 0\}$ 

$\bullet$~ a Cantor action $\Phi_{\vp} \colon X_{\vp} \to X_{\vp}$
 
 $\bullet$~ a contraction map  $\whvp \colon \whGamma_{\vp} \to \whGamma_{\vp}$ with open image.

\bigskip

We then have a result from previous works with Lukina:

{\bf Theorem:} $\ds \cD_{\vp} =   \bigcap_{n > 0} \ \whvp_0^n(\whGamma_{\vp}) \subset \whGamma_{\vp}$
 
 This connects the invariants for Cantor actions, with invariants for contraction profinite groups.
 
  
  
\vfill   

}


  
    
 
    
\frame % 1
{
  \frametitle{}

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

 
 \medskip
 
 {\bf Theorem:}
  Let $\vp$ be a renormalization of the finitely generated group $\G$. 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} \ \vp^{\ell}(\whg) = \whe \} \\
\cD_{\vp} & = &  \bigcap_{n > 0} \ \whvp_0^n(\whGamma_{\vp}) \subset \whGamma_{\vp}
\end{eqnarray*}
Moreover,  the contraction factor $\cN_{\vp}$ is pro-nilpotent.

 \medskip
 

\vfill   

}


  
    
 
    
\frame % 1
{
 
 {\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

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.
  
\vfill   

}


    
    
 
    
\frame % 1
{
  \frametitle{}

{\bf Basic Problem:}   Let $\vp$ be a renormalization of the finitely generated group $\G$. If    $K(\vp)$ is finite,  show that $\cD_{\vp}$ must be virtually nilpotent.
\smallskip

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

\bigskip

{\bf Future Work:} Let $\G$ be a renormalizable group. Characterize the space of renormalizations of $\G$.
\smallskip

This is related to questions about research into properties of open subgroups of profinite groups, and scale invariant groups of Benjamini, and Nekrashevych \& Pete.
\vfill   

}


    
    
 
    
\frame % 1
{
  \frametitle{}
 {\scriptsize    
 
 [BW2004] 
{U.~Baumgartner and G.~Willis}, {\it Contraction groups and scales of automorphisms of totally disconnected locally compact groups}, {\bf  Israel J. Math.}, 2004.
  
 [DHL2015] {J.~Dyer, S.~Hurder and O.~Lukina},
 {\it The discriminant invariant of Cantor group actions},
 {\bf Topology and Its Applications},  2017.
 
  
[DHL2016] {J.~Dyer, S.~Hurder and O.~Lukina},
 {\it Molino theory for matchbox manifolds},
 {\bf Pac. Jour. Math.}, 2017.
 
[G1981] {M.~Gromov},  {\it Groups of polynomial growth and expanding maps}, 
 {\bf Inst. Hautes \'{E}tudes Sci. Publ. Math.},   1981.


 [HL2017]  {S.~Hurder and O.~Lukina},
 {\it Wild solenoids},
 {\bf Trans. A.M.S.}, 2019.
 
 
[HL2018] {S.~Hurder and O.~Lukina},
 {\it Orbit equivalence and classification of weak solenoids},
 {\bf Indiana Univ. Math. J.}, to appear; {arXiv:1803.02098}.


[HLvL2020] {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.


[vL2020] {W.~Van Limbeek},
 {\it Towers of regular self-covers and linear endomorphisms of tori},
 {\bf Geom. Topol.}, 2018.   


 }  

 
 
 

}

 
 
  


\end{document}

