% Midwest Dynamical Systems
% Northwestern University 
% November 13, 2021 - final
 
 
% slides for talk "Arboreal Dynamics"
 
 
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\date[February 18, 2020]


  
\title{Arboreal Dynamics \\  \phantom{.} \\\small{Midwest Dynamical Systems}\\Northwestern University\\\small{November 13, 2021}}


\author{\small{Steve Hurder, University of Illinois at Chicago\\   \phantom{.} \\ {\footnotesize  Joint works with}\\  \phantom{.}  \\ Olga Lukina, University of Vienna \\ Wouter Van Limbeek, University of Illinois at Chicago}} 

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

 \section{MWDS@50}


  \frame % 1
{
  \frametitle{}

\begin{center}
{\underline{Fifty Years of Midwest Dynamics}}
\end{center}
First meetings were  small conferences organized by 

 $\bullet$~ [Spring 1970]~ Ken Meyer, Sheldon Newhouse, Clark Robinson, Don Saari,  Carl Simon,  Bob Williams
 

 $\bullet$~ [Spring 1971]~  plus   John Franks, Charles Conley,  Joel Robbin 

\bigskip

Steve Batterson gives a history of these early days of   MWDS   in:
 
  {\it Steve {S}male and the prehistory of the MWDS {C}onference},  2009.

 \bigskip
 
 We will give two applications of arboreal dynamics today, one from 50 years ago, and the other much more recent about Galois actions.

 \vfill   

}


  \frame % 1
{
  \frametitle{}

Recall a result  from ``The Beginning'' by Mike Shub:

{\bf Theorem:} $M$ a compact manifold without boundary. If there exists expansive diffeomorphism $\phi \colon M \to M$, then    $\G = \pi_1(M,x)$ admits a self-embedding  $\vp \colon \G \to \G$ and $\G$ has polynomial growth type.

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

\bigskip

 $\star$~  C.~Pugh \& M.~Shub, {\it Ergodicity of Anosov actions}, \\{\bf Invent. Math.}, 1972.
 
  $\star$~ {M.~Hirsch, C.~Pugh \& M.~Shub}, {\it Invariant manifolds},  \\
 {\bf Lecture Notes in Mathematics, Vol. 583}, 1977.
 
 
 \vfill   

}

 
  \frame % 1
{
  \frametitle{}
  

{\bf Definition:}  A finitely generated group $\G$ is said to be \underline{strongly scale-invariant} if there exists    a proper   finite index inclusion  $\phi \colon \G \to \G$, with finite intersection in the limit. That is, 
  for  the descending chain of  proper subgroups $\G_0 \supset \G_1  \supset \cdots$  
$$\G_0 = \G , \G_1 = \phi(\G_0) , \ldots , \G_{\ell +1} = \phi(\G_{\ell}), \ldots $$
then also require   that $\cap_{\ell > 0} ~ \G_{\ell}$ is a finite group. 

\bigskip

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

 
  \bigskip
The fundamental group of a manifold   in Shub's Theorem is strongly scale invariant, as the uniformly expanding condition forces 
$\cap_{\ell > 0} ~ \G_{\ell}$ to be the trivial group. 

 \vfill   

}

 
  \frame % 1
{
  \frametitle{}
  
  We then have the celebrated result of Gromov:
  
{\bf Theorem:} Let $\G$ be a finitely generated group with polynomial growth type, then $\G$ is \emph{virtually nilpotent}; that is, $\G$ has a  finite-index  nilpotent subgroup.

\medskip

$\star$~   {M.~Gromov},  {\it Groups of polynomial growth and expanding maps},
 {\bf Inst. Hautes \'{E}tudes Sci. Publ. Math.},   1981.

\bigskip
 
 
{\bf Conjecture:} [Benjamini 1996]  If $\G$ is strongly scale-invariant, then it contains a finitely-generated nilpotent subgroup of finite index.

 
\smallskip

Is this true, without assuming the existence of an expanding map?

 \vfill   

}

 
  \frame % 1
{
  \frametitle{}

 \bigskip

 \emph{Arboreal dynamics} \&   results for \emph{profinite groups} are used to  show:

\medskip




 {\bf Theorem:}
  Let $\G$   be  strongly scale-invariant, and assume   that the induced map on the profinite completion is strongly scale-invariant. Then  $\G$ is  virtually nilpotent. 

 
\bigskip

 $\star$~ {S.~Hurder, O.~Lukina and W.~Van Limbeek},
 {\it Cantor dynamics of renormalizable groups},
 {\bf Groups Geom. Dyn.}, to appear, 2021; {arXiv:2002.01565}.

\bigskip

First review  some basics of arboreal actions.
  \vfill   

}

 
 
 \section{Arboreal actions}

  


  \frame % 1
{
  \frametitle{}

{\bf Arboreal actions:}
\medskip

$\bullet$ ~  $\cT$ a connected tree with root vertex $v_0$ and bounded valence

$\bullet$ ~  Level of vertex $v$    is number of edges in unique path  to $v_0$

$\bullet$ ~ $V_n$ denotes the set of vertices at level $n$; $V_0 = \{v_0\}$


\bigskip

$\bullet$ ~ $\G$   a   countable group

$\bullet$ ~  $\phi \colon \G \times \cT \to \cT$ action preserving root vertex $v_0$

$\bullet$ ~  $\phi_n \colon \G \to  {\rm Perm}(V_n)$, $G_n = {\rm image}(\phi_n)$


$\bullet$ ~   $\G_{v_n} = \{ \gamma \in \G \mid \phi_n(\gamma)(v_n) = v_n\}$ for choice of $v_n \in V_n$  


\medskip

{\bf Definition:} $\phi$ is \underline{transitive} if   for each  $n$, 
$\phi_n$ acts transitively on the vertices at level $n$.

 \vfill   

}

 
 
  \frame % 1
{
  \frametitle{}

 

\begin{figure}[ht]
\begin{center}
 rooted tree and chain of isotropy groups
 
\begin{tikzpicture}[scale=0.4]
\filldraw[black] (4,4) circle (4pt) node[anchor=south]{root vertex $v_0$};

\draw[blue, thick] (4,4) -- (-2,0);
\draw[blue, thick] (4,4) -- (10,0);

\filldraw[black] (-2,0) circle (4pt) node[anchor=east]{ Level 1  \qquad \quad$\G_{v_1}$};
\filldraw[black] (10,0) circle (4pt);

\draw[red, thick] (-2,0) -- (-5,-3);
\draw[red, thick] (-2,0) -- (-2,-3);
\draw[red, thick] (-2,0) -- (1,-3);

\filldraw[black] (-5,-3) circle (4pt)  node[anchor=east]{Level 2 \quad \quad};
\filldraw[black] (-2,-3) circle (4pt)  node[anchor=west]{ $\G_{v_2}$};
\filldraw[black] (1,-3) circle (4pt);

\draw[red, thick] (10,0) -- (7,-3);
\draw[red, thick] (10,0) -- (10,-3);
\draw[red, thick] (10,0) -- (13,-3);

\filldraw[black] (7,-3) circle (4pt);
\filldraw[black] (10,-3) circle (4pt);
\filldraw[black] (13,-3) circle (4pt);

\draw[green, thick] (-5,-3) -- (-6,-6);
\draw[green, thick] (-5,-3) -- (-4,-6);

\draw[green, thick] (-2,-3) -- (-3,-6);
\draw[green, thick] (-2,-3) -- (-1,-6);

\draw[green, thick] (1,-3) -- (0,-6);
\draw[green, thick] (1,-3) -- (2,-6);


\draw[green, thick] (7,-3) -- (6,-6);
\draw[green, thick] (7,-3) -- (8,-6);

\draw[green, thick] (10,-3) -- (9,-6);
\draw[green, thick] (10,-3) -- (11,-6);

\draw[green, thick] (13,-3) -- (12,-6);
\draw[green, thick] (13,-3) -- (14,-6);

\filldraw[black] (-6,-6) circle (4pt) node[anchor=east]{Level 3 \quad};
\filldraw[black] (-4,-6) circle (4pt);
\filldraw[black] (-3,-6) circle (4pt)  node[anchor=west]{ $\G_{v_3}$};
\filldraw[black] (-1,-6) circle (4pt);
\filldraw[black] (0,-6) circle (4pt);
\filldraw[black] (2,-6) circle (4pt);

 \filldraw[black] (6,-6) circle (4pt);
\filldraw[black] (8,-6) circle (4pt);
\filldraw[black] (9,-6) circle (4pt);
\filldraw[black] (11,-6) circle (4pt);
\filldraw[black] (12,-6) circle (4pt);
\filldraw[black] (14,-6) circle (4pt);

\draw[dotted,thick] (-6,-6) -- (-6,-8);
\draw[dotted,thick] (-4,-6) -- (-4,-8);
\draw[dotted,thick] (-3,-6) -- (-3,-8);
\draw[dotted,thick] (-1,-6) -- (-1,-8);
\draw[dotted,thick] (0,-6) -- (0,-8);
\draw[dotted,thick] (2,-6) -- (2,-8);
\draw[dotted,thick] (6,-6) -- (6,-8);
\draw[dotted,thick] (8,-6) -- (8,-8);
\draw[dotted,thick] (9,-6) -- (9,-8);
\draw[dotted,thick] (11,-6) -- (11,-8);
\draw[dotted,thick] (12,-6) -- (12,-8);
\draw[dotted,thick] (14,-6) -- (14,-8);

 \filldraw[black] (-3,-10) circle (4pt) node[anchor=north]{ $x_{\sigma} \in \fX$};

%%%%
%%%%
\end{tikzpicture}
\label{fig:lochness}
\end{center}
\end{figure} 


 \vfill   

}

  
  
  \frame % 1
{
  \frametitle{}

 
Boundary of $\cT$ is totally disconnected space $\fX$

 $\bullet$~ Infinite path $\sigma$  in $\cT$ $\leftrightarrow$ a sequence of vertices $\{v_0, v_1, v_2, \ldots \}$ where
  $v_{n} \in V_n$   \&  each pair $\{v_n , v_{n+1}\}$   connected by an edge  

 $\bullet$~ $\sigma$ defines point $x_{\sigma} \in \fX$ of the boundary of $\cT$

\medskip
 
Tree action $\phi$     induces boundary action 
 $\Phi \colon \G \times \fX \to \fX$

 
\bigskip

{\bf Proposition:} Arboreal action $\phi$ is transitive then  $\Phi \colon \G \times \fX \to \fX$ is minimal equicontinuous Cantor action.

\medskip

   
   

 \vfill   

}

  
 
\frame % 1
{

{Conversely:}  

$\bullet$~ $\Phi \colon \G \times \fX \to \fX$ an equicontinuous minimal action 

$\bullet$~  clopen set $U \subset \fX$ is \emph{adapted} if  set of ``return times'' is a group
$$\G_U = \{\gamma \in \G \mid \Phi(\gamma)(U) \cap U \ne \emptyset\}$$ 

$\bullet$~  Collection of adapted sets  $\cA$ are partially ordered by inclusion

$\bullet$~  Each $U \in \cA$ corresponds to a vertex in tree  $\cT_{\cA}$



$\bullet$~  Action on $\cA$ gives an arboreal action on $\cT_{\cA}$.
 
 
 \vfill   

}



  
    \section{Models}
 
\frame % 1
{

  \underline{Group Chain Model}
  
  \smallskip
Choose a path $\tau$ in $\cT$ with sequence of vertices $\{v_0, v_1, \ldots\}$

Get chain of finite index subgroups $\G_{\ell} = \G_{v_{\ell}}$ where 
$$\G = \G_0 \supset \G_1   \supset \G_2 \supset \cdots$$


$\G$ acts on finite set $X_{\ell} = \G/\G_{\ell} \cong V_{\ell}$, so $\G$ acts on infinite product of $X_{\ell}$ and so on the Cantor space
$$\fX_{\infty} =  \varprojlim  ~ \{p_{\ell+1} \colon X_{\ell+1} \to X_{\ell} \mid \ell \geq 0\} \subset \prod_{\ell > 0} ~ X_{\ell}$$

The action  $\Phi_{\infty} \colon \G \times \fX_{\infty} \to \fX_{\infty}$  is conjugate to the boundary action $\Phi \colon \G \times \fX \to \fX$ for the tree model.
  
 \vfill   

}

 
\frame % 1
{
 
\bigskip

{Conversely:}  

$\bullet$~ given  group chain $\cG = \{G_0 \supset \G_1 \supset \G_2 \cdots \}$ 

$\bullet$~    vertices $V_n = \G/\G_n$ with 
 quotient map $\G/\G_{n+1} \to \G/\G_n$ 
 
$\bullet$~   coset inclusion  defines edge from $V_n$ to $V_{n+1}$

$\bullet$~  obtain tree $\cT_{\cG}$ with $\G$ action.



 \vfill   

}

\frame % 1
{
\underline{Profinite Model}

\smallskip

Let $C_{\ell} \subset \G_{\ell}$ be \emph{normal core}, largest normal subgroup: 

$\G/C_{\ell} \cong G_{\ell}\subset {\rm Perm}(V_{\ell})$ is finite group with order $m_{\ell}$
$$\G_{\ell+1} \subset \G_{\ell}~ \Longrightarrow ~ C_{\ell+1} \subset C_{\ell}  ~ \Longrightarrow~ p_{\ell+1} \colon G_{\ell+1} \to G_{\ell}~{\rm is \ onto}$$
$$\fG(\Phi) = \varprojlim  ~ \{p_{\ell+1} \colon G_{\ell+1} \to G_{\ell} \mid \ell \geq 0\} \subset \prod_{\ell > 0} ~ G_{\ell}$$
 
 

For each $\ell > 0$ have $G_{\ell} = \G/C_{\ell}$ and $X_{\ell} = \G/\G_{\ell}$ so get 


action $G_{\ell} \times X_{\ell} \to X_{\ell}$ with isotropy group $D_{\ell} = \G_{\ell}/C_{\ell}$.

Induces action $\whPhi \colon \fG(\Phi) \times \fX_{\infty} \to \fX_{\infty}$, and restriction to image of $\G$ in $\fG(\Phi)$ is conjugate to boundary action.


 \vfill   

}

  
 
\frame % 1
{

  
  
 The profinite action includes the action of the isotropy subgroup
 $$D_{\infty} \cong \{ \whg \in \fG(\Phi) \mid \whPhi(\whg)(x_{\infty}) = x_{\infty} \}$$
  This is a ``hidden aspect'' of an arboreal action -- the action can be free yet the group $D_{\infty}$ is non-trivial, even be a Cantor group.
 
 \bigskip
 
$\bullet$~  $\fX_{\infty} \cong \fG(\Phi)/D_{\infty} $ is profinite homogeneous space

$\bullet$~ action of $D_{\infty}$ on $\fX_{\infty}$ is induced from adjoint action on $\fG(\Phi)$

$\bullet$~  isotopy at $x_{\infty}\in \fX_{\infty}$ is 
$\displaystyle D_{\infty} = \varprojlim  ~ \{D_{\ell +1}  \to D_{\ell} \mid \ell \geq 0\}$.

\bigskip

{\bf Principle:} Dynamical properties of  $\Phi \colon \G \times \fX  \to \fX$  are determined by adjoint action of $D_{\infty}$ on profinite group $\fG(\Phi)$.

 \vfill   

}

  
  
  

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

Many sources of examples with variety of dynamical properties:

\bigskip

$\bullet$~ Towers of finite covers of compact manifold $M$ correspond to group chains in $\G = \Pi_1(M, x)$

$\bullet$~ Solutions of   polynomial equations $f^{\circ n}(z) = \alpha$ generate the tree of iterated roots, with action of its absolute Galois group, the inverse limit of Galois groups of $f^{\circ n}(z)$ for $n \geq 1$

$\bullet$~ Actions on rooted trees, generated by finite automata; weakly branch groups \& self-similar groups
  
  $\bullet$~ Renormalization of groups generate Cantor actions with contraction mappings



  \bigskip
  
We focus on the  case of  renormalizable groups: $\G$ is finitely generated, and   $\phi \colon \G \to \G$ is injection with finite-index image.


 \vfill   

}


  
   \frame % 1
{
  \frametitle{}

{\bf Abelian odometers:}

$\G = \mZ^k$ and $\phi \colon \mZ^k \to \mZ^k$ is multiplication  by a matrix $M \in GL(k,\mZ)$ with determinant $d = |M| > 1$.

\bigskip

For example, $k=1$ then $M = [d]$ for $d > 1$ and so 
$$\fG_{\phi} =  \varprojlim_{\ell \to \infty}   ~ \mZ/d^{\ell}\mZ \cong \widehat{\mZ}_d$$

is a classical odometer.  $D_{\phi}$ is the trivial group, as $\G_{\ell} = C_{\ell}$.

\bigskip

For $k > 1$ things get more complicated, but not much more so.

In any case, the dynamics is quite boring.

 \vfill   

}


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

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

 

 
 $\bullet$~  Let $m > 1$. Define 
$\phi_1[x,y,z] = [mx,my,m^2z]$

$$\G_{\ell}  \ = \ \left\{ [x,y,z] = \left[ \begin{array}{ccc} 1 & m^{\ell}x & m^{2\ell}z \\ 0 & 1 & m^{\ell}y \\ 0 & 0 & 1 \end{array}\right] \mid x,y,z \in \mZ \right\}$$

Fact: $\G_{\ell}$ is not normal in $\G$, as $m^{2\ell}$ does not divide $m^{\ell}$.

$\bullet$~  None-the-less, the inverse limit $D_{\phi_1} $ is the trivial group.

 $\bullet$~  $\fX_{\phi_1} = \fG_{\phi_1}/D_{\phi_1} = \fG_{\phi_1}$ is a     nilpotent group. 

\bigskip


 \vfill   

}


  
   \frame % 1
{
  \frametitle{}


{\bf Heisenberg group:} {\bf Example 2:}
 
 $\bullet$~    $p,q > 1$   distinct primes. Define  $\phi_2[x,y,z] = [px,qy,pqz]$

$$\G_{\ell}  \ = \ \left\{ [x,y,z] = \left[ \begin{array}{ccc} 1 & p^{\ell}x & (pq)^{\ell}z \\ 0 & 1 & q^{\ell}y \\ 0 & 0 & 1 \end{array}\right] \mid x,y,z \in \mZ \right\}$$

$$C_{\ell}  \ = \ \left\{ [x,y,z] = \left[ \begin{array}{ccc} 1 & (pq)^{\ell}x & (pq)^{\ell}z \\ 0 & 1 & (pq)^{\ell}y \\ 0 & 0 & 1 \end{array}\right] \mid x,y,z \in \mZ \right\}$$

$\bullet$~ The inverse limit $D_{\phi_2}  \cong \widehat{\mZ}_q \times \widehat{\mZ}_p$, so action has large isotropy


$\bullet$~  $\ds \fX_{\phi_2}  \cong    \{[x,y,z] \mid x \in \widehat{\mZ}_{p} \ , \ y \in \widehat{\mZ}_{q} \ ,  \ z \in \widehat{\mZ}_{pq} \}$.

 
 
 \vfill   

}


 
 \section{Pro-nilpotent}

  \frame % 1
{
  \frametitle{}

{\bf Renormalizable Groups:}

Given $\phi \colon \G \to \G$ with finite index image, define  $\G_{\ell} = \phi^{\ell}(\G)$

 
$$\fX_{\phi} =  \varprojlim  ~ \{p_{\ell+1} \colon \G/\G_{\ell+1} \to \G/\G_{\ell} \mid \ell \geq 0\}$$

  $\phi$ induces the shift (or contraction) map
$\tau_{\phi} \colon \fX_{\phi}  \to \fX_{\phi}$.

\bigskip
For the proof of Gromov, the contraction on the universal covering of the manifold intertwines with the group action, which is used to show the action of $\G$ is nilpotent.  

 \vfill   

}
  

\frame % 1
{


$\bullet$~ For the renormalizable case, we want a contraction map 
 $\widehat{\phi} \colon \fG(\phi) \to \fG(\phi)$ that intertwines with the contraction map $\tau_{\phi}$
 
 \smallskip
 
$\bullet$~  It is immediate that $\phi \colon \G \to \G$ induces a map of the \emph{full profinite completion} $\widehat{\Gamma}$ of $\G$, so we get a contraction mapping 
$\widehat{\phi} \colon \widehat{\G} \to \widehat{\Gamma}$.

\smallskip

$\bullet$~  The problem is that $\widehat{\Gamma}$ is huge, and the isotropy of the action $\widehat{\Gamma} \times \fX_{\phi} \to \fX_{\phi}$ is also huge, and may also be highly ineffective.
\smallskip

$\bullet$~  The dynamics of the contraction  $\widehat{\phi}$ need not be closely connected to  the action of $\tau_{\phi}$. Need   for image $\whPhi(\whGamma) \subset {\rm Homeo}(\fX_{\phi})$

$\bullet$~  Need to show that $\widehat{\phi} \colon \widehat{\G} \to \widehat{\Gamma}$ descends to a contraction
$\widehat{\tau}_{\phi} \colon \fG_{\phi}   \to \fG_{\phi} $ that intertwines the contraction $\tau_{\phi}$ 

$\bullet$~  Show that   $\whPhi  \colon \fG_{\phi} \times \fX_{\phi} \to \fX_{\phi}$ satisfies a \emph{regularity property}.


 \vfill   

}

  
   \frame % 1
{
  \frametitle{}
   
     
  {\bf Definition:}    An action $\Phi \colon H \times \fX \to \fX$    is \underline{topologically free}  if  $$ \Iso(\Phi) = \{ x \in \fX \mid \exists ~ g \in \G ~ , ~ g \ne id ~, ~g \cdot x = x    \} = \bigcup_{e \ne g \in \G} \ \Fix(g) $$
  is meager in $\fX$.  

 


  {\bf Definition:}    For $H$ is a topological group and    $\fX$ is a Cantor space, an action $\Phi \colon H \times \fX \to \fX$   is  \underline{quasi-analytic}    
  if for each non-empty clopen set $U \subset \fX$  and $g \in H$ such that $\Phi(g)(U) = U$:
  
  $\bullet$ ~  if  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, the quasi-analytic  and     topologically free properties for an action are equivalent.
 \vfill   

}
  

\frame % 1
{

The first step in the proof of the theorem is the technical claim. 

It   says  what happens locally,  determines what happens globally.

\medskip


{\bf Theorem:}   Let $\G$ be a finitely generated group and $\phi \colon \G \to \G$ a renormalization. Then   $\whPhi  \colon \fG_{\phi} \times \fX_{\phi} \to \fX_{\phi}$ is quasi-analytic.   

\medskip

The proof of this result in the {\bf GGD} paper uses ideas that appear in a variety of formats in study of renormalizable groups.


\bigskip

{\bf Corollary:} Let $\G$ be a finitely generated group and $\phi \colon \G \to \G$ a renormalization. Then $\phi$ induces a contraction mapping $\widehat{\phi} \colon \fG_{\phi}  \to \fG_{\phi} $ that intertwines with the contraction $\tau_{\phi} \colon \fX_{\phi} \to \fX_{\phi}$.

 \vfill   

}
  

\frame % 1
{


We can now take advantage of   works in the literature on open contraction mappings of profinite groups by:

$\star$~ Baumgartner, Caprace, Gl\"{o}ckner, Reid, Willis:

\medskip

The result we need follows from the ``elementary'' result

{\bf Lemma:} A finite $p$-group is nilpotent.


\medskip

See the papers:
 
$\star$~ {C.~Reid},  {\it Endomorphisms of profinite groups}, {\bf Groups Geom. Dyn.},  2014.   



$\star$~ {W.~van Limbeek}, 
  {\it Structure of normally and finitely non-co-Hopfian groups}, {\bf Groups Geom. Dyn.},  2021.   
 



\vfill   

}
  

\frame % 1
{

{\bf Theorem:} 
 Let $\phi \colon \G \to \G$ be a renormalization for the  finitely generated group $\G$, with associated Cantor action $(X_{\phi}, \G, \Phi_{\phi})$.   
 
   Let  $\widehat{\phi} \colon \fG_{\phi} \to \fG_{\phi}$ be the   embedding induced from $\phi$. Then
  there exists a closed \emph{pro-nilpotent normal} subgroup  $\widehat{N}_{\phi} \subset \fG_{\phi}$ so that:
\begin{enumerate}
\item $\fG_{\phi} \cong \widehat{N}_{\phi} \rtimes D_{\phi}$ is a semi-direct product;
\item $\widehat{N}_{\phi}$ is $\widehat{\phi}$-invariant, and $\widehat{\phi}$ restricts to an open contracting embedding on $\widehat{N}_{\phi}$;
\item $D_{\phi}$ is $\widehat{\phi}$-invariant, and $\widehat{\phi}$ restricts to an automorphism of $D_{\phi}$.
\end{enumerate}
Moreover, let $\whe \in \fG_{\phi}$ be the identity element, then we have
$$
\widehat{N}_{\phi} = \{g \in  \fG_{\phi} \mid \lim_{n \to \infty} \ \widehat{\phi}^n(g) = \whe\} \quad , \quad D_{\phi} = \bigcap_{n > 0} \ \widehat{\phi}^n(\fG_{\phi}) ~ .
$$



\vfill   

}
  

\frame % 1
{

Here is the precise form of our main result:

{\bf Theorem:}  Let $\phi$ be a renormalization of the finitely generated group $\G$. Suppose that  
$ \bigcap_{\ell > 0} \phi^{\ell}(\G)  \subset \G$ and $D_{\phi}$
are both finite groups.  Then $\G$ is  virtually nilpotent. If both groups  are trivial, then $\G$ is nilpotent.

\bigskip

{\bf Problem:} Analyze the induced automorphism   $ \widehat{\phi} \colon D_{\phi} \to D_{\phi}$. \\
Show that   $D_{\phi}$ is always   nilpotent in this case.

\bigskip

{\bf Problem:} Suppose that $\phi$ is induced from a Riemannian expanding map of compact manifold $M$. How is the isotropy group $D_{\phi}$ related to the geometry of the manifold $M$?

 
\vfill   

}
  
 
  \frame % 1
{
  \frametitle{}
   
   

Arboreal actions are studied in terms of their invariants:

$\bullet$~ is the action \emph{stable} or \emph{wild}? 

{\small \phantom{-} $\star$~ {Hurder, Lukina},  {\it Wild solenoids}, {\bf Trans. AMS},  2019.   }

$\bullet$~ is the    \emph{prime spectrum} finite? 

{\small  \phantom{-} $\star$~  Similar to the Steiniz numbers, see

 \phantom{-} {Hurder, Lukina},  {\it Prime spectrum of solenoidal manifolds}, preprint,  2021.}

$\bullet$~ is the action \emph{turbulent}?

{\small \phantom{-} $\star$~ Work in progress, based on the ideas in 

 \phantom{-} {Gr\"oger, Lukina},  {\it Measures and stabilizers in group Cantor actions}, {\bf DCDS},  2021.   }

Renormalized actions are always stable, with finite prime spectrum and not turbulent.

  
 \vfill   

}


   
\section{Galois actions}    
 
 
  \frame % 1
{
  \frametitle{}

 {\bf Arboreal representations of Galois groups}  
 
 \medskip
 
 $\bullet$ Studied in Arithmetic Dynamics starting with 
 
  $\star$~  {R.W.K. Odoni},
  {\it The {G}alois theory of iterates and composites of polynomials},
  {\bf Proc. London Math. Soc. (3)}, 1985.
  
  \medskip
  $\bullet$ Much work has been done, see the survey in
  
    $\star$~  R. Jones, Galois representations from pre-image trees: an arboreal survey,  ``Th\'eorie de Nombres et Applications'', 2013,
    
and also later works   by Benedetto, Ferraguti, Ingram, Jones, Juul, Looper, Pagano, Tucker,   and others ...
    
    \medskip
    
    $\bullet$ Applications of topological dynamics to study arboreal representations   in the works of Cortez and Lukina

 \vfill   
 

}

\frame % 1
{
  \frametitle{}

  
   
Let $K$ be a number field, and let $f(x)$ be a polynomial of degree $d\geq 2$ with coefficients in the ring of integers of $K$.

\medskip
Let $\alpha \in K$, and let $f^n (x) = f \circ f^{n-1}(x)$ be the $n$-th iteration.

\medskip
Suppose $f^n(x) - \alpha$ is irreducible over $K$ and has $d^n$ distinct roots.

\medskip
Adjoin these roots to $K$ to obtain a finite extension $K(f^{-n}(\alpha))/K$.

\medskip
Then the action of the Galois group 
  $$H_n = {\rm Gal}(K(f^{-n}(\alpha))/K)$$ 
permutes the roots of $f^n-\alpha$ and this action is transitive.

 \vfill   

}



  \frame % 1
{
  \frametitle{}

To obtain a tree $\cT$, let 

\medskip
$V_n = \{\textrm{ roots of }f^n(x) - \alpha \}$.

\medskip
Draw an edge from $b$ to $a$ \\ if $f(b) = a$.
 \pgfputat{\pgfxy(4., -1.2)}{\pgfbox[left,base]{\pgfimage[width=4cm]{D-ary-tree-2}}}\pgfputat{\pgfxy(6., 1.3)}{\footnotesize $\alpha$}\pgfputat{\pgfxy(7.4, 0.7)}{\footnotesize $f^{-1}(\alpha)$}\pgfputat{\pgfxy(7.9, 0.2)}{\footnotesize $f^{-2}(\alpha)$}

  
  \vfill   

}

  \frame % 1
{
  \frametitle{}


Finite Galois groups $H_n$ act transitively on vertices   $V_n = f^{-n}(\alpha)$.

\medskip
There are homomorphisms $H_n \to H_{n-1}$, which preserve $\cT$.

\medskip
Thus there is a profinite group
  $$\mathcal{G} = \lim_{\longleftarrow}\{H_n \to H_{n-1}\} \subset Aut(\cT). $$
This group is the image of the arboreal representation
  $$\rho_{f,\alpha}: {\rm Gal}(K^{sep}/K) \to Aut(\cT),$$
where $K^{sep}$ is a separable closure of $K$.

 
 \vfill   

}

  \frame % 1
{
  \frametitle{}

 {\bf Question:}  
 For which polynomials is the image group $\mathcal{G}$ finite index in the full automorphism group $ Aut(\cT)$?
 
 $\star$~  {R.W.K. Odoni},
  {\it The {G}alois theory of iterates and composites of polynomials},
  {\bf Proc. London Math. Soc. (3)}, 1985.

\bigskip

{\bf Question:}  What can you say about the image group  $\mathcal{G}$  for 
specific classes of polynomials, for instance, for PCF quadratic polynomials?


 
 %$\star$~ {P.~Ingram},
 %{\it Arboreal {G}alois representations and uniformization of
    %          polynomial dynamics},
 %{\bf Bull. Lond. Math. Soc.}, 2013.

 % $\star$~  R. Jones, Galois representations from pre-image trees: an arboreal survey,  ``Th\'eorie de Nombres et Applications'', 2013,.
  
 % $\star$~   N. Boston and R. Jones, The image of an arboreal Galois representation, Pure and Applied Math. Quart.,  2009.
  
 \medskip
 We introduce another dynamical property of arboreal actions related to this question, its local regularity.
 
  
 \vfill   

}



  \frame % 1
{
  \frametitle{}

  

  {\bf Definition:}    For $H$ is a topological group and    $\fX$ is a Cantor space, an action $\Phi \colon H \times \fX \to \fX$   is  \underline{locally quasi-analytic}    
  if there exists $\e > 0$,  such that for each  clopen set $U \subset \fX$  with $\diam(U) < \e$:
    
  $\bullet$ ~  for  $g \in H$ and adapted clopen set $V \subset U$,  if the restriction $\Phi(g) | V$ is the identity map on $V$, 
  then $\Phi(g)$ is the identity on  $U$.  

 \bigskip
 
   
{\bf Definition:}  An equicontinuous  Cantor action $\Phi \colon H \times \fX \to \fX$ is   \underline{stable} if the associated profinite action  $\whPhi \colon \whGamma \times \fX \to \fX$ is locally quasi-analytic. The action  is said to be \underline{wild} otherwise. 



 \vfill   

}


  \frame % 1
{
  \frametitle{}
  
  \bigskip

{\bf Theorem:} [Lukina 2018] If  $\mathcal{G}$  has finite index in $ Aut(\cT)$, then the action of $\mathcal G$ on the tree boundary $\fX$ is wild.


{\bf Theorem:} [H-Lukina 2019] Suppose that  $\mathcal{G} \subset Aut(\cT)$ is abelian, then the action is not wild.

\bigskip
{\bf Corollary:} If $\mathcal{G}$ is abelian, then it must have infinite index in $Aut(\cT)$ (for any $d \geq 2$).


\bigskip
{\bf Remark:} This statement was obtained for quadratic polynomials ($d = 2$) by different arguments in 

$\star$ Ferraguti, Pagano, {\it Constraining images of quadratic arboreal representations}, {\bf IMRN}, 2020.



 \vfill   

}


  \frame % 1
{
  \frametitle{}
  

\bigskip

{\bf Theorem:} [H-Lukina 2021]  Suppose that  $\mathcal{G} \subset Aut(\cT)$ is topologically finitely generated, virtually nilpotent and has finite prime spectrum, then the action is not wild.

\bigskip
{\bf Problem:} How to relate the dynamics of the image of the arboreal representation with its number theory properties?
 \vfill   
 
 

}


 

\frame % 1
{
  \begin{center}
 
 {\large 
 Thank you for your attention!
 }
 \end{center}
 


}

 
    
\frame % 1
{
  \frametitle{References}
 {\scriptsize    
 
   
 
  M. I. Cortez and O. Lukina, {\it Settled elements in profinite groups}, {arXiv:2106.00631}.

 
 {S.~Hurder and O.~Lukina},
 {\it Wild solenoids},
 {\bf Trans. A.M.S.},  371:4493--4533,  2019; {arXiv:1702.03032}
 
 
 {S.~Hurder and O.~Lukina},
 {\it Orbit equivalence and classification of weak solenoids},
 {\bf Indiana Univ. Math. J.},  69:2339--2363, 2020; {arXiv:1803.02098}.

 {S.~Hurder and O.~Lukina},
 {\it Limit group invariants for non-free Cantor actions},
 {\bf Ergodic Theory Dynamical Systems}, 41:1751--1794, 2021; {arXiv:1904.11072}.

 {S.~Hurder and O.~Lukina},
 {\it Nilpotent Cantor actions},
 {\bf Proc. A.M.S.}, 150, 289--304, 2022; {arXiv:1905.07740}.

{S.~Hurder, O.~Lukina and W.~Van Limbeek},
 {\it Cantor dynamics of renormalizable groups},
 {\bf Groups Geom. Dyn.}, to appear, 2021; {arXiv:2002.01565}.

 {S.~Hurder and O.~Lukina},
 {\it The prime spectrum of solenoidal manifolds},
 {preprint}; {arXiv:2103.06825}.

{O.~Lukina}, {\it Arboreal Cantor actions}, 
{\bf J. Lond. Math. Soc. (2)}, 99:678--706, 2019; {arXiv:1801.01440}.

     
{O.~Lukina}, {\it Galois groups and {C}antor actions},
 {\bf Trans. A.M.S.}, 374:1579--1621, 2021; {arXiv:1809.08475}.
 
  
 \vfill   
}
}



  


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