% Fall Southeastern Sectional Meeting
% Spring Central and Western Joint Sectional Meeting
% University of Hawaii at Manoa, Honolulu, HI
% March 22-24, 2019 (Friday - Sunday) 
% Meeting # 1147, Abstract Issue: 40/2

 
% slides for talk "Foliation index theory for weak solenoids"
 % John Roe, October 6, 1959 -- March 9, 2018

 
\documentclass{beamer}

\usepackage{amsfonts}

\usepackage{graphicx}

\usepackage{pgf,pgfarrows,pgfnodes}
\usepackage{wrapfig}

 \usepackage{pdfpages}
 
\input xy
\xyoption {all}



% \usetheme{default}
 \usetheme{Singapore}
% \usetheme{CambridgeUS}
% \usetheme{Boadilla}
% \usetheme{Antibes}
% \usetheme{Berkeley}
% \usetheme{Berlin}
% \usetheme{Copenhagen}
% \usetheme{Darmstadt}
% \usetheme{Dresden}
% \usetheme{Frankfurt}
% \usetheme{Goettingen}
% \usetheme{Hannover}
% \usetheme{Ilmenau}
% \usetheme{JuanLesPins}
% \usetheme{Luebeck}
% \usetheme{Madrid}
% \usetheme{Malmoe}
% \usetheme{Marburg}
% \usetheme{Montpellier}
% \usetheme{PaloAlto}
% \usetheme{Pittsburgh}
% \usetheme{Rochester}
% \usetheme{Szeged}
% \usetheme{Warsaw}
 
% \usecolortheme{crane}
 % \beamertemplatesolidbackgroundcolor{craneorange!25}
 
 \parskip=4pt
 

% math notation shortcuts

\newcommand{\ds}{\displaystyle}
\newcommand{\sop}{{\bf Proof:} } % replaced by command \proof
\newcommand{\eop}{\;\;\; \Box} % replaced by command \endproof - still used inside theorem environment

%   bolds
\newcommand{\bZ}{{\bf Z}}
\newcommand{\bC}{{\bf C}}

\newcommand{\bK}{{\bf K}}
\newcommand{\F}{{\mathcal F}} % our favorite foliation
\newcommand{\e}{{\epsilon}} % our favorite constant

\newcommand{\wmW}{\widehat{\mW}}

\newcommand{\cGF}{\cG_{\F}} % denotes the pseudogroup of F
\newcommand{\cGM}{\cG_{\fM}} %
\newcommand{\cGN}{\cG_{\fN}} %
\newcommand{\cGX}{\cG_{\fX}} %
\newcommand{\cGY}{\cG_{\fY}} %
\newcommand{\Dom}{{\rm Dom}} %
\newcommand{\dX}{d_{\fX}} % leafwise metric
\newcommand{\dXp}{d_{\fX}'} % leafwise metric
\newcommand{\dY}{d_{\fY}} % leafwise metric
\newcommand{\diamX}{{\rm diam}_{\fX}} %
\newcommand{\G}{\Gamma}
\newcommand{\ovx}{\overline{x}} % 
\newcommand{\ovy}{\overline{y}} %  
\newcommand{\ovz}{\overline{z}} %  
\newcommand{\oU}{\overline{U}}
\newcommand{\ovgamma}{\overline{\gamma}} %  


% blackboard bolds
\newcommand{\mB}{{\mathbb B}}
\newcommand{\mC}{{\mathbb C}}
\newcommand{\mD}{{\mathbb D}}
\newcommand{\mE}{{\mathbb E}}
\newcommand{\mF}{{\mathbb F}}
\newcommand{\mG}{{\mathbb G}}
\newcommand{\mK}{{\mathbb K}}
\newcommand{\mN}{{\mathbb N}}
\newcommand{\mP}{{\mathbb P}}
\newcommand{\mQ}{{\mathbb Q}}
\newcommand{\mR}{{\mathbb R}}
\newcommand{\mS}{{\mathbb S}}
\newcommand{\mT}{{\mathbb T}}
\newcommand{\mW}{{\mathbb W}}
\newcommand{\mZ}{{\mathbb Z}}



% cal script letters
\newcommand{\cA}{{\mathcal A}}
\newcommand{\cB}{{\mathcal B}}
\newcommand{\cC}{{\mathcal C}}
\newcommand{\cD}{{\mathcal D}}
\newcommand{\cE}{{\mathcal E}}
\newcommand{\cG}{{\mathcal G}}
\newcommand{\cH}{{\mathcal H}}
\newcommand{\cI}{{\mathcal I}}
\newcommand{\cJ}{{\mathcal J}}
\newcommand{\cK}{{\mathcal K}}
\newcommand{\cL}{{\mathcal L}}
\newcommand{\cM}{{\mathcal M}}
\newcommand{\cN}{{\mathcal N}}
\newcommand{\cO}{{\mathcal O}}
\newcommand{\cP}{{\mathcal P}}
\newcommand{\cQ}{{\mathcal Q}}
\newcommand{\cR}{{\mathcal R}}
\newcommand{\cS}{{\mathcal S}}
\newcommand{\cT}{{\mathcal T}}
\newcommand{\cU}{{\mathcal U}}
\newcommand{\cV}{{\mathcal V}}
\newcommand{\cW}{{\mathcal W}}
\newcommand{\cX}{{\mathcal X}}
\newcommand{\cY}{{\mathcal Y}}
\newcommand{\cZ}{{\mathcal Z}}




% fraktur letters

\newcommand{\fA}{{\mathfrak{A}}}
\newcommand{\fB}{{\mathfrak{B}}}
\newcommand{\fC}{{\mathfrak{C}}}
\newcommand{\fD}{{\mathfrak{D}}}
\newcommand{\fG}{\mathfrak{G}}
\newcommand{\fH}{\mathfrak{H}}
\newcommand{\fK}{{\mathfrak{K}}}
\newcommand{\fL}{{\mathfrak{L}}}
\newcommand{\fM}{{\mathfrak{M}}}
\newcommand{\fN}{{\mathfrak{N}}}
\newcommand{\fO}{{\mathfrak{O}}}
\newcommand{\fP}{{\mathfrak{P}}}
\newcommand{\fQ}{{\mathfrak{Q}}}
\newcommand{\fS}{{\mathfrak{S}}}
\newcommand{\fT}{{\mathfrak{T}}}
\newcommand{\fU}{{\mathfrak{U}}}
\newcommand{\fV}{{\mathfrak{V}}}
\newcommand{\fX}{{\mathfrak{X}}}
\newcommand{\fY}{{\mathfrak{Y}}}
\newcommand{\fZ}{{\mathfrak{Z}}}
\newcommand{\fW}{{\mathfrak{W}}}

% greek letters

\newcommand{\vp}{{\varphi}}
\newcommand{\whvp}{\widehat{\varphi}}
\newcommand{\whPhi}{\widehat{\Phi}}
\newcommand{\whe}{\widehat{e}}
\newcommand{\whg}{\widehat{g}}
\newcommand{\whh}{\widehat{h}}
\newcommand{\whU}{\widehat{U}}
\newcommand{\whV}{\widehat{V}}
\newcommand{\whG}{\widehat{G}}
\newcommand{\whH}{\widehat{H}}
\newcommand{\whC}{\widehat{C}}

\newcommand{\wtSigma}{\widetilde{\Sigma}}

\newcommand{\bRt}{{\bf R}_{0}}
\newcommand{\whcG}{\widehat{\cG}}
 \newcommand{\psg}{{\rm pseudo}{\star}{\rm group}}

  \newcommand{\Aut}{{\bf Aut}}
  \newcommand{\Homeo}{{\bf Homeo}}
\newcommand{\wtM}{\widetilde{M}} %  
\newcommand{\mH}{{\mathbb H}}

 \newcommand{\oG}{{\mathfrak{G}(\Phi)}}

\date[April 12, 2018]


  
\title[Wild things]{Foliation index theory for weak solenoids}


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

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

 \section{Intro}

{
\setbeamercolor{background canvas}{bg=}
\includepdf[pages=1]{John_Roe+Nathan.jpg}
}

 \frame % 1
{
  \frametitle{}


[1983] ~~ A long time ago...

   \vfill   

}

{
\setbeamercolor{background canvas}{bg=}
\includepdf[pages=1]{Roe-Letter19830730c.pdf}
}


{
\setbeamercolor{background canvas}{bg=}
\includepdf[pages=1]{Roe-conference.jpg}
}


  

 \frame % 1
{
  \frametitle{}

 
\bigskip

{\bf Problem [Folklore 1974]:} ~ Geometry of leaves of   foliations.
\bigskip

{\bf Problem [Atiyah 1986]:} ~ Index theory for coverings.

\bigskip
{\bf Problem [Connes 1980]:} ~ Index theory of foliations.

\bigskip

{\bf Problem [Roe 1983]:} ~ Index theory of   complete open manifolds.

 
 \vfill   

}

 

 \frame % 1
{
  \frametitle{}


{\bf Goal of Talk:} ~ Weak solenoids =  class of foliated spaces whose leaf geometries remain unclassified.

\medskip


{\bf Goal of Program} 

 \medskip
 $\bullet$ ~ Structure of weak solenoids. 

 \medskip
 $\bullet$ ~ Index Theory  for weak solenoids. 

\medskip
 $\bullet$ ~ Spectral geometry of leaves in  weak solenoids 
 
   
 \medskip
 $\bullet$ ~ Invariants from $C^*$-algebras associated to Cantor actions.

 \bigskip
 $\star$ ~ Spectral flow, L\"{u}ck approximation for irregular coverings.


 \medskip
 $\star$ ~   Bowdoin Lecture [1988] on $\eta$-invariants and spectral flow.

 \vfill   

}


\section{Solenoids}


 \frame % 1
{
  \frametitle{}

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

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

 
 \vfill   

}
  
   

 \frame % 1
{
  \frametitle{}

 {\bf Example:} Vietoris solenoid is given by chain of subgroups of $\mZ$
    \begin{align*}  \xymatrix{
 \mS^1   &   \ar[l]_{m_1} \mS^1    &  \ar[l]_{m_2} \mS^1   & \ar[l]_{m_3} \mS^1  & \ar[l]_{m_4} \mS^1    \cdots     
 } 
\end{align*}

$\star$ ~ each $m_{\ell} > 1$ is an integer 

$\star$ ~ $m_{\ell} \colon \mS^1 \to \mS^1$ is an $m_{\ell}$-fold covering map.

  
  

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

  

 \vfill   

}
 
   
   
 \frame % 1
{
  \frametitle{}


\begin{center}
Inverse limit space = Vietoris solenoid [1927]
   \end{center}
 \begin{eqnarray*}
\cS   & \equiv & \varprojlim \,\{\, m_{\ell+1} \colon  \mS^1 \to \mS^1 \mid     \ell \geq 0\}\\
& & \\
& \equiv & \{(x_0, x_1, \ldots ) \in \cS   \mid m_{\ell+1}(x_{\ell + 1}) =  x_{\ell} ~ {\rm for ~ all} ~ \ell \geq 0 ~\} \\
& & \\
&  \subset & \prod_{\ell \geq 0} ~ \mS^1
\end{eqnarray*}

$\star$ ~ Give the product space the product or Tychonoff topology;

$\star$ ~  $\cS$ has the restricted topology;

$\star$ ~ $\cS$ is a compact, connected metric space, so a continuum.

 \vfill   
  
}


   
 \frame % 1
{
  \frametitle{}

 Properties of the space $\cS$
  
 
$\star$~ $\cS$ is an uncountable union of lines, each dense in $\cS$. 

    
 
 $\star$~ Whatever  stage $k \geq 0$ you start at, the limit is the same:
$$\cS  \cong  \varprojlim \,\{\, m_{\ell+1} \colon  \mS^1 \to \mS^1 \mid     \ell \geq k\}$$

 \smallskip
 
 $\star$~ $\cS$ is homogeneous - 
  for any pair of points $x,y \in \cS$ there is a homeomorphism $h \colon \cS \to \cS$ with $h(x) = y$.
 
 

 \vfill   
  
}

  
 \frame % 1
{
  \frametitle{}
  \bigskip
   
 $\bullet$~ R.H.~Bing,    Canad. J. Math., 1960\\
 $\bullet$~ M.C.~McCord, Transactions A.M.S, 1965

  {\bf Theorem:}   Let $\vec{p} = \{p_1 , p_2, p_3, \ldots \}$ and $\vec{q} = \{q_1 , q_2, q_3, \ldots \}$ be sequences of prime numbers. Say that  $\vec{p} \sim  \vec{q}$ if there is $\ell_p , \ell_q \geq 0$ so that the two sets 
 $$\{ p_{\ell} \mid \ell \geq \ell_p \} = \{ q_{\ell} \mid \ell \geq \ell_q \}$$
 Then the solenoids they generate, call them $\cS(\vec{p})$ and $\cS(\vec{q})$, are homeomorphic if and only if $\vec{p} \sim  \vec{q}$.
 
 \bigskip
 
 {\bf Question:} Is there a similar result in higher dimensions?
    
 \vfill   
  
}


  

   
 \frame % 1
{
  \frametitle{}

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

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

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

\medskip

The path connected components of $M_{\infty}$ are manifolds, which are non-compact covering spaces for $M_0$, so that    $M_{\infty}$ is a generalized lamination, or foliated space with Cantor transversals.
     

 \vfill   

}
 
  
  
    
  
 \frame % 1
{
  \frametitle{}

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


{\bf Proposition:} The monodromy  is    an \emph{equicontinuous  group action}.

$\bullet$  ~  A Cantor action $\vp \colon G  \times \fX \to \fX$ is \underline{equicontinuous} if for some metric $\dX$ on $\fX$,   for every $\e > 0$ there exists $\delta > 0$ such that 
 $$\dX(x,y) < \delta ~ \Longrightarrow ~ \dX(\vp(g)(x) , \vp(g)(y)) < \e \quad {\rm for ~ all } ~ g \in G   .$$
   
  When $G = \mZ$, then a minimal equicontinuous Cantor action $\vp \colon G  \times \fX \to \fX$ is conjugate to a classical odometer. 

  \vfill   

}

 
\section{Profinite models}


  \frame % 1
{
  \frametitle{ }

 
  Cantor action $(\fX, G, \Phi)$ is minimal equicontinuous action. 

\medskip

 {\bf Problem:} Classify Cantor actions up to topological conjugacy, or continuous orbit equivalence, or return equivalence.
 
 
  
  
 {\bf Definition:}  $\oG = \overline{H_{\Phi}} =$       closure of $H_{\Phi} = \Phi(G) \subset \Homeo(\fX)$   \\
   in the \emph{uniform topology on maps}.  $\oG$ is profinite group.


\medskip

  
 For $x \in \fX$,  $\cD_x = \{ \whh \in \overline{H_{\Phi}} \mid \whh \cdot x = x \} = \oG_x$     (isotropy group)
 
  \medskip
 
 {\bf Lemma:} Transitive left action of $\oG$ on $\fX \cong \oG/\cD_x$. 

\medskip

Closed subgroup $\cD_x \subset \oG$   is independent of the choice of basepoint $x$, up to topological isomorphism. 

\medskip

The action $(\fX, \oG, \whPhi)$ is called the  \underline{profinite model} for $(\fX, G, \Phi)$. 
 
  \vfill   

}

  

  \frame % 1
{
  \frametitle{ }

 \begin{align*}  
 \xymatrix{
   &             \oG \ar[d] \\
   G =  \pi_1(M_0 , x_0)  \ar[ru]^{\whPhi}    \ar[r]^{\qquad \Phi}    &   \fX       
} 
\end{align*}
  
  
{\bf Theorem:}  $\cD_x$ is trivial if and only if  $\fX$ is   a Cantor group.

  
 \medskip
 
So for $\cD_x$   trivial,   we have a  tower of normal coverings, as in Cheeger-Gromov [1985], L\"uck [1994], $\ldots$

\medskip

 On the other hand, the case when $\cD_x$   is non-trivial seems to be almost completely unknown in the literature.
 
  \vfill   

}


  \frame % 1
{
  \frametitle{ }

  \medskip
{\bf Problem:}  Relate asymptotic geometry \& analysis of leaves in weak solenoids to algebraic properties of subgroup $\cD_x \subset \oG$.


 
 \medskip
{\bf Proposition:} $\cD_x$ is \emph{totally not normal}: for any $\whh \in \cD_x$ there exists $\whg \in \oG$ such that $\whg^{-1} \ \whh \ \whg \not\in \cD_x$.

  
\bigskip
  For group chain $\cG = \{G_{\ell} \mid \ell \geq 0\}$    the  \underline{normal core} of $G_{\ell}$ in $G$
     $$ C_{\ell} =  \bigcap_{g \in G} gG_{\ell} g^{-1}  \subset G_{\ell}$$

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

\vfill   

}

  
 \frame % 1
{
  \frametitle{}

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

\medskip

The proof of $3$ uses ideas of   Lubotzky [1993] on torsion elements in the profinite completion of torsion free subgroups of ${\bf SL}(n,\mZ)$, and a construction due to {\bf Lenstra}.

 
  \vfill   

}

 \section{Wild actions} 
   
 
   




 \frame % 1
{
  \frametitle{}

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

\medskip

   
 Choose a neighborhood basis $\{\whC_{\ell} \mid \ell \geq 1\}$ of the identity $\whe \in \oG$. Each $\whC_{\ell}$ is a normal subgroup of $\oG$.
 
 \medskip
 
 Nested: $\whC_{\ell +1} \subset \whC_{\ell}$, with $\cap_{\ell \geq 1} \  \whC_{\ell} = \{\whe\}$.
 
 \medskip
 
 Given $\cD_x \subset \oG$, define $\whU_{\ell} = \cD_x \cdot \whC_{\ell}$. Then $U_{\ell} = \whU_{\ell}/\cD_x$ is clopen neighborhood of basepoint $x \in \fX$.

\medskip

 $\Phi(G) \subset \oG$ is dense subgroup;   $G_{\ell} = \{g \in G \mid \Phi(g) \in  \whU_{\ell}\}$.
 
 \medskip
 
   $\Phi_{\ell} \colon G_{\ell} \times U_{\ell} \to U_{\ell}$ is minimal action;  $\whPhi_{\ell} \colon \whG_{\ell} \times U_{\ell} \to U_{\ell}$.
 
 \medskip
 
  $\cD_x \in \whU_{\ell}$ for all $\ell \geq 1$, $\cD_x =  \cap_{\ell \geq 1} \ \whU_{\ell}$.  
 
   \vfill   

}

 
 \frame % 1
{
  \frametitle{}

$\whPhi_{\ell}$ induces map $\rho_{\ell} \colon \cD_x \to \Homeo(U_{\ell})$; 

Set $K_{\ell} \equiv \ker \{\rho_{\ell}\}$ for $\ell \geq 1$. Then $K_1 \subset K_2 \subset \cdots$

\medskip
 
 {\bf Theorem:} The isomorphism class of the direct limit group 
 $$\Upsilon(\Phi) = \varinjlim \ \{ K_{\ell} \subset K_{\ell+1} \mid \ell \geq 1\}$$
 is a conjugacy invariant of a Cantor action  $(\fX, G, \Phi)$.
 
\medskip
 A Cantor action  $(\fX, G, \Phi)$ is:
 
 $\bullet$ ~  \underline{stable} if the  chain $\{K_{\ell} \mid \ell \geq 1\}$  is bounded.
 
That is,  if there exists $\ell_0$ so that $K_{\ell} = K_{\ell+1}$ for $\ell \geq \ell_0$. 

\medskip

 $\bullet$ ~    \underline{wild} if the  chain $\{K_{\ell} \mid \ell \geq 1\}$  is unbounded.  
 
  \vfill   

}

    
  
 \frame % 1
{
  \frametitle{}

{\bf Theorem [Hurder-Lukina 2019]:} The property that a Cantor action is wild is invariant by a continuous orbit equivalence.

\bigskip

$\star$ ~ The   examples of group actions on trees generated by automata studied by Nekrashevych, Bartholdi, Grigorchuk \emph{et al}  typically induce wild actions on the Cantor boundary of a tree.

\bigskip
 
 {\bf Theorem [Lukina 2018]:}
Let $p$ and $d$ be distinct odd primes, let $K=\mQ_p$ be the field of $p$-adic numbers. Let
  $ f(x) = (x+p)^d - p$.
Then the action of ${\rm Gal}_\infty(f)$ is stable.
 
\bigskip

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

 \vfill   

}



   

  

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

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

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

\medskip

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

\vfill   

}



 \frame % 1
{
  \frametitle{ }

Let $\fX$ be a Cantor set, and let $\cH \subset {\bf Homeo}(\fX)$ be a group.

\medskip
An element $h \in \cH$ is \emph{non-Hausdorff}  if there exists $x \in \fX$ and a collection of open sets $\{U_n\}_{n \geq 1}$ with $\bigcap U_n = \{x\}$, such that

\begin{enumerate}
\item $h(x) = x$,
\item $h|_{U_n}$ is non-trivial,
\item for $n \geq 1$, there exists an open set $O_n \subset U_n$ with $h|_{O_n} = id$.
\end{enumerate}

\medskip

{\bf Theorem:} [Winkelnkemper, 1983] 
The germinal \'etale groupoid $\Gamma(\fX,G, \Phi)$ associated to a Cantor action $(\fX, G, \Phi)$ is non-Hausdorff if and only if $\cH = \Phi(G)$ contains a non-Hausdorff element.
 



  
\vfill   

}


 \frame % 1
{
  \frametitle{}

{\bf Definition:} A Cantor action $(\fX, G, \Phi)$ has \underline{finite type} if each group in the chain  $\{K_{\ell} \mid \ell \geq 1\}$ of isotropy groups is  finite. 

\medskip

{\bf Theorem [Hurder-Lukina, 2019]:} 
$(\fX, G, \Phi)$  a Cantor action: 

$\bullet$~ If $\oG$ contains a  non-Hausdorff element $\Rightarrow$   action is wild.

$\bullet$~  $G$ is finitely generated and   action is wild of finite type $\Rightarrow$  $\oG$ contains a non-Hausdorff element. 

 
\bigskip
{\bf Example [Lukina 2018]:} This is false if we only assume that $G$ is   a countable group.
There are examples of absolute Galois groups for function fields, whose   arboreal representations are wild, but do not have a non-Hausdorff element.

 
  
\vfill   

}


   
 \frame % 1
{
  \frametitle{}
 
 Here are two further results:
 \medskip
 
{\bf Theorem [Hurder-Lukina 2018]:} There exists \underline{uncountably many wild actions} of torsion-free finite index subgroups of ${\bf SL}(n,\mZ)$ with distinct asymptotic discriminants. 

\bigskip

 {\bf Theorem [Hurder-Lukina 2018]:}  Let  $\Phi \colon G \times \fX \to \fX$ be a Cantor action with $G$ a finitely generated nilpotent group. Then the action is stable. Moreover, any Cantor action which is continuously orbit equivalent must be return equivalent. That is, their germinal \'etale groupoids   are Morita equivalent.


    
  
  
\vfill   

}
  


  \section{Index Theory}
 
   
 \frame % 1
{
  \frametitle{}
 
Let $C_r^*(\fX, G, \Phi)$ be the reduced $C^*$-algebra associated to the Cantor action $(\fX, G, \Phi)$.

\medskip

Let $\fM$ be   a weak solenoid whose monodromy action is return equivalent to  the Cantor action $(\fX, G, \Phi)$. 

\medskip

Let $\F$ be the foliation on $\fM$, and let $D$ be a leafwise elliptic operator along the leaves of $\F$.

\medskip


Then $D$ induces a KK-class $[D] \in KK(C(\fM), C_r^*(\fX, G, \Phi))$. 
 
  
 
 
     
  
\vfill   

}
  

  
   
 \frame % 1
{
  \frametitle{}

 Let $\fX = \oG/\cD_x$ be the profinite model for the action.
 
 A representation $\rho \colon \cD_x \to U(N)$ gives $[\rho] \in KK^1(\mC , C(\fM))$
 
 Form the index pairing 
 $$ [\rho] \colon KK(C(\fM) , C_r^*(\fX, G, \Phi)) \longrightarrow K_*(C_r^*(\fX, G, \Phi))$$
 


 {\bf Question:} Do the values of the index pairing suffice for showing that a Cantor action $(\fX, G, \Phi)$ is wild?
 
 \bigskip
 \pause
 
 \begin{center}
 
 {\large 
 Thank you for your attention!
 }
 \end{center}
 
  
\vfill   

}
  
  
     

  \section{References}

    
\frame % 1
{
  \frametitle{References}
 {\tiny    
 
 
{N.~Boston and R.~Jones},
 {\it Arboreal {G}alois representations},
 {\bf Geom. Ded.},  124, 2007.

{A.~Clark and S.~Hurder},
 {\it Homogeneous matchbox manifolds},
 {\bf Trans. A.M.S.},   365:3151--3191, 2013.
 
  {J.~Dyer, S.~Hurder and O.~Lukina},
 {\it Molino theory for matchbox manifolds},
 {\bf Pac. Jour. Math.},  289:91-151, 2017.
 
 
{S.~Hurder},
 {\it Eta invariants and the odd index theorem for coverings},
 {\bf Contemp. Math. A.M.S.},   105:47--82, 1990.
 
{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 submitted},  2018; {arXiv:1803.02098}.
 
{S.~Hurder and O.~Lukina},
 {\it Dynamics and rigidity for Cantor actions},
 {\bf preprint},  2019.
 
{R. Jones}, {\it Galois representations from pre-image trees: an arboreal survey},
  in {\bf Actes de la {C}onf\'erence ``{T}h\'eorie des {N}ombres et
 {A}pplications''}, 107-136, 2013.

 
{A.~Lubotzky},  {\it Torsion in profinite completions of torsion-free groups},
 {\bf Quart. J. Math.} 44:327--332, 1993.


{O.~Lukina},
 {\it Arboreal {C}antor actions}, {\bf Jour. L.M.S.},  to appear,  arXiv:1801.01440.

{O.~Lukina},
 {\it Galois Groups and Cantor actions}, {preprint},   arXiv:1809.08475.

  {M.C.~McCord},
  {\it Inverse limit sequences with covering maps},
  {\bf Trans. Amer. Math. Soc.}, 114:197--209, 1965

{V.~Nekrashevych}, {\it Palindromic subshifts and simple periodic groups of intermediate growth},
 {\bf Ann. of Math. (2)}, 187:667--719, 2018.  {arXiv:1601.01033}.



{R.W.K. Odoni},
  {\it The {G}alois theory of iterates and composites of polynomials},
  {\bf Proc. L.M.S.}, 51:385-414, 1985.


 {L.~ Vietoris}, {\it  \"{U}ber den h\"oheren {Z}usammenhang kompakter {R}\"aume und eine {K}lasse von zusammenhangstreuen {A}bbildungen},
 {\bf Math. Ann.}, 97:454--472, 1927.



}  
}

 \section{HBD}
 
 

 
  


\end{document}

