% The Second Australia-China-Japan-Singapore-U.S. Index Theory Conference
% Noncommutative Geometry and K-Theory
% June 2, 2023
 
 
% slides for talk "Non-commutative geometry of solenoidal manifolds"
 
 
\documentclass{beamer}
  \usetheme{Singapore}

\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}
 
  

 \usepackage{wasysym}
 
 \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{\ovp}{\overline{p}} %  
\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}}
\newcommand{\cCO}{{\mathcal CO}}




% 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{\whtau}{\widehat{\tau}}
\newcommand{\whe}{\widehat{e}}
\newcommand{\whg}{\widehat{g}}
\newcommand{\whh}{\widehat{h}}
\newcommand{\whx}{\widehat{x}}
\newcommand{\whU}{\widehat{U}}
\newcommand{\whV}{\widehat{V}}
\newcommand{\whG}{\widehat{G}}
\newcommand{\whH}{\widehat{H}}
\newcommand{\whC}{\widehat{C}}
\newcommand{\whD}{\widehat{D}}

\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{\wtx}{\widetilde{x}} %  
\newcommand{\wtz}{\widetilde{z}} %  
\newcommand{\mH}{{\mathbb H}}

 \newcommand{\oG}{{\mathfrak{G}(\Phi)}}
\newcommand{\diam}{{\rm diam}} 
\newcommand{\vol}{{\rm Vol}} 

\newcommand{\whgamma}{{\widehat{\gamma}}}
\newcommand{\whGamma}{{\widehat{\Gamma}}}
 \newcommand{\Fix}{{\rm Fix}} 
\newcommand{\Iso}{{\rm Iso}} 
\newcommand{\Image}{{\rm Image}} 

\newcommand\mor{\mathrel{\stackrel{\makebox[0pt]{\mbox{\normalfont\tiny a}}}{\sim}}}

\newcommand{\CO}{{\rm CO}} 

\newcommand{\whmZ}{\widehat{\mZ}}

\newcommand{\ova}{\overline{a}}
\newcommand{\ovb}{\overline{b}}
\newcommand{\ovc}{\overline{c}}
\newcommand{\ovq}{\overline{q}}

 
\newcommand{\whcH}{{\widehat{\cH}}}


 \newcommand{\lcm}{{\rm lcm}} 
 
\newcommand\comm{\mathrel{\stackrel{\makebox[0pt]{\mbox{\normalfont\tiny c}}}{\sim}}}



\date[June 2, 2023]


  
\title{Non-commutative geometry\\ of solenoidal manifolds}


\author{Steve Hurder\\ University of Illinois at Chicago} 
 
 
\begin{document}
 
   \frame{\titlepage} % # 1

\section{Solenoidal manifolds}

\frame
{
  \frametitle{}

 A \emph{solenoidal manifold} is a compact connected space, defined as the inverse limit of   a tower of proper covering maps,
$$
   \cS_{\cP} \equiv  \varprojlim \,\{\, q_{\ell} \colon  M_{\ell} \to M_{\ell -1} \mid  \ell \geq 1\} ~  , 
$$
where each $M_\ell$ is a compact connected manifold, and $q_{\ell}$ is a  covering map of finite degree greater than one.

\medskip

{\bf Theorem:}   $\cS_{\cP}$ is a foliated space with foliation $\F_{\cP}$ whose leaves have dimension $n = \dim(M_0)$

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

\medskip

$\Longrightarrow$ ~ Leaves of $\F_{\cP}$ have smooth structure, partitions of unity, Riemannian metric, and all associated objects, such as leafwise elliptic operators and Hermitian vector bundles.
 
 \vfill   
}
 
\frame
{
  \frametitle{}

{\bf Theme:} 
  Study    $\cS_{\cP}$    as a generalized manifold.
      \bigskip
      
$\star$ ~       {D.~Sullivan},
  {\it Linking the universalities of {M}ilnor-{T}hurston, {F}eigenbaum and {A}hlfors-{B}ers},
  in {\bf Topological methods in modern mathematics ({S}tony {B}rook,  {NY}, 1991)}, 
  {Publish or Perish, Houston, TX}, 1993, pages 543--564.


      
$\star$ ~ {D.~Sullivan},
 {\it Solenoidal manifolds},
   {\bf J. Singul.}, 9:203--205, 2014.

 $\star$ ~ 
{A.~Verjovsky},
 {\it Low-dimensional solenoidal manifolds},
  {\bf arXiv:2203.10032}


\medskip

{\bf Problem:} Investigate the (non-commutative)  geometric properties of solenoidal manifolds - their invariants and symmetries. 



 \vfill   
}
 
 
\frame
{
  \frametitle{}

 
{\bf Example:} ~  The {Vietoris - van Dantzig Solenoid}
$$\cS(\vec{m}) =  \underleftarrow{\lim} ~ \{\, q_{\ell}:\mS^1 \to \mS^1 \mid  \ell \geq 1\} 
$$
where $q_{\ell}$ is  a covering map  of the circle $\mS^1$   of degree  $m_{\ell} > 1$, and   $\vec{m} = (m_1, m_2,\dots)$  denotes the collection of covering degrees. 

\medskip
When $m_i =2$ for all $i \geq 1$ we get the \emph{Smale attractor}, which is a minimal set for a $C^{\infty}$-foliation:
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=.4\textwidth]{pix/Smale-Williams_Solenoid_Large}
\label{figure-originalhirsch}
\end{center}
\end{figure}

 \vfill   
}


\frame
{
  \frametitle{}


{\bf Example:} ~    {Solenoidal Riemann Surface (after Sullivan)}


$$\widehat{\Sigma_g} =  \underleftarrow{\lim} ~ \{\,  q  \colon \Sigma_{g'} \to \Sigma_g\} 
$$
where $M_0  = \Sigma_g$ is a surface with genus $g \geq 1$, and   the 
inverse limit is over all finite coverings of $\Sigma_g$.

 \begin{figure}[htbp]
\begin{center}
\includegraphics[width=.8\textwidth]{pix/Surface_solenoid}
\label{figure-sullivan}
\end{center}
\end{figure}
 
 
 \vfill   
}


\frame
{
  \frametitle{}

Homeomorphism $\vp \colon \cS_{\cP} \to \cS'_{\cP'}$ maps leaves of $\F_{\cP}$ to leaves of $\F_{\cP'}$

\medskip

In both examples above, the space $\cS_{\cP}$ is homogeneous - the group of homeomorphisms act transitively.
\bigskip

{\bf Problem:} Find invariants of solenoidal manifolds that do not change by a homeomorphism -  a solenoidal manifold does not have a preferred presentation as a tower of coverings. Also, most solenoidal manifolds are not homogeneous.

\bigskip
Discuss two applications:

\medskip

$\star$ ~ ``Gap Labeling'' for leafwise elliptic operators

\medskip

$\star$ ~ ``Spectrum'' of elements $\gamma \in \G = \pi_1(M_0)$

 \vfill   
}


 
\frame
{
  \frametitle{}

Given the   solenoidal manifold
$$
\cS_{\cP} \equiv \lim_{\longleftarrow} ~ \{ q_{\ell } \colon M_{\ell } \to M_{\ell -1}\} ~ \subset \prod_{\ell \geq 0} ~ M_{\ell} \  .  
$$
Projections $p_{\ell} \colon \cS_{\cP} \to M_{\ell}$ whose restriction to leaves of $\F_{\cP}$ are covering maps.

Covering maps $\ovq_{\ell} = q_{\ell} \circ \cdots \circ q_1\colon M_{\ell} \to M_0$.

Choose basepoint $x \in \cS_{\cP}$ and set $x_{\ell} = p_{\ell}(x) \in M_{\ell}$.

The fiber $\fX = p_0^{-1}(x_0)$ is a Cantor set.

Set $\G_{\ell} = \Image \ \{(\ovq_{\ell})_{\#} \colon \pi_1(M_{\ell}, x_{\ell}) \to  \pi_1(M_0 , x_0) \} = \G$
 
 Obtain a nested chain of subgroups of finite index, 
 $$\G = \G_0 \supset \G_1 \supset \cdots \supset \G_{\ell} \supset \cdots$$
 \smallskip
   
 
 \vfill   
}
 
\frame
{
  \frametitle{}

 
  
 Set $X_{\ell} = \G/\G_{\ell}$ which is finite set with transitive $\G$-action, and $\G$-projections $\ovp_{\ell} \colon X_{\ell} \to X_{\ell -1}$.  Obtain $\G$-Cantor space
 
 $$
X_{\infty}  \equiv \lim_{\longleftarrow} ~ \{ \ovp_{\ell } \colon X_{\ell } \to X_{\ell -1}\} ~ \subset \prod_{\ell \geq 0} ~ X_{\ell} \  .  
$$
Action of $\G$ on $X_{\infty}$ is conjugate to the  monodromy along leaves of the foliation $\F_{\cP}$ of $\cS_{\cP}$  for the fiber  $\fX = p_0^{-1}(x_0)$.

 The action $\Phi \colon \G \times X_{\infty} \to X_{\infty}$ is minimal and equicontinuous.
 
 \medskip
 
 {\bf Theorem:} $\cS_{\cP}$ is homeomorphic to the suspension foliation for the action $(X_{\infty},\G,\Phi)$. That is,
 $\cS_{\cP} \cong \wtM \times X_{\infty}/(\gamma \cdot \wtz) \sim (z, \Phi(\gamma)(x))$ where $\wtM$ is the universal covering space for $M$.

 
 
 \vfill   
}



 \section{Dynamics}

\frame
{
  \frametitle{}
Quick review of some results from 

$\star$ ~ Hurder \& Lukina, 
{\it Wild solenoids}, {\bf Trans. Amer. Math. Soc.}, 371:4493-4533, 2019; {arXiv:1702.03032}.

\medskip

Let $(\fX,\G,\Phi)$  be a Cantor   action.
That is, for each $\gamma \in \G$,  $\Phi(\gamma) \colon \fX \to \fX$ is a homeomorphism of the Cantor space $\fX$.

\bigskip

We say that $(\fX,\G,\Phi)$ is:

   $\bullet$ ~ a \emph{minimal} action if $\cO(x) = \{\gamma \cdot  x \mid \gamma \in \G\}$ is dense in $\fX$ for each $x \in \fX$.

 
   
   $\bullet$ ~  \emph{equicontinuous} with respect to a metric $\dX$ on $\fX$, if for all $\e >0$ there exists $\delta > 0$, such that for all $x , y \in \fX$ and $\gamma \in \G$ we have  that 
 $\ds  \dX(x,y) < \delta$ implies   $\dX(\gamma \cdot x, \gamma \cdot y) < \e$.
 
 
 \vfill   
}
  
\frame
{
  \frametitle{}

Let  $\Phi(\G) \subset \Homeo(\fX)$ denote the image subgroup. 

\medskip

$\oG = \overline{\Phi(\G)}   \subset \Homeo(\fX)$ in the \emph{uniform topology of maps}. 

\medskip


$\oG$  is a separable profinite group. 

 \medskip



  $\whPhi \colon \oG \times \fX \to \fX$ denotes the induced action of $\oG$ on $\fX$. 

\medskip


$(\fX,\G,\Phi)$  minimal implies the group $\oG$ acts transitively on $\fX$.  

\medskip


Isotropy group $\fD(\Phi,x) = \{ \whgamma \cdot x = x \mid \whgamma \in \oG\}$

\medskip


Then $\fX \cong \oG/\fD(\Phi, x)$ as $\oG$-spaces.  That is, $\fX$ is a homogeneous $\oG$-space.
 
 \medskip
  \vfill   
}
 
\frame
{
  \frametitle{}


$\fX$ is profinite group $\Longleftrightarrow$   isotropy $\fD(\Phi, x)$ is trivial.
\medskip

A  minimal equicontinuous Cantor action $(\fX,\G,\Phi)$, then either:

$\star$ ~  it is homogeneous if $\fD(\Phi, x)$ is trivial, or

$\star$ ~  it has  $\whgamma  \in \oG$ with $\whPhi(\whgamma)(x) = x$ and $\whgamma \ne id$ 

\bigskip


McCord answered a question of Bing, showing:

{\bf Theorem:}  The solenoidal manifold $\cS_{\cP}$ is homogeneous if and only if the action of $\oG$ on $\fX$ has trivial isotropy.

  \vfill   
}
 
\frame
{
  \frametitle{}

{\bf Problem:} Study the properties of solenoidal manifolds, and their corresponding Cantor monodromy actions, when the isotropy group $\fD(\Phi, x)$ is non-trivial.

\bigskip


If  $\fD(\Phi, x)$ is non-trivial, then have non-trivial adjoint action
$${\rm Adj} \colon   \fD(\Phi, x)  \times  \fX \to \fX$$

{\bf Principle:} The ``geometry'' of the $\G$-space $\fX$ is strongly influenced by the properties of the adjoint action of $\fD(\Phi, x)$.

\medskip

{\bf  Theme:} For a solenoidal manifold $\cS_{\cP}$, invariants of $(X_{\infty},\G,\Phi)$  determine invariants of the solenoidal manifold.

 \vfill   
}
 
 
   
 \section{Types}
 
 \frame
{
  \frametitle{}

 
 The \emph{Steinitz number} defined by   a sequence of integers $\vec{m} = (m_1, m_2, \ldots)$ with $m_{\ell} > 1$ is 
the \emph{supernatural number}
   \begin{equation*}
\xi(\vec{m}) = \lcm \{ m_1   m_2 \cdots m_{\ell} \mid  \ell > 0\} \ ,
\end{equation*}
where   $\lcm$ denotes the least common multiple of the collection of integers.  A Steinitz number $\xi$ can be written uniquely as the formal product over the set of primes   $\Pi$, 
\begin{equation*}
\xi  = \prod_{p \in \Pi} ~ p^{\chi_{\xi}(p)} 
\end{equation*}
 The  \emph{characteristic function} $\chi_{\xi} \colon \Pi \to \{0,1,\ldots, \infty\}$ counts the multiplicity with which a prime $p$ appears in the infinite product $\xi$. 

 \vfill   
}
 
 
 
 
\frame
{
  \frametitle{}

 


Two Steinitz numbers  $\xi$ and $\xi'$ are said to be \emph{asymptotically equivalent} if there exists finite integers $m, m' \geq 1$ such that $m \cdot \xi = m' \cdot \xi'$, and we then write $\xi \mor \xi'$

\bigskip

{\bf Definition:}  The \emph{type} associated to a Steinitz number  $\xi$ is the asymptotic equivalence class of $\xi$,     denoted by  $\tau[\xi]$.

\bigskip

{\bf Lemma.}
 $\xi$ and $\xi'$ satisfy $\xi \mor \xi'$ if and only if their characteristic functions $\chi, \chi'$ satisfy
 \begin{itemize}
\item $\chi(p) = \chi'(p)$ for all but finitely many primes $p \in \Pi$
\item $\chi(p) = \infty$ if and only iff $\chi'(p) = \infty$ for all primes $p \in \Pi$.
\end{itemize}
 


 \vfill   
}
 
 
 
 
\frame
{
  \frametitle{}

 {\bf   Steinitz order}
 
 Let  $\fG$ be a separable profinite group, and  $\fU \subset \fG$  an open subgroup. Then $\fU$ has finite index.

 
 
The \emph{Steinitz order} of $\fG$ is  the supernatural number associated to a presentation of $\fG$ as an inverse limit of finite groups, defined by:
  
  
  $$\xi(\fG) =  \lcm \{\# \ \fG/\fN  \mid \fN \subset \fG~ \text{open normal subgroup}\}$$
  
  The \emph{type} of $\fG$ is the asymptotic equivalence class $\tau[\fG]$ of $\xi(\fG)$.
  
  \bigskip
  
$\star$ ~ Wilson,  {\bf  Profinite groups}, {London Mathematical Society Monographs. New Series}, Vol. 19,    1998.  

 \vfill   
}
 

 
\frame
{
  \frametitle{}
 
 {\bf Relative Steinitz order}
 
Let $\fD \subset \fG$ be a closed subgroup, and $\fN \subset \fG$ is an open normal subgroup, then $\fN \cdot \fD$ is   an open subgroup of $\fG$. 

\medskip


{\bf Definition:}
Let $\fD \subset \fG$ be a closed subgroup of   profinite group $\fG$. The \emph{Relative Steinitz order} of the pair $(\fG , \fD)$ is defined by :
$$\xi(\fG : \fD) =  \lcm \{\# \ \fG/(\fN \cdot \fD)  \mid  \fN \subset \fG~ \text{open normal subgroup}\}$$
 


  Steinitz orders  satisfy the Lagrange identity, where the multiplication is taken in the sense of supernatural numbers
$$ \xi(\fG)  = \xi(\fG : \fD) \cdot \xi(\fD)  \ . $$
 
$\tau[\fG: \fD]$ is the asymptotic equivalence class of $\xi(\fG : \fD)$.


 \vfill   
}
 
 
 
 
\frame
{
  \frametitle{}

 {\bf  Typesets}

Let $\fD \subset \fG$ be a closed subgroup.

For   $\whgamma  \in \fG$,   let $\fG_{\whgamma} = \overline{ \{\whgamma^n  \mid n \in \mZ\} } \subset \fG$. 

Then $\fG_{\whgamma}$ is an abelian Cantor group, and define
$$\xi(\fG: \fD, \whgamma)    =  \lcm \{\# \  (\fG_{\whgamma}/ \fG_{\whgamma} \cap \fD \cdot \fN)  \mid    \fN  ~ \text{open normal subgroup}\}$$
  $\tau[\fG: \fD, \whgamma]$ is the asymptotic equivalence class of $\xi(\fG: \fD, \whgamma)$.

\medskip

{\bf Definition:} The \emph{typeset} of $(\fG, \fD)$ is the  collection of types
$$\tau[\fG: \fD] = \{\tau[\fG: \fD, \whgamma] \mid \whgamma \in \fG \}  \  .$$

Find more discussion and results in the work

$\star$ ~ Hurder \& Lukina, 
{\it Type invariants for  solenoidal manifolds}, arXiv:2305.00863
\medskip



 \vfill   
}
 
 
 
\frame
{
  \frametitle{}
{\bf Application to solenoidal manifolds.}

\medskip

 Given a nested chain of subgroups of finite index, 
 $$\G = \G_0 \supset \G_1 \supset \cdots \supset \G_{\ell} \supset \cdots$$
Let $C_{\ell} \subset \G_{\ell}$ be the normal core; that is, the largest normal subgroup in $\G_{\ell}$. Then define the Cantor groups
$$\whGamma_{\infty}  \equiv \lim_{\longleftarrow} ~ \{ \ovp_{\ell } \colon \G/C_{\ell} \to \G/C_{\ell -1}\} ~ \subset \prod_{\ell \geq 0} ~ \G/C_{\ell} \  .  $$
$$\whD_{\infty}  \equiv \lim_{\longleftarrow} ~ \{ \ovp_{\ell } \colon \G_{\ell}/C_{\ell} \to \G_{\ell -1}/C_{\ell -1}\} ~ \subset \prod_{\ell \geq 0} ~ \G_{\ell }/C_{\ell} \  .  $$


 \vfill   
}
 
 
 
\frame
{
  \frametitle{}

{\bf Proposition:} The relative Steinitz order satisfies
\begin{eqnarray*}
\xi(\whGamma_{\infty} : \whD_{\infty}) & = &   \lcm \ \{ \# X_{\ell} \mid \ell = 1, 2, \cdots \}\\
& = &   \lcm \ \{ \deg \{ \ovp_{\ell} \colon M_{\ell} \to M_0\} \mid \ell = 1, 2, \cdots \}
\end{eqnarray*}

That is, the Steinitz order of a tower of coverings is equal to a relative Steinitz order of profinite groups associated to the tower.

 \bigskip
 
 We give an example in the case of Vietoris solenoids.


 \vfill   
}
 
 
  

\frame
{
  \frametitle{}

  Choose a sequence of integers $\vec{m} = (m_1, m_2, \ldots)$ with $m_{\ell} > 1$.
  
  Form the chain of coverings of the circle
  
$$
 \xymatrix{
\mS^1    &   \ar[l]_{m_1}  \mS^1    &  \ar[l]_{m_2}  \mS^1   &  \ar[l]_{m_3} \mS^1        &  \ar[l]_{m_4}       \cdots     
} 
$$
$$
 \cS(\vec{m})  = \lim_{\longleftarrow} ~ \{ m_{\ell} \colon \mS^1 \to \mS^1\}  \subset \prod_{\ell \geq 0} ~ \mS^1
$$
{\bf Definition:} The type $\tau[\cS(\vec{m})]$ is the asymptotic equivalence class of $\xi(\cS(\vec{m})) = \lcm \{ m_1   m_2 \cdots m_{\ell} \mid  \ell > 0\}$.


For example, the type of the Smale solenoid is $\{2^{\infty}\}$.

\bigskip

 
  {\bf Theorem.} [Bing, 1960],  [McCord, 1965]
   $1$-dimensional solenoids $\cS(\vec{m})$ and $\cS(\vec{m}')$ are homeomorphic  if  and only if $\xi(\vec{m}) \mor \xi(\vec{m}')$.
   
      
   \medskip
   
   
$\Longrightarrow$ ~ 
 A Vietoris solenoid is completely determined up to homeomorphism by the type $\tau[\cS(\vec{m})]$.


 \vfill   
}
 
 
 
  
\frame
{
  \frametitle{}


 The integer Heisenberg group  is the simplest non-abelian nilpotent group,  represented as the upper triangular matrices in ${\rm GL(3,\mZ)}$.  
$$
\G =   \left\{  \left[ {\begin{array}{ccc}
   1 & a & c\\
   0 & 1 & b\\
  0 & 0 & 1\\
  \end{array} } \right] \mid a,b,c  \in \mZ\right\} .
$$


{\bf Theorem:}
Given any function $\chi  \colon \Pi \to \{0,1,\ldots, \infty\}$, there is a minimal equicontinuous action $(\fX,\G,\Phi)$ of the Heisenberg group with Steinitz order  $\xi[\fX,\G,\Phi]$ having characteristic function $\chi$.

\medskip
The proof uses the fact that a profinite nilpotent group is isomorphic to the product of its Sylow $p$-subgroups, and the judicious construction of subgroup chains in $\G$.

$\star$ ~ Hurder \& Lukina, {\it Prime spectrum and dynamics for nilpotent Cantor actions}, {arXiv:2305.00896}


 \vfill   
}
 
 
 
  

  
 \section{Gap Labeling}


\frame
{
  \frametitle{}

``Gap Labeling'' refers to the evaluation of index classes for elliptic operators on the tiling space for an aperiodic tiling on $\mR^n$

Bellisard gave talks on   ``Gap-Labeling'' in the 1980's and proposed the problem of showing that the labeling system was exactly a subgroup of $\mR$ determined by the dynamics of the foliation.

\medskip 
{\small

$\star$ ~ Bellissard,  {\it Gap labeling theorems for {S}chr\"{o}dinger operators},
  {\bf From number theory to physics ({L}es {H}ouches, 1989)}, {Springer}, 1992.
  
  \smallskip
        
         $\star$ ~   Benameur \&  Oyono-Oyono,
{\it Gap-labelling for quasi-crystals (proving a conjecture by {J}. {B}ellissard)},
 {\bf Operator algebras and mathematical physics ({C}onstan\c{t}a, 2001)},
 {Theta, Bucharest}, {2003}.

 \smallskip

 $\star$ ~   Kaminker \&  Putnam,
 {\it A proof of the gap labeling conjecture}, {\bf Michigan Math. J.},  51:537--546, 2003.
    
     \smallskip

    
 $\star$ ~  Bellissard,  Benedetti \& Gambaudo,
  {\it Spaces of tilings, finite telescopic approximations and gap-labeling},
 {\bf Comm. Math. Phys.}, 261:1--41, 2006.
 
 }
 
 \vfill   
}
 
\frame
{
  \frametitle{}

Again, consider a solenoidal manifold $\cS_{\cP}$ which determines a nested chain of subgroups of finite index, 
 $$\G = \G_0 \supset \G_1 \supset \cdots \supset \G_{\ell} \supset \cdots$$
Define the finite set  $X_{\ell} = \G/\G_{\ell}$ with projections $\ovp_{\ell} \colon X_{\ell} \to X_{\ell -1}$,    
then we have the Cantor space 
$$
X_{\infty}  \equiv \lim_{\longleftarrow} ~ \{ \ovp_{\ell } \colon X_{\ell } \to X_{\ell -1}\} ~ \subset \prod_{\ell \geq 0} ~ X_{\ell} \  .  
$$
There is a unique $\G$-invariant  probability measure $\mu$ on $X_{\infty}$ 

which induces a trace ${\rm Tr}_{\mu} \colon K_0(C(X_{\infty})) \to \mR$.

 
 \vfill   
}
 
\frame
{
  \frametitle{}

For the Cantor dynamical system $(X_{\infty}, \G, \Phi)$ define: 
$$\xi(\whGamma_{\infty} : \whD_{\infty})^{-1}   = \varinjlim  \ \{\frac{1}{m} \mQ \mid m \ {\rm divides} \ \xi(\whGamma_{\infty} : \whD_{\infty})\} \subset \mQ \ .$$

\bigskip

 
{\bf Theorem:}  
$\ds {\rm Tr}_{\mu} \colon K_0(C(X_{\infty})) \to \xi(\whGamma_{\infty} : \whD_{\infty})^{-1} \subset \mR$.

\medskip

The proof follows from the observation that an idempotent in $C(X_{\infty})$ has support in a clopen subset of $X_{\infty}$, and a clopen set in $X_{\infty}$ is a finite union of adapted clopen sets.

\bigskip

The Gap Labeling Theorems are   extensions of this result from  $C(X_{\infty})$ to the cross product algebra $C^*(\F_{\cP}) \cong C(X_{\infty}) \rtimes \G$.
 \vfill   
}
 
\frame
{
  \frametitle{}



 Let $\cS_{\cP}$ be solenoidal manifold with even-dimensional leaves, with projection $p_0 \colon \cS_{\cP} \to M_0$.
 
 \medskip
 
Assume that $M_0$ is a ${\rm Spin}_{\mC}$-manifold, with Dirac operator $\cD$.

\medskip

 Then $\cD$ lifts to a  leafwise Dirac operator  $\cD_{\F}$ for $\F_{\cP}$.
 
 \medskip
 
 Given a K-theory class $[E] \in K^0(\cS_{\cP})$ we can form the pairing $\cD_{\F} \otimes [E]$ then take the leafwise  index class $${\rm Ind}_{\F}(\cD_{\F} \otimes [E]) \in K_0(C^*(\F_{\cP})) \ .$$
 
 The  invariant measure $\mu$ on $X_{\infty}$ induces a holonomy invariant transverse measure  $\mu_{\F}$ for $\F_{\cP}$, which defines a trace    
 $${\rm Tr}_{\mu} \colon C^*(\F_{\cP}) \to \mR \ .$$ 
 
 
 
 \medskip
 
  
 \vfill   
}
 
\frame
{
  \frametitle{}

{\bf Theorem:} (Gap Labeling for Solenoidal Manifolds)
$${\rm Tr}_{\mu}({\rm Ind}_{\F}(\cD_{\F} \otimes [E])) \in \xi(\whGamma_{\infty} : \whD_{\infty})^{-1} \subset \mR$$

\bigskip

  The basic idea of the proof is to use 
  $$K^0(\cS_{\cP}) = K^0( \ \varprojlim \,\{  p_{\ell} \colon  M_{\ell} \to M_{\ell -1}\}) =  \varinjlim K^0(M_{\ell})$$
  so that $[E] \in K^0(\cS_{\cP})$ is the lift to $\cS_{\cP}$ of a class $[E_{\ell}] \in K^0(M_{\ell})$.
 
 The index pairing ${\rm Ind}(\cD_{\ell} \otimes [E_{\ell}]) \in K_0(M_{\ell})$ and the result follows.
 
 
 \vfill   
}
 
\frame
{
  \frametitle{}

 
 There is an analogous result  for odd-dimensions:
 
 {\bf Theorem:} (Gap Labeling for Spectral Flow)
 $${\rm Tr}_{\mu} \circ {\rm Ind}_{\F} \colon K_1(\cS_{\cP}) \otimes K^1(\cS_{\cP}) \to \xi(\whGamma_{\infty} : \whD_{\infty})^{-1} \ .$$

 \bigskip
 
 {\bf Remark:} 
Previous  work proved that the odd foliation index theorem measures the spectral flow for the leafwise Dirac operator.
      The above result is an analog for towers of coverings of the results for flat ${\rm U(n)}$-bundles in the work:

  
 $\star$ ~ Hurder,   {\it Eta invariants and the odd index theorem for coverings},
 {\bf Geometric and topological invariants of elliptic operators, 1988)},
  {Contemp. Math.}, Vol. 105, Amer. Math. Soc.,  1990.
        
        
   \vfill   
}
 
 

 \section{Spectra}

 
\frame
{
  \frametitle{}

{\bf Aside:} Let $M$ be a closed Riemannian manifold,  with basepoint $x_0 \in M_0$ and fundamental group $\G = \pi_1(M_0, x_0)$.
Each $\gamma \in \G$ is represented by a closed curve $c_{\gamma} \colon \mS^1 \to M_0$ with $c_{\gamma}(0) = x_0$. 

\medskip

Let $\|\gamma\|$ denote the shortest curve in the free homotopy class of $c_{\gamma}$. 

\medskip

{\bf Definition:} The unmarked  length spectrum of $M$ is the collection
$$\sigma(M) = \{ \|\gamma\| \mid \gamma \in \G\}$$.
 
 The collection $\sigma(M)$ does not determine $M$ up to isometry, even in the case where $\Sigma_g$ is a Riemann surface with genus $g \geq 2$,  though it does severely restrict the possibilities.
 
  
   \vfill   
}
 
 
\frame
{
  \frametitle{}

{\bf Question:}  Are there invariants for solenoidal manifolds analogous to the length spectra for compact manifolds?

\bigskip


Let  $\cS_{\cP}$ be a solenoidal manifold with base Riemannian manifold $M$. We are interested in invariants of $\cS_{\cP}$ up to homeomorphism. 

In many cases, the leaves of $\F_{\cP}$ are simply connected, so it does not make sense to consider minimal length closed curves in $\cS_{\cP}$.

   \vfill   
}
 
 
\frame
{
  \frametitle{}


The space  $\cS_{\cP}$ is compact, so the lift of a geodesic $\sigma \colon \mR \to \cS_{\cP}$ does satisfy a recurrence property, so one can consider the length of approximately closed paths. 

\medskip

Since the length of a curve in $\cS_{\cP}$ is not a diffeomorphism invariant, we must measure the ``return times'' to an open set in the transversal $\fX$ in terms of an algebraic quantity that is invariant.

\bigskip

This is the idea behind the \emph{typeset spectrum} of $\cS_{\cP}$.

   \vfill   
}
 
 
\frame
{
  \frametitle{}
Consider a solenoidal manifold $\cS_{\cP}$ which determines a nested chain of subgroups of finite index, 
 $$\G = \G_0 \supset \G_1 \supset \cdots \supset \G_{\ell} \supset \cdots$$
with the associated  group action $\Phi \colon \G \times X_{\infty} \to X_{\infty}$.

\medskip
Obtain a homomorphism $\Phi \colon \G \to \Phi(\G) \subset \whGamma_{\infty} \subset \Homeo(X_{\infty})$ and the isotropy subgroup $\whD_{\infty} \subset \whGamma_{\infty}$.

\medskip

For $\whgamma \in \whGamma$,  the  closure of the orbit, $X_{\whgamma} = \overline{\{\whgamma^n \cdot x \mid n \in \mZ\} }$, has an action of $\mZ$, and so by the suspension construction we obtain  a ``solenoidal 1-cycle'', $\phi_{\whgamma} \colon \fT_{\whgamma} \to \cS_{\cP}$.

\medskip
The relative Steinitz order $\xi(\whGamma_{\infty} : \whD_{\infty}, \whgamma)$ is the Steinitz order of the 1-dimensional solenoid  $\fT_{\gamma}$.

 
 \vfill   
}
 
 
\frame
{
  \frametitle{}


{\bf Definition:} The \emph{typeset spectrum} of $\cS_{\cP}$ is the collection of types
$$\tau[\cS_{\cP}] = \{ \tau[\whGamma_{\infty} : \whD_{\infty}, \whgamma] \mid \whgamma \in \whGamma\} \  .$$

 \bigskip
 
 {\bf Question:} 
 What aspects of the non-commutative geometry of a solenoidal manifold $\cS_{\cP}$ are determined by its type spectrum $\tau[\cS_{\cP}] $?
 
   
 
   
 \vfill   
}
 
 
\frame
{
  \frametitle{}

 The solenoidal manifolds with base the $n$-torus $\mT^n$ are determined up to homeomorphism  by a dense subgroup of $\mQ^n$. 
 
 $\star$ ~  Giordano, Putman \& Skau,
  {\it {$\mathbb Z^d$}-odometers and cohomology},
  {\bf Groups Geom. Dyn.}, 13:909--938, 2019.  

\medskip

The notion of typesets was introduced by Baer for the classification of dense subgroups of ${\mQ}^n$, and studied in depth in works by D.~Arnold, M.C.R.~Butler and L.~Fuchs. 

For $n > 1$ this problem is ``unsolvable'' in the sense of Descriptive Set Theory, but the special case of Butler subgroups are reducible to a countable equivalence using the typeset invariants above:

 $\star$ ~   Thomas,
 {\it The classification problem for finite rank {B}utler groups},
   in  {\bf Models, modules and abelian groups}, {Walter de Gruyter, Berlin}, 2008

 



 \vfill   
}
 
 
\frame
{
  \frametitle{}

Let $\cS_{\cP}$ be a solenoidal manifold with base manifold  $M = \mT^n$. Then for each $\gamma \in \mZ^n \subset \whGamma$ we obtain a  solenoidal 1-cycle  $\phi_{\gamma} \colon \fT_{\gamma} \to \cS_{\cP}$ with a foliation denoted by $\F_{\gamma}$.
 
The leafwise Dirac operator $\partial_{\gamma}$ on the 1-dimensional solenoid  $\fT_{\gamma}$ defines an odd index class $[\partial_{\gamma}] \in K_1(\F_{\gamma})$. The foliation index theorem for odd degree operators   yields a pairing 
$${\rm Tr}_{\mu} \circ {\rm Ind}_{\F_{\gamma}} \colon  K_1(\F_{\gamma}) \times K^1(\cS_{\cP}) \to \mR$$

{\bf Theorem:} For each $[u] \in K^1(\cS_{\cP})$ we have
$${\rm Tr}_{\mu} \circ  {\rm Ind}_{\F_{\gamma}}(\partial_{\gamma} \otimes [u]) \in \tau[\cS_{\cP}]$$
\medskip


 \vfill   
}
 
    

\frame
{
  \frametitle{}


{\bf Definition:} The \emph{non-commutative type}  of $\gamma \in \G$ is the collection 
$$\tau_{nc}[\cS_{\cP}, \gamma] = \tau[\{ {\rm Ind}_{\F_{\gamma}}(\partial_{\gamma} \otimes [u]) \mid [u] \in K^1(\cS_{\cP})\}] \  .$$

The \emph{non-commutative typeset} of $\cS_{\cP}$ is the collection of types:
$$\tau_{nc}[\cS_{\cP}] = \{ \tau[\{ {\rm Ind}_{\F_{\gamma}}(\partial_{\gamma} \otimes [u]) \mid [u] \in K^1(\cS_{\cP})\}] \mid \gamma \in \G \} \  .$$

\bigskip

{\bf Problem:} Study the  properties of the non-commutative types for solenoidal manifolds.

 

 \vfill   
}
 
    

\frame
{
  \frametitle{}

{\bf Final Remark:}  For $M = \Sigma_2$ a Riemann surface of genus $g =2$, for any $m \geq 1$, the fundamental group $\G_g$ contains a subgroup of finite index which is isomorphic to the free group on $m$-generators, $\mF_m \cong \mZ \star \cdots \star \mZ$.

\medskip

Given any minimal equicontinuous Cantor action $(\fX, \G, \Phi)$ where $\G$ is finitely generated with $m$-generators, then there is a surjection $\mF_m \to \G$. We can then then use this to define a finite covering $M_0$ of $\Sigma_2$ over which there is a solenoidal manifold $\cS_{\cP}$ with base manifold $M_0$ whose associated monodromy action is   $(\fX, \G, \Phi)$.

 \vfill   
}
 
    

\frame
{
  \frametitle{}


{\bf Theorem:} Every minimal equicontinuous Cantor action $(\fX, \G, \Phi)$ with $\G$   finitely generated can be realized by a solenoidal manifold with base $\Sigma_2$. 

\bigskip

{\bf Corollary:} Every type and typeset invariants for Cantor actions can be realized by 2-dimensional solenoidal manifolds.

\bigskip
\bigskip

{\it Thank you for your attention!}


 \vfill   
}
 
    


\end{document}

