% Topology Seminar - UWM
% Monday, April 17, 2023

% slides for talk "Homeomorphisms of solenoidal manifolds"
 
 
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\date[April 17, 2023]


  
\title[Solenoidal manifolds]{Homeomorphisms of solenoidal manifolds}


\author{Steve Hurder\\ {\footnotesize April 17, 2023}}

\institute[UIC] {University of Illinois at Chicago\\www.math.uic.edu/$\sim$hurder}

  
 
\begin{document}

\frame{\titlepage} % # 1


    
\frame % 1
{
  \frametitle{ }
  
  A report on   joint works with 
  
  Olga Lukina, Mathematical Institute,  Leiden University
  
 

  \bigskip
  {\footnotesize 
 $\star$~  {Clark \& Hurder},
  {\it Homogeneous matchbox manifolds},
  {\bf Trans. A.M.S.}, 365, 2013.

 $\star$~ {Clark, Hurder \& Lukina},
  {\it Classifying matchbox manifolds},
  {\bf Geom. \& Top.},  23, 2019

 $\star$~ {Dyer, Hurder \& Lukina},
  {\it Molino theory for matchbox manifolds},
  {\bf Pac. J. Math.}, 289, 2017

 $\star$~ {Hurder \& Lukina},
  {\it Wild solenoids},
  {\bf Trans. A.M.S.}, 371, 2019

 $\star$~ {Hurder, Lukina \& van Limbeek},
 {\it Cantor dynamics of renormalizable groups},
 {\bf Groups, Geom., and Dynamics}, 15, 2021

 $\star$~ {Hurder \& Lukina},
  {\it  Type invariants for  solenoidal manifolds},
  {preprint}, 2023
}



    \vfill   

}


  \section{Example}
 


\frame % 1
{
  \frametitle{ }
 
  A \emph{continuum} is a compact, connected,   non-empty
metric space. 

\medskip

Examples include compact manifolds, finite CW complexes, laminations of compact manifolds, Hawaiian Earrings, etc

\medskip

A topological  space $X$ is \emph{homogeneous} if for
every $x, y \in X$, there exists a homeomorphism $h \colon X \to X$
such that $h(x) = y$. 

\bigskip

{\bf Theorem.} [Bing, 1960] 
Let $X$ be    a homogeneous, circle-like
continuum that contains an arc. Then either $X$ is homeomorphic to a circle, or to   a Vietoris solenoid.
 
\bigskip

  \vfill   

}



\frame % 1
{
  \frametitle{Vietoris solenoids}
  
  Choose a sequence of integers $\vec{m} = (m_1, m_2, \ldots)$ with $m_{\ell} > 1$.
  
  Form the chain of coverings of the circle
  
      \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}       \cdots     
} 
\end{align*}
\begin{align*}   
 \cS(\vec{m})  = \lim_{\longleftarrow} ~ \{ m_{\ell} \colon \mS^1 \to \mS^1\}  \subset \prod_{\ell \geq 0} ~ \mS^1
\end{align*}

 \hspace{100pt} \includegraphics[width=0.3\textwidth]{Smale-Williams_Solenoid_Large} 

\vfill

}

 

\frame % 1
{
  \frametitle{ }

 The \emph{supernatural number}, or \emph{Steinitz number}, defined by $\vec{m}$
   \begin{equation*}
\xi(\vec{m}) = \lcm \{ m_1   m_2 \cdots m_{\ell} \mid  \ell > 0\} \ ,
\end{equation*}
  $\lcm$ 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 % 1
{
  \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 $\xi \mor \xi'$

The type associated to a Steinitz number  $\xi$ is the asymptotic equivalence class of $\xi$,     denoted by  $\tau[\xi]$.

{\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 % 1
{
  \frametitle{ }
  
  {\bf Theorem.} [Bing, 1960]
   $1$-dimensional solenoids $\cS(\vec{m})$ and $\cS(\vec{m}')$ are homeomorphic  if  $\xi(\vec{m}) \mor \xi(\vec{m}')$.
   
   \bigskip
   
 {\bf Theorem.} [McCord, 1965]
   If $1$-dimensional solenoids $\cS(\vec{m})$ and $\cS(\vec{m}')$ are homeomorphic  then  $\xi(\vec{m}) \mor \xi(\vec{m}')$.
   
   \bigskip
   
 {\bf Conclusion:}
 A Vietoris solenoid is completely determined up to homeomorphism by the type $\tau[\xi(\vec{m})]$.

\bigskip

{\bf Exercise.} Write down two strings $\vec{m}$ and $\vec{m'}$ with $\xi(\vec{m}) \mor \xi(\vec{m'})$.

Explicitly construct a homeomorphism between $\cS(\vec{m})$ and $\cS(\vec{m}')$.

 \vfill   

}



 
  \section{Towers of Covers}
 
 
\frame % 1
{
  \frametitle{ }
  
{\bf Theorem.} [Clark \& H, 2013]    Let $X$ be a homogeneous continuum, and suppose that every for every $x \in X$, the connected component of a neighborhood of $x$ is an $n$-disk.   Then $M$ is homeomorphic to a McCord solenoid of dimension $n$. 

\bigskip

There is a \emph{presentation} $\cP$ consisting of:

$\star$~ $M_{\ell}$ is   compact, closed, connected, $n$-dimensional manifold,

$\star$~ $p_{\ell+1} \colon M_{\ell +1} \to M_{\ell}$ is a   \emph{proper} covering map.
$$M \cong \cS(\cP) = \lim_{\longleftarrow} ~ \{p_{\ell+1} \colon M_{\ell +1} \to M_{\ell}\}  \subset \prod_{\ell \geq 0} ~ M_{\ell}$$
$M$ is called  

$\bullet$~ a \emph{weak solenoid} (McCord) or

$\bullet$~  a \emph{solenoidal manifold} (Sullivan)
  \vfill   

}


  
 
\frame % 1
{
  \frametitle{ }
  
  {\bf Questions:} 
  
  \medskip
  
  (1) What is $\cS(\cP)$, for a general covering sequence  $\cP$?


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

 $\star$~  {A.~Verjovsky}, {\it Low-dimensional solenoidal manifolds}, {arXiv:2203.10032v2}

 \bigskip
  

(2) When are $\cS(\cP)$ and $\cS(\cP')$ homeomorphic? 

 \medskip
  

(3) What is the structure of $\Homeo(\cS(\cP))$?
 
 \bigskip
 
 

   \vfill   

}


  
 
\frame % 1
{
  \frametitle{ }
  
 
$q_{\ell} \colon M_{\ell} \to M_0$ is covering map, $m_{\ell} = {\rm degree}(q_{\ell})$

$\xi(\cP)  = \lcm \{ m_1   m_2 \cdots m_{\ell} \mid  \ell > 0\}$

\bigskip

   
 {\bf Theorem.} [H \& Lukina, 2023]
If  solenoidal manifolds $\cS(\cP)$ and $\cS(\cP')$ are homeomorphic  then  $\xi(\cP)  \mor  \xi(\cP')$.

\bigskip

{\bf Remark.} For $n \geq 2$, the converse is false in so many ways, and opens the door to many   questions

\bigskip

Need to take a ``deep dive'' into what homeomorphism between solenoidal manifolds means.
 \vfill   

}

  
 
\frame % 1
{
  \frametitle{ }
  
 
  
  Projections $\whp_{\ell} \colon \cS(\cP) \to M_{\ell}$ 
  
  $q_{\ell} = p_{\ell} \circ \cdots \circ p_1 \colon M_{\ell} \to M_0$
  
   Choose a basepoint $x \in \cS(\cP)$,  set $x_{\ell} = \whp_{\ell}(x) \in M_{\ell}$

Fundamental groups  $\pi_1(M_{\ell}, x_{\ell})$

$(q_{\ell})_{\#} \colon \pi_1(M_{\ell}, x_{\ell}) \to \G_{\ell} \subset \G = \pi_1(M_0 , x_0)$

Group chain $\cG(\cP) = \{\G \supset \G_1 \supset \G_2 \supset \cdots \}$

 \vfill   

}

  
 
\frame % 1
{
  \frametitle{ }
  


$\bullet$~ Each inclusion $\G_{\ell} \subset \G$ has finite index

$\bullet$~ $\G_{\ell}$ is not assumed to be normal in $\G$

$\bullet$~ $\G$ acts transitively on finite set $X_{\ell} = \G/\G_{\ell}$

$\bullet$~ $C_{\ell} \subset \G_{\ell}$ is kernel of action map $\Phi_{\ell} \colon \G \to \Aut(X_{\ell})$

$\bullet$~ $C_{\ell}$ is normal in $ \G$ 

$\bullet$~  $Q_{\ell} = \G/C_{\ell}$ is finite group, acts on $X_{\ell}$.

 \vfill   

}

  
 
\frame % 1
{
  \frametitle{ }
  
$$
X_{\infty}   \equiv    \lim_{\longleftarrow} ~ \{ p_{\ell +1} \colon X_{\ell +1} \to X_{\ell}  \mid \ell \geq 0 \}    \subset    \prod_{\ell \geq 0} ~ X_{\ell} \  .   
$$


$$
\whG_{\infty}   \equiv   \lim_{\longleftarrow} ~ \{ p_{\ell +1} \colon Q_{\ell +1} \to Q_{\ell}  \mid \ell \geq 0 \}    \subset    \prod_{\ell \geq 0} ~ Q_{\ell} \  .   
$$

The  fiber $q_{\ell}^{-1}(x_0) \subset M_{\ell}$ is 
 identified with  $X_{\ell} = \G/\G_{\ell}$ as a   $\G$-space. 
 
 Action $\Phi \colon \G \times X_{\infty} \to X_{\infty}$ is identified with monodromy of fibration $\whp \colon \cS(\cP) \to M_0$.
  
  
 $\bullet$~  $X_{\infty}$ and $\whG_{\infty}$ are  Cantor sets
  
$\bullet$~   Action $\Phi$ is equicontinuous and minimal - a Cantor action.
  
  
 \vfill   

}


\frame % 1
{
  \frametitle{ }
  
{\bf Theorem.} Suppose that $\cS(\cP)$ and $\cS(\cP')$ are homeomorphic, then   $\Phi \colon \G \times X_{\infty} \to X_{\infty}$ and $\Phi' \colon \G' \times X'_{\infty} \to X'_{\infty}$ are \emph{return equivalent}. 

\bigskip

{\bf Theorem.} [Clark, H \& Lukina, 2019] Suppose that  $\cS(\cP)$ and $\cS(\cP')$ are solenoidal manifolds of the same dimension, and their monodromy actions are return equivalent. If the base manifolds $M_0$ and $M'_0$ are \emph{strongly Borel} (ie all finite coverings satisfy the Borel Conjecture), and each space contains a simply connected leaf, then $\cS(\cP)$ and $\cS(\cP')$  are homeomorphic.

\bigskip

{\bf Remark.} There are examples which show this is about as sharp of converse as can be expected.

$\star$~ The study of the homeomorphism problem   reduces to the study return equivalence between minimal equicontinuous Cantor actions.

 \vfill   

}


 
  \section{Dynamics}
 
 
\frame % 1
{
  \frametitle{ }
  
  Let      $\Phi \colon \G \times \fX \to \fX$   a minimal equicontinuous Cantor action

 {\bf Properties:}
 
  $U \subset \fX$ clopen means closed and open. 
  
  $U$ \emph{adapted} if $U \ne \emptyset$, and  for all $\gamma \in \G$,  $\gamma \cdot U \cap U \ne \emptyset$ then $\gamma \cdot U = U$
  
  $\G_U = \{\gamma \in \G \mid \gamma \cdot U = U\}$
  
  $\G_U$ is subgroup of finite index in $\G$, as $\G$ translates $U$ to give a partition of $\fX$
      
   $\G_U$ defines finite covering of $M_0$ when $\G = \pi_1(M_0, x_0)$
   
   \bigskip
   
   {\bf Fact.} Action $\Phi$ is equicontinuous iff action of $\G$ on collection of clopen subsets has finite orbits.
   
$\Rightarrow$ ~ adapted clopen sets form a subbasis for the topology of $\fX$.
    \vfill   

}

 
 
\frame % 1
{
  \frametitle{ }

 {\bf Definition.} Actions  $\Phi \colon \G \times \fX \to \fX$ and  $\Phi' \colon \G' \times \fX' \to \fX'$ are \emph{return equivalent} if there exists adapted sets $U \subset \fX$ and $U' \subset \fX'$ and homeomorphism $h \colon U \to U'$ that conjugates the subgroups
 \begin{eqnarray*}
\cH(U) & = & {\rm Image}\{\Phi_U \colon \G_U \to  \Homeo(U)\} \\
\cH'(U') & = &  {\rm Image}\{\Phi'_{U'} \colon \G'_{U'} \to  \Homeo(U')\}
\end{eqnarray*}

 The subtlety in this definition is that the map $\Phi_U \colon \G_U \to \cH(U)$ may have kernel, and likewise for $\Phi'_{U'}$.
 
That is, return equivalence is \emph{lossy}.
 
 
  \vfill   

}


 
\frame % 1
{
  \frametitle{ }

{\bf Definition.} Define the group of germs of return equivalences
$$\cE_x(\fX, \G, \Phi) = \{h \colon U \to V \mid x \in U, V \subset \fX  \} / \sim  $$

 

{\bf Example.} Let $\fG(\G)$ be the full profinite completion of $\G$ with respect to all normal subgroups of  finite index,   basepoint $\whe \in \fG(\G)$.
 
 \medskip
 
 {\bf Theorem.}   $\cE_{\whe}(\fG(\G), \G, \Phi) \cong {\rm Comm}(\G)$,  
 where ${\rm Comm}(\G)$ is the group of abstract commensurators of $\G$.
 
 \medskip
 
  $\star$~ {E.~Bering, IV and D.~Studenmund},
 {\it Topological Models of Abstract Commensurators}, {arXiv:2108.10586v1}.

\medskip

{\bf Problem.} Calculate $\cE_x(\fX, \G, \Phi)$ in general.


  \vfill   

}


 
\frame % 1
{
  \frametitle{ }

The action $\Phi$ is equicontinuous, so isometric on $\fX$

$\Rightarrow$ ~ closure $\fG(\Phi) = \overline{\Phi(\G)} \subset \Homeo(\fX)$ is profinite group
    
Action $\Phi$ is minimal $\Rightarrow$  induced action $\whPhi \colon \fG(\Phi) \times \fX \to \fX$ is transitive

\medskip
Isotropy group $\fD_x = \{ \whphi \in \fG(\Phi) \mid \whphi(x) = x\}$ is closed subgroup

\medskip

$\fD_x$ is called \emph{discriminant} of action by H \& Lukina, or \emph{parabolic subgroup} by Nekrashevych

\bigskip

{\bf Guiding Principle.} 
Dynamics of action $\Phi \colon \G \times \fX \to \fX$  determined by   action of $\fD_x$ on $\fX$

 \vfill   

}


 
\frame % 1
{
  \frametitle{ }

The action $\Phi$ is:

$\bullet$~ \emph{free} if $\fD_x$ is identity $\Rightarrow$ $\fX = \fG(\Phi)/\fD_x$ is group
    
$\bullet$~ \emph{topologically free} if $id \ne \whg \in \fX_x$ fixes no open set in $\fX$
    
$\bullet$~ \emph{stable} if $\exists ~ \e > 0$ such that action $\whPhi$ on $\e$-neighborhood of $x$ is topologically free
    
$\bullet$~ \emph{wild} if $\forall ~ \e > 0$ there exists $id \ne \whg \in \fD_x$ and $U \subset \fX$ with $\diam(U) < \e$ so that action $\whPhi(\whg)$ on $U$ fixes   open subset $V \subset U$
    
\bigskip
{\bf Theorem.} The property that an action is wild or stable is an invariant of return equivalence.

\medskip

{\bf Remark.} For each of these cases, the study of $\cE_x(\fX, \G, \Phi)$  has special techniques \& invariants.
 \vfill   

}


 

\frame % 1
{
  \frametitle{ }

\begin{itemize}
\item The  \emph{asymptotic discriminant}  of an equicontinuous action $(\fX,\G, \Phi)$,   distinguishes the actions up to local conjugacy.

\medskip
\item In particular, using the asymptotic discriminant, one divides all actions into  \emph{stable}  and  \emph{wild}.

 \medskip
\item Direct limit group invariants, which distinguish different classes of wild actions.

\medskip
\item Prime spectrum of actions (\emph{type} and \emph{typeset}).


\end{itemize}


 \vfill   

}


\frame % 1
{
  \frametitle{ }

{\bf   Abundance of actions on trees/Cantor sets}

\medskip
{\bf Theorem.} [Dyer, H. \& Lukina, 2017]
Every finite group, and every separable profinite group, can be realized as the stable discriminant of an action of a torsion-free finite index subgroup of ${\rm SL}(n, \mathbb{Z})$, for $n \geq 5$, on a Cantor set.
 
 

\bigskip
{\bf Theorem.} [H. \& Lukina, 2019]
There exists uncountably many wild actions on Cantor sets of the same torsion-free subgroup of ${\rm SL}(n, \mathbb{Z})$, for $n \geq 5$, with distinct asymptotic discriminants.
 

 \vfill   

}


 

\frame % 1
{
  \frametitle{ }


The study of $\cE_x(\fX, \G, \Phi)$    depends on the algebraic structure of $\G$:


$\star$~   $\G$ is torsion-free abelian  
  
$\star$~ $\G$ is torsion-free nilpotent
  
$\star$~ $\G$ is torsion-free arithmetic lattice in higher rank linear group
  
 $\star$~  $\G$ is an automatic group acting on a tree boundary
  
 $\star$~  $\G$ is a branch group acting on a tree boundary
  
 $\star$~  $\G$ is an absolute Galois group with arboreal action on  roots
   
\bigskip

Each of these cases deserves its own discussion.  
 \vfill   

}



 
  \section{Typesets}
 
 
\frame % 1
{
  \frametitle{ }
The notions of type and typeset for $(\fX, \G, \Phi)$ are inspired by the case for $\G$ abelian.
   
   Let   $\cG(\cP) = \{\G \supset \G_1 \supset \G_2 \supset \cdots \}$ be a group chain for action. 
   
   Set $m_{\ell} = \#\{\G_{\ell}/\G_{\ell+1}\}$

Set $\xi(\cG) =   \lcm \{ m_1   m_2 \cdots m_{\ell} \mid  \ell > 0\}$

\bigskip

{\bf Theorem.} [H. \& Lukina, 2023] The type $\tau[\xi(\cG)]$ depends only on the return equivalence class of the action $(X_{\infty} , \G, \phi)$ it determines.


 \vfill   

}



\frame % 1
{
  \frametitle{ }


Let  $\Pi = \{2,3,5, \ldots\}$ denote the set of   primes. 

For   $\ds \xi = \prod_{p \in \Pi} \ p^{\chi(p)}$, define:
\begin{eqnarray*}
\pi(\xi) ~ & = & ~  \{ p \in \Pi \mid  \chi(p) > 0  \}    ,   {\rm prime \ spectrum \ of} \ \xi  \ ;  \\
\pi_f(\xi) ~ & = & ~  \{ p \in \Pi \mid 0 < \chi(p) < \infty \}  ,  {\rm   finite \ prime \ spectrum of} \ \xi \ ;   \\
\pi_{\infty}(\xi) ~ & = & ~  \{ p \in \Pi \mid  \chi(p) = \infty \} \,    {\rm   infinite \ prime \ spectrum \ of} \ \xi \  \end{eqnarray*}

  Note that if $\xi \mor \xi'$, then $\pi_{\infty}(\xi) = \pi_{\infty}(\xi')$. 
  
  The property that $\pi_f(\xi)$ is an \emph{infinite} set is 
   preserved by asymptotic equivalence of Steinitz numbers, so is an invariant of type.
 


 \vfill   

}



\frame % 1
{
  \frametitle{ }

The \emph{typeset} for $(\fX, \G, \Phi)$. 

$\fG(\Phi)$ the profinite closure of the action.

For $\gamma \in \G$ we obtain a subgroup $\langle \gamma \rangle \subset \fG(\Phi)$ whose closure $\cA(\gamma) = \overline{\langle \gamma \rangle}$ is a compact abelian group,

Get an abelian action $(\cA(\gamma), \mZ, \Phi_{\gamma})$
\medskip


{\bf Definition.} $\tau[\gamma] = \tau[\xi(\cA(\gamma), \mZ, \Phi_{\gamma})]$

\bigskip

{\bf Definition.} The \emph{typeset} for  $(\fX, \G, \Phi)$ is the collection of   types
$$\cT[\fX, \G, \Phi] = \{ \tau[\gamma] \mid \gamma \in \G\}$$

{\bf Theorem.} [H. \& Lukina, 2023] The \emph{commensurable equivalence class} of the typeset $\cT[\fX, \G, \Phi]$ is an invariant of the return equivalence class of $(\fX, \G, \Phi)$.

 \vfill   

}



\frame % 1
{
  \frametitle{ }

The type and typeset were introduced for the classification of dense subgroups of $\mQ^n$.

 $\star$~ [Baer, 1937], [Butler, 1965], [Arnold, 1982], [Fuchs, 2015]

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This problem is equivalent, via Pontrjagin Duality, to the classification of profinite groups defined by a   
group chain  $\cG =\{\G_{\ell} \mid \ell > 0\}$   in $\G_0 = \mZ^n$. That is, for the study of homeomorphisms between 
 solenoidal manifolds  with base   $\mT^n$. 
 
 \bigskip
 
 It is   known that this problem is not ``solvable'':
 
 $\star$~ {S.~Thomas},
 {\it The classification problem for torsion-free abelian groups of finite rank},
   {\bf J. Amer. Math. Soc.}, 16, 2003.

  
 
 
   
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The classification problem for dense subgroups of $\mQ^n$  has been solved for   special cases.

{\bf Definition.} $A \subset \mQ^n$ is a \emph{Butler group} if it can be written as a sum $A = A_1 + \cdots + A_s$ of rank $1$ subgroups.
   
  In this case the classification problem is much more tractable:
    
    
  
 $\star$~   {D.~Arnold and C.~Vinsonhaler},
  {\it Isomorphism invariants for abelian groups},
  {\bf Trans. A.M.S.}, 330, 1992.



 $\star$~  {S.~Thomas},
 {\it The classification problem for finite rank {B}utler groups},
   in  {\bf Models, modules and abelian groups},  2008.
 
\bigskip

{\bf Problem.} Find classes of Cantor actions $(\fX, \G, \Phi)$ for which the return equivalence group $\cE(\fX, \G, \Phi)$ can be calculated.

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