% Huntonfest - Durham, UK
% Tuesday, September 9, 2025

% slides for talk "Mapping class groups of solenoidal manifolds"
 
 
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\date[September 9, 2025]


  
\title[Solenoidal manifolds]{Mapping class groups of solenoidal manifolds}


\author{Steve Hurder\\ {\footnotesize September 9, 2025}\\{\footnotesize Joint work with Olga Lukina}}

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

  
 
\begin{document}

\frame{\titlepage} % # 1

  \section{Main}
 
    
\frame % 1
{
  \frametitle{ }
  In the beginning...
  
   {\footnotesize 
 $\star$~  {Fokkink \& Oversteegen}, 
  {\it Homogeneous weak solenoids},
  {\bf Trans. A.M.S.}, 354, 2002.
}



  {\bf Definition:} A continuum $\fM$ is homogeneous if for any $x,y \in \fM$ there is $h \in \Homeo(\fM)$ with $h(x) = y$.
  
  \medskip
  
   


  {\bf Theorem.}  
A weak solenoid $\fM$ is homogeneous iff  it is a normal solenoid (i.e. it is transversally modeled on a profinite group).
 
 
\bigskip
 
 
 {\bf Problem:} Let $\fM$ be a weak solenoid. Are there further relations between  the structure of $\fM$ and $\Homeo(\fM)$?

    \vfill   

}


 


\frame % 1
{
  \frametitle{ }
    $\Homeo^+(\fM)$ denotes the  orientation preserving homeomorphisms.



 Let $\Homeo_0(\fM)$ denote the closed subgroup of $\Homeo^+(\fM)$  which are isotopic to the identity map. This is a \emph{very large} group, and is contractible.  A better problem is then:
 
 {\bf Problem:} Let $\fM$ be a weak solenoid. Calculate the \emph{Mapping Class Group},   $\MCG(\fM)  = \Homeo^+(\fM)/\Homeo_0(\fM)$, and relate it to the structure of $\fM$.

\medskip

{\bf Background:}  Suppose that   $\Sigma_g$  is a closed surface of genus $g \geq 2$.
  
  $\MCG(\Sigma_g)$ is independent of the genus  for $g \geq 2$, and is often called just ``the Mapping Class Group''.   
  The book 
  
   {\footnotesize 
 $\star$~  {Farb \& Margalit}, 
  {\bf A Primer on Mapping Class Groups},
 {Princeton Mathematical Series}, Vol. 49, 2012, 
}

 studies  $\MCG(\Sigma_g)$, and gives many examples and applications.


   \vfill   

}

 
\frame % 1
{
  \frametitle{ }
 
When the compact surface $\Sigma_g$ is replaced by a weak solenoid $\fM$, new techniques are needed.  The goal is to develop methods for calculating $\MCG(\fM)$ when $\fM$ is a weak solenoid.
 
One motivation  for this work comes from:

 {\footnotesize 
 $\star$~  {Odden}, 
  {\it The baseleaf preserving mapping class group of the universal hyperbolic solenoid},
 {\bf Trans. Amer. Math. Soc.}, 357:1829--1858, 2005.
}


  Let  $\whX$ be the universal hyperbolic solenoid. That is, $\whX$ is the solenoid defined by a 
 cofinal   collection $\cP$ in the set of all finite oriented normal coverings of $\Sigma_g$.
   
  
  
Let $\whG$ be the full profinite completion of $\G = \pi_1(\Sigma_g , x)$.
\medskip

{\bf Theorem [Odden]:} $\MCG(\whX) \cong \Comm(\G) \times \whG$  




 

  \vfill   

}

 
  

\frame % 1
{
  \frametitle{ }

 

  {\bf Definition:} Let $\G$ be a countable group.  The \emph{abstract commensurator group} $\Comm(\G)$    is the collection of
isomorphisms $\phi \colon H \to K$ between  finite-index subgroups $H, K \subset \G$, modulo the equivalence   which identifies isomorphisms that agree on a finite-index subgroup $L \subset H \cap K$. 

\medskip

{\bf Remark:} Let $\G' \subset \G$ be a finite index subgroup. Given    a commensurator $\phi \colon H \to K$ in $\Comm(\G)$, observe that $\phi \colon H \cap \G' \to K \cap \G'$ is a commensurator in $\Comm(\G')$. The converse clearly holds, so there is a natural isomorphism $\Comm(\G') \simeq \Comm(\G)$.

\medskip

The group  $\Comm(\G)$ can be intuitively viewed as the group of ``germs''
of isomorphisms between finite-index subgroups of G.   

 \vfill   

}
 
 
\frame % 1
{
  \frametitle{ }

  The work of Odden was extended by 

  {\footnotesize 
 $\star$~  {Belk \& Forrest}, 
  {\it Compact aspherical solenoids},
   https://arxiv.org/abs/1009.5716v4}.

  {\footnotesize 
 $\star$~  {Bering \& Studenmund}, 
  {\it Topological models of abstract commensurators},
{\bf Groups Geom. Dyn.}, 18:1403--1425, 2024. 
}

\smallskip

Let $M$ be a closed aspherical manifold; that is, the universal covering of $M$ is contractible.
Let $\whM$ be the universal solenoid, defined as  the inverse limit of the system of   all finite oriented normal coverings of $M$.
Let $\cE(\whM)$ be the group of homotopy self-equivalences of $\whM$. 
 Let $\whG$ be the full profinite completion of $\G$.

\medskip

{\bf Theorem:} There is an isomorphism $\cE(\whM) \to \Comm(\G) \times \whG$.

\medskip
 
{\bf Problem:} Extend these results to a larger class of solenoidal manifolds, beyond the universal solenoids, and   calculate $\MCG(\fM)$.
 \vfill   

}


 


  \section{Towers}

\frame % 1
{
  \frametitle{}
  What is a solenoidal manifold? 
  
  Here is the most basic example:
  
  Choose a sequence of integers $\vec{m} = (m_1, m_2, \ldots)$ with $m_{\ell} > 1$.
  Form the tower of coverings 
      \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*}   
 \fM_{\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{}
  
 Solenoidal manifolds: choose a tower of finite index coverings:
     \begin{align*}   
 \xymatrix{
\cP  \equiv  &    \{  M_0    &   \ar[l]_{p_1}  M_1    &  \ar[l]_{p_2}  M_2   &  \ar[l]_{p_3} M_3        &  \ar[l]_{p_4}       \cdots    \}  
} 
\end{align*}
$\star$~ For $\ell \geq 0$, $M_{\ell}$ is   closed, connected, $n$-dimensional manifold,

$\star$~ $p_{\ell+1} \colon M_{\ell +1} \to M_{\ell}$ is a   \emph{proper} covering map.
$$\fM_{\cP} = \lim_{\longleftarrow} ~ \{p_{\ell+1} \colon M_{\ell +1} \to M_{\ell}\}  \subset \prod_{\ell \geq 0} ~ M_{\ell}$$
${\widehat{q}}_0 \colon \fM_{\cP} \to M_0$ is fibration defined as concatenation of the  maps $\{p_{\ell} \mid \ell \geq 0\}$, with  fiber $\fX = {\widehat{q}}_0^{-1}(x_0)$   a Cantor space.
$\fM_{\cP}$ is:   

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

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

$\bullet$~ \emph{normal}   if every composition $M_{\ell} \to M_0$ is a normal covering, which implies that  $\fX$ is a profinite group.

 \vfill

}
 

\frame % 1
{
  \frametitle{ }
  
  {\bf First Basic Fact:} There is a canonical homeomorphism between $\fM_{\cP}$ and $\ds \fM_{\cP_k} = \lim_{\longleftarrow} ~ \{p_{\ell+1} \colon M_{\ell +1} \to M_{\ell} \ , \ \ell \geq k\}$.
   
   
 {\bf Second Basic Fact:}
 
  {\footnotesize 
 $\star$~  {Rogers \& Tollefson}, 
  {\it Homeomorphisms homotopic to induced homeomorphisms of weak solenoidal spaces},
  {\bf Colloq. Math.} 25:81--87, 1971/72.
}

  {\bf Theorem:} Let $\cP$ be a tower of coverings, and   $M_0$   a \emph{strongly Borel} manifold. Then a self-homeomorphism $h \colon \fM_{\cP} \to \fM_{\cP}$ is isotopic to a self-homeomorphism  induced by a \emph{shuffle map}, where $i_{\ell} \geq \ell$ is an increasing sequence, and each $q_{i_\ell}$ is a covering map:
 \begin{align*} \xymatrix{ M_{0}   & \ar[l] M_{i_0} \ar[ld]^{q_{i_0}}& \ar[l] M_{i_1} \ar[ld]^{q_{i_1}} & \ar[l] M_{i_2} \ar[ld]^{q_{i_2}}  & \ar[l] \cdots \\
M_0 & \ar[l] M_1  & \ar[l] M_2 & \ar[l] M_3   & \ar[l]\cdots } \end{align*}

 \vfill   

}

  
 
  
   
\frame % 1
{
  \frametitle{ }
  
Pointed maps between towers of coverings are determined by maps between fundamental groups.
The problem is   to understand maps between fundamental groups in a chain of subgroups.

 Given a presentation $\cP$ there are fibration maps   $\whp_{\ell} \colon \fM_{\cP} \to M_{\ell}$ and covering maps    
 $q_{\ell} = p_{\ell} \circ \cdots \circ p_1 \colon M_{\ell} \to M_0$.

 
    Choose a basepoint $\whx \in \fM_{\cP}$,  set 
    
    $\bullet$~ $x_{\ell} = \whp_{\ell}(\whx) \in M_{\ell}$ 

      $\bullet$~  $\G_{\ell} = (q_{\ell})_{\#} \{ \pi_1(M_{\ell}, x_{\ell})\} \subset \G = \pi_1(M_0, x_0)$
  
   
    $\bullet$~  $\cG_{\cP} = \{\G \supset \G_1 \supset \G_2 \supset \cdots \}$ 

\medskip
{\bf Question:} How to  construct shuffle maps for group chains?

If the shuffle maps are all inclusions, then the First Basic Fact implies the induced map of $\fM_{\cP}$ is the identity. So one needs the shuffle   maps to act non-trivially on fundamental groups.

 \vfill   

}

     
    \section{Commensurators}

  
 
\frame % 1
{
  \frametitle{ }
  

 The  fiber $q_{\ell}^{-1}(x_0) \subset M_{\ell}$ is 
 identified with  $X_{\ell} = \G/\G_{\ell}$ as a   $\G$-space. 

$\bullet$~ $\G$ 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, acting transitively  on $X_{\ell}$.
$$
X_{\cP}   \equiv    \lim_{\longleftarrow} ~ \{ p_{\ell +1} \colon X_{\ell +1} \to X_{\ell}  \mid \ell \geq 0 \}    \subset    \prod_{\ell \geq 0} ~ X_{\ell} \  .   
$$
$$
\whG_{\cP}   \equiv   \lim_{\longleftarrow} ~ \{ p_{\ell +1} \colon Q_{\ell +1} \to Q_{\ell}  \mid \ell \geq 0 \}    \subset    \prod_{\ell \geq 0} ~ Q_{\ell} \  .   
$$
   $\Phi \colon \G \times X_{\cP} \to X_{\cP}$ isomorphic to  monodromy of   ${\widehat{q}}_0 \colon \fM_{\cP} \to M_0$.
  
 Action is a (generalized) \emph{odometer}. That is, it is a minimal, equicontinuous action on a Cantor space.

 \vfill   

}

  
\frame % 1
{
  \frametitle{ }
  
 
 Clopen set  $U \subset X_{\cP}$ is \emph{adapted} if $U \ne \emptyset$, and  for all $\gamma \in \G$,  $\gamma \cdot U \cap U \ne \emptyset$ then $\gamma \cdot U = U$
  
  $\G_U = \{\gamma \in \G \mid \gamma \cdot U = U\}$   is a subgroup of finite index in $\G$.
  
 \medskip
   
 {\bf Definition.} Odometers  $\Phi \colon \G \times \fX \to \fX$ 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 $h_U \colon U \to U'$ 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   map $\Phi_U \colon \G_U \to \cH_U$ may have kernel, and likewise for $\Phi'_{U'}$.
 
That is, return equivalence loses some of the information about the group action, and it is necessary to control this loss. 


   \vfill   

}

 
 
\frame % 1
{
  \frametitle{ }
  
  {\bf Definition:}
 Let  $(\fX, \G, \Phi)$   be an odometer action, and $\whx \in \fX$. 
 
 \smallskip
 
$\bullet$~ A \emph{dynamical commensurator}   is a homeomorphism $h_U \colon U \to V$, where $U$ and $V$ are adapted subsets with $\whx \in U \cap V$, such that $h_U(\whx) = \whx$, and $h_U$ induces an isomorphism $\Theta_U \colon \cH_U \to \cH_V$.

\smallskip
$\bullet$~  Commensurators $h_U \colon U \to V$ and $h_{U'} \colon U' \to V'$ are equivalent if there exists and adapted set $U''$ with $\whx \in U'' \subset U \cap U'$ such that $h_U|U'' = h'_{U'}|U''$. We   write $(h_U,U,V) \comm (h'_{U'},U',V')$.


\smallskip
The \emph{dynamical commensurator group} of $(\fX, \G, \Phi)$ at $\whx$ is the set of germs of dynamical commensurators,
$$
\Comm(\fX, \G, \Phi, \whx) = \{ h \colon U \to V \mid \whx \in U \cap V\} \slash  \comm  $$

 The  {dynamical commensurator group} is implicitly studied in
 
   {\footnotesize 
 $\star$~  {Hurder \& Lukina}, 
  {\it Limit group invariants  for non-free Cantor actions},
 {\bf Ergodic Theory Dynam. Systems}, 41:1751-1794, 2021.
 }


  \vfill   

}

  
\frame % 1
{
  \frametitle{ }
 
 
Recall a result on classifying weak solenoids from 
 
   {\footnotesize 
 $\star$~  {Clark, Hurder \& Lukina}, 
  {\it Classifying matchbox manifolds},
{\em Geom. Topol.}, 23(1):1--27, 2019.
}

 
 
\medskip



{\bf Theorem.} Suppose that $\fM_{\cP}$ and $\fM'_{\cP'}$ are   weak solenoids.
  If $\fM_{\cP}$ and $\fM'_{\cP'}$ are homeomorphic,  then their monodromy odometers  $\Phi \colon \G \times X_{\cP} \to X_{\cP}$ and $\Phi' \colon \G' \times X'_{\cP'} \to X'_{\cP'}$ are \emph{return equivalent}. 

 


 
 \medskip
 
 {\bf Corollary:} For $\whx \in \fM_{\cP}$ there is a well-defined map
 $$\sigma_{\cP,  \whx} \colon \MCG(\fM_{\cP}, \whx) \longrightarrow    \Comm(X_{\cP}, \G, \Phi, \whx)$$
 
 \medskip
 
To understand the image of this map, the strategy is to construct an embedding  $\chi_{\Phi} \colon \Comm(X_{\cP}, \G, \Phi, \whx) \longrightarrow \Comm(\G)$.
 \vfill   

}


    \section{Coherence}

 

\frame % 1
{
  \frametitle{ }

Let $\Phi \colon \G \times \fX \to \fX$ be an odometer. The action is:

 $\bullet$~    \emph{quasi-analytic} if for each clopen set $U \subset \fX$, 
  if  the action of $g \in \G$ satisfies $\Phi(g)(U) = U$ and the restriction $\Phi(g) | U$ is the identity map on $U$, 
  then $\Phi(g)$ acts as the identity on $\fX$. 
  
 $\bullet$~   \emph{topologically free} if it is effective, and the quasi-analytic condition holds for $U = \fX$.

 $\bullet$~    \emph{locally quasi-analytic}  if there exists $\e > 0$ such that for any non-empty open set $U \subset \fX$ with $\diam (U) < \e$,  and  for any non-empty open subset $V \subset U$,  if the action of $g \in \G$ satisfies $\Phi(g)(V) = V$ and the restriction $\Phi(g) | V$ is the identity map on $V$,    then $\Phi(g)$ acts as the identity on   $U$.  

 $\bullet$~    \emph{coherent} if for every adapted set $U \subset \fX$, the restricted holonomy action map $\Phi_U \colon \G_U \to Homeo(U)$ has finite kernel.

 \vfill   

}


 
\frame % 1
{
  \frametitle{ }

 
 {\bf Example:} 
If $\G$ is residually finite, and $\whG$ is any profinite completion for which the canonical map $\G \to \whG$ is an embedding, then the odometer action of $\G$ on $\fX = \whG$ is coherent and   free.

\smallskip

{\bf Example:} 
If $\G$ is virtually nilpotent, that is, $\G$ contains a nilpotent subgroup $\G' \subset \G$ with finite index, then  an odometer  action of $\G$ is coherent  and  locally quasi-analytic. 
\smallskip

{\bf Example:}  If $\G$ is a weakly branch group, and $\fX$ is the boundary of the tree on which the group acts, then the odometer action of $\G$ on $\fX$ is inherently not coherent, and not locally quasi-analytic. 

  \vfill   

}


 
\frame % 1
{
  \frametitle{ }
  
  


{\bf Proposition:} Let $\Phi \colon \G \times \fX \to \fX$ be  an effective and coherent odometer action, and  $U \subset \fX$ an adapted set. Then there exists a  subgroup $\G_U' \subset \G_U$ of finite index in $\G$ such that the restriction  $\Phi_U \colon \G_U' \to \Homeo(U)$ is   injective.

{\it Proof.}   The action map $\Phi \colon \G \to \Homeo(\fX)$ is injective.
\smallskip



  $\G_U = \{\gamma \in \G \mid \gamma \cdot U = U\}$   is subgroup of finite index in $\G$.

\smallskip

$K_U = \ker \{ \Phi_U \colon \G_U \to \Homeo(U)\} \subset \G$ is a finite group.

\smallskip

Let $U' \subset U$ be adapted and sufficiently small such that $K_U$ acts effectively on the finite set of translates
$X_U = \{ \gamma \cdot U' \mid \gamma \in \G \}$.

\smallskip

Then $\ds \G'_U =  \bigcap_{\gamma \in \G} \G_{\gamma \cdot U'}$ has finite index in $\G_U$ hence also in $\G$.

 

$\G'_U$ acts trivially on the set of translates $X_{U'}$.
It follows that   the restriction $\Phi_U \colon \G'_U \to \Homeo(U)$ is injective.
\hfill $\Box$




   \vfill   

}


\frame % 1
{
  \frametitle{ }

{\bf Theorem:} Let   $\Phi \colon \G \times \fX \to \fX$ be  an effective, coherent and locally quasi-analytic  odometer.   Let  $\whx \in \fX$. Then there exists an injection 
$$\chi_{\Phi} \colon \Comm(\fX, \G, \Phi, \whx) \longrightarrow \Comm(\G)$$
 

{\it Proof.}
Let $h \colon U \to V$ with $h(\whx) = \whx$, and  $h_*$  conjugates the images   $\cH_U  =\Phi_U(\G_U) \subset \Homeo(U)$ with  $\cH_{V} =\Phi_{V}(\G_V)  \subset \Homeo(V)$.

\smallskip

Let $\G'_{U} \subset \G_U$ be finite index and  $\Phi_U \colon \G'_U \to \Homeo(U)$ is injective

\smallskip

Let $\G'_{V} \subset \G_V$ be finite index and  $\Phi_V \colon \G'_V \to \Homeo(V)$ is injective

Then $\G''_U = \Phi_U^{-1} \{ \Phi_U(\G'_U) \cap h_*^{-1}(\Phi_V(\G'_V)) \}$ has finite index in $\G$

Then $\vp = \Phi_V^{-1} \circ h_* \circ \Phi_U \colon \G''_U \to \G''_V$ is a commensurator.

Set $\chi_{\Phi}[h,U,V]  = \vp$. \hfill $\Box$



   \vfill   

}

 
 
 \frame % 1
{
  \frametitle{ }
  
     
A finite $CW$-complex  $Y$ is \emph{aspherical} if it is connected  and its universal covering space  is  contractible.      Equivalently, $Y$ is aspherical if all homotopy groups $\pi_{\ell}(Y, y_0) = 0$ for $\ell > 1$.

 A manifold $M$ is \emph{Borel}   that if it is a closed  aspherical   manifold, and a homotopy self-equivalence   is homotopic to a  self-homeomorphism.
  We require a somewhat stronger condition:
 
 {\bf Definition:}  A closed connected manifold $M$ is said to be \emph{strongly Borel} if every finite  covering  of $M$ is  a Borel manifold.
  
   The following classes of closed manifolds are  strongly Borel:     



$\bullet$~     infra-nilmanifolds,\\
$\bullet$~      closed Riemannian manifolds $M$ with negative sectional curvatures,\\
$\bullet$~      closed Riemannian manifolds $M$ of dimension $n \ne 3,4$ with non-positive sectional curvatures. 
 

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}

 
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We combine the above results to obtain a method to analyze $\MCG(\fM_{\cP})$ when $M_0$ is a strongly Borel manifold. 
We require another result from

 {\footnotesize 
 $\star$~  {CHL}, 
  {\it Classifying matchbox manifolds},
{\em Geom. Topol.}, 23(1):1--27, 2019.
}

\bigskip

{\bf Theorem.} Suppose that $\fM_{\cP}$ and $\fM'_{\cP'}$ are   weak solenoids, and

1) $\fM_{\cP}$ and $\fM'_{\cP'}$ have  the same dimension, \\
2)  their monodromy actions are return equivalent, \\
3) the base manifolds $M_0$ and $M'_0$ are \emph{strongly Borel}, \\
4) each space contains a simply connected leaf;

then $\fM_{\cP}$ and $\fM'_{\cP'}$  are homeomorphic.

 

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}

    
\frame % 1
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  Above results   applied to the self-homeomorphisms of $\fM_{\cP}$ yields:
 
 \medskip
 
 
 {\bf Theorem:} Let $\cP$ be a presentation such that $M_0$ is strongly Borel, and the associated odometer $(X_{\cP}, \G, \Phi)$ is effective, coherent and locally quasi-analytic.
 Then for $\whx \in \fM_{\cP}$ we have the composition  
 $$ \MCG(\fM_{\cP}, \whx) \stackrel{\sigma_{\cP, \whx}}{\longrightarrow}    \Comm(X_{\cP}, \G, \Phi, \whx) \stackrel{\chi_{\Phi}}{\longrightarrow} \Comm(\G)$$
where

$\bullet$~ $\sigma_{\cP, \whx}$  is surjective

$\bullet$~ $\chi_{\Phi}$  is injective

\medskip

The next step is to analyze   the map 
$\chi_{\Phi}$  in examples.
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}
   
    \section{Dim = 1}

\frame % 1
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  The case when $M_0 =   \mS^1$ is   classical. Note that  $\G = \mZ$.
  
It is an exercise that  $\Comm(\mZ) \cong \mQ^*$.
  
  \medskip
  
  
  {\bf Example 1:} Let $\cP$ be given by the maps $p_{\ell}(z) = z^{\ell+1}$, for  $\ell \geq 0$.

  $p_{\ell}$ is covering corresponding to subgroup $(\ell +1) \cdot \mZ \subset \mZ$.
 
     
  The fiber $\fX$ of ${\widehat{q}}_0 \colon \fM_{\cP}  \to \mS^1$ is   the full  profinite completion $\whmZ$

     $\fM_{\cP}$ is the \emph{universal solenoid} over $\mS^1$.

\medskip
We can calculate  $\MCG(\fM_{\cP})$ directly. 

For   $m/n \in \mQ^*$ define $\vp_{m/n}(x) = \frac{m}{n} \cdot x$ which induces $\ovp_{m/n} \colon M_n = \mR/n\mZ \to M_m = \mR/m\mZ$ and $h_{m/n} \colon \fM_{\cP} \to \fM_{\cP}$

For any $\whz \in \whmZ$ also have map $\whz \colon \fM_{\cP} \to \fM_{\cP}$ where $\whz   (y) = y \cdot \whz$

\medskip

{\bf Theorem:}  $\MCG(\fM_{\cP}) \cong \mQ^* \times \whmZ$ ,  $\MCG(\fM_{\cP}, \whx) \cong \mQ^*$


\bigskip

  

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}

  
\frame % 1
{
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 {\bf Example 2:} Let $\cP$ be coverings defined by the chain of subgroups
 $$\G = \mZ \supset \G_1 = m_1 \cdot \mZ \supset \G_2 = m_1 m_2 \cdot \mZ \supset \G_3 = m_1 m_2 m_3 \cdot \mZ \supset \cdots $$
 where each $m_i > 1$. Set $\vec{m} = \{m_1, m_2, m_3, \ldots\}$,  $X_{\cP} = X_{\vec{m}}$.
 
 The odometer $(X_{\vec{m}}, \mZ, \Phi)$ is free, and $X_{\vec{m}}$ is a profinite completion of $\mZ$.
 We then have, for any $\whx \in X_{\vec{m}}$:
\smallskip

$\bullet$~  $\sigma_{\cP, \whx} \colon \MCG(\fM_{\vec{m}}, \whx) \to    \Comm(X_{\vec{m}}, \mZ, \Phi, \whx) $ is an isomorphism

$\bullet$~  $\chi_{\Phi} \colon \Comm(X_{\vec{m}}, \mZ, \Phi, \whx) \to   \Comm(\mZ)$ is an inclusion.

  
  
 Almost everything to say about these solenoids is well-known: see

 {\footnotesize 
 $\star$~  {Kwapisz}, 
  {\it Homotopy and dynamics for homeomorphisms of solenoids and {K}naster continua},
 {\bf Fund. Math.}, 168(3):251--278, 2001 
}

and   for entropy calculations, see:


  {\footnotesize 
 $\star$~  {Lind \& Ward}, 
  {\it Automorphisms of solenoids and {$p$}-adic entropy},
{\bf Ergodic Theory Dynam. Systems}, 8:411--419, 1988.
}



  \vfill   

}


\frame % 1
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{\bf Detour:} 
Given $\vec{m} = \{m_1, m_2, m_3, \ldots\}$, the \emph{supernatural number} (or \emph{Steinitz number}) defined by $\vec{m}$ is:
   \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 $\xi$. 

 
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}


\frame % 1
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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 \emph{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}
 

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}

 
   
\frame % 1
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   Given  $\displaystyle \xi = \prod_{p \in \Pi} \ p^{\chi(p)}$, define:
\begin{eqnarray*}
\pi(\xi) ~ & = & ~  \{ p \in \Pi \mid  \chi(p) > 0  \}   \ , ~  {prime \ spectrum \ of} \ \xi   \\
\pi_f(\xi) ~ & = & ~  \{ p \in \Pi \mid 0 < \chi(p) < \infty \} \ , ~   {finite \ prime \ spectrum \ of} \ \xi  \\
\pi_{\infty}(\xi) ~ & = & ~  \{ p \in \Pi \mid  \chi(p) = \infty \} \ , ~   {infinite \ prime \ spectrum \ of} \ \xi  
\end{eqnarray*}
 
 {\bf End of detour.}
 \bigskip
  
  Now return to $\chi_{\Phi} \colon \Comm(X_{\vec{m}}, \mZ, \Phi, \whx) \to   \Comm(\mZ) \cong \mQ^*$

{\bf Proposition:} Let $\xi$ be the type of $\vec{m}$. Then multiplication by $p$ $\times_p \in {\rm Image}(\chi_{\Phi})$ if and only if $p \in \pi_{\infty}(\xi)$  

 

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}


 
    \section{Dim $>$ 1}

 
 
\frame % 1
{
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 {\bf Theorem:} Let $\cP$ be a presentation such that   $M_0 = \Sigma_g$ for $g \geq 2$. Assume that the associated odometer $(\fX, \G, \Phi)$ is effective, coherent and locally quasi-analytic.
 Consider   the composition  
 $$ \MCG(\fM_{\cP}, \whx) \stackrel{\sigma_{\cP, \whx}}{\longrightarrow}    \Comm(X_{\cP}, \G, \Phi, \whx) \stackrel{\chi_{\Phi}}{\longrightarrow} \Comm(\G)$$
 The map $\sigma_{\cP, \whx}$ is an isomorphism, and the map $\chi_{\Phi}$ is injective.  



\bigskip

When  $\cP$ is  the tower of all finite coverings of $\Sigma_g$, the map
$\chi_{\Phi} \circ \sigma_{\cP, \whx} \colon  \MCG(\fM_{\cP}, \whx) \to  \Comm(\G)$ 
is an isomorphism, and so   this recovers the result of Odden.

\medskip

{\bf Problem:} For other presentations $\cP$ with base $\Sigma_g$ calculate the image of 
$\chi_{\Phi} \colon \Comm(X_{\cP}, \G, \Phi, \whx) \to  \Comm(\G)$.


$\bullet$~ The case when   $M_0 = \mT^2$  is an interesting exercise.
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}

 
\frame % 1
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 {\bf Theorem:}  Let $M_0$ be a closed strongly Borel manifold. Assume that $\G = \pi_1(M_0, x)$ is residually finite.
Let $\cP$ be a presentation such that   the associated odometer $(\fX, \G, \Phi)$ is effective, coherent and locally quasi-analytic.
 Consider   the composition  
 $$ \MCG(\fM_{\cP}, \whx) \stackrel{\sigma_{\cP, \whx}}{\longrightarrow}    \Comm(X_{\cP}, \G, \Phi, \whx) \stackrel{\chi_{\Phi}}{\longrightarrow} \Comm(\G)$$
 The map $\sigma_{\cP, \whx}$ is a surjection, and the map $\chi_{\Phi}$ is injective.  

\medskip

{\bf Remark:} The problem is thus   to calculate the image of $\chi_{\Phi}$.
If the transversal $X_{\cP}$ is a profinite group, this calculation is independent of the choice of basepoint $\whx$. Otherwise, the calculation may depend on the choice of $\whx$, which determines the group chain  $\cG_{\cP}$.
 
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}
  
\frame % 1
{
  \frametitle{ }

  
  
 
We then have the following extension of the results of Belk \& Forrest, and Edgar \& Studenmund:

\medskip

{\bf Theorem:} Let $M_0$ be a closed strongly Borel manifold. Assume that   $\cP$ is a   presentation such that the action $(X_{\cP}, \G, \Phi)$ is effective, and $\fM_{\cP}$ is a normal solenoid. 
Then  
$$\MCG(\fM_{\cP}) \cong \MCG(\fM_{\cP}, \whx) \times \whG_{\cP}$$

and  for   the composition  
 $$ \MCG(\fM_{\cP}, \whx) \stackrel{\sigma_{\cP, \whx}}{\longrightarrow}    \Comm(X_{\cP}, \G, \Phi, \whx) \stackrel{\chi_{\Phi}}{\longrightarrow} \Comm(\G)$$
 the map $\sigma_{\cP, \whx}$ is a surjection, and the map $\chi_{\Phi}$ is injective.  

\medskip
 
   

 
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}
  
\frame % 1
{
  \frametitle{ }


  
Let $\cP = \{p_{\ell+1} \colon M_{\ell +1} \to M_{\ell} \ , \ \ell \geq k\}$ be a presentation, and let $m_{\ell} = \deg (p_{\ell+1})$. Let $\vec{m} = \{m_1, m_2, m_3, \ldots\}$ and let $\tau(\cP)$ be the type of the Steinitz number $\xi(\cP)$ associated to $\vec{m}$. 

\bigskip

{\bf Theorem:} Suppose that $\cP$ and $\cP'$ are presentations such that  $\fM_{\cP}$ and $\fM'_{\cP'}$ are homeomorphic. Then $\tau(\cP) = \tau(\cP')$.


\medskip

  {\footnotesize 
 $\star$~  {Hurder \& Lukina}, 
  {\it Type invariants for  solenoidal manifolds},
 {\bf Ergodic Theory Dynam. Systems}, 2025.
}

 
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}
  
\frame % 1
{
  \frametitle{ }



{\bf Problem:}  Let $M_0$ be a closed strongly Borel manifold, with $\G = \pi_1(M_0, x)$   residually finite. For which types $\tau$

$\bullet$~   does there exists   $\cP$ with base $M_0$ such that $\tau(\cP) = \tau$?

$\bullet$~ is the map $\chi_{\Phi} \colon \Comm(X_{\cP}, \G, \Phi, \whx) \to \Comm(\G)$ injective?

\bigskip

 The case when $M_0$ is a nilmanifold is treated in the papers
 
   {\footnotesize 
 $\star$~  {Hurder \& Lukina}, 
  {\it Mapping class groups  of solenoidal manifolds},
 {in preparation}, 2025.
}

  {\footnotesize 
 $\star$~  {Hurder \& Lukina}, 
  {\it Mapping class groups  of nilpotent solenoidal manifolds},
 {in preparation}, 2025.
}


 
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\end{document}

