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\date[September 9, 2025]


  
\title[Solenoidal manifolds]{Dynamical Commensurator Groups}


\author{Steve Hurder\\ {\footnotesize October 29, 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{Introduction}

   
\frame % 1
{
  \frametitle{ }
 
  Let $\fM$ be a compact, connected metric space (a continuum).   
  
   $\Homeo(\fM)$ denotes the    homeomorphisms of $\fM$.

  $\Homeo_0(\fM)$ denotes the closed subgroup of $\Homeo(\fM)$  which are isotopic to the identity map, a  contractible space. 

\smallskip

{\bf Definitions:} 

$\bullet$~ [Mapping Class Group]    $\MCG(\fM)  \equiv \Homeo(\fM)/\Homeo_0(\fM)$

\smallskip

$\bullet$~  [Pointed Mapping Class Group:]    $\whx \in \fM$, set   
$$\MCG(\fM, \whx)  \equiv \Homeo(\fM, \whx )/\Homeo_0(\fM, \whx)$$



{\bf Problem:} For $\fM$ a \emph{solenoidal manifold}, calculate 
$$\MCG(\fM) ~ , ~\MCG(\fM, \whx)$$

     \vfill   

}

 
\frame % 1
{
  \frametitle{ }
 
{\bf Example:}  For 1-dimensional solenoids:  

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


 
{\bf Example:} Let $\Sigma_g$ be a closed surface of genus $g \geq 2$ with basepoint $x \in \Sigma_g$.  Let $\whG$ denote   the profinite completion of $\G = \pi_1(\Sigma_g, x)$.
  Let  $\whSigma$ be the \emph{universal hyperbolic solenoid}. That is, $\whSigma$ is the solenoidal manifold defined by a 
 cofinal   collection $\cP$ in the set of all finite oriented normal coverings of $\Sigma_g$.  
  
{\bf Theorem [Odden]:} 
$$\MCG(\whSigma , \whx) \cong \Comm(\G) ~ , ~ \MCG(\whSigma) \cong \Comm(\G) \times \whG$$  



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



\vfill

}


 \frame % 1
{
  \frametitle{ }
  
 

{\bf Theorem:} [Bering \& Studenmund] Let $M$ be a closed aspherical manifold, and $\whM$ denote  the universal solenoid over $M$.   Let $\cE(\whM, \whx)$ be the group of pointed homotopy self-equivalences of $\whM$. 
Then there  are   isomorphisms 
$$\cE(\whM, \whx) \cong  \Comm(\G)  ~ , ~ \cE(\whM) \cong  \Comm(\G) \times \whG$$

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

\bigskip

{\bf Question:} What about $\MCG(\fM_{\cP} , \whx)$ for a general solenoid $\fM_{\cP}$? 
 

\vfill

}

\section{Solenoids}
 
 
\frame % 1
{
  \frametitle{ }
  
$\bullet$ ~ Let $M$ be a compact connected manifold without boundary.

$\bullet$ ~ Let $\G = \pi_1(M,x)$ for choice of $x \in M$

 
$\bullet$ ~ Let $\cG = \{\G = \G_0 \supset \G_1 \supset \G_2 \supset \cdots\}$ be a subgroup chain.

   
The chain $\cG$ and basepoint $x\in M$ determine a tower of finite index coverings by closed manifolds:
     \begin{align*}   
 \xymatrix{
\cP_{\cG}  \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*}

where  $q_{\ell} = p_{\ell} \circ \cdots \circ p_1 \colon M_{\ell} \to M_0$ induces the map
$$(q_{\ell})_{\#} \colon \pi_1(M_{\ell} , x_{\ell}) \to \G_{\ell} \subset \pi_1(M_0, x) = \G_0$$
with image $\G_{\ell}$.

    \vfill   

}

 
 
 \frame % 1
{
  \frametitle{}

$$\fM_{\cP} = \lim_{\longleftarrow} ~ \{p_{\ell+1} \colon M_{\ell +1} \to M_{\ell}\}  \subset \prod_{\ell \geq 0} ~ M_{\ell}$$
A point $\whz \in \fM_{\cP}$ is a sequence $\whz =(z_0, z_1, z_2, \ldots)$ where $z_{\ell} \in M_{\ell}$ and $p_{\ell +1}(z_{\ell  +1}) = z_{\ell}$ for $\ell \geq 0$.

 \medskip

$\bullet$ ~ $\fM_{\cP}$  is a compact connected space.  

$\bullet$ ~ $\whq_{\ell} \colon \fM_{\cP} \to M_{\ell}$ defined by $\whq_{\ell}(\whz) = z_{\ell}$ is a fibration.

$\bullet$ ~ Each fiber   $\fX_{\ell} = \whq_{\ell}^{-1}(x_{\ell})$   a Cantor space.

 $\bullet$ ~    If each $\G_{\ell}$ is a normal subgroup of $\G$, then the fiber  $\fX_0$ is a profinite group.

\medskip

 {\bf Definition:} 
$\fM_{\cP}$ is a \emph{weak solenoid} (McCord, 1985) or  a \emph{solenoidal manifold} (Sullivan, 2014)

 

   \vfill   

}

  


\frame % 1
{
  \frametitle{ }


{\bf Theorem:} Let $h \colon \fM_{\cP} \to \fM_{\cP}$ be a homeomorphism preserving a basepoint $\whx \in \fM_{\cP}$.
There exists $n(i) \geq i$ and  $h_i \colon M_{n(i)} \to M_i$ which induces $h_* \colon \fM_{\cP} \to \fM_{\cP}$ that is $\e$-homotopic to $h$.

 \medskip

{\bf Corollary:} Let $h \colon \fM_{\cP} \to \fM_{\cP}$ be a homeomorphism preserving a basepoint $\whx \in \fM_{\cP}$. Then $h$ induces an injection $h_{\#} \colon \G_{n(i)}   \to \G_i$ whose image has finite index.

\medskip

{\bf Remark:}  There is no a priori bound on $n(i)$
 
 \bigskip
  
 $\star$~   {\footnotesize  {Rogers \& Tollefson}, 
  {\it Homeomorphisms homotopic to induced homeomorphisms of weak solenoidal spaces},
  {\bf Colloq. Math.} 25:81--87, 1971/72.
}



\vfill

}

  


  \section{Commensurators}



\frame % 1
{
  \frametitle{ }

 
  Let $\G$ be a countable group.  
  
  A \emph{commensurator} of $\G$ is a pair of finite-index subgroups $H, K \subset \G$ and an isomorphism $\phi \colon H \to K$.
  
  Two commensurators $\phi_1 \colon H_1 \to K_1$ and $\phi_2 \colon H_2 \to K_2$ are equivalent, $\phi_1 \sim \phi_2$, if there exists a finite index subgroup $H_3 \subset H_1 \cap H_2$ such that $\phi_1 | H_3 = \phi_2 | H_3$.
  
  \medskip
  
  {\bf Definition:}  The \emph{abstract commensurator group} $\Comm(\G)$    is the collection of
commensurators $\phi \colon H \to K$, modulo   $\sim$.
 
  
 \vfill   

}

 

  


\frame % 1
{
  \frametitle{ }


{\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, hence  $\Comm(\G') \cong \Comm(\G)$.

\medskip

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

   \bigskip
   
{\bf Example:} $\Comm(\mZ) = {\bf GL}(\mQ) = \mQ^*$ the non-zero rationals

 \medskip

{\bf Example:} $\Comm(\mZ^2) = {\bf GL}(\mQ^2)$ = invertible $2 \times 2$ rational matrices

 

 \vfill   

}

   
\frame % 1
{
  \frametitle{ }


  Let $h \colon \fM_{\cP} \to \fM_{\cP}$ be a homeomorphism preserving a basepoint $\whx \in \fM_{\cP}$.
Let $\G = \pi_1(M_0 , x)$, then by Rogers \& Tollefson, the homeomorphism $h$ determines  $\phi_h \in \Comm(\G)$. This map does not depend on the isotopy class of $h$, so we obtain:

\medskip

{\bf Theorem:} There is a  map $\chi \colon \MCG(\fM_{\cP}, \whx) \to \Comm(\G)$.

\bigskip

$\bullet$ ~ The map $\chi$ need not be onto, as $\phi_h$ must preserve the group chain $\cG_{\cP}$

\medskip

$\bullet$ ~ The map $\chi$ need not be injective.

\medskip
  
$\bullet$ ~ Use ideas from dynamical systems to study  $\chi$.

 \vfill   

}

 
     


    
 
 \section{Dynamics}

 
   
  
 
\frame % 1
{
  \frametitle{ }
  
Let $\cG = \{ \G = \G_0 \supset \G_1 \supset \G_2  \supset \cdots\}$ be a group chain in $\G$.

For $\ell \geq 0$ let   $X_{\ell} = \G/\G_{\ell}$ as a left  $\G$-space. 

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


$$
X_{\cG}   \equiv    \lim_{\longleftarrow} ~ \{ p_{\ell +1} \colon X_{\ell +1} \to X_{\ell}  \mid \ell \geq 0 \}    \subset    \prod_{\ell \geq 0} ~ X_{\ell} \  .   
$$
By the definition of the inverse limit,  
\begin{equation*}\label{eq-presentationinvlim3}
x = (x_0, x_1, \ldots ) \in X_{\cG}   ~ \Longleftrightarrow  ~ p_{\ell +1}(x_{\ell +1}) =  x_{\ell} ~ {\rm for ~ all} ~ \ell \geq 0 ~. 
\end{equation*}
$\G$ acts on $X_{\cG}$ by acting on each factor $X_{\ell}$.

The action  $\Phi \colon \G \times X_{\cG} \to X_{\cG}$   is a (generalized) \emph{odometer}. That is, it is a minimal, equicontinuous action on a Cantor space.

 \vfill   

}

  
    
\frame % 1
{
  \frametitle{ }
  
 
 Clopen set  $U \subset X_{\cG}$ 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 $\Phi'_{U'}$. 
 
 Thus,  a homeomorphism $h_{U} \colon U \to U'$ which conjugates   $\cH_U$ with $\cH'_{U'}$ need not induce an isomorphism $\phi_U \colon \G_U \to \G'_{U'}$.

 
 

   \vfill   

}

 
 
\frame % 1
{
  \frametitle{ }
  
  {\bf Definition:}
 Let  $(\fX, \G, \Phi)$   be an odometer action, and $x \in \fX$. 
 
 \smallskip
 
$\bullet$~ A \emph{dynamical commensurator}   is a homeomorphism $h_U \colon U \to V$, where $U$ and $V$ are adapted subsets with $x \in U \cap V$, such that $h_U(x) = x$, 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 $x \in U'' \subset U \cap U'$ such that $h_U|U'' = h'_{U'}|U''$. We   write $(h_U,U,V) \comm (h'_{U'},U',V')$.


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

\medskip
 
 
{\bf Problem:} How is   $\Comm(\fX, \G, \Phi, x)$ related to $\Comm(\G)$?

   \vfill   

}

  
 
\frame % 1
{
  \frametitle{ }

The action $\Phi \colon \G \times X_{\cG} \to X_{\cG}$   is \emph{effective} if the action map $\Phi \colon \G \to \Homeo(\fX)$ has trivial kernel.

The literature contains a notion of the   commensurator group relative to $\Homeo(\fX)$:
$$\Comm_{\Homeo(\fX)}(\G) = \{h \in \Homeo(\fX) \mid h \colon \Phi(\G) \cong \Phi(K); H,K \subset_f \G\}$$

  
 
{\bf Problem:} How are   these groups related?
$$\Comm(\fX, \G, \Phi, x) ~ , ~  \Comm_{\Homeo(\fX)}(\G)  ~ , ~ \Comm(\G) $$

 

For an effective action $\Phi$, clearly  $\Comm_{\Homeo(\fX)}(\G) \subset \Comm(\G)$.

The distinction between $\Comm(\fX, \G, \Phi, x)$ and $\Comm_{\Homeo(\fX)}(\G)$ is in the domains of the conjugating maps. Also, for $\Comm(\fX, \G, \Phi, x)$ we   need not  assume  the action is effective.

 
   \vfill   

}

  

\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 $\fX$. 
  
  (The dynamical analog of the \emph{Unique Root Property}.)
  
  \medskip
  
 $\bullet$~   \emph{topologically free} if it is effective, and the quasi-analytic condition holds for $U = \fX$.

\medskip

 $\bullet$~    \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   $U$.  

 (A  localized  \emph{Unique Root Property}.)

 
 \vfill   

}


  
\frame % 1
{
  \frametitle{ }

  {\bf Definition:}  An odometer  $(\fX, \G, \Phi)$  is \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.

\bigskip

 

{\bf Example:} 
If $\G$ is virtually nilpotent, that is, $\G$ contains a nilpotent subgroup $\G' \subset \G$ with finite index, then  an effective 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. 


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

    
 
  
 
  \vfill   

}


   
  
\frame % 1
{
  \frametitle{}
     



{\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}. 

 

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


 
 \medskip
 
 {\bf Corollary:} For $\whx \in \fM_{\cP}$ there is a well-defined map
 $$\sigma_{\cP,  \whx} \colon \MCG(\fM_{\cP}, \whx) \longrightarrow    \Comm(X_{\cP}, \G, \Phi, x)$$
 
 \medskip
 
 
 
  
 
  \vfill   

}


 \section{Results}

 


\frame % 1
{
  \frametitle{ }

  
 
 {\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  
 $$ \MCG(\fM_{\cP}, \whx) \stackrel{\sigma_{\cP, \whx}}{\longrightarrow}    \Comm(X_{\cP}, \G, \Phi, x) \stackrel{\chi_{\Phi}}{\longrightarrow} \Comm(\G)$$
where  $\sigma_{\cP, \whx}$  is surjective, and   $\chi_{\Phi}$  is injective.

 
\medskip

  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. 

    \vfill   

}

  

\frame % 1
{
  \frametitle{ }

{\bf Example:}   $\Sigma_g$   a closed surface of genus $g \geq 2$,  $\G = \pi_1(\Sigma_g, x)$.

Let $\cG = \{\G = \G_0 \supset \G_1 \supset \G_2 \supset \cdots\}$ with $\cap \G_{\ell} = \{1\}$.


  Let  $\fM_{\cP}$ be the  solenoidal manifold over $\Sigma_g$ defined by $\cG$. 
  
  \medskip
  
   
{\bf Theorem [H \& Lukina, 2025]:} If the odometer $(\G, X_{\cG}, \Phi)$ is   coherent and locally quasi-analytic, then
$$\MCG(\fM_{\cP} , \whx) \cong     \Comm(X_{\cG}, \G, \Phi, x) \subset \Comm(\G)$$  

 $\star$~   {\footnotesize  {H \& Lukina}, 
  {\it Mapping class groups  of solenoidal manifolds},
 {\it preprint}, 2025.}




\vfill

}


  
 
 
 
 
  
  
  


 
 \frame % 1
{
  \frametitle{ }
  
  
 Let $\G = \pi_1(M,x)$ let $\cG = \{\G = \G_0 \supset \G_1 \supset \G_2 \supset \cdots\}$ be a group chain with $\cap \G_{\ell} = \{1\}$.

 \medskip

{\bf Theorem:} [H \& Lukina, 2025] Assume that $M$   is strongly Borel. 
If the odometer $(\G, X_{\cG}, \Phi)$ is   coherent and locally quasi-analytic, then 
$$\MCG(\fM_{\cG} , \whx) \twoheadrightarrow \Comm(X_{\cG}, \G, \Phi, x) \subset \Comm(\G)  $$
The first map is onto, but may have kernel.

\medskip

$\star$ ~  These results reduce the calculation of $\MCG(\fM_{\cG} , \whx)$ to calculating the image $Comm(X_{\cG}, \G, \Phi, x) \subset \Comm(\G)$.  

$\star$ ~ 
That is, we look for subgroups $H,K \subset \G$ and isomorphisms $\phi \colon H \to K$ which map  the group chain $\cG$ into itself.

 \vfill   

}

  
  
    
     
   
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{\bf Example:}   The integer Heisenberg group:
$${\footnotesize 
\Gamma_{\mZ} =   \left\{  \left[ {\begin{array}{ccc}
   1 & a & c\\
   0 & 1 & b\\
  0 & 0 & 1\\
  \end{array} } \right] \mid a,b,c  \in {\mathbb Z}\right\} .
  }
$$
 We denote a $3 \times 3$ matrix in $\Gamma_{\mZ}$ by the coordinates as $(a,b,c)$.
   Let $\cH$ denote the real Heisenberg group,     $a,b,c$ are real numbers.  
 
 
For distinct  primes $p, q \geq 2$, define the self-embedding $\varphi_{p,q} \colon \Gamma \to \Gamma$ by  
$\varphi(a,b,c) = (pa, qb, pqc)$. Define 
$$\Gamma_{\ell} = \varphi_{p,q}^{\ell}(\Gamma) = \{(p^{\ell} a, q^{\ell}b, (pq)^{\ell}c) \mid a,b,c \in {\mathbb Z}\} $$
which yields a group chain $\cG_{p,q}$ in $\G_{\mZ}$. The subgroups $\G_{\ell}$ are not normal, and the limit Cantor space $X_{p,q}$ is not a group.
The order of the subgroups generated by $a$ and $b$ are invariants.

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Let $\cH_{p,q}$ be the solenoidal manifold defined by the tower of coverings $\cH/\G_{\ell}$ of $\cH/\Gamma_{\mZ}$. Then:
   $$\MCG(\cH_{p,q}, \whx) \cong \mZ \times \mZ \times \{\pm 1\} \times \{\pm 1\}$$

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Many more examples of nilpotent odometer actions to which the theorem applies are given in the papers

 $\star$~   {\footnotesize  {H \& Lukina}, 
  {\it Type invariants for non-abelian odometers},
 {\bf Ergodic Theory Dynamical Systems}, to appear, 2025.}


 $\star$~   {\footnotesize  {H \& Lukina}, 
  {\it Prime spectrum and dynamics for nilpotent Cantor actions},
 {\bf  Pacific J. Math.}, 327:107--128, 2023.}

 
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