% Huntonfest - Durham, UK % Tuesday, September 9, 2025 % slides for talk "Mapping class groups of solenoidal manifolds" \documentclass{beamer} \usepackage{amsfonts} \usepackage{graphicx} \input xy \xyoption {all} % \usetheme{default} \usetheme{Singapore} % \usetheme{CambridgeUS} % \usetheme{Boadilla} % \usetheme{Antibes} % \usetheme{Berkeley} % \usetheme{Berlin} % \usetheme{Copenhagen} % \usetheme{Darmstadt} % \usetheme{Dresden} % \usetheme{Frankfurt} % \usetheme{Goettingen} % \usetheme{Hannover} % \usetheme{Ilmenau} % \usetheme{JuanLesPins} % \usetheme{Luebeck} % \usetheme{Madrid} % \usetheme{Malmoe} % \usetheme{Marburg} % \usetheme{Montpellier} % \usetheme{PaloAlto} % \usetheme{Pittsburgh} % \usetheme{Rochester} % \usetheme{Szeged} % \usetheme{Warsaw} % \usecolortheme{crane} % \beamertemplatesolidbackgroundcolor{craneorange!25} \parskip=4pt % math notation shortcuts \newcommand{\ds}{\displaystyle} \newcommand{\sop}{{\bf Proof:} } % replaced by command \proof \newcommand{\eop}{\;\;\; \Box} % replaced by command \endproof - still used inside theorem environment % bolds \newcommand{\bZ}{{\bf Z}} \newcommand{\bC}{{\bf C}} \newcommand{\bK}{{\bf K}} \newcommand{\F}{{\mathcal F}} % our favorite foliation \newcommand{\e}{{\epsilon}} % our favorite constant \newcommand{\wmW}{\widehat{\mW}} \newcommand{\cGF}{\cG_{\F}} % denotes the pseudogroup of F \newcommand{\cGM}{\cG_{\fM}} % \newcommand{\cGN}{\cG_{\fN}} % \newcommand{\cGX}{\cG_{\fX}} % \newcommand{\cGY}{\cG_{\fY}} % \newcommand{\Dom}{{\rm Dom}} % \newcommand{\dX}{d_{\fX}} % leafwise metric \newcommand{\dXp}{d_{\fX}'} % leafwise metric \newcommand{\dY}{d_{\fY}} % leafwise metric \newcommand{\diamX}{{\rm diam}_{\fX}} % \newcommand{\Homeo}{\rm Homeo}% \newcommand{\G}{\Gamma} \newcommand{\ovx}{\overline{x}} % \newcommand{\ovy}{\overline{y}} % \newcommand{\ovz}{\overline{z}} % \newcommand{\oU}{\overline{U}} % blackboard bolds \newcommand{\mB}{{\mathbb B}} \newcommand{\mC}{{\mathbb C}} \newcommand{\mD}{{\mathbb D}} \newcommand{\mE}{{\mathbb E}} \newcommand{\mF}{{\mathbb F}} \newcommand{\mG}{{\mathbb G}} \newcommand{\mK}{{\mathbb K}} \newcommand{\mN}{{\mathbb N}} \newcommand{\mP}{{\mathbb P}} \newcommand{\mQ}{{\mathbb Q}} \newcommand{\mR}{{\mathbb R}} \newcommand{\mS}{{\mathbb S}} \newcommand{\mT}{{\mathbb T}} \newcommand{\mW}{{\mathbb W}} \newcommand{\mZ}{{\mathbb Z}} % cal script letters \newcommand{\cA}{{\mathcal A}} \newcommand{\cB}{{\mathcal B}} \newcommand{\cC}{{\mathcal C}} \newcommand{\cD}{{\mathcal D}} \newcommand{\cE}{{\mathcal E}} \newcommand{\cG}{{\mathcal G}} \newcommand{\cH}{{\mathcal H}} \newcommand{\cI}{{\mathcal I}} \newcommand{\cJ}{{\mathcal J}} \newcommand{\cK}{{\mathcal K}} \newcommand{\cL}{{\mathcal L}} \newcommand{\cM}{{\mathcal M}} \newcommand{\cN}{{\mathcal N}} \newcommand{\cO}{{\mathcal O}} \newcommand{\cP}{{\mathcal P}} \newcommand{\cQ}{{\mathcal Q}} \newcommand{\cR}{{\mathcal R}} \newcommand{\cS}{{\mathcal S}} \newcommand{\cT}{{\mathcal T}} \newcommand{\cU}{{\mathcal U}} \newcommand{\cV}{{\mathcal V}} \newcommand{\cW}{{\mathcal W}} \newcommand{\cX}{{\mathcal X}} \newcommand{\cY}{{\mathcal Y}} \newcommand{\cZ}{{\mathcal Z}} % fraktur letters \newcommand{\fA}{{\mathfrak{A}}} \newcommand{\fB}{{\mathfrak{B}}} \newcommand{\fC}{{\mathfrak{C}}} \newcommand{\fD}{{\mathfrak{D}}} \newcommand{\fG}{\mathfrak{G}} \newcommand{\fH}{\mathfrak{H}} \newcommand{\fK}{{\mathfrak{K}}} \newcommand{\fL}{{\mathfrak{L}}} \newcommand{\fM}{{\mathfrak{M}}} \newcommand{\fN}{{\mathfrak{N}}} \newcommand{\fO}{{\mathfrak{O}}} \newcommand{\fP}{{\mathfrak{P}}} \newcommand{\fQ}{{\mathfrak{Q}}} \newcommand{\fS}{{\mathfrak{S}}} \newcommand{\fT}{{\mathfrak{T}}} \newcommand{\fU}{{\mathfrak{U}}} \newcommand{\fV}{{\mathfrak{V}}} \newcommand{\fX}{{\mathfrak{X}}} \newcommand{\fY}{{\mathfrak{Y}}} \newcommand{\fZ}{{\mathfrak{Z}}} \newcommand{\fW}{{\mathfrak{W}}} % greek letters \newcommand{\whp}{{\widehat{p}}} \newcommand{\Aut}{{\rm Aut}} \newcommand{\vp}{{\varphi}} \newcommand{\ovp}{\overline{\vp}} \newcommand{\ovq}{\overline{q}} \newcommand{\whe}{\widehat{e}} \newcommand{\whg}{\widehat{g}} \newcommand{\whq}{\widehat{q}} \newcommand{\whx}{\widehat{x}} \newcommand{\why}{\widehat{y}} \newcommand{\whz}{\widehat{z}} \newcommand{\whphi}{\widehat{\phi}} \newcommand{\whmZ}{\widehat{\mZ}} \newcommand{\bRt}{{\bf R}_{0}} \newcommand{\whcG}{\widehat{\cG}} \newcommand{\whG}{\widehat{\G}} \newcommand{\whM}{\widehat{M}} \newcommand{\whX}{\widehat{X}} \newcommand{\whPhi}{\widehat{\Phi}} \newcommand{\lcm}{{\rm lcm}} \newcommand{\diam}{{\rm diam}} \newcommand{\MCG}{{\rm Mod}} \newcommand{\Comm}{{\rm Comm}} \newcommand\mor{\mathrel{\stackrel{\makebox[0pt]{\mbox{\normalfont\tiny a}}}{\sim}}} \newcommand\comm{\mathrel{\stackrel{\makebox[0pt]{\mbox{\normalfont\tiny c}}}{\sim}}} \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. \vfill } \frame % 1 { \frametitle{ } 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. \vfill } \frame % 1 { \frametitle{ } 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. \vfill } \section{Dim = 1} \frame % 1 { \frametitle{ } 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 \vfill } \frame % 1 { \frametitle{ } {\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 { \frametitle{ } {\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 $\xi$. \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 \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} \vfill } \frame % 1 { \frametitle{ } 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 $p \in \pi_{\infty}(\xi)$ \vfill } \section{Dim $>$ 1} \frame % 1 { \frametitle{ } {\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. \vfill } \frame % 1 { \frametitle{ } {\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 $\cG_{\cP}$. \vfill } \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 \vfill } \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. } \vfill } \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. } \vfill } \end{document}