% AMS Virtual Meeting % October 25-26, 2025 % "Dynamical Commensurator Groups" \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{\whSigma}{\widehat{\Sigma_g}} \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]{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 } \frame % 1 { \frametitle{ } {\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. \vfill } \frame % 1 { \frametitle{ } \medskip Let $\cH_{p,q}$ be the solenoidal manifold defined by the tower of coverings $\cH/\G_{\ell}$ of $\cH/\Gamma_{\mZ}$. Then: $$\MCG(\cH_{p,q}, \whx) \cong \mZ \times \mZ \times \{\pm 1\} \times \{\pm 1\}$$ \bigskip 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.} \vfill } \end{document}