% Group Theory Working Seminar
% Purdue University
% April 16, 2024
 
 
% slides for talk "Applications of the Contraction Mapping Theorem: Toral actions"
 
 
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\date[April 16, 2024, 2023]


  
\title{Applications of the Contraction Mapping Theorem: Toral actions}


\author{Steve Hurder\\ \small University of Illinois at Chicago\\April 16, 2024} 
 
 
\begin{document}
 
   \frame{\titlepage} % # 1

 
\frame
{
  \frametitle{}

  \section{Prelude}

 $\G$ is a finitely generated group.


{\bf Question.} What are the ``essential'' actions of $\G$ on a compact manifold   $M$? On a compact metric space $X$?

\bigskip
 
 {\bf Strategy.}
 For an  essential  action, find relations between 
 
 $\bullet$~ The algebraic properties of $\G$ (e.g. nilpotent, higher rank, etc)
 
 $\bullet$~ The dynamical properties of the action (e.g. minimal, ergodic, expansive, positive entropy, etc)
 
 $\bullet$~ The cohomological properties of the action.
 
\bigskip

Two cases that are better understood:

$\bullet$~ The action is by isometries

$\bullet$~ The action has some aspects of hyperbolic behavior

 \vfill   
}
 
 \section{Toral Actions}


\frame
{
  \frametitle{}
For $A \in SL(n,\mZ)$ get transformation $\phi_A \colon \mR^n \to \mR^n$ which restricts to $\phi_A \colon \mZ^n \to \mZ^n$. The quotient $\mT^n = \mR^n/\mZ^n$ is the standard n-torus, and we get induced map $\phi_A \colon \mT^n \to \mT^n$.
 
\bigskip
{\bf Arnold's Cat Map.} Let  $\ds A =  \left[\begin{array}{cc}2 & 1 \\1 & 1\end{array}\right]$,  $\phi_A \colon \mT^2 \to \mT^2$

 \begin{figure}[htbp]
\begin{center}
\includegraphics[width=.5\textwidth]{pix/Arnoldcatmap}
\label{figure-disk}
\end{center}
\end{figure}

 
 \vfill   
}
 

\frame
{
  \frametitle{}

 
 \begin{figure}[htbp]
\begin{center}
\includegraphics[width=.7\textwidth]{pix/poor-cat}
\end{center}
\end{figure}



 
 \vfill   
}
 

\frame
{
  \frametitle{}

 From order to chaos and back. Use a sample mapping on a picture of 150x150 pixels. The number shows the iteration step; after 300  iterations, the original image returns.


 \begin{figure}[htbp]
\begin{center}
\includegraphics[width=.4\textwidth]{pix/Arnold_cat}
\end{center}
\end{figure}

  
 \vfill   
}
 
 
\frame
{
  \frametitle{}
 {\bf Proposition.} The periodic points of $\phi_A \colon \mT^n \to \mT^n$ are dense.
 
 \medskip
 
 {\it Proof.} Let $\mZ[1/m] \subset \mQ$ denote the rational points with denominator $1/m$.  Then we obtain an induced map $$\phi_A \colon \mZ[1/m]^n/\mZ^n  \to \mZ[1/m]^n/\mZ^n$$ 
 $\phi_A$ acts as permutation of the finite set $\mZ[1/m]^n/\mZ^n$.  
 
 $\Rightarrow$  a finite power of $\phi_A$ fixes this set. 
 
 $\mT^n_{\mQ} = \mQ^n/\mZ^n \subset \mT^n$ is dense  
 
 $\Rightarrow$    periodic points of $\phi_A$ are dense.
   
  \vfill   
}
 

  \section{Anosov}

\frame
{
  \frametitle{}
 
 
{\bf Definition.}   A diffeomorphism    $f \colon M \to M$ is \emph{Anosov} if there exists a direct sum decomposition $TM =  E^+ \oplus E^-$ where    
 
$\bullet$~ $Df$ uniformly expands   the distribution $E^+$

$\bullet$~ $Df$ uniformly contracts   the distribution $E^-$

That is, there exists $\lambda > 1$ such that 

$\bullet$~ for all $\vec{v} \in E^+$, $Df(\vec{v}) \in E^+$ and $\|Df (\vec{v}) \| \geq \lambda \|\vec{v}\|$   

$\bullet$~ for all $\vec{v} \in E^-$, $Df(\vec{v}) \in E^-$ and $\|Df (\vec{v}) \| \leq \lambda^{-1} \|\vec{v}\|$   

$\Rightarrow$~ the distributions $E^+$ and $E^-$ are uniquely integrable, defining foliations $\F^+$ and $\F^-$ of $M$. The leaves of these foliations are \emph{smoothly immersed submanifolds}.

\bigskip

 {\bf Definition.} $A \in SL(n,\mZ)$ is hyperbolic $\Leftrightarrow$   all eigenvalues   $| \lambda| \ne 1$.
 
 {\bf Observation.} $A \in SL(n,\mZ)$ hyperbolic $\Leftrightarrow$ $\phi_A \colon \mT^n \to \mT^n$  is Anosov.
 
  

  \vfill   
}
 


\frame
{
  \frametitle{}
 
  
 {\bf Definition.} A  smooth  action $\phi \colon \G \times M \to M$ is \emph{Anosov} if there exists $\gamma \in \G$ such that $\phi(\gamma)$ is Anosov diffeomorphism of $M$.
 
 \bigskip
 
   {\bf Revised Problem.} Classify the Anosov actions of $\G$ on $M$.
   
 \bigskip
 
 {\bf Conjecture.}   Let   $f \colon M \to M$ be Anosov action, then $M$ is an infra-nil manifold. That is, $\Lambda = \pi_1(M, x)$ has a nilpotent subgroup of finite index and the universal covering $\widetilde{M}$ is contractible.
 
 \bigskip
 
 {\bf Definition.} $x \in M$ is \emph{wandering} for action $f \colon M \to M$ if there is an open neighborhood $x \in U$ such that the translates of $U$ are disjoint, and is \emph{non-wandering} otherwise.
 
 \bigskip
 {\bf Theorem.} If  $f \colon M \to M$ be Anosov action and the non-wandering set  $\Omega(f) = M$, then the conjecture is true.
 
 \bigskip
 
  

  \vfill   
}
 


\frame
{
  \frametitle{}
 
  This motivates the working assumption that  $\phi \colon \G \times M \to M$ is an Anosov action and $M$ is a nil-manifold. In fact, let  $M = \mT^n$. 
  
  For  $\gamma \in \G$, we have 
  $\phi(\gamma)_* \in \Aut \{H_1(\mT^n ; \mZ) \} \subset \Aut \{H_1(\mT^n ; \mR) \}$ which gives an affine representation
  
  $$\rho \colon \G \to \Aut \{ H_1(\mT^n ; \mR)/H_1(\mT^n ; \mZ) \} \subset \Homeo (\mT^n )$$
    
   This is called the \emph{standard action} for $\phi$.
   
   
  \bigskip
  
  {\bf Problem.} Find conditions on an Anosov  action  $\phi \colon \G \times M \to M$, for $M$ a nil-manifold, which are sufficient to imply there exists a subgroup $\G' \subset \G$ of finite index such that the restriction of $\phi$ to $\G'$  is conjugate to the standard action of $\G'$.
  
  
    \vfill   
}
 


\frame
{
  \frametitle{}
 



  {\bf Theorem.} Let $f \colon \mT^n \to \mT^n$ be Anosov, then $f$ is \emph{topologically} conjugate to a linear hyperbolic automorphism $\phi_A$. 

  \bigskip
  
  $\star$~   J.~Franks, {\it Anosov diffeomorphisms on tori}, 
 {\bf Trans. Amer. Math. Soc.}, 145:117--124, 1969.

 $\star$~ A.~Manning, {\it There are no new {Anosov} diffeomorphisms on tori}, 
  {\bf American Jour. Math.}, 96:422--429, 1974.

 

\bigskip

 Assume we have an  Anosov action $\phi \colon \G \times \mT^n \to \mT^n$, then   
for some $\gamma \in \G$, the action $\phi(\gamma)$ is conjugate to a hyperbolic $\rho(\gamma) \in \Aut(\mT^n)$
  
  \medskip
  
 Does this imply that the full action $\phi$ is a  standard linear action?

  


  \vfill   
}
 


  \section{A Deformation}


\frame
{
  \frametitle{}
 
     
{\bf Theorem.}    [Folkert Tangerman,  1990]
 There exists an analytic family $\{\varphi_t \mid  0 \leq t \leq 1\}$ of volume-preserving real analytic  actions  of $SL(2,{\mZ})$ on ${\mT}^2$,   with $\varphi_0 = \varphi$ the standard action, such that the \emph{action} $\varphi_t$ is not    topologically conjugate to $\varphi$ for all $0 < t \leq 1$.

\smallskip

Observe that the deformed actions are Anosov.

{\it Sketch of construction.}

{\bf Lemma.}  For the generators of $SL(2,{\mZ})$: \\
$\bullet$~    $A = \left[\begin{array}{cc} 0 & -1\\1 & 0\end{array}\right]
$ and $B = \left[\begin{array}{cc}1 & -1\\1 & 0\end{array}\right]$     \\
$\bullet$~  $A$ has order 4, $B$ has order 6, and  $A^2 = B^3 = -I$. \\
$\bullet$~  $SL(2,{\mZ})$ is isomorphic to the   amalgamated product 
$$ SL(2,{\mZ}) \cong ({\mZ}/4{\mZ}) \times_{\mZ/2\mZ} ({\mZ}/6{\mZ})$$ 
generated by  $\{A, B\}$.
 


  \vfill   
}
 


\frame
{
  \frametitle{}
  Let $T \colon \mR^2 \times \mR^2 \to \mR^2$ be the translation action.
  
 Let $\vec{Z}_1 = x \frac{\partial}{\partial y} -  y \frac{\partial}{\partial x}$ be the rotational vector field about the origin.     

Let  $\psi \colon [0,1] \to [0, 1]$ be a smooth function such that $\psi(0) = 1$, $\psi(s) \geq 0$ for all $s$, and $\psi(s) = 0$ for $s \geq 10^{-4}$.  

Define the divergence-free vector field $\vec{Z}_{\psi} = \psi(x^2+y^2) \cdot \vec{Z}_1$.

Form the  translate $Z_+   =  DT_{(1/2, 0)}(Z_{\psi})$ of  the vector field  $\vec{Z}_{\psi}$,   centered at the point $(1/2, 0)  \in {\mR}^2$, and  the   vector fields 
$$Z_- = D(\phi_A^2)(Z_+) = D(\phi_{-I})(Z_+) \quad , \quad Z = Z_+ + Z_-$$  
Note that $D(\phi_A^2)(Z) = Z$.  
  \vfill   
}
 


\frame
{
  \frametitle{}
 Form the infinite sum
  $$
  \widetilde{Z} = \sum_{(a,b) \in {\mZ}^2} DT_{(a,b)}(Z)
$$
  which is well-defined since the supports of the translates are disjoint. $\widetilde{Z}$ is invariant under the action of $\mZ^2$.  


 Let $\xi(t) \colon \mR^2 \to \mR^2$ be the flow of $\widetilde{Z}$, then for $(a,b) \in \mZ^2$, 
 $$\xi(t) \circ \phi_A^2 = \phi_A^2 \circ \xi(t) \quad , \quad T_{(a,b))} \circ \xi(t) = \xi(t) \circ T_{(a,b)}$$ 
 Let $\widetilde{\xi}(t) \colon \mT^2 \to \mT^2$ be the induced flow. Set
 $$\phi_t(A) = \widetilde{\xi}(t)^{-1} \circ \phi_A \circ \widetilde{\xi}(t) \quad , \quad \phi_t(B) = \phi_B$$
 Then $\phi_t(A)^2 = \phi_{-I}$ so we get an action $\phi_t \colon SL(2, \mZ) \times \mT^2 \to \mT^2$.
 
   \vfill   
}
 


\frame
{
  \frametitle{}

 {\bf Proposition.} If there exists a homeomorphism $h \colon \mT^2 \to \mT^2$ conjugating the action $\phi_t$ to $\phi_1$ then $t = 0$.
 \bigskip
    
 The proof  uses   fundamental domains for the actions of $A$ and $B$. 
 
   \bigskip
   
   There is a companion result:
   
   {\bf Theorem.} Let $h \colon \mT^2 \to \mT^2$ be a homeomorphism which conjugates the actions $\phi, \phi' \colon SL(2,\mZ) \times \mT^2 \to \mT^2$, where $\phi$ is the standard action. Then $h$ is smooth.
   
   \medskip
   
   $\star$~ Elise Cawley, {\it The {T}eichm\"{u}ller space of the standard action of {${\rm
              SL}(2,\mZ)$} on {${\mT}^2$} is trivial},
 {\bf Internat. Math. Res. Notices},  {International Mathematics Research Notices}, 7:135--141, 1992.


  \vfill   
}
 
\section{Stability}

 
\frame
{

  \frametitle{}

The key to the above example is that $H^1(SL(2,\mZ) ; \mR) \ne 0$, which is   used implicitly in the construction of the deformation.

\bigskip
 
 
  ``The notion of hyperbolicity has proved the key to questions of stability in the cases of actions by $\mZ$ or $\mR$. For actions by other groups, we would hope to find conditions analogous to hyperbolicity in the sense that they facilitate analysis to a comparable extent. However, we should not expect these conditions to resemble hyperbolicity too closely, for they should reflect the algebra of the particular group being studied.''

\medskip

$\star$~ D.~Stowe
{\it Stable orbits of differentiable group actions}, 
  {\bf Trans. Amer. Math. Soc.}, 277:665--684, 1983.
  

 \vfill
}


\frame
{



{\bf Theorem:}   Let $\phi \colon \G \times M \to M$ be a smooth action with isolated fixed-point $x \in M$. Suppose that $H^1(\G ; \mV_{\phi}) = 0$ for all finite dimensional modules over the action of $D_x\phi$. Then for a nearby action  $\phi'$ there is an isolated fixed-point $x'$ near to $x$.

 

$\star$~  D.~Stowe, 
  {\it The stationary set of a group action}, 
  {\bf Proc. Amer. Math. Soc.}, 79:139--146, 1980.

\bigskip



{\bf  Definition.}[SVC]
 $\G$ satisfies the {\em strong  vanishing cohomology} condition   if  $H^1( \Gamma' ; {\mR}^N_{\rho'})  = \{0\}$  for every  subgroup  $\Gamma' \subset \Gamma$ of finite index        and representation $\rho' \colon \Gamma' \rightarrow GL(N,{\mR})$, $N \geq 1$.
 
 \medskip
 
 $\Gamma$ satisfies SVC  $\Longrightarrow$  $\Gamma' \subset \Gamma$   finite index   satisfies SVC.

  \vfill
}


\frame
{

 
 
{\bf Theorem.}[Margulis] 
Let $\Gamma \subset G$ be an irreducible lattice in a connected semi-simple algebraic ${\mR}$-group G.  Assume that the $\mR$-split rank of each factor of G is at least 2,  and that $G_{\mR}^0$ has no compact factors.  Then  $\Gamma$ satisfies condition SVC.  
 
 \medskip
  
$\star$~ Theorem 2.1 in  {G. A.} Margulis, 
  {\bf Discrete Subgroups of Semisimple Lie Groups}.
  Springer-Verlag, 1991.

\medskip

{\bf Example.}   For $n \geq 3$, a lattice  $\Gamma \subset SL(n, {\bf Z})$ satisfies SVC.


\vfill
}

\section{Rigidity}

\frame
{

{\bf Theorem.} Let $\phi \colon \G \times \mT^n \to \mT^n$, for  $n \geq 3$ and $\G \subset SL(n,\mZ)$  finite index. Then a deformation $\phi_t$ of $\phi$ is smoothly conjugate to $\phi_t$.

 
\bigskip

The proof has three parts, and uses:

$\bullet$~ Action is Anosov and the periodic points of the action are dense.  

$\bullet$  $\G$ satisfies SVC condition so the periodic points are stable.

$\bullet$  $\G$ contains maximal abelian semi-simple subgroup, hence the action is \emph{trellised}, and so the conjugation must be smooth.

\bigskip
 
   
  $\star$~  {S.~Hurder}, {\it Rigidity for Anosov actions of higher rank lattices},
 {\bf Annals of Math. (2)}, 135:361-410, 1992.


\vfill
}


\frame
{

{\bf Step 1.} Let $\phi_t \colon \G \times \mT^n \to \mT^n$ be a 1-parameter family of actions.

\medskip

Let $\gamma_0 \in \G$ so that $\phi_0(\gamma_0)$ is Anosov, hence a hyperbolic matrix.

\medskip

The diffeomorphism $\phi_0(\gamma_0)$ is structurally stable by Anosov, so there exists  homeomorphisms $h_t \colon \mT^n \to \mT^n$ for $0 \leq t \leq \e$   conjugating $\phi_t$ and $\phi_0$.  

\medskip

The periodic points of $\phi_0(\gamma_0)$ are dense, so the same holds for $\phi_t(\gamma_0)$ for $0 \leq t \leq \e$.

\medskip

Next, must show that $h_t$ conjugates the full action of $\G$.

\vfill
}



\frame
{

{\bf Step 2.}  
Let $y \in \mT^n_{\mQ}$ be a rational point, so is periodic for $\phi_0(\gamma_0)$.

Set $y_t = h_t(y)$ which is isolated periodic for $\phi_t(\gamma_0)$ for $0 \leq t \leq \e$.

Let $\G_y = \{\gamma \in \G \mid \phi_0(\gamma)(y) = y\}$ a  finite index subgroup of $\G$.

$\G$ satisfies the SVC condition, so isolated periodic points of the action of $\G_y$ are stable.

There exists $0 < \e_y' \leq \e$ such that $\phi_t(\G_y)(y_t) = y_t$ for   $0 \leq t < \e_y'$

So $\phi_t(\G_y)(y_t) = y_t$ for $t = \e_y'$

There is $n \geq 1$ such that $\gamma_0^n \in \G_y$ which is again Anosov, hence $y_t$ is isolated fixed point for $\phi_t(\G_y)$.

Hence $y_t$ is isolated fixed point for $\phi_t(\G_y)$ for all $0 \leq t \leq \e$.

Thus, $h_t$ conjugates $\phi_t$ to $\phi_0$ on the dense set $\mT^n_{\mQ} \subset \mT^n$

and so conjugates the action of $\phi_t$ to $\phi_0$  for all $0 \leq t \leq \e$.

 


\vfill
}



\frame
{

{\bf Step 3.}  We    show that the conjugating homeomorphism is smooth. 

That is, the ``Teichm\"{u}ller space'' of deformations of the standard action $\phi_0 \colon \G \times \mT^n \to \mT^n$ is trivial.

\medskip

{\bf Definition.} A $C^1$-action $\phi \colon \G \times \mT^n \to \mT^n$ is \emph{Cartan} if there exists an abelian subgroup $\cA \subset \G$ generated by $\{\gamma_1, \ldots , \gamma_n\} \subset \cA$ such that each $\phi(\gamma_i)$ is Anosov with 1-dimensional stable foliation $\F_i^-$ and the foliations $\{\F_1^- , \ldots , \F_n^-\}$ are completely transverse. 

That is, their tangent spaces define  a framing of $T\mT^n$ --  a trellis

 

 \begin{figure}[htbp]
\begin{center}
\includegraphics[width=.3\textwidth]{pix/trellis}
\end{center}
\end{figure}




\vfill
}



\frame
{

{\bf Theorem.}  
Let $G$ be a semi-simple analytic Lie group and $\G \subset G$ a lattice.  Let $H$ be a Cartan subgroup of $G$, then there exists $g \in G$ such that $\Gamma_H = \Gamma \cap g^{-1}Hg$ is a uniform lattice in $g^{-1}Hg$.
 
 \medskip
 
$\star$~ G.~Prasad and {M. S.} Raghunathan, 
 {\it Cartan subgroups and lattices in semi-simple groups}, 
  {\bf Annals of Math.}, 96:296--317, 1972.

\bigskip

{\bf Corollary.} Let $\G \subset SL(n, \mZ)$ be a lattice. Then the standard action  $\phi \colon \G \times \mT^n \to \mT^n$ is Cartan.




\vfill
}



\frame
{
Let $\cA = \langle \gamma_1, \ldots , \gamma_n \rangle \subset \G \subset SL(n,\mZ)$ be a Cartan subgroup.

The stable manifolds of  Anosov diffeomorphisms are preserved by a conjugacy, so the conjugacy $h_t \colon \mT^n \to \mT^n$ preserves the stable foliations $\{\F_1^-, \ldots , \F_n^-\}$. That is, $h_t$ preserves the lines  in the trellises for the actions.

 \medskip
 
 Stowe's Theorem implies that the actions of $\G_y$ are conjugated at a periodic orbit $y$, which implies that the exponents of contraction are equal for the actions $\phi_0(\gamma_i)$ and $\phi_t(\gamma_i)$ at $y$ and $h_t(y)$, respectively.
 
 \medskip
The Livsic Theorem then implies that the restriction of $h_t$ to the stable foliations is a smooth map of 1-dimensional manifolds.
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\frame
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We   have  homeomorphisms $h_t \colon \mT^n \to \mT^n$ whose restrictions to a transverse family of 1-dimensional foliations of $\mT^n$ are leafwise smooth. These 1-manifolds are smoothly embedded in $\mT^n$.

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{\bf Theorem.} Let $\F_s$ and $\F_u$ be two transverse foliations with uniformly smooth leaves, of some manifold $M$. If $f$ is uniformly smooth along the leaves of $\F_s$ and $\F_u$, then $f$ is smooth.

\medskip

$\star$~ J.-L Journ\'{e}, 
 {\it A regularity lemma for functions of several variables}, 
  {\bf Rev. Mat. Iberoamericana}, 4:187--193, 1988.

 \bigskip
 
 Thus, the conjugating maps $h_t$ are smooth, as was to be shown.
 
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 The above results apply to a wide variety of other lattice actions:
 
  
  $\star$~   Section 7, {\it Rigidity for Anosov actions of higher rank lattices}.
  
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\section{Zimmer Program}


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Let $\phi \colon \G \times M \to M$ be a $C^{\infty}$-action on a compact $n$-manifold $M$.
Choose a \emph{measurable} framing $TM \cong M \times \mR^n$ then the derivative defines a measurable cocycle, or ``virtual homomorphism'',
  $$D\phi \colon \G \times M \to GL(n,\mR)$$
$$D\phi(\gamma_2 \gamma_1, x) = D\phi(\gamma_2, \gamma_1 \cdot x) \cdot D\phi(\gamma_1, x)$$
This is just the chain rule for group actions. 

 
$\bullet$~ \emph{Margulis Rigidity} gives conditions on a lattice $\G \subset G$ which imply that a homomorphism $\rho \colon \Gamma \to H$     extends to a homomorphism $\widehat{\rho} \colon G \to H$.

$\bullet$~ 
 \emph{Zimmer Superrigidity} gives conditions on a lattice $\G \subset G$  and measure-preserving action $\phi \colon \G \times M \to M$ which imply that a cocycle $\alpha \colon \G \times M \to H$ is \emph{measurably} conjugate to a constant cocycle, which extends to  a homomorphism  $\widehat{\rho} \colon G \to H$.
 
 
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\frame
{


In the 1980's,  Zimmer used superrigidity to study volume-preserving actions of higher rank lattices on compact manifolds.
  He posed the question whether a group action, given by a map $\phi \colon \G \to \Diff(M)$, must behave like   its cocycle $D\phi$?

{\bf Conjecture.} [Zimmer] Let $\G \subset G$ be a higher rank lattice, and suppose there are no non-trivial representations $\widehat{\rho} \colon G \to GL(n,\mR)$, then an action $\phi \colon \G \times M \to M$ factors through a finite action.

   
For a   discussion of the Zimmer Program, see

$\star$~ D.~Fisher, 
 {\it Groups acting on manifolds: around the {Z}immer program}, 
 in  {\bf Group actions in ergodic theory, geometry, and
              topology---selected papers}, Univ. Chicago Press, Chicago, IL, 2020, pages 609--683.



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\frame
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 There has been remarkable progress towards establishing the Zimmer Conjecture in special cases. 
 
\medskip
{\bf Hypothesis.} Suppose $G$ is a connected semisimple Lie group with finite center, all of whose noncompact almost-simple factors have $\mR$-rank 2 or higher, and suppose $\G$  is a lattice in $G$.

\medskip

{\bf Theorem.} 
Let $G$  and $\G$ be as in the Hypothesis. 
Let $\alpha$  be a $C^{\infty}$-action of $\G$ on a compact nilmanifold $M = N/\Lambda$. Suppose $\alpha$ can be lifted to an action on the universal cover $\widetilde{M}$ of $M$, and let $\rho$  be the associated linear data of $\alpha$. 
If $\alpha(\gamma)$ is hyperbolic for some element $\gamma \in \G$, then there are a finite-index subgroup $\G' \subset \G$  and a $C^{\infty}$-diffeomorphism  $h \colon M \to M$, homotopic to identity, such that $\rho(\gamma) \circ h = h \circ \alpha(\gamma)$  for all $\gamma \in \G'$.  

 
$\star$~ A.~Brown, F.~Rodriguez Hertz and Z.~Wang,
 {\it Global smooth and topological rigidity of hyperbolic lattice actions}, 
  {\bf Ann. of Math. (2)}, 186:913--972, 2017.
 

 
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When the dimension $m$ of $M$ is less than $n$,  there are no non-trivial representations $\rho \colon SL(n, \mZ) \to GL(m, \mR)$. The Zimmer Conjecture then becomes a generalized version of Witte's Theorem for $\mS^1$.

\medskip

{\bf Theorem.}   Given a subgroup $\G \subset SL(n, \mZ)$  of finite index, and a closed manifold $M$ with dimension $m < n-1$, then  every $C^2$-action $\alpha \colon \G \times M \to M$ is finite.
     
\medskip

 

$\star$~ A.~Brown, D.~Fisher and S.~Hurtado, 
 {\it Zimmer's conjecture for actions of $SL(m, \mZ)$}, 
  {\bf Invent. Math.}, 221:1001--1060,  2020.

  

 

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There have been many more results in the period between 1990 and today. The introduction to the paper by Brown, Rodriguez Hertz, and  Wang gives a nice overview, as of 2015. 

\medskip

Moreover, the action continues with the conference:

$\bullet$~ Group Actions and Rigidity: Around the Zimmer Program\\
Introductory school: Rigidity, Dynamics and Geometric Structures\\
April 15 to 19, 2024 - CIRM, Marseille Luminy\\
https://indico.math.cnrs.fr/event/9764/

\bigskip

This is part of a thematic semester 

  
$\bullet$~  Group Actions and Rigidity: Around the Zimmer Program, Paris\\
April 15th to July 5th, 2024\\
https://indico.math.cnrs.fr/category/619/

 
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