% Group Theory Working Seminar
% Purdue University
% April 15, 2024
 
 
% slides for talk "Applications of the Contraction Mapping Theorem: Circle actions"
 
 
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\date[June 2, 2023]


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


\author{Steve Hurder\\ \small University of Illinois at Chicago\\April 15, 2024} 
 
 
\begin{document}
 
   \frame{\titlepage} % # 1

\section{Prelude}

\frame
{
  \frametitle{}

 {\bf Definition.} Let $(X,d)$ be a metric space. Then a map $T \colon X \to X$ is called a \emph{contraction mapping} on $X$ if there exists $0 \leq C < 1$ such that $d(T(x), T(y)) \leq C \cdot d(x,y)$ for all $x,y \in X$.
 
 \bigskip\bigskip
 
 {\bf Banach Fixed-Point Theorem.} [Banach, 1922]  Let $(X,d)$ be a non-empty complete metric space with a contraction mapping $T \colon X \to X$. Then $T$ admits a unique fixed-point $x^* \in X$.

\bigskip

In these   lectures, we give three applications of this theorem to the study of group actions on compact spaces.
 \vfill   
}

\section{Circle Actions}

\frame
{
  \frametitle{}

$\bullet$~ $\G$ will always denote a finitely generated group.

{\bf Question:} What are the non-trivial actions of $\G$ on the circle $\mS^1$?

\bigskip

{\bf Theorem.} [Margulis] Let a group $\G$ act by homeomorphisms on $\mS^1$. Then, either there is a $\G$-invariant probability measure on $\mS^1$,\\ or $\G$ contains a free nonabelian group.

\bigskip

$\star$~  {Gregory Margulis}, {\it Free subgroups of the homeomorphism group of the circle},  {\bf C. R. Acad. Sci. Paris S\'{e}r. I Math.}, {331},   {669--674},1980. 

\medskip

The proof uses a ``local contraction principle'' on the space of probability measures on $\mS^1$ to construct an invariant Cantor set, and a ping-pong game for the action on this Cantor set.

 \vfill   
}
 
\frame
{
  \frametitle{}

{\bf Theorem.} [Witte-Morris]  Let $\G$ be an arithmetic subgroup of a $\mQ$-simple algebraic $\mQ$-group $G$, with $\mQ -{\rm rank}(G) \geq 2$. Then every continuous action of $\G$ on the circle $\mS^1$ or the real line $\mR$ factors through the action of a finite quotient of $\G$.
\medskip

{\bf Example.}  Let $\G \subset SL(n, \mR)$ be a lattice for $n \geq 3$, then an action of $\G$ on $\mS^1$ factors through a finite action.

\bigskip

$\star$~  {Dave Witte}, {\it Arithmetic groups of higher {${\bf Q}$}-rank cannot act on
              {$1$}-manifolds},  {\bf Proc. Amer. Math. Soc.}, {122},   {669--674}, {1994}. 

\bigskip
The proof uses the \emph{antithesis} of a contraction mapping principle, as it shows that a $\mZ$-extension of $\G$ is left orderable.
 
 
 \vfill   
}
 

\frame
{
  \frametitle{}


{\bf Definition.} 
A group $\G$ is called (left) orderable if there exists a total ordering $\leq$ on $\G$ which is compatible with the group law, in the sense that $g\leq h$ and $k\in \G$ implies $k\cdot g \leq k\cdot h$. 

\bigskip

  {\bf Theorem.}  $\G$ acts faithfully on $\mR$ if and only if $\G$  is left orderable.

\medskip

{\bf Example.} Every torsion  free nilpotent group $\G$ acts faithfully on $\mR$.

\bigskip

$\star$~  {Andr\'{e}s Navas}, {\bf Groups of circle diffeomorphisms},  {Chicago Lectures in Mathematics}, {University of Chicago Press}, {2011}. 

 \vfill   
}
 
  \section{Fuchsian Actions}


\frame
{
  \frametitle{}
 
 Let $\G \subset PSL(2,\mR)$ be a torsion-free cocompact subgroup; so a discrete subgroup such that the quotient $M = PSL(2,\mR)/\G$ is a compact 3-manifold.
 
 Let $\mD^2 \subset \mC^2$ be the unit disk with the Poincar\'e metric, 
 $$ds^2 = \frac{ 4 (dx^2 + dy^2 )}{ \left( 1 - (x^2 + y^2)^2  \right) } $$
 The geodesics for this model are the circular acrs
 \begin{figure}[htbp]
\begin{center}
\includegraphics[width=.3\textwidth]{pix/disk}
\label{figure-disk}
\end{center}
\end{figure}

  \vfill   
}
 

\frame
{
  \frametitle{}

 For a matrix $A =  \left[\begin{array}{cc}a & b \\c & d\end{array}\right]$, the action on $z = x + iy \in \mC$  by 
 $\ds A \cdot z = \frac{az + b}{cz + d}$
maps the boundary $\mS^1 = \{  \|z\| =1\}$ to itself, so defines an action $\Phi_0 \colon \G \times \mS^1 \to \mS^1$.

Alternately, each point $z \in \mS^1$ is identified with the point at $+\infty$ for a family of geodesics in $\mD^2$, and the action $\Phi_0$  is  described by its action on geodesics in $\mD^2$.

Then the quotient $\Sigma = \mD^2/\G$ is a compact Riemann surface with a metric $g_0$ of constant negative curvature.
 \vfill   
}
 

\frame
{
  \frametitle{}

 Consider a metric $g$ on $\Sigma$ with negative curvature - for example a metric which is $C^3$-close to $g_0$,

The lift $\widetilde{g}$ of $g$ to the universal cover $\mD^2 \to \Sigma$ is $\G$-invariant, so the deck transformations preserve the geodesics of $\widetilde{g}$ in $\mD^2$. The action of $\G$ then acts on the endpoints of the geodesics at $+\infty$, so defines an action $\Phi_{g} \colon \G \times \mS^1 \to \mS^1$. Let $\fM^-$ denote the space of metrics of negative curvature on $\Sigma$. Thus we obtain

$$\Phi \colon \fM^- \to {\Hom}\{ \G \to \Homeo(\mS^1)\}$$

where $\Phi(g) = \Phi_{g}$.
 \vfill   
}
 
    

  \section{Anosov Flows}


\frame
{
  \frametitle{}
 
 The geodesic flow   $\phi_t \colon M \to M$  for the Riemann surface $\Sigma$ of  constant negative curvature has a  simple description when lifted to $SL(2, \mR)$, given by 
  $$\widetilde{\phi_t}\left[\begin{array}{cc}a & b \\c & d\end{array}\right] = \left[\begin{array}{cc}e^{t/2} & 0 \\0 & e^{-t/2} \end{array}\right] \cdot \left[\begin{array}{cc}a & b \\c & d\end{array}\right] = \left[\begin{array}{cc} e^{t/2}a & e^{t/2} b \\ e^{-t/2}c & e^{-t/2} d\end{array}\right]$$
  
  The action $\widetilde{\phi_t}$
  
  $\bullet$~ expands the subgroup
  $\ds U^+ = \left[\begin{array}{cc}1& b \\0 & 1\end{array}\right]$
  
  $\bullet$~  contracts  the subgroup
  $\ds U^- = \left[\begin{array}{cc}1& 0 \\c & 1\end{array}\right]$
  
  \vfill   
}
 

\frame
{
  \frametitle{}
The infinitesimal model for this action is given by the commutator actions on Lie the algebra 
${\mathfrak{s}\mathfrak{l} (2,\mR)} = \left[\begin{array}{cc}r& s \\t & -r\end{array}\right] \quad, r,s,t \in \mR$
 
Let $X = \left[\begin{array}{cc}0& 1 \\0 & 0\end{array}\right]$, $Y = \left[\begin{array}{cc}0& 0 \\1 & 0\end{array}\right]$, $Z= \left[\begin{array}{cc}1& 0 \\0 &  -1\end{array}\right]$

then $[Z,X] = 2X$, $[Z,Y] = -2Y$ and $[X,Y]= Z$.

\bigskip

Then we have two Lie subalgebras   $L^+ = \langle Z, X \rangle$ and $L^- = \langle Z, Y \rangle$ which generate subgroups of $SL(2,\mR)$. The left translates of these subgroups define codimension one foliations $\widetilde{\F}^+$ and $\widetilde{\F}^-$ which are invariant under the left action of $\G$, so descend to foliations $\F^+$ and $\F^-$ on $M$. Moreover, they are transverse to the $\mS^1$ subgroup of $SL(2,\mR)$ generated by the Lie vector $\vec{\theta} = X-Y$.


 \vfill   
}
 
 
\frame
{
  \frametitle{}
 
{\bf Conclusion.} There is a direct sum decomposition $TM = \langle Z \rangle  \oplus \langle X \rangle \oplus \langle Y \rangle$
where the flow of $\phi_t$ is:

$\bullet$~ tangent to the distribution $\langle Z \rangle$

$\bullet$~ expands   the distribution $\langle X \rangle$

$\bullet$~ contracts   the distribution $\langle Y \rangle$

\medskip

{\bf Definition.} Let $M$ be a compact manifold of dimension $n$.\\
 A smooth flow $\phi_t$ of $M$ is Anosov if there exists a direct sum decomposition $TM = E_0 \oplus E^+ \oplus E^-$ where the flow of $\phi_t$ is
  
  $\bullet$~ tangent to the distribution $E_0$

$\bullet$~ uniformly expands   the distribution $E^+$

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

  \vfill   
}
 
  
\frame
{
  \frametitle{}
 
 
{\bf Theorem.} [Anosov]  The \emph{weak unstable} and \emph{weak stable} distributions $E_0 \oplus E^+$ and $E_0 \oplus E^-$ are both integrable, and integrate to codimension one  foliations denoted $\F^+$ and $\F^-$ on $M$.


\medskip

{\bf Theorem.} [Anosov] Anosov flows are structurally stable. If $\phi'_t$ is $C^1$-close to   $\phi_t$, then $\phi'_t$ is also  an Anosov flow, and there is a homeomorphism of $M$ conjugating   orbits of $\phi_t$ to   orbits of $\phi'_t$.
 
 \bigskip
 
$\star$~ {D.V. Anosov}, 
  {\it Tangent fields of transversal foliations in {U}-systems}, 
  {\bf Math. Notes Acad. Sci., USSR}, 2:818--823, 1967.

 
$\star$~ {D.V. Anosov}, 
  {\it Geodesic Flows on Closed Riemannian Manifolds with Negative
  Curvature},  90,  {\bf Proc. Steklov Inst. Math.},  1967.
  
    \vfill   
}
 
  
\frame
{
  \frametitle{}

The proof of structural stability by Anosov uses the \emph{shadowing principle} for Anosov flows.

John Mather gave an alternate proof using the Banach Fixed-Point Theorem for the space of sections of a vector bundle.


$\star$~ {Mather}, {\it Anosov diffeomorphisms}, appendix to S.~Smale, {\it Differentiable Dynamical Systems}, {Bulleting A.M.S.}, 73, 1967, pages 792-795.
\bigskip

  
    \vfill   
}
 
  
\frame
{
  \frametitle{}


In Section~3 of  work with Katok, we refined the argument of Mather to show the contraction mapping was locally defined on a space of sections with a norm defined by the Zygmund condition:

{\bf Definition.} The Zygmund norm of $f \colon [a,b] \to \mR$  is given by
$$\Lambda_*(f) = \sup_{a < x < b} \ \limsup_{h \to 0} \ \frac{f(x+h) + f(x-h) - 2f(x)}{|h|}$$

{\bf Remark.} $\Lambda_*(f) < \infty$ implies that $f$ is $\alpha$-H\"{o}lder for all $\alpha < 1$.
\medskip

$\star$~  {S.~Hurder and A.~Katok}, 
  {\it Differentiability, rigidity and {Godbillon-Vey} classes for {Anosov} flows}, 
  {\bf Publ. Math. Inst. Hautes Etudes Sci.}, 72:5--61, 1990.



  \vfill   
}
 
  
\frame
{
  \frametitle{}


{\bf Theorem.} [Hurder-Katok] If $\phi_t$ is a volume preserving Anosov flow of a 3-manifold, then the foliations $\F^+$ and $\F^-$ are $C^1$ and their derivatives have bounded   Zygmund norm. 

{\bf Corollary.} The induced action at infinity for a geodesic flow $\phi_t$ is $C^{1+\alpha}$ for all $\alpha < 1$. That is, the flow induces  a homomorphism   
$$\Phi \colon \G \to \Diff^{1+\alpha}(\mS^1) \subset \Homeo(\mS^1)$$

 
\bigskip
$\Rightarrow$  variations of metrics of negative curvature on $\Sigma$ generate variations of $C^{1+\alpha}$-actions of $\G = \pi_1(\Sigma, x_0)$ on $\mS^1$.
  \vfill   
}
 


  \section{Godbillon-Vey}


\frame
{
  \frametitle{}
 
Let $\F$ be a codimension one $C^2$-foliation of a compact    manifold $M$.    
Assume that the normal bundle to $\F$ is orientable, then there exists a 1-form $\omega$ such that ${\rm ker}(\omega) = T\F$.

\medskip

By the Frobenius Theorem, the distribution $T\F$ is integrable implies that there is a 1-form $\eta$ such that $d\omega = \eta \wedge \omega$. Then a calculation shows that $\eta \wedge d\eta$ is closed, and that its cohomology class $[\eta \wedge d\eta] \in H^3(M ; \mR)$ is independent of the choice of  $\omega$.

\bigskip

{\bf Theorem.} [Godbillon-Vey] The class $GV(\F) \in H^3(M ; \mR)$  is a diffeomorphism invariant of the foliation $\F$, and depends only on the concordance class of $\F$.


  \vfill   
}
 

\frame
{
  \frametitle{}

Roussarie calculated the Godbillon-Vey invariant for the weak unstable foliation of $M= SL(2, \mR)/\G$.

\bigskip

 Let $X^*, Y^*, Z^*$ be the dual 1-forms for the framing $\{X,Y,Z\}$ of $TM$. Then $T\F^+ = {\rm ker}(Y^*)$ so take $\omega = Y^*$. Then 
$d\omega = -2 Z^* \wedge Y^*$ and set $\eta = -2 Z^*$, so we get
$$\eta \wedge d\eta  = -2Z^* \wedge -2dZ^* = 4 Z^* \wedge X^* \wedge Y^*$$
is a volume form on $M$. Hence $0 \ne GV(\F^+)  \in H^3(M ; \mR)$.

\bigskip

$\star$~ C.~Godbillon and J.~Vey, 
  {\it Un invariant des feuilletages de codimension 1}, 
  {\bf C.R. Acad. Sci. Paris}, 273:92--95, 1971.

 \vfill   
}
 

\frame
{
  \frametitle{}
Suppose that $\F$ is a codimension one foliation on a circle bundle $\mS^1 \to M \to B$ which is transverse to the circle fibers.
Then there is a map, ``integration over the fibers''
$$\int_{\mS^1} \colon H^3(M; \mR) \to H^2(B ; \mR)$$
 which yields a class $\ds gv(\F) = \int_{\mS^1} \ GV(\F) \in H^2(B ; \mR)$.
 
 Suppose the base $B = \Sigma$ as for the case   $M= SL(2, \mR)/\G$ then $gv(\F) \in H^2(\Sigma ; \mR) \cong H^2(\G ; \mR)$.
 
 
 \vfill   
}
 

\frame
{
  \frametitle{}

There is a formula for this class, given by Thurston:

Let $\gamma_1, \gamma_2 \in \G$ and let $\alpha(\gamma_i) \in \Diff(\mS^1)$. Set 
$$f= \log \{\alpha(\gamma_2)'\} \quad , \quad g =\log \{\alpha(\gamma_1 \gamma_2)'\}$$
$$ gv(\gamma_1, \gamma_2) = \int_{\mS^1} \ f \ d g = {\bf Area}\{(f,g) \colon \mS^1 \to \mR^2\}  $$

This last equality follows from Stokes' Theorem for planar curves.

\bigskip

$\star$~ {R.~Bott},
  {\it On some  formulas for the characteristic classes of group actions},
  in {\bf Differential topology, foliations and Gelfand-Fuks cohomology (Proc. Sympos., Pontif\'\i cia Univ. Cat\'olica, Rio de Janeiro, 1976)},  {Lect. Notes in Math. Vol.  652},   1978:25--61.
  
  
 \vfill   
}
 
    

\frame
{
  \frametitle{}
{\bf  Observations.}  

$\bullet$~ The area integral $\ds \int_{\mS^1} \ f \ d g $ is well defined when the image of $\mS^1$ has Lebesgue measure 0.

$\bullet$~ If the maps $f,g$ are $\alpha$-H\"{o}lder   continuous for $\alpha > 1/2$ then the image circle in $\mR^2$ has Lebesgue measure 0.
  
  \bigskip
  
  {\bf Theorem.} [Hurder \& Katok, 1991]  Let $g$ be a metric of negative curvature on a compact surface $\Sigma$. Then there is a well-defined Godbillon-Vey class $gv(\F_{g}^+)  \in H^2(\G ; \mR)$.
  
  \bigskip
  
 $\star$~  {T.~Tsuboi},  {\it On the {H}urder-{K}atok extension of the {G}odbillon-{V}ey invariant},
 {\bf J. Fac. Sci. Univ. Tokyo}, 37:255--262, 1990.



  \vfill   
}


  \section{Mitsumatsu Defect}


\frame
{
  \frametitle{}
 
 {\bf Theorem.} [Formula of Mitsumatsu] 
 Let $(\Sigma, g)$ be a closed Riemann surface with strictly negative curvature and Euler characteristic $\chi(\Sigma)$. Then
 $$\langle GV(\F_g^+) , [M]\rangle =  4\pi^2 \cdot \chi(\Sigma) - 3 \cdot \int_M \ (\frac{\partial}{\partial \theta} H)^2 dvol $$
  where $\frac{\partial}{\partial \theta}$ is the unit tangent vector to the circle fibers of $M \to \Sigma$, and $H$ is the unique bounded positive solution to the Riccati equation along the geodesic flow on $M$.
  
  \bigskip
  
  {Y.~Mitsumatsu},
  {\it A relation between the topological invariance of the {G}odbillon-{V}ey invariant and the differentiability of {A}nosov foliations},
  in {\bf Foliations ({T}okyo, 1983)},  {Adv. Stud. Pure Math.}, Vol. 5, {North-Holland, Amsterdam}, 1985, pages 159--167.


  \vfill   
}
 

\frame
{
  \frametitle{}
Mitsumatsu showed this formula under the assumption that the foliation $\F_g^+$ is $C^2$. In 
 Section 9 of the paper with Katok we extended this formula to the   case when $\F_g^+$ is $C^{1+\alpha}$.
 
 \bigskip
 
  This is fortuitous because of the observation
  
  {\bf Corollary.} [Mitsumatsu] $\langle GV(\F_g^+) , [M]\rangle =  4\pi^2 \cdot \chi(\Sigma)$ if and only if $g$ has constant negative curvature.

\bigskip
Thus, if $g$ a metric with variable negative curvature on $\Sigma$, then 
$$\langle GV(\F_g^+) , [M]\rangle <  4\pi^2 \cdot \chi(\Sigma)$$
 
 \vfill   
}
 


\frame
{
  \frametitle{}

The space of metrics with negative curvature on $\Sigma$ is locally connected, so we obtain for the group  cohomology:
 
{\bf Theorem.}  Let $\Diff^{1+\alpha}(\mS^1)$ denote the group of diffeomorphisms of the circle with $\alpha$-H\"{o}lder first derivatives.  
For $\alpha > 1/2$:

$\bullet$~   There is a well-defined   class
$gv \in H^2(\Diff_{\delta}^{1+\alpha}(\mS^1) ; \mR)$.

$\bullet$~ There exists continuous families of cycles $f_t \colon \Sigma \to B\Diff_{\delta}^{1+\alpha}(\mS^1)$ such that the evaluation $\langle gv , [f_t\Sigma]\rangle$ varies continuously and non-trivially with $t$.

 \vfill   
}
 
\section{Two Problems}

\frame
{
  \frametitle{}

{\bf Problem 1.} Thurston extended his construction of foliations on $\mS^3$ with variable Godbillon-Vey class in an unpublished work (see below). Show that the Thurston construction can also be deformed with holonomy in the group $\Diff_{\delta}^{1+\alpha}(\mS^1)$.

\bigskip

{\bf Problem 2.} Heitsch modified Thurston's construction to show that the higher codimension Godbillon-Vey classes   are also variable.    Show that the Heitsch  constructions can also be deformed with holonomy in the group $\Diff_{\delta}^{1+\alpha}(\mS^1)$.

\bigskip

$\star$~ {J.~Heitsch},
  {\it Independent variation of secondary classes},
  {\bf Ann. of Math. (2)}, 108:421--460, 1978.

\medskip
$\star$~ {T.~Mizutani},
  {\it On {T}hurston's construction of a surjective homomorphism {$H_{2n+1}(B\Gamma_n,\Bbb Z)\to\Bbb R$}},
  in {\bf Geometry, dynamics, and foliations 2013}, 
  {Math. Soc. Japan, Tokyo}, 2017,   211--219.


 \vfill   
}
 



\end{document}

