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\title{$L^p$-cohomology}
\author{P. Pansu}
\date{\today}

\newtheorem{thm}{Theorem}
\newtheorem{lem}{Lemma}
\newtheorem{prop}[lem]{Proposition}
\newtheorem{cor}[lem]{Corollary}
\newtheorem{defi}[lem]{Definition}
\newtheorem{rem}[lem]{Remark}
\newtheorem{nota}[lem]{Notation}
\newtheorem{ques}[lem]{Conjecture}
\newtheorem{exa}[lem]{Example}
\newtheorem{ex}[lem]{Examples}
\def\proof{\par\medskip\noindent {\bf Proof.}{\hskip1em}}
\def\qed{~q.e.d.}

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\def\Z{\mathbb{Z}}
\def\H{\mathbb{H}}
\def\O{\mathbb{O}}
\def\n#1{{\parallel #1 \parallel}}
\def\sp{{\rm sp}\,}
\def\dim{{\rm dim}}
\def\ker{{\rm ker}}
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\begin{document}

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\titlepage
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\begin{frame}
Advertisement: http://analysis-situs.org
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\begin{frame}

\frametitle{What is it ?}

\begin{eqnarray*}
topological~space\quad&\to&\quad cohomology\\
manifold\quad&\to&\quad de~Rham~cohomology\\\pause
metric~space\quad&\to&\quad cohomology~with~decay~condition\\
Riemannian~manifold\quad&\to&\quad de~Rham~cohomology~with~decay~condition\pause
\end{eqnarray*}


\begin{defi}
Let $M$ be a Riemannian manifold. Let $p>1$. $L^p$-{\em cohomology} of $M$ is the cohomology of the complex of $L^p$-differential forms on $M$ whose exterior differentials are $L^p$ as well,
\begin{eqnarray*}
H^{k,p}&=&\textrm{closed  }k \textrm{-forms in }L^p /d((k-1) \textrm{-forms in }L^p ),\\\pause
R^{k,p} & = &\textrm{closed  } k \textrm{-forms in }L^p/\textrm{closure of }d((k-1) \textrm{-forms in }L^p )
,\\ 
T^{k,p}
& = & \textrm{closure of }d((k-1) \textrm{-forms in }L^p )/ d((k-1) \textrm{-forms in }L^p ).
\end{eqnarray*}
$R^{k,p}$ is called the {\em reduced cohomology}. $T^{k,p}$ is called the
{\em torsion}.
\end{defi}

\end{frame}
\begin{frame}

\frametitle{Example : the real hyperbolic plane $H^2_{\R}$}


Here $H^{0,p} = 0 = H^{2,p}$ for all $p$. \pause 

If $p=2$, since the Laplacian on $L^2$ functions is bounded below, $T^{1,2} = 0$. Therefore 
\begin{eqnarray*}
H^{1,2} &=& R^{1,2}\\ \pause
&=&
\{L^2\textrm{ harmonic 1-forms}\} \\ \pause
&=& \{\textrm{harmonic functions } h
\textrm{ on } H^2_{\R}\textrm{ with } \nabla h\in L^2\}/\R \,.
\end{eqnarray*}
\pause
Using conformal invariance, switch from hyperbolic metric to euclidean metric on the disk $D$.
\begin{eqnarray*} 
H^{1,2} & = & \{ \textrm{harmonic functions }h\textrm{ on } D \textrm{
with } \nabla h\in L^2\}/\R\\ \pause
& = & \{ \textrm{Fourier series } \Sigma
a_n e^{in\theta } \textrm{ with }
  a_0 = 0, \Sigma \vert n\vert \, \vert a_n\vert ^2 < +\infty \},\pause
\end{eqnarray*} 
which is Sobolev space $H^{1/2}(\R/2\pi\Z)$ mod constants.
\vfill\pause
More generally, for $p>1$, $T^{1,p} = 0$ and $H^{1,p}$ is equal to the Besov space $B^{1/p}_{p,p}(\R/2\pi\Z)$ mod constants.


\end{frame}
\begin{frame}


\frametitle{Example : the real line $\R$}

$H^{0,p}=0$. \pause

\bigskip

$R^{1,p}= 0$, \pause 

since every function in $L^p (\R)$ can be approximated in $L^p$ with derivatives of compactly supported
functions. Therefore $H^{1,p}$ is only torsion. \pause

\bigskip

$T^{1,p}$  {\em is non zero and thus infinite dimensional}. \pause 

Indeed, the 1-form $\frac{dt}{t}$ (cut off near the origin) is in $L^p$ for all $p>1$ but it is not the differential of a function in $L^p$.


\end{frame}
\begin{frame}

\frametitle{What are our favourite spaces ?}

\begin{itemize}
  \item $L^p$-cohomology has been used (L. Saper, S. Zucker) to study manifolds with \textcolor{blue}{thin ends}. The answer is related to the topology of a compactification.

\bigskip

  \item In this talk: manifolds with \textcolor{blue}{large ends}, e.g. groups. $L^p$-cohomology is related to analytic features of a compactification.
\end{itemize}

\end{frame}


\begin{frame}

\frametitle{Functoriality}
\begin{eqnarray*}
\textrm{cohomology}&\to&\textrm{continuous maps}\\\pause
L^p \textrm{-cohomology}&\to&\textrm{uniform maps}.\pause
\end{eqnarray*}

\begin{defi}
A map $f:X\to Y$ between metric spaces is {\em uniform} if $d(f(x),f(x'))$ is bounded from above in terms of $d(x,x')$ only. \pause
\end{defi}

\begin{ex}
The obvious map $\Z\to \R$ is uniform. Any homomorphism between groups (with left invariant metrics) is uniform. The parametrization of a cusp by a punctured disk is not uniform.\pause
\end{ex}

\begin{center}
\includegraphics[width=2in]{cusp.pdf}
\end{center}

\begin{prop}
Among contractible Riemannian manifolds admitting a cocompact isometric group action, $L^p$-cohomology is natural under uniform maps.
\end{prop}


\end{frame}
\begin{frame}

\frametitle{$L^p$-cohomology of discrete groups}

$L^p$-cohomology can be discretized. \pause
It makes sense for discrete groups, and cannot see any difference between a cocompact lattice in a semi-simple Lie group $G$, the Lie group $G$ itself or the Riemannian homogeneous space $G/K$, $K$ compact. \pause

\bigskip

In conclusion,

\begin{itemize}
  \item $L^p$-cohomology is a tool to investigate discrete groups.
  \item It shares nearly all properties of usual cohomology.
  \item Nevertheless, it is not easy to calculate it.
  \item In the case of cocompact lattices in Lie groups, it can probably be computed by analytic means.\pause
\end{itemize}


\bigskip

$L^2$-invariants are a huge subject. In this course, we explain a few applications of $L^p$-cohomology, $p\not=2$, to negatively curved Riemannian manifolds and groups.
\begin{enumerate}
  \item Hopf Euler characteristic conjecture
  \item Cannon conjecture on hyperbolic groups with boundary a 2-sphere
  \item Classification of hyperbolic compactly generated groups
  \item Coarse and large scale conformal embeddings 
  \item Curvature pinching
\end{enumerate}

\end{frame}
\begin{frame}

\frametitle{Outline}

\begin{enumerate}
  \item \textcolor{red}{Hopf Euler characteristic conjecture}
  \item Cannon conjecture on groups with boundary a 2-sphere
  \item Classification of hyperbolic compactly generated groups
  \item Coarse and large scale conformal embeddings 
\item Curvature pinching
\end{enumerate}

\end{frame}
\begin{frame}

\frametitle{Hopf conjecture}

\section{Euler characteristic}

\begin{rem}
\begin{itemize}
  \item Compact 2-dimensional negatively curved manifolds have negative Euler characteristic. 
  \item $2m$-dimensional compact hyperbolic manifolds have Euler characteristic proportional to $(-1)^m$. 
  \item This generalizes to all compact negatively curved locally symmetric spaces (Gauss-Bonnet). \pause 
\end{itemize}
\end{rem}

\begin{ques}[\textcolor{red}{H. Hopf}] 
If $M$ is $2m$-dimensional compact negatively curved, then $(-1)^m \chi(M)>0$. 
\end{ques}

\smallskip

The conjecture is often extended to all aspherical manifolds. See M. Davies and B. Eckman's surveys in \emph{Guido's book of conjectures}, and W. Lück's survey in \emph{Bulletin of the Manifold Atlas}.\pause 

\begin{thm}
\textcolor{red}{(M. Gromov, 1991).} This is true provided $M$ also admits a {\em Kähler} metric. 
\end{thm}


\end{frame}
\begin{frame}



\frametitle{Role of the Kähler condition}

\begin{defi}
A Riemannian manifold $M$ is {\em Kähler} if it admits a parallel complex structure.  \pause 
\end{defi}
Then $M$ is a complex manifold. Every complex submanifold in complex projective space admits a Kähler metric. \pause 

\begin{prop}\textcolor{red}{(Part of hard Lefschetz theorem).} Let $M^{2m}$ be a compact Kähler manifold with {\em Kähler form} $\omega$. Then wedging with $\omega$ maps harmonic forms to harmonic forms,  \pause and this induces an {\em injection} in cohomology $H^{k}(M,\R)\to H^{k+2}(M,\R)$ for all $k<m$. \pause 
\end{prop}

\begin{cor}
Let $M^{2m}$ be a complete Kähler manifold with Kähler form $\omega$. Then wedging with $\omega$ maps $L^2$-harmonic forms to $L^2$-harmonic forms, and this induces an {\em injection} in reduced $L^2$-cohomology $R^{k,2}(M)\to R^{k+2}(M)$ for all $k<m$. 
\end{cor}


\end{frame}
\begin{frame}



\frametitle{Role of negative curvature}

\begin{prop}
\textcolor{red}{(M. Gromov).} Let $\tilde{M}$ be a complete simply connected negatively curved Riemannian manifold. Let $k\geq 2$. 
\begin{itemize}
  \item \textcolor{red}{(Coning of cycles).} Every $k-1$-cycle $z$ spans a $k$-chain $c$ with $vol(c)\leq\textrm{const. }vol(z)$. \pause 
  \item \textcolor{red}{(Coning of forms).} Every closed bounded differential $k$-form $\alpha$ on $\tilde{M}$ is the differential of a bounded $(k-1)$-form $\beta$ with $\n{\beta}_{L^{\infty}}\leq\textrm{const. }\n{\alpha}_{L^{\infty}}$. \pause 
\end{itemize}
\end{prop}

\begin{cor}
Assume $\tilde{M}^{2m}$ covers both a compact Kähler manifold and a compact negatively curved Riemannian manifold. Then $R^{k,2}(\tilde{M})=0$ for all $k\not=m$. \pause Furthermore, $T^{*,2}(\tilde{M})=0$. \pause 
\end{cor}

\textcolor{blue}{Proof.} Lift Kähler form to universal cover $\tilde{M}$.   Write $\omega=db$ with $b$ bounded.  \pause Let $k<m$. For $\alpha$ a closed $k$-form in $L^2$, 
\begin{eqnarray*}
\omega\wedge\alpha=d(b\wedge\alpha)\quad \textrm{and} \quad b\wedge\alpha\in L^2,  \pause 
\end{eqnarray*}
thus $\omega\wedge\alpha=0$ in $R^{k+2,2}(\tilde{M})$. If $\alpha$ is harmonic, conclude that $\alpha=0$ in $R^{k,2}(\tilde{M})$.


\end{frame}
\begin{frame}

\frametitle{Vanishing of torsion}

Let $\alpha$ be a $k$-form in $L^2$, $k<m$. Let us show that $\|\alpha\|_2^2\leq C\,\langle\Delta\alpha,\alpha\rangle$. 
\begin{eqnarray*}
\|\alpha\|^2&=&\|\omega\wedge\alpha\|^2=\langle\omega\wedge\alpha,db\wedge\alpha\rangle=\langle\omega\wedge\alpha,d(b\wedge\alpha)\rangle+\langle\omega\wedge\alpha,b\wedge d\alpha\rangle\\
&\leq& \|b\|_{\infty}\|\alpha\|(\|d^*(\omega\wedge\alpha)\|+\|d\alpha\|).
\end{eqnarray*}\pause
$$
\|d^*(\omega\wedge\alpha)\|^2+\|d(\omega\wedge\alpha)\|^2=\langle\Delta(\omega\wedge\alpha),\omega\wedge\alpha\rangle=
\langle\omega\wedge\Delta\alpha,\omega\wedge\alpha\rangle=
\langle\Delta\alpha,\alpha\rangle.
$$\pause
thus
\begin{eqnarray*}
\|\alpha\|^2\leq 4\|b\|_{\infty}^2\langle\Delta\alpha,\alpha\rangle= C^2\,(\|d\alpha\|^2+\|d^*\alpha\|^2).
\end{eqnarray*}\pause
We show that $T^{k,2}=0$ by induction on $k$. On $0$-forms, $d^\alpha=0$, hence $\|\alpha\|\leq C\,\|d\alpha\|$, and $dL^2\Omega^0$ is closed in $L^2\Omega^1$. If $\alpha\in L^2\Omega^2$, $\alpha=d\beta$ is an exact 2-form, there exists $\beta'\in L^2\Omega^1$ of minimal norm such that $d\beta'=\alpha$. Since $d^*\beta'=0$, $\|\beta'\|\leq C\,\|d\beta'\|=\|\alpha\|$, and $dL^2\Omega^1$ is closed in $L^2\Omega^2$, and so on.\pause

\medskip

Furthermore $\Delta$ is invertible on $(\mathrm{ker}\Delta)^\perp$. Indeed, Hodge decomposition $L^2\Omega^m=$ $\mathrm{ker}\Delta\oplus\overline{dL^2\Omega^{m-1}}\oplus\overline{d^*L^2\Omega^{m+1}}$, which holds in general for complete Riemannian manifolds, reads $L^2\Omega^m=\mathrm{ker}\Delta\oplus\overline{dL^2\Omega^{m-1}}\oplus\star\overline{dL^2\Omega^{m-1}}$. Since $\Delta$ commutes with $d$ and $\star$, inequality $\|\alpha\|^2\leq C^2\langle\Delta\alpha,\alpha\rangle$ persists on $(\mathrm{ker}\Delta)^\perp$ in degree $m$.

\end{frame}
\begin{frame}


\frametitle{$L^2$-Betti numbers}

Let $\tilde{M}$ cover a compact manifold $M$. If nonzero, $R^{k,2}(\tilde{M})$ is infinite dimensional.  \pause Nevertheless, \textcolor{red}{M. Atiyah} has defined a {\em von Neumann dimension} 
$$
b^{k,2}(M)=dim_{vN}R^{k,2}(\tilde{M}),
$$
called the $k$-th {\em $L^2$-Betti number} of $M$. \pause 

\begin{exa}[\textcolor{red}{W. Lück}] 
If $M$ admits a tower of finite degree $d_j$ normal coverings $M_j$ such that $\bigcap_j \pi_1 (M_j )=\{1\}$, then
\begin{eqnarray*}
b^{k,2}(M)=\lim_{j\to\infty}\frac{b^k (M_j ,\R)}{d_j}. \pause 
\end{eqnarray*}
\end{exa} 

\begin{prop}
Let $\tilde{M}$ cover a compact manifold $M$. Then $$\chi(M)=\sum_{k}(-1)^{k}b^{k,2}(M).$$ 
\end{prop}


\end{frame}
\begin{frame}

\frametitle{Index theorem}

\begin{prop}\textcolor{red}{(Relative index theorem, M. Gromov-B. Lawson).} Let $\tilde{M}$ be a simply connected nonpositively curved Riemannian manifold. Then there exists $k$ such that $H^{k,2}(\tilde{M})\not=0$. \pause 
\end{prop}
See Pansu's notes on $L^2$ Betti numbers.

\bigskip

\textcolor{blue}{Proof of Gromov's theorem.} Assume $M$ is compact and admits both a negatively curved metric and a Kähler metric. Then all $b^{k,2}(M)$ vanish except $b^{m,2}(M)$, which is nonzero, thus $(-1)^{m}\chi(M)=b^{m,2}(M)>0$. \pause 

\bigskip\vfill

In conclusion, we have used
\begin{itemize}
   \item Lefschetz mechanism, $L^2$-Betti numbers. \pause 
   \item Vanishing of $L^{\infty}$-cohomology. \pause 
  \item Cup-product $H^{k,2}\otimes H^{2,\infty}\to H^{k+2,\infty}$.
\end{itemize}

\end{frame}
\begin{frame}
\frametitle{Further remarks}

Gromov calls Kähler manifolds $(M,\omega)$ such that $\omega=db$, $b\in L^{\infty}$, \emph{Kähler hyperbolic manifolds}. Hermitian symmetric spaces are Kähler hyperbolic. Teichmüller space is Kähler hyperbolic (\textcolor{red}{S. Kruskal}).

\medskip

For such manifolds, Gromov's theorem determines the sign of the Euler characteristics of all the sheaves $\Omega^q$ of exterior powers of the complex cotangent bundle.

\bigskip

Nonvanishing of $L^2$-cohomology is equivalent to $0$ belonging to the $L^2$ spectrum of $\Delta$. J. Lott's \emph{zero in the spectrum conjecture} states that this should be the case for the universal coverings of all compact aspherical manifolds. There is a non-aspherical counterexample (\textcolor{red}{M. Farber and S. Weinberger}).

\end{frame}


\begin{frame}
\frametitle{Outline}

\begin{enumerate}
  \item Hopf Euler characteristic conjecture
\vskip2pt
\textcolor{red}{Interlude: more on $L^{\infty}/L^p$ cohomology of hyperbolic groups}

  \item Cannon conjecture on groups with boundary a 2-sphere
  \item Classification of hyperbolic compactly generated groups
  \item Coarse and large scale conformal embeddings 
\item Curvature pinching
\end{enumerate}

\end{frame}
\begin{frame}

\frametitle{More on $L^\infty$ cohomology of hyperbolic groups}

\begin{rem}
$H^{0,\infty}$ counts connected components. For unbounded metric spaces, $H^{1,\infty}\not=0$. 
\pause
\end{rem}

\begin{thm}[\textcolor{red}{M. Gromov} 1986] 
For a finitely generated group $G$,

$G$ is hyperbolic $\Leftrightarrow H^{2,\infty}(G)=0\Leftrightarrow H^{k,\infty}(G)=0$ for all $k\geq 2$.
\end{thm}\pause

\medskip

\textbf{Step 1. Duality Lemma}. For a manifold $M$,

$H^{k,\infty}(M)=0\Leftrightarrow$ linear filling in dimension $k$, 

i.e. every $k$-cycle $S$ bounds a $k+1$-current $T$ with $\mathrm{Mass}(T)\leq C\,\mathrm{Mass}(S)$.
\pause

\medskip

\textbf{Step 2}. Filling inequalities are quasi-isometry invariant. Pass from current to disk...\pause

\medskip

\textbf{Step 3}. \textcolor{red}{Bonk-Schramm} show that $G$ hyperbolic is quasi-isometric to a convex subset of $H^n_\R$, $n$ large.\pause

\medskip

\textbf{Step 4}. In a convex subset of $H^n_\R$, the cone over $S$ has Mass $\leq C$ Mass$(S)$.\pause

\medskip

\textbf{Conclusion}.  $H^{2,\infty}(M)=0\Rightarrow$ linear filling of loops $\Rightarrow G$ hyperbolic. 

Conversely, $G$ hyperbolic $\Rightarrow$ linear filling in all dimensions $k\geq 2\Rightarrow H^{k,\infty}(M)=0$.

\end{frame}

\begin{frame}

\frametitle{Proof of Duality Lemma}

\begin{lem}
$H^{k,\infty}=0\Leftrightarrow \forall$ bounded closed $k$-form $\alpha$, $\exists \beta$ with $\|\beta\|_{\infty}\leq C\,\|\alpha\|_{\infty}$ s. t. $d\beta=\alpha$.
\end{lem}
\textbf{Proof}. Use norm $\|\alpha\|_{\infty}+\|d\alpha\|_{\infty}$ to complete smooth differential forms into a Banach space $\mathcal{C}$. Then $d: \mathcal{C}\to\mathcal{C}$ is continuous, with kernel $\mathcal{Z}$. It descends to a continuous operator $\bar d:\mathcal{C}/\mathcal{Z}\to\mathcal{Z}$. If $L^{\infty}$-cohomology vanishes, this is a continuous bijection between Banach spaces, hence an isomorphism. Get a smooth $\beta$...

\bigskip

\textbf{Proof of Duality Lemma}.

A $k$-cycle $S$ of finite mass $M(S):=\sup\{\langle S,\beta\rangle\,;\,\|\beta\|_{\infty}\leq 1\}$ defines a functional on smooth bounded $k$-forms, which vanishes on exact forms. If $L^{\infty}$-cohomology vanishes, one can define a functional $T$ on smooth bounded exact $k+1$-forms by $\langle T,\alpha\rangle=\langle S,\beta\rangle$ for some $\beta$ such that $d\beta=\alpha$. Then $T$ is bounded above by $C\,M(S)\|\cdot\|_{\infty}$. By \textcolor{red}{Hahn-Banach}, $T$ extends to a linear functional on smooth $k+1$-forms, i.e. a current, of finite mass $M(T)\leq C\,M(S)$ and such that $\partial T=S$. 

\bigskip

The converse follows from the symmetry between differential forms and currents, $\|\alpha\|_{\infty}=\sup\{\langle T,\alpha\rangle\,;\,M(T)\leq 1\}$.


\end{frame}

\begin{frame}

\frametitle{More on $L^p$ cohomology of hyperbolic groups}

\begin{exa}
Let $M$ be $n$-dimensional, 1-connected, curvature $-a^2\leq K\leq -1$. Then $H^{1,p}(M)\not=0$ provided $p>(n-1)a$.
\end{exa}
In horospherical coordinates $(r,\theta)$, $u=u(\theta)$ has $|\nabla u|\leq C\,e^{-r}$, $vol\leq e^{(n-1)ar}$.\pause

\bigskip

This is sharp. Functions with $\|\nabla u\|_p<\infty$ on $H^n_\R$ have a limit $u_\infty$ along almost every ray, $u\in L^p \Leftrightarrow u_\infty=0$. If $p\leq n-1$, $\|\nabla u_{\partial B(r)}\|_p$ tends to 0, hence $H^{1,p}(H^n_\R)=0$ if $p\leq n-1$. \pause

\bigskip

\begin{thm}[\textcolor{red}{Bourdon-Pajot}]
Let $G$ be a hyperbolic group. 
\begin{enumerate}
  \item There exists $p(G)$ such that $H^{1,p}(G)\not=0$ for $p>p(G)$.\pause
  \item $1,p$-cohomology classes on hyperbolic groups have well-defined boundary values, so $H^{1,p}(G)$ identifies with a function space on $\partial G$.
\end{enumerate}
\end{thm}
Fix a visual metric on $\partial G$. Pick a maximal ball packing of $\partial G$ at each scale $2^{-k}$ and work on the incidence graph $X$ whose vertices $v$ correspond to balls $B(x_v, r_v)$. Then $X$ is quasi-isometric to $G$. Given a Lipschitz function $u$ on $\partial G$, set $u(v)=u(x_v)$. Then $|\nabla u|(v)\sim r_v$. If metric is $Q$-Ahlfors-regular, $\|\nabla u\|_p<\infty$ if $p>Q$. 

\end{frame}

\begin{frame}
\frametitle{Outline}

\begin{enumerate}
  \item Hopf Euler characteristic conjecture
  \item \textcolor{red}{Cannon conjecture on groups with boundary a 2-sphere}
  \item Classification of hyperbolic compactly generated groups
  \item Coarse and large scale conformal embeddings 
\item Curvature pinching
\end{enumerate}

\end{frame}
\begin{frame}


\frametitle{Cannon conjecture}

\begin{thm}[\textcolor{red}{W. Thurston 1979, G. Perelman 2002}] 
A closed 3-manifold group which is infinite, does not split as a free product and does not contain $\Z^2$ is a cocompact lattice in $SO(3,1)$. \pause 
\end{thm}

\bigskip

If $G$ is a lattice in $SO(n,1)$, then $G$ is hyperbolic and its ideal boundary is an $n-1$-sphere. \pause 

\bigskip

\begin{ques}[\textcolor{red}{J. Cannon}, circa 1988]
Let $G$ be a hyperbolic group whose ideal boundary is a topological 2-sphere. Then $G$ is virtually a cocompact lattice in $SO(3,1)$.
\end{ques}

\bigskip

Before Perelman, it was thought of as an alternative to Thurston's methods.

Avatar of Riemann mapping theorem: uniqueness of a (coarse) conformal structure on the 2-sphere with a large conformal group.

\end{frame}
\begin{frame}

\frametitle{Conformal dimension}

\begin{rem}
The ideal boundary of a hyperbolic group carries a natural conformal structure (technically, a {\em quasi-M\"obius structure}), but no canonical metric. \pause
\end{rem}

\begin{defi}[\textcolor{red}{Bourdon-Pajot}]
Define the {\em conformal dimension} of a hyperbolic group as the least Hausdorff dimension of an Ahlfors-regular metric in the natural quasi-M\"obius structure. \pause 
\end{defi}

\bigskip

\textcolor{red}{Strategy (B. Kleiner).} Prove that this infimum is achieved. Then prove that dimension minimizing metrics are Riemannian if boundary is 2-dimensional. Then apply a result of \textcolor{red}{D. Sullivan (1978)}: {\em every uniformly quasiconformal group of the standard 2-sphere is conjugate to a subgroup of $SO(3,1)$.} \pause

\begin{thm}[\textcolor{red}{S. Keith-T. Laakso, M. Bonk-B. Kleiner 2005}] Let $G$ be a hyperbolic group whose ideal boundary is a 2-sphere. If conformal dimension is achieved, then $G$ is virtually a cocompact lattice in $SO(3,1)$.
\end{thm}

\end{frame}
\begin{frame}

\frametitle{$L^p$-dimension}

\begin{rem}
For hyperbolic groups, $H^{1,p}$ is nonzero for $p$ large, but zero for $p$ small.
\end{rem}

\begin{defi}
Define the {\em $L^p$-dimension} of a group as the least $p>1$ such that its $H^{1,p}$ is nonzero.
\end{defi} 
For instance, $L^p$-dimension of hyperbolic $n$-space is $n-1$. \pause 


\begin{thm}[\textcolor{red}{Same people + M. Bourdon-H. Pajot 2003}] Let $G$ be a hyperbolic group. Then $L^p$-dimension is less than or equal to conformal dimension. \pause If conformal dimension is achieved, then $G$ does not split over a virtually cyclic subgroup, and $L^p$-dimension and conformal dimension coincide.
\end{thm}\pause

\begin{ex}[\textcolor{red}{M. Bourdon-H. Pajot}] There exist hyperbolic groups for which conformal dimension $>2\geq L^p$-dimension (and which do not split over virtually cyclic subgroups). For such groups, conformal dimension cannot be achieved.
\end{ex}

\end{frame}
\begin{frame}

\frametitle{Bourdon-Pajot's examples}

Let $G=A\star_C B$ where $A$ and $B$ are hyperbolic, $C$ is quasi-convex and malnormal in both $A$ and $B$. Then $G$ is hyperbolic, $A$ and $B$ are quasi-convex in $G$, hence 
$$
\mathrm{ConfDim}(G)\geq \max\{\mathrm{ConfDim}(A),\mathrm{ConfDim}(B)\}.
$$
Mayer-Vietoris implies that
\begin{eqnarray*}
b^{1,2}(G)\geq b^{1,2}(A)+b^{1,2}(B)-b^{1,2}(C)>0,
\end{eqnarray*}
if $b^{1,2}(A)>b^{1,2}(C)$. For instance, let $A$ and $B$ be lattices of isometries of Bourdon buildings, $A$ of large covolume and negative Euler characteristic, $B$ of large conformal dimension, and $C$ a free group on 2 generators. Then $H^{1,2}(G)\not=0$, hence $L^p-\mathrm{dim}(G)\leq 2<\mathrm{ConfDim}(G)$.

\smallskip

For a Bourdon building with $p$-sided right-angled faces and thickness $q$, Euler characteristic is proportional to $q^2-p(q-1)$, conformal dimension equals $1+\frac{\log q}{\arg\cosh(\frac{p-2}{2})}$.
\pause

\bigskip
\bigskip

In conclusion, we have used
\begin{itemize}
  \item Mayer-Vietoris and $L^2$-Betti numbers.
  \item Expression of $H^{1,p}$ as a function space on the ideal boundary.
\end{itemize}

\end{frame}
\begin{frame}

\frametitle{Further remarks}

\textcolor{red}{Bourdon-Kleiner} show that conformal dimension is equal to the infimum of $p$ such that the continuous part of $L^p$ cohomology separates points of the ideal boundary.

\smallskip

Continuous part means continuous functions on $\partial G$ which continuously extend to functions on $G$ with finite $\ell^p$ norm of gradient.

\bigskip

\begin{thm}[\textcolor{red}{Bourdon} 2008]
Consider the continuous part of $L^p$ cohomology of $G$ as a Banach algebra $\mathcal{A}_p(G)$ with norm $\|\cdot\|_{\infty}+\|\cdot\|_{H^{1,p}(G)}$. 

Let $G_1$, $G_2$ be hyperbolic groups which do not split over finite groups. Let $p>\max\{\mathrm{ConfDim}(G_1),\mathrm{ConfDim}(G_1)\}$. Then any Banach algebra isomorphism $\mathcal{A}_p(G_1)\to \mathcal{A}_p(G_2)$ is induced by a quasi-isometry.
\end{thm}\pause

\bigskip

\textbf{Question}. Is $\partial G$ retrievable from the whole $H^{1,p}(G)$, and not its continuous part ?

\bigskip

In 2015, \textcolor{red}{Bourdon-Kleiner} implemented succesfully Kleiner's strategy for Coxeter groups. They fully describe the equivalence relations on $\partial G$ induced by $H^{1,p}(G)$ when $L^p-\mathrm{dim}(G)\leq p\leq ConfDim(G)$ in this special case.


\end{frame}
\begin{frame}

\frametitle{Outline}

\begin{enumerate}
  \item Hopf Euler characteristic conjecture
  \item Cannon conjecture on groups with boundary a 2-sphere
  \item Classification of hyperbolic compactly generated groups
  \item Coarse and large scale conformal embeddings 
  \item \textcolor{red}{Curvature pinching}
\end{enumerate}


\end{frame}
\begin{frame}

\frametitle{Curvature pinching}
\section{Curvature pinching}

\begin{rem}
Rank one symmetric spaces are hyperbolic spaces over the reals $H^{n}_{\R}$, the complex numbers $H^{m}_{\C}$, the quaternions $H^{m}_{\H}$, and the octonions $H^{2}_{\O}$.

Real hyperbolic space has sectional curvature $-1$. Other rank one symmetric spaces are $-\frac{1}{4}$-{\em pinched}, i.e. their sectional curvature ranges between $-1$ and $-\frac{1}{4}$. \pause
\end{rem}

\begin{defi}
Define the {\em optimal pinching} $\delta(G)$ of a discrete (or Lie) group $G$ as the least $\delta>-1$ such that $G$ is bi-uniformly equivalent to a $\delta$-pinched Riemannian manifold. \pause 
\end{defi}

\begin{ques}
The optimal pinching of $SU(m,1)$, $Sp(m,1)$ ($m\geq 2$) and $F_{4}^{-20}$ is $-\frac{1}{4}$.
\end{ques}

\end{frame}
\begin{frame}

\frametitle{$L^p$-cohomology of real hyperbolic spaces}

\begin{center}
%\includegraphics[width=3in]{rhn.pdf}
\end{center}

\end{frame}
\begin{frame}

\frametitle{$L^p$-cohomology and pinching}

\begin{thm}
If $M^{n}$ is simply connected and $\delta$-pinched for some $\delta\in[-1,0)$, then
\begin{eqnarray*}
p<1+\frac{n-k}{k-1}\sqrt{-\delta} \quad\Rightarrow\quad T^{k,p}(M)=0.\pause
\end{eqnarray*}
\end{thm}
This is sharp. For instance, consider the semidirect product $G=\R^3 \rtimes_{\alpha}\R$ where $\alpha=diag(1,1,2)$. 
\begin{itemize}
  \item It admits a $-\frac{1}{4}$-pinched left-invariant Riemannian metric, therefore $\delta(G)\leq -\frac{1}{4}$.
  \item It has $T^{2,p}(G)\not=0$ for $2<p\leq 4$. This implies that $\delta(G)=-\frac{1}{4}$.\pause
\end{itemize} 

\begin{rem}
Complex hyperbolic plane $H^{2}_{\C}$ is isometric to $G'=Heis^3 \rtimes_{\alpha}\R$ where $\alpha=diag(1,1,2)$ and $Heis$ denotes the Heisenberg group. Therefore it is very close to $G$.\pause
\end{rem}

\begin{thm}
$T^{2,p}(H^{2}_{\C})=0$ for $2<p< 4$.
\end{thm}


\end{frame}
\begin{frame}

\frametitle{Proofs of vanishing theorems}

\textcolor{blue}{Proof of torsion comparison theorem}

Use the gradient vectorfield $\xi$ of a Busemann function and its flow $\phi_t$, whose derivative is controlled by sectional curvature. For $\alpha$ a closed $k$-form in $L^p$, 
$$
\phi_{t}^{*}\alpha=\alpha+d\left(\int_{0}^{t}\phi_{s}^{*}\iota_{\xi}\alpha\,ds\right)
$$ 
has a limit as $t\to+\infty$ under the assumptions of the theorem. This boundary value map injects $H^{k,p}$ into a function space of closed forms on the ideal boundary, showing that $H^{k,p}$ is Hausdorff.\pause

\bigskip

\textcolor{blue}{Proof of torsion vanishing for $H^{2}_{\C}$}

For $p\notin\{4/3,2,4\}$, differential forms $\alpha$ on $H^{2}_{\C}$ split into components $\alpha_{+}$ and $\alpha_{+}$ which are contracted (resp. expanded) by $\phi_{t}$. Then
$$
B_t :\alpha\mapsto\int_{0}^{t}\phi_{s}^{*}\iota_{\xi}\alpha_{+}\,ds-\int_{-t}^{0}\phi_{s}^{*}\iota_{\xi}\alpha_{-}\,ds
$$
converges as $t\to +\infty$ to a bounded operator $B$ on $L^p$. $P=1-dB-Bd$ retracts the $L^p$ de Rham complex onto a complex of differential forms on $Heis^3$ with missing components and weakly regular coefficients. If $2<p<4$, this complex is nonzero in degrees 1 and 2, but it is so small that its cohomology can be shown to be Hausdorff.

\end{frame}
\begin{frame}

\frametitle{Non-vanishing of torsion}

Use Poincaré duality. Let $p'=p/p-1$ denote the conjugate exponent. In order to prove that a closed $k$-form $\alpha$ is nonzero in cohomology, it suffices to construct a sequence $\psi_{j}$ of $(n-k)$-forms such that $\n{d\psi_j}_{L^{p'}}$ tends to zero but $\int\alpha\wedge\psi_{j}$ does not tend to zero.\pause

\bigskip

In conclusion, we have used
\begin{itemize}
  \item Poincaré duality.
  \item A deformation retraction of space onto a subspace, with controlled effect on the $L^p$-norms of forms. For certain ranges of $p$, this provides a boundary value.\pause
\end{itemize}

\bigskip

\begin{ques}
\begin{itemize}
  \item For rank 1 symmetric spaces, $T^{k,p}=0$ except for at most 1 value of $p$ in each degree.
  \item For higher rank symmetric spaces, $H^{k,p}=0$ for $k<rank$, $T^{k,p}=0$ for $k=rank$.
  \item For $k=rank$, $R^{k,p}\not=0$ for $p$ large, and $R^{k,p}$ is a function space on the maximal boundary.
  \item For each $p>1$, there exists $k$ such that $H^{k,p}\not=0$.
\end{itemize}
\end{ques}

\end{frame}
\begin{frame}

\frametitle{$L^p$-cohomology of $H^{2}_{\C}$}

\begin{center}
%\includegraphics[width=3in]{table0.pdf}
\end{center}

\end{frame}
\begin{frame}

\frametitle{$L^p$-cohomology of $G$}

\begin{center}
%\includegraphics[width=3in]{table1.pdf}
\end{center}

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\end{document}


