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\title{Large scale conformal geometry}
\author{Pierre Pansu, Université Paris-Sud}
\date{July 24th, 2018}


\mode<presentation>{\usetheme{Warsaw}}

\begin{document}

\begin{frame}
\titlepage
\end{frame}

\begin{frame}

Goal: perform conformal geometry on discrete groups.\pause

\bigskip

\begin{defi}
$X$, $X'$ metric spaces. Map $f:X\to X'$ is a coarse embedding if 
\begin{eqnarray*}
\alpha(d(x,x'))\leq d(f(x),f(x'))\leq \omega(d(x,x')),
\end{eqnarray*}
where $\alpha\to+\infty$, $\omega<+\infty$.
\end{defi}

\textbf{Example}.
$X$, $X'$ finitely generated or Lie groups, viewed as metric spaces. Every injective homomorphism is a coarse embedding.
\pause

\bigskip

New input: sharp invariants which take continuously many values.

\medskip

Sample new result: there can be no \textbf{coarse conformal map} between Jorge Lauret's quasi-abelian groups $S_A$ and $S_B$ if
\begin{eqnarray*}
\frac{\Re e(\mathrm{trace}(A))}{\min\Re e(\mathrm{sp}(A))}> \frac{\Re(\mathrm{trace}(B))}{\min\Re e(\mathrm{sp}(B))}>0.
\end{eqnarray*}


\end{frame}

\section{Infinitesimal conformal geometry}

\subsection{Conformal cartography}

\begin{frame}

\begin{minipage}{5cm}
\includegraphics[width=2in]{Gerard_Mercator.jpg}
\end{minipage}\hskip1cm
\begin{minipage}{4cm}
1569 : Gerard de Kremer (aka \emph{Mercator}) creates a new world map. \pause

\vskip.5cm

Mercator's map is \emph{conformal} :  angles between curves are preserved.

\vskip.5cm

Distance distorsion blows up at poles.
\end{minipage}

\end{frame}

\subsection{Holomorphic functions}

\begin{frame}

\begin{thm}[Schwarz Lemma]
Let $D\subset\C$ denote the unit disk. Let $f:D\to D$ be a holomorphic function such that $f(0)=0$. Then $|f'(0)|\leq 1$.
\end{thm}\pause

\begin{cor}[Schwarz-Pick Lemma]
Let $f:D\to D$ be a holomorphic function. Then $f$ is 1-Lipschitz with respect to hyperbolic distance.
\end{cor}\pause

\begin{cor}[Liouville]
Let $f:\C\to D$ be a holomorphic function. Then $f$ is constant.
\end{cor}

\end{frame}

\subsection{Conformal diffeomorphisms}

\begin{frame}

In higher dimensions, 

\begin{defi}
A diffeomorphism between Riemannian manifolds is \emph{conformal} if its differential maps infinitesimal spheres to infinitesimal spheres.
\end{defi}\pause

\textbf{Examples}
\begin{enumerate}
  \item Euclidean space is conformal to the sphere with one point deleted.
  \item Hyperbolic space is conformal to a Euclidean ball.
  \item Euclidean balls are not conformal to Euclidean space.
\end{enumerate}\pause

\bigskip

\textbf{Disappointing} : conformal diffeomorphisms are rare. 

In dimension $n\geq 3$, every conformal diffeomorphism between Euclidean domains is the restriction of a global conformal diffeomorphism of the sphere $S^n$, i.e. an element of the Möbius group $O(n+1,1)$.

\end{frame}

\subsection{Quasi-conformal homeomorphisms}

\begin{frame}

\begin{defi}
A diffeomorphism between Riemannian manifolds is \emph{quasiconformal} if its differential maps infinitesimal spheres to infinitesimal ellipsoids of bounded excentricity.
\end{defi}

\begin{ex}
$\displaystyle z\mapsto z|z|^{K-1}$ is quasiconformal $\C\to\C$.
\end{ex}\pause

\begin{thm}[H. Grötsch 1928]
Euclidean balls are not quasiconformal to Euclidean space.
\end{thm}\pause

\textbf{Proof} : quasi-Schwartz Lemma. A quasiconformal diffeomorphism $f$ of the ball satisfies
\begin{eqnarray*}
\frac{1}{L}d(x,x')-C\leq d(f(x),f(x'))\leq Ld(x,x')+C
\end{eqnarray*}
with respect to hyperbolic distance. It is a \emph{quasiisometry}, i.e. a large scale biLipschitz map.

\end{frame}

\section{Microscopic conformal geometry}

\subsection{Quasisymmetry}

\begin{frame}

\emph{Quasisymmetric} : homeomorphism between metric spaces such that
\begin{eqnarray*}
\forall x,\,\forall x',\,\forall x'',\quad\frac{d(f(x),f(x'))}{d(f(x),f(x''))}\leq \eta(\frac{d(x,x')}{d(x,x'')}),
\end{eqnarray*}
where $\eta\in Homeo(\R_+)$.

\begin{center}
\includegraphics[width=1.5in]{qs.pdf}
\end{center}\pause

\bigskip

\textbf{Remark}. Quasisymmetric $\implies$ quasiconformal.

\begin{thm}[H. Grötsch 1928]
Quasiconformal homeomorphisms of Euclidean spaces are quasisymmetric.
\end{thm}

\end{frame}



\subsection{Boundaries of hyperbolic groups}

\begin{frame}

A \emph{hyperbolic group} has an ideal boundary $\partial G$, a compact space equipped with a family of \emph{visual distances}. 

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

Paulin:\hskip2em     quasiisometries of $G\iff$ quasisymmetric homeomorphisms of $\partial G$.\pause

\smallskip

Furthermore, visual distances are Ahlfors-regular.

\smallskip

\emph{$p$-Ahlfors-regular metric space}: there exists a probability measure $\mu$ such that $\mu(B(x,R))\simeq R^p$ pour $R<R_0$.\pause

\smallskip
Whence the notion of \emph{quasisymmetric gauge}: the equivalence class of Ahlfors-regular distances on $\partial G$ which are quasisymmetric to a visual distance.

\smallskip

This ``microscopic conformal structure'' determines the large scale geometry of $G$.

\end{frame}

\section{Mesoscopic conformal geometry}

\subsection{Sphere packings}

\begin{frame}

\emph{Sphere packing} : collection of balls with disjoint interiors.

\smallskip

\emph{Incidence graph} : one vertex per ball, an edge when two balls touch.

\begin{center}
\includegraphics[width=1in]{packing.pdf}
\end{center}

\begin{defi}
A graph is \emph{packable in $\R^d$} if it is the incidence graph of a sphere packing of $\R^d$.
\end{defi}\pause

\begin{thm}[Koebe 1931]
A graph is packable in $\R^2$ if and only if it is simple and planar (i.e. embeddable into $\R^2$).
\end{thm}

Interpretation : mesoscopic version of Riemann's conformal mapping theorem. 

\end{frame}

\subsection{A theorem by Benjamini et Schramm}

\begin{frame}

\textbf{Notes}. 
\begin{itemize}
  \item There are fast algorithms to compute Koebe's packing. 
  \item Apply Koebe's theorem to the equal disk packing of a plane domain. The obtained discrete map converges to Riemann's conformal mapping as radius tends to 0.
  \item This is an efficient algorithmic way to compute an approximation of Riemann's conformal mapping.
  \item A variant of Koebe's theorem, Andreev's theorem, is the basic building block of Thurston's 1982 hyperbolization theorem.
\end{itemize}\pause

\bigskip

In higher dimensions, little is known.

\begin{thm}[Benjamini-Schramm 2013]
The grid in $\R^n$ is packable in $\R^d$ iff $n\leq d$. There exist  lattices of hyperbolic $n$-space whose Cayley graphs are packable in $\R^d$ only if $n\leq d$.
\end{thm}

\end{frame}

\section{Coarse conformal maps}

\subsection{Packings}

\begin{frame}[label=packings]

\textbf{$(N,\ell,R,S)$-packings}: This means a countable collection of balls $B_j$ of radii $\in[R,S]$ such that the collection of concentric balls $\ell B_j$ has multiplicity $\leq N$.

\bigskip

\textbf{Example}: A usual disk or ball packing is a $(1,1,0,\infty)$-packing.\pause

\bigskip

\begin{defi}
A \emph{coarse conformal map} is a map $f:X\to X'$ between metric spaces, enriched with a correpondance between balls $B\mapsto B'$ such that $f(B)\subset B'$, and there exist $R>0$ such that $\forall S\geq R$ and $\ell'\geq 1$, there exist $N'\geq 1$ and $\ell\geq 1$ such that $f$ maps $(1,\ell,R,S)$-packings to $(N',\ell',0,\infty)$-packings.
\end{defi}\pause

\bigskip

Coarse conformal maps can be precomposed with coarse embeddings and postcomposed with quasisymmetric homeomorphisms.

\bigskip

\textbf{Examples}. 

\begin{itemize}

   \item Every nilpotent group c.c. maps to some $\R^N$ (Assouad).
  
  \item Every hyperbolic group c.c. maps to  some $O(N,1)$ (Bonk-Schramm).
  
  \item $z\mapsto z|z|^{K-1}$ is coarsely conformal on $\R^N$.
  
  \item For every hyperbolic group $G$, the Poincar\'e model $G\to O(1,1)\times\partial G$ is coarsely conformal.

\end{itemize}

\end{frame}

\subsection{Questions}

\begin{frame}[label=questions]


\textbf{Questions}.
What is the optimal $N$ such that 
\begin{itemize}
  \item Heisenberg group c.c. maps to $\R^N$?
  \item Lauret's quasi-abelian group $S_A$ c.c. maps to $O(N,1)$?
\end{itemize}
\pause

\bigskip

\textbf{Partial answer}. A necessary condition is that some dimension increases. ``Non-squeezing theorem''.

\medskip

For nilpotent groups $G$, relevant dimension $d_1(G)=d_2(G)$ is exponent of volume growth.

\medskip

For hyperbolic groups $G$, two candidates, conformal dimension of ideal boundary $d_2(G)$, or cohomological dimension (least $p$ such that $\ell^p$-cohomology does not vanish) $d_1(G)$.\pause

\bigskip

\textbf{Open problem}. Find more numerical invariants that must increase under coarse conformal maps.

\end{frame}

\subsection{Results}

\begin{frame}[label=rough]

\begin{thm}
Let $G$, $G'$ be Lie or finitely generated groups, nilpotent or hyperbolic. Assume that $G$ c.c. maps to $G'$. Then
\begin{eqnarray*}
d_1(G)\leq d_2(G').
\end{eqnarray*}
\end{thm} 
\pause

\medskip

For instance, no c.c. map
\begin{itemize}
  \item of Heisenberg group $\mathbb{H}^{2m-1}$ into $\R^n$ if $n<2m$.
  \item of $U(m,1)$ into $O(n,1)$ for $n\leq 2m$.
\end{itemize}\pause

\bigskip

Lauret's quasi-abelian groups $S_A$ with $\min\Re e(\mathrm{sp}(A))>0$ satisfy 
\centerline{$d=d_1(S_A)=d_2(S_A)=\frac{\Re e(\mathrm{trace}(A))}{\min\Re e(\mathrm{sp}(A))}$} 
which takes all values in $[1,+\infty)$. Theorem states that $d$ must increase under c.c. maps.\pause

\bigskip

Isometry groups of Bourdon buildings satisfy $d=d_1(G)=d_2(G)$ that cover a dense subset of $[1,+\infty)$. Weird coarse (in fact, quasiisometric) embeddings exist among them. Theorem states that $d$ must increase. Alternate proof of this due to Hume-Mackay-Tessera 2017, based on $p$-separation.

\end{frame}

\subsection{$p$-energies}

\begin{frame}

\textbf{$p$-energy}: if $u$ is a map to a metric space, 
$$E_{p,\ell,R,S}(u)=\sup_{(1,\ell,R,S)-packings \,\{B_j\}} \sum_{j}osc(u_{|B_j})^p.$$

\medskip

\textbf{Example}: On a $d$-Ahlfors-regular metric space, Lipschitz maps have finite $d$-energy. \pause

\smallskip

\textbf{$p$-modulus} of a curve family $\mathcal{F}$: it is the inf of $p$-energies of maps $u$ to metric spaces such that $\mathrm{length}(u\circ\sigma)\geq 1$ for each curve $\sigma\in \mathcal{F}$.\pause

\smallskip

\textbf{$p$-almost every curve}: means for all unbounded curves starting from some compact set but a family of vanishing $p$-modulus.\pause

\smallskip

\textbf{$p$-parabolicity}: means that the $p$-modulus of the family of unbounded curves starting in a compact set is equal to 0.\pause

\begin{thm}[Troyanov, based on Coulhon/Saloff-Coste and Gromov]
A finitely generated group $G$ is $p$-parabolic iff the volume growth is polynomial of degree $d(G)\leq p$.
\end{thm}\pause

%\emph{Proof}. According to Coulhon and Saloff-Coste, volume growth $> R^p$ $\Rightarrow$ isoperimetry is better that of $\R^p$ $\Rightarrow$ (Ahlfors) non $p$-parabolic.
%\medskip

%Combined with Gromov, parabolic implies virtually nilpotent.

\textbf{Corollary}. No c.c. map or coarse embedding $G\to G'$ if $G'$ is nilpotent and $G$ hyperbolic, or $G$ nilpotent with $d(G)>d(G')$.

\end{frame}

%\subsection{\texorpdfstring{$\ell^p$}{} cohomology for metric spaces}

\begin{frame}

If $G'$ is hyperbolic, compose the c.c. map $G\to G'$ with the \emph{generalized Poincar\'e model} $G'\to\R\times \partial G'$. The resulting map is roughly conformal. Furthermore, $\R\times \partial G'$ is $p$-Ahlfors regular for every $p>\mathrm{dim}_{AR}(G')$. 

\medskip

\begin{thm}[Benjamini-Schramm 2013 revisited]
Let $Y$ be a compact, $Q$-Ahlfors-regular metric space. If there exists a roughly conformal map $f:X\to Y$, then
\begin{enumerate}
  \item either $X$ is $Q$-parabolic,
  \item or $\ell^Q \bar{H}^1(X)\not=0$.
\end{enumerate}
\end{thm}

\medskip

If $G$ is hyperbolic, it is $p$-parabolic for no $p$, so the obstruction comes from reduced $\ell^p$ cohomology, whence $d_1(G)$.

\medskip

If $G$ is nilpotent, it has vanishing reduced $\ell^p$ cohomology for all $p$, so the obstruction comes from $p$-parabolicity.

\end{frame}

\begin{frame}[label=lpcoh]

\textbf{Cochains} are functions $\kappa:X^{k+1}\to\R$. The \emph{$\ell^p$ norm} of $\kappa$ is the sup of sums $(\sum_{j}\sup|\kappa_{|B_j}|^p)^{1/p}$ over all $(1,\ell,R,S)$-packings (parameters $\ell,R,S$ are fixed). 

\begin{defi}
The $\ell^p$ cohomology of metric space $X$ is
$\displaystyle \ell^p H^k=\{k\textrm{-cocycles }\ell^p\}/d\{\ell^p\,(k-1)\textrm{-cochains}\}$. Reduced $\ell^p$ cohomology is
$\displaystyle \ell^p \bar{H}^k=\{k\textrm{-cocycles }\ell^p\}/\overline{d\{\ell^p\,(k-1)\textrm{-cochains}\}}$.
\end{defi}

\begin{prop}
For bounded geometry uniformly contractible Riemannian manifolds or simplicial complexes, for every $0<R\leq S<\infty$, this coincides with previous definitions.
\end{prop}\pause

If $G$ is a hyperbolic group, $\ell^p H^1=\ell^p \bar{H}^1$ vanishes for $p<d_1(G)$ and does not vanish for $p>d_2(G)$.

\end{frame}

%\subsection{Packing into Ahlfors regular spaces}

\begin{frame}

\begin{thm}[Benjamini-Schramm 2013 revisited]
Let $Y$ be a compact, $Q$-Ahlfors-regular metric space. If there exists a roughly conformal map $f:X\to Y$, then
\begin{enumerate}
  \item either $X$ is $Q$-parabolic,
  \item or $\ell^Q \bar{H}^1(X)\not=0$.
\end{enumerate}
\end{thm}\pause

\emph{Proof}.

\begin{enumerate}
\item Naturality : $f$ transports finite energy maps. In particular, if $Y$ is $Q$-Ahlfors-regular, $E_Q(f)<\infty$.\pause

\item If $\ell^Q H^1(X)=0$, every finite $Q$-energy map has a limit at infinity. Thus $f$ has a limit $y\in Y$.\pause

\item On $Y\setminus\{y\}$, there exists a finite energy function $v_y$ that tends to $+\infty$ at $y$. Thus $v_y\circ f$ has finite energy and tends to $+\infty$, contradiction.\pause

\item Finite energy maps have limits along almost every curve. So $f$ has limits along $Q$-almost every curve.\pause

\item If $\ell^Q \bar{H}^1(X)=0$, these limits are $Q$-almost always the same, $y$. \pause

\item Then $v_y\circ f$ has finite energy and tends to $+\infty$ along $d$-almost every curve. Thus $Q$-almost every curve = no curve, i.e. $X$ is $Q$-parabolic.
\end{enumerate}

\end{frame}

\end{document}

\begin{frame}[label=embed]

A \emph{coarse embedding} $X\to Y$ between metric spaces maps points at distance $T$ to points at distance at least $\eta(T)$ and at most $\phi(T)$.

\bigskip

\textbf{Example}. Any injective group homomorphism is a coarse embedding.

%A homeomorphism $X\to Y$ between metric spaces is \emph{quasi-symmetric} if for all $x\in X$, points $x',x''\in X$ at comparable distances from $x$ are mapped to points $y',y''$ at comparable distances from the image $y$ of $x$.

%\begin{center}
%\includegraphics[width=1.5in]{qs.pdf}
%\end{center}

\medskip

\hyperlink{rough<1>}{\beamerreturnbutton{Back}}

%\textbf{Example}. Stereographic projection $\R^d\to S^d$ is quasi-symmetric.

\end{frame}

\end{document}

\subsection{Proof of Benjamini-Schramm's theorem}

\begin{frame}

Let $G$ be the incidence graph of a sphere packing of $\R^d$.
\begin{enumerate}
  \item Stereographically map $\R^d$ to $S^d$.
  \item Resulting vertex map $f:V\to S^d$ has finite \emph{$d$-energy}.
  \item Assume that every function of finite $d$-energy has a limit at infinity (vanishing of \emph{$\ell^d$-cohomology}). Then $f$ converges to point $y\in S^d$.
  \item Let $v$ be a finite $d$-energy function on $S^d\setminus\{y\}$ that tends to infinity at $y$. Then $v\circ f$ has finite $d$-energy.
  \item Therefore it has a limit along $d$-almost every unbounded path.
  \item Contradiction, unless $d$-almost every path is no path at all, i.e. $G$ is \emph{$d$-parabolic}.
\end{enumerate}\pause

\begin{prop}
If graph $G$ is packable in $\R^d$ and $\ell^d$-cohomology of $G$ vanishes, then $G$ is $d$-parabolic.
\end{prop}

\end{frame}

\begin{frame}

Let $G$ be the incidence graph of a sphere packing of $\R^d$.
\begin{enumerate}
  \item Stereographically map $\R^d$ to $S^d$.
  \item Resulting vertex map $f:V\to S^d$ has finite \emph{$d$-energy}.
  \item Assume that every function of finite $d$-energy \textcolor{red}{is a limit of functions having} a limit at infinity (vanishing of \textcolor{red}{reduced} $\ell^d$-cohomology). Then $f$ converges to point $y\in S^d$ \textcolor{red}{along $d$-almost every unbounded path}.
  \item Let $v$ be a finite $d$-energy function on $S^d\setminus\{y\}$ that tends to infinity at $y$. Then $v\circ f$ has finite $d$-energy.
  \item Therefore it has a limit along $d$-almost every unbounded path.
  \item Contradiction, unless $d$-almost every path is no path at all, i.e. $G$ is \emph{$d$-parabolic}.
\end{enumerate}

\begin{prop}
If graph $G$ is packable in $\R^d$ and \textcolor{red}{reduced} $\ell^d$-cohomology of $G$ vanishes, then $G$ is $d$-parabolic.
\end{prop}\pause

\textbf{Example}. $\ell^d \bar{H}^1(\mathbb{Z}^n)=0$ for all $n$. $\mathbb{Z}^n$ is $d$-parabolic iff $n\leq d$.

\end{frame}



