## Notes of Emmanuel Trelat’s lecture

Asymptotique du spectre sous-riemannien

Avec Luc Hillairet et Yves Colin de Verdière.

Nos résultats sont très partiels. Le cas équirégulier marche mais on aimerait aller plus loin. Cela conduit à une nouvelle notion de volume.

1. Les laplaciens sous-riemanniens

1.1. Notations

Etant donnée une structure sous-riemannienne ${(M,D,g)}$ (par exemple, une forme quadratique positive non définie sur le cotangent) et une mesure lisse ${\mu}$, on définit le laplacien

$\displaystyle \begin{array}{rcl} \Delta_{\mu,g}=div_\mu \nabla^g. \end{array}$

C’est l’extension de Friedrichs associée à l’intégrale de Dirichlet

$\displaystyle \begin{array}{rcl} \int |d\phi|_{g^*}\,d\mu. \end{array}$

Si ${(X_i)}$ est un champ de repères orthonormé, alors

$\displaystyle \begin{array}{rcl} \Delta_{\mu,g}=-\sum_i X_i^* X_i=-\sum_i X_i^2 +div_\mu(X_i)X_i. \end{array}$

Sous l’hypothèse de Hörmander, ${\Delta_{\mu,g}}$ est hypoelliptique. Cela signifie que son inverse gagne un chouïa de régularité dans l’échelle de Sobolev. Par conséquent, sur une variété compacte, ${\Delta_{\mu,g}}$ a un spectre discret, i.e. une suite de valeurs propres ${\lambda_1<\lambda_2\leq\cdots\leq \lambda_j}$ (avec multiplicité) et de fonctions propres ${\phi_j}$.

1.2. Parenthèse microlocale

${\phi_j^2\,d\mu}$ est une suite de mesures de probabilité qui possède des limites faibles. On les appelle les mesures quantiques. Ces mesures possèdent des relèvements canoniques dans le cotangent. En effet, à chaque fonction ${a}$ sur le cotangent est associée un opérateur pseudodifférentiel ${Op(a)}$ sur les fonctions sur la base (il y a plusieurs choix, faisons-en un). On définit une mesure sur le cotangent par

$\displaystyle \begin{array}{rcl} \mu_j(a)=\langle Op(a)\phi_j,\phi_j\rangle_{L^2}. \end{array}$

Le comportement des limites (concentration éventuelle) est largement inconnu. Il y a quelques exemples intégrables qui sont bien connus. Lorsque le flot géodésique est ergodique, on s’attendait à ce que toute mesure limites soit Lebesgue, mais ce n’est pas vrai.

1.3. Invariants spectraux

La fonction de comptage ${N(\Lambda)}$ est le nombre de valeurs propres ${\leq\lambda}$.

On espère qu’en faisant des moyennes de Cesaro, on arrive à faire converger les mesures associées aux fonctions propres.

Definition 1 La mesure locale de Weyl est la mesure de probabilité sur ${M}$ définie par

$\displaystyle \begin{array}{rcl} \int f\,dw_{\Delta}=\lim_{\lambda\rightarrow\infty}\frac{1}{N(\lambda)}\sum_{\lambda_j\leq\lambda}\int f|\phi_j|^2\,d\mu. \end{array}$

lorsqu’elle existe.

La mesure microlocale de Weyl est la mesure de probabilité sur ${S^*M=S(T^*M)}$ définie par

$\displaystyle \begin{array}{rcl} \int a\,dW_{\Delta}=\lim_{\lambda\rightarrow\infty}\frac{1}{N(\lambda)}\sum_{\lambda_j\leq\lambda}\langle Op(a)\phi_j,\phi_j\rangle_{L^2}, \end{array}$

lorsqu’elle existe.

1.4. La conjecture d’ergodicité quantique

La conjecture d’ergodicité quantique : est-ce que, pour un ensemble d’indices de densité un, ${\phi_j^2\,d\mu}$ converge ?

La conjecture d’ergodicité quantique unique : est-ce que la suite ${\phi_j^2\,d\mu}$ converge ? Paraît hors de portée.

La densité facilite la vie, comme le montre le Lemme suivant.

Lemma 2 (von Neumann-Koopman) Si, étant donnée une suite bornée de nombres positifs, la moyenne de Cesaro temps vers 0, alors la suite tend vers 0 le long d’un ensemble d’indices de densité 1.

On aimerait montrer davantage : pour tout opérateur ${Op(a)}$, l’espérance coïncide avec l’intégrale selon ${W}$, et la variance vaut 0.

2. Résultats

2.1. Nos résultats de 2014

Theorem 3 ${M}$ variété compacte de dimension 3. Alors ${w_\Delta}$ coincide avec la mesure de Popp normalisée. Quant à ${W_\Delta}$, c’est la moyenne des deux mesures ${\hat{\nu}_1}$ est ${\hat{\nu}_{-1}}$ obtenues en relevant Popp normalisée aux deux sous-variétés du cotangent, graphes de ${\pm}$ la forme de contact.

2.2. Travaux récents

Theorem 4 (Loi locale de Weyl) On suppose la structure équirégulière (i.e. le drapeau engendré par les crochets de sections de ${D}$ est de rangs constants). La mesure de Weyl ${w_\Delta}$ s’exprime en fonction de la mesure de Popp,

$\displaystyle \begin{array}{rcl} dw_\Delta=e(1,0,0)dP, \end{array}$

${e}$ est le noyau de la chaleur pris au temps 1 et à l’origine pour un certain laplacien sur le groupe de Carnot tangent, et une certaine mesure sur ce groupe, les nilpotentisés de ${\Delta}$ et de ${\mu}$.

Question. Quand est-ce que la densité ${f(q)=e(1,0,0)}$ est constante ?

Lemma 5 ${f}$ est constante si et seulement si les cones tangents sont deux isométriques, i.e. si et seulement si le groupe d’isométries est transitif.

C’est le cas pour toutes les structures équirégulières de dimension ${\leq 5}$ sauf un, le groupe d’Heisenberg de dimension 5.

2.3. Cas équisingulier

Cela signifie qu’il existe une stratification compatible avec les chutes de dimensions des espaces du drapeau (c’est automatiquement le cas si la structure est analytique réelle).

Lorsque les rangs ne chutent que d’une unité à la fois (e.g. Martinet, Grushin), on montre que ${w_\Delta}$ est proportionnelle avec la mesure de Popp adaptée au cas singulier.

Exemple. Pour Martinet, ${N(\lambda)\sim \lambda^2\log\lambda}$. Pour Grushin, ${N(\lambda)\sim \lambda\log\lambda}$.

Remarque. ${W_\Delta}$ founirait un choix naturel de mesure sur l’espace des géodésiques.

## Notes of lectures given at the Nonpositive curvature and infinite dimensions workshop in Nancy in august 2015

Conference Nonpositive curvature and infinite dimension, Nancy, august 24-28, 2015

1. Alain Valette: A-T-menability

A locally compact group (or: is a-(T)-menable) if it admits a proper affine isometric action on a Hilbert space. It has the Haagerup property if it admits a ${C^0}$ unitary representation which almost admits invariant vectors. Haagerup property is equivalent to a-T-menability for ${\sigma}$-compact groups.

The Haagerup property is a weak form of amenability: the class of Haagerup groups contains amenable groups, but also free groups, Coxeter groups, closed subgroups of ${SO(n,1)}$ and ${SU(n,1)}$.

It is equivalent to existence of a proper action on a median metric space (Chatterji-Drutu-Haglund). Examples are spaces with (measured) walls. For ${SU(n,1)}$, still no direct way (eg an action on a space with measured walls) of proving Haagerup property.

A-T-menability is not stable under semi-direct products ${G\times_\rho H}$. It works when one of the factors is amenable. It fails in general, as expressed by the recent criterion for permutational wreath products.

2. Stefan Wenger: Minimal disks in metric spaces

Plateau’s problem (minimizing area among disks with a given boundary) has a solution in arbitrary proper metric spaces. The solutions are ${\sqrt{2}}$-quasiconformal maps. In case space satisfies a quadratic isoperimetric inequality, their Hölder exponent (or ${W^1,p}$ class) is controlled sharply by the constant in the quadratic isoperimetric inequality.

It follows that filling disks can be constructed in asymptotic cones. One recovers Papasoglu’s result that asymptotic cones are simply connected when a quadratic isoperimetric inequality holds. Furthermore, if constant is ${<1/4\pi}$, asymptotic cones are trees, thus space is hyperbolic. If constant is ${=1/4\pi}$, minimal disks are Lipschitz, see further applications in Lytchak’s talk.

Question: are there examples of spaces where filling by disks and filling by Lip disks differ ?

3. Adrien Le Boudec: Groups acting on trees with almost prescribed local action

3.1. Neretin’s group

Neretin’s group ${N}$ is the group of homeos of the boundary of a regular tree T which are piecewise tree automorphisms.

Caprace-de Mets: let ${G}$ be a profinite group. Let ${Comm(G)}$ be the group of isomorphisms between open subgroups of ${G}$ (identify two suchmaps if they coincide on some open subgroup). Take for ${G}$ an arbitrary compact open subgroup of ${Aut(T)}$. Then ${Comm(G)=N}$.

Kapoudjian: ${N}$ is simple and compactly generated.

Bader-Caprace-Gelander-Mozes: ${N}$ does not have lattices. ${N}$ has infinite asymptotic dimension (it contains ${{\mathbb Z}^d}$ for all ${d}$).

3.2. Groups defined by local conditions

I prefer to study much smaller groups, for which I will establish similar properties.

Fix a finite set ${S}$ and fix a bijection of each link of vertex of T with ${S}$. Then any ${G}$ in ${Aut(T)}$ and any vertex, one has a permutation ${\sigma(g,v)}$ of ${S}$. Fix a subgroup ${F}$ of permutations of ${S}$.

Burger-Mozes: ${U(F)=\{g \in Aut(T) \,;\, \sigma(g,v) \in F \textrm{ at every vertex v}\}}$.

Bader-Caprace-Gelander-Mozes: ${G(F)=\{g \in Aut(T) \,;\, \sigma(g,v) \in F \textrm{ for all but finitely many vertices}\}}$.

${U(F)}$ is a closed subgroup of ${Aut(T)}$, but ${G(F)}$ is not. The action of ${G(F)}$ on ${T}$ is not proper.

Definition 1 Given subgroups ${F, define

$\displaystyle G(F,F')=G(F) \cap U(F').$

Theorem 2 ${G(F,F')}$ is compactly generated, and its asymptotic dimension equals 1 (it is thus much smaller than ${N}$). Furthermore, ${G(F,F')}$ also appears as a group of commensurators.

Assume that ${F}$ has index 2 in ${F'}$, and that ${F}$ is generated by its point stabilizers. Then ${G(F,F')}$ has a simple subgroup of index 8.

Assume that F acts transitively on ${S}$. Then ${G(F,F')}$ has Haagerup property. Assume further that for every point ${s \in S}$, the stabilizer ${F_s}$ is essential in ${F'_s}$, and ${|F'_s|>|F_s|^{d-1/d-2}}$. Then ${G(F,F')}$ has no lattices.

Example: ${F=PSL(2,q)}$, ${F'=PGL(2,q)}$ actig on the projective line ${P^1(F_q)}$.

corollary Among compactly generated simple groups, having lattices is not a quasi-isometry invariant.

Examples of groups with lattices are obtained when ${G(F_1,F'_1) < G(F,F')}$ is discrete and cocompact (${F}$ simply transitive).

4. Pierre-Emmanuel Caprace: From amenability to buildings

Joint work with Nicolas Monod. We are interested in amenable ${CAT(0)}$ groups, i.e. pairs ${(X,A)}$ where ${X}$ is a proper ${CAT(0)}$ space and ${A}$ an amenable group acting isometrically and cocompactly on ${X}$.

4.1. An ancient subject

Theorem 3 (Adams-Ballmann 1998 (Avez 1970 for manifolds)) If ${A}$ acts properly discontinuously, then ${(\partial X,Tits metric)}$ is a round sphere. Furthermore, ${A}$ is virtually ${{\mathbb Z}^d}$. If, in addition, ${X}$ is geodesically complete, then ${X}$ is Euclidean ${d}$-space.

Question: what if one removes the assumption that the action is properly discontinuous ? I.e. we want to allow ${A}$ to be possibly non discrete.

4.2. Motivation

Let ${M}$ be a closed manifold. Assume ${M}$ carries a locally ${CAT(0)}$ metric. Does ${M}$ also admit nonpositively curved Riemannian metrics? In 2 and 3 dimensions, the answer is yes (it follows from classification results). The answer is no in dimensions ${\ge 4}$ (Davis-Januskiewicz 1991, Davis-Januskiewicz-Lafont 2013 for dimension 4).

Switch from manifolds to groups. Let ${G}$ be a connected Lie group. Assume ${G}$ carries a locally ${CAT(0)}$ left-invariant metric. Does ${M}$ also admit nonpositively curved left-invariant Riemannian metrics? Answer is yes if ${G}$ is Gromov-hyperbolic (Cornulier-Tessera). The general case is still open. The following remark is a starting point.

Proposition 4 Let ${G}$ be a connected, locally ${CAT(0)}$ Lie group. Then ${G}$ is solvable. In fact, more generally, if a connected Lie group ${G}$ acts freely and cocompactly on a a proper ${CAT(0)}$ space ${X}$, then ${G}$ is solvable.

This is proved by induction on the dimension of the solvable radical. The initialization step deals with semi-simple groups. ${G}$ having no compact subgroups, it must be a product of copies of ${\widetilde{SL(2,{\mathbb R})}}$, so its center ${Z}$ is infinite. One can assume that ${G}$ acts minimally (no proper invariant closee convex subset). Then any nontrivial element of ${Z}$ has constant displacement function, so ${X}$ splits a Euclidean factor ${{\mathbb R}^k}$ on which ${G}$ acts cocompactly. This happens only if ${k=0}$, so ${G}$ is trivial. I need more tools to prove the induction step, I will continue later on.

The point I wanted to make is that is raises the question of cocompact actions of solvable groups on ${CAT(0)}$ spaces. I note that dimension will play a role.

4.3. Geometric dimension

I will use Kleiner’s geometric dimension.

Definition 5 (Kleiner) Let ${Z}$ be a ${CAT(1)}$ space. At each point ${z\in Z}$, the space ${S_z}$ of directions is the set of germs of geodesics emanating from ${z}$. It is again a ${CAT(1)}$ space. Say ${Z}$ has geometric dimension 1 if all ${S_z}$‘s are discrete, and so on, inductively.

Fact: geometric dimension does not increase when passing from a ${CAT(0)}$ space to its asymptotic cones, and then to its ideal boundary. The converse does not hold.

4.4. Main result

Theorem 6 Let ${X}$ be a proper ${CAT(0)}$ space. Assume that isometries of ${X}$ do not have a common fixed point on the ideal boundary of ${X}$. Let ${A}$ be an amenable locally compact group acting cocompactly on ${X}$. Then the ideal boundary ${\partial X}$, in its Tits metric, is a spherical building. Furthermore, each irreducible factor of ${\partial X}$ is isometric to the 2-point space ${S^0}$ or to the spherical building pf a simple algebraic group over a local field (possibly ${{\mathbb R}}$ or ${\mathbb{C}}$).

Note that, in that case, ${\partial X}$ has geometric dimension 0 if and only if ${X}$ is hyperbolic. In that case, the theorem does not say much. Here is what we know.

Theorem 7 (Caprace-Cornulier-Monod-Tessera) Let ${X}$ be a propeer geodesic hyperbolic space. Assume that isometries of ${X}$ do not have a common fixed point on the ideal boundary of ${X}$. If ${X}$ admits a cocompact action of an amenable group. Then, modulo a compact normal subgroup, ${Isom(X)}$ is either a rank one simple Lie group, or a cloased subgroup ofnthe isometry group of a tree that acts 2-transitively on the boundary, or a subgroup of ${Isom({\mathbb R})}$.

Corollary 8 In main Theorem, if ${X}$ is geodesically complete, then ${X}$ is a product of flats, summetric spaces, semi-regular trees and Bruhat-Tits buildings.

4.5. Tools

The key tool is a result of Bernhard Leeb who proved that if ${X}$ is a proper geodesically complete ${CAT(0)}$-space and ${\partial X}$ is an irreducible spherical building of dimension ${\geq 1}$, then ${X}$ is a symmetric space or a Euclidean building Note Leeb does not assume any group action. Bruhat-Tits is a subclass of Euclidean buildings.

An other tool will be an other result of Adams-Ballmann.

Theorem 9 (Adams-Ballmann 1998 (Burger-Schroeder 1987 for manifolds)) If ${A}$ be an amenable locally compact group, acting continuously on a proper${CAT(0)}$ space ${X}$, then ${A}$ either stabilizes a flat in ${X}$ or fixes a point at infinity.

The conclusion is rather antinomic with being cocompact, whence the strong consequences I will draw.

Here is a convenient terminology.

Definition 10 A group ${H}$ is an AB group if it satisfies the conclusion of the above theorem, i.e. any action on a proper ${CAT(0)}$ space stabilizes a flat or fixes a boundary point.

Fact. If ${X}$ is a proper ${CAT(0)}$ space with a cocompact isometry group, then a closed subgroup ${H is amenable if and only if it is an AB group. Warning. This equivalence fails in general. For instance, ${F_2\wr{\mathbb Z}}$ is an AB group but it is not amenable. Thompson’s group ${F}$ is an AB group as well.

4.6. Fixed points sets

Cocompact groups can hardly have fixed points on the boundary. Indeed,

Proposition 11 (Burger-Schroeder 1987 for manifolds) Let ${\Gamma}$ be a discrete group acting properly cocompactly on a proper ${CAT(0)}$ space ${X}$. Then the fixed point set ${\partial X^\Gamma}$ is contained in the maximal spherical joint factor of ${\partial X}$. If ${\Gamma}$ action on ${X}$ is minimal, this factor is the boundary of the maximal flat factor of ${X}$.

Proof. We provide a proof that illustrate ideas that are extendable to the non discrete case. Let ${\xi}$ be a fixed point, ${\rho}$ a ray converging to ${\xi}$. By cocompactness, there is a radial sequence for ${\xi}$, i.e. a sequence ${\gamma_n\in\Gamma}$ such that ${d(\rho(0),\gamma_n\rho(n))}$ stays bounded. The sequence ${\gamma_n\rho}$ subconverges to a line ${\ell}$ with endpoints ${\xi}$ and ${\xi'}$. Conjugate the picture with ${\gamma\in \Gamma}$. Since ${g}$ fixes ${\xi}$, ${d(\rho(0),\gamma_n g\gamma_n^{-1}\rho(0))}$ stays bounded. Hence ${\gamma_n g\gamma_n^{-1}}$ stays bounded in ${\Gamma}$, i.e. is constant (up to extracting), ${\gamma_n g\gamma_n^{-1}=h}$. ${h}$ must fix ${\gamma_n^{-1}\xi'}$ for ${n}$ large enough. Since ${\Gamma}$ is finitely generated, every element of ${\Gamma}$ fixes some ${\eta=\gamma_n^{-1}\xi'}$. ${\Gamma}$ stabilizes the union of all lines with endpoints ${\{\xi,\eta\}}$, therefore ${X}$ is isometric to ${{\mathbb R}\times X'}$, contradicting minimality.

4.7. Proof of Adams-Ballmann’s first theorem

Assuming Adams-Ballmann’s second theorem. Replace ${X}$ with a minimal invariant subspace ${Y}$. ${Y}$ has a canonical splitting ${Y={\mathbb R}^k\times Y'}$. ${A}$ acts on ${Y'}$ without fixed points on ${\partial Y'}$. Adams-Ballmann’s second theorem implies that ${A}$ stabilizes a flat in ${Y'}$. By minimality, ${Y'}$ is a point.

Arguments in the proof of the proposition lead to the following

Lemma 12 Let ${X}$ be a proper ${CAT(0)}$ space. Let ${G be a closed subgroup which is cocompact on ${X}$, and fixes a point ${\xi\in\partial X}$. Then

$\displaystyle \begin{array}{rcl} Op(\xi)=\{\eta\in\partial X\,;\,\eta\textrm{ opposite }\xi\} \end{array}$

is nonempty and ${G}$ is transitive on it.

4.8. Geometric Levi Decomposition

Theorem 13 (Geometric Levi Decomposition) Let ${X}$ be a proper ${CAT(0)}$ space. Let ${G be a closed subgroup which is cocompact on ${X}$, and fixes a point ${\xi\in\partial X}$. Pick ${\xi'\in Op(\xi)}$. Then

1. ${G=G^u G_{\xi'}}$ where ${G^u}$ is the set of ${g}$ whose displacement tends to zero along rays converging to ${\xi}$.
2. ${G^u}$ is normal in ${G}$ and ${G^u\cap G_{\xi'}}$ is compact.
3. ${G_{\xi'}}$ is cocompact on the set of lines with endpoints ${\xi}$ and ${\xi'}$.
4. ${G^u}$ is amenable. In fact, it is compactible, i.e. there exists a compact subgroup ${K and a sequence ${g_n\in G}$ that conjugates every element of ${G^u}$ into a bounded sequence all of whose accumulation points are in ${K}$.

In other words, as soon as a group of isometries fixes a point at infinity ${\xi}$, two new actions arise, the Busemann character (an action on ${{\mathbb R}}$), and the action on the transverse space ${X_\xi}$, the quotient set of rays converging to ${\xi}$, which inherits a quotient ${CAT(0)}$ metric. The kernel of the action on ${{\mathbb R}\times X_\xi}$ is ${G^u}$. A homomorphism to ${G_{\xi'}}$ is produced like in the proof of the Proposition above, by taking coherently (using an ultrafilter) limits of subsequences of ${(g_n g g_n^{-1})}$.

Corollary 14 Assume further that ${G}$ is unimodular. Then ${\partial X^G}$ is again in the maximal spherical factor of ${\partial X}$.

Indeed, compactibility and unimodularity force ${G^u}$ to be compact, and ultimately ${Op(\xi)}$ is a single point.

4.9. Spherical buildings

Recall that our goal is to prove that the boundary of a ${CAT(0)}$ space with an amenable cocompact locally compact group of isometries is a spherical building.

We can take the following (remarkable) theorem as a definition of spherical buildings.

Theorem 15 (Balser-Lytchak) Let ${Z}$ be a ${CAT(1)}$-space. Assume that

1. ${\mathrm{gdim}(Z)=d<\infty}$.
2. ${Z}$ contains a pair ${\{z,z'\}}$ of antipodal points, i.e. ${d(z,z')\geq\pi}$.
3. Every pair of antipodal points lie in a common ${d}$-sphere.
4. There is a special point ${z_0\in Z}$ with a compact neighborhood.

Then ${Z}$ is a spherical building.

The special point can be in fact any regular point.

Examples.

• If ${d=0}$, this merely means that all distance points are at infinite distance.
• Round spheres are buildings.
• Spherical joins of buidlings are buildings.

For instance, the spherical join of a 3 point set and a sphere is a “trisphere”, made of 3 spheres that meet along a codimension 1 sphere.

Recall (Kleiner) that if a proper cocompact ${CAT(0)}$ space has geometric dimension ${(\partial X)=d}$, then there is ${d+1}$-dimensional flat in ${X}$. Leeb proved that every ${d}$-sphere isometrically embedded in ${\partial X}$ bounds a ${d+1}$-flat.

In particular, antipodal points contained in a sphere in ${\partial X}$ are opposite, i.e. bound a line.

4.10. Set-up for the main theorem

The fourth axiom will be the hardest to check. We need find a regular point. The center of a Weyl chamber fixed by ${A}$ (see Adams-Ballmann’s theorem) is our candidate. For this, we need that the fixed point set be not too large (eg contain

Proposition 16 Let ${X}$ be a proper ${CAT(0)}$ space, ${H cocompact. Then

1. ${(\partial X)^H}$ is contained in some ${d}$-sphere, ${d=\mathrm{gdim}(\partial X)}$.
2. Its radius is at most ${\pi/2}$, unless ${\partial X}$ has a nonempty spherical factor.
3. If, in addition, ${H}$ is amenable, then radius is at most ${\pi/2}$, unless ${\partial X}$ is a sphere.

4.11. Proof of (1)

By induction on ${d=\mathrm{gdim}(\partial X)}$.

If ${d=0}$, i.e. ${X}$ is hyperbolic, there is at most 1 fixed point (otherwise, ${H}$ would stabilize a point or a line of ${X}$).

Let ${d>0}$. Pick a fixed point ${\xi}$. We know that it has at least one opposite point ${\xi'\in Op(\xi)}$. According to the geometric Levi decomposition, ${H_{\xi'}}$ acts cocompactly on the union of parallel lines ${P(\xi,\xi')={\mathbb R}\times X_\xi}$. Its boundary ${\partial P(\xi,\xi')}$ is the spherical join of the pair ${\{\xi,\xi'\}}$ and ${\partial X_\xi}$.

Claim. ${(\partial X)^{H_{\xi'}}\subset \partial P(\xi,\xi')}$.

Indeed, given ${\eta\in (\partial X)^{H_{\xi'}}}$, by cocompactness, one finds a point in ${X}$ from where the angle of ${\xi,\eta}$ equals the Tits distance, thus getting a flat sector. Again by cocompactness, one produces a flat half-plane whose boundary contains ${\xi,\xi',\eta}$, which is contained in ${P(\xi,\xi')}$.

Since ${\mathrm{gdim}(\partial X_\xi)<\mathrm{gdim}(\partial X)}$, conclude by induction (requires more work to produce a ${d}$-sphere…).

4.12. Main step

It consists in proving that ${\mathrm{gdim}(\partial X^A)=d}$, and that every interior point ${\xi}$ is“regular” in the sense that ${\partial X_\xi}$ is a round sphere. (It is likely that this statement holds for all amenable ${CAT(0)}$ groups having boundary fixed points. We are able to prove this only if ${A}$ is totally disconnected.)

Consequence. Pick such an interior point ${\xi}$. Then ${A}$ is transitive on the set of ${d}$-spheres containing ${\xi}$.

Indeed, aconsider antipodes of ${\xi}$ in such spheres. An element ${g\in A}$ that maps an antipode to another must send whole sphere to whole sphere, since it fixes a ${d}$-dimensional subset of these spheres.

4.13. Tri-spheres

Again, ${\xi}$ is an interior point of the fixed point set. One shows that two ${d}$-spheres through ${\xi}$ form a tri-sphere.

Now we use the assumption that the full isometry group has no common fixed point to show that the ${G}$-orbit of ${\xi}$ intersects each half-sphere of the tri-sphere. This allows to produce a reflection fixing a given ${d}$-sphere ${S_0}$. Such reflections suffice to show that ${G}$-translates of ${(\partial X)^A}$ cover ${S_0}$.

5. Yves Cornulier: Large scale geometry of Lie groups

For finitely generated groups, polycyclic ${\Leftrightarrow}$ solvable and all subgroups are finitely generated. Nilpotent groups are automatically polycyclic.

Example: Baumslag-Solitar group ${{\mathbb Z}[\frac{1}{n}]\times_n{\mathbb Z}}$ is finitely generated, solvable but not polycyclic.

Tits alternative: A finitely generated subgroup of ${GL(n,{\mathbb R})}$ either contains a free subgroup or is virtually solvable.

For Lie groups, solvable implies polycyclic. Therefore, a virtually finitely generated solvable group embeds in some ${GL(n,{\mathbb R})}$ if and only if it is virtually polycyclic.

A virtually solvable subgroup of ${GL(n,{\mathbb R})}$ is contained and cocompact in a closed subgroup ${G}$ with finitely many connected components (beware that ${G}$ need not be the Zariski closure). This is proved by reduction to the nilpotent case. For groups ${\Gamma}$ with unipotent Zariski closure ${F}$, Malcev theory shows that ${\Gamma}$ is cocompact in ${F}$. In general, pass to derived subgroups ${\Gamma'}$ and ${F'}$. ${F/F'\Gamma}$ is abelian, ${\Gamma}$ is cocompact in ${F'\Gamma}$,

Therefore every virtually polycyclic group is a lattice in a virtually connected solvable Lie group.

5.2. Structure theory of Lie groups

The idea is to define a normal form for Lie groups. I.e. replace every Lie group with a simpler Lie group that reflects its large scale geometry.

Theorem 17 (Iwasawa, Mostow) In a virtually connected Lie group, all maximal compact subgroups are conjugate, they contain all topology: ${G}$ is diffeomorphic to ${K\times G/K}$, and ${G/K}$ is diffeomorphic to ${{\mathbb R}^k}$.

Nevertheless, it is ${G/K}$ which contains the large scale geometry.

Theorem 18 Every connected Lie group has a closed cocompact subgroup of the form ${H\times {\mathbb Z}^k}$ where ${H}$ is solvable and connected.

For every solvable connected Lie group ${R}$, there exist proper and cocompact homomorphisms

1. ${R\rightarrow L}$ where ${L}$ is solvable connected,
2. ${T\rightarrow L}$ where ${T}$ is triangulable (i.e. a subgroup of upper triangular matrices).

Therefore, we have the following diagram of proper and cocompact homomorphisms

$\displaystyle \begin{array}{rcl} G\leftarrow H\times{\mathbb Z}^k\rightarrow L\leftarrow T\times{\mathbb R}^k. \end{array}$

Example. ${G=\widetilde{SL(2,{\mathbb R})}}$. Then ${T=}$ upper triangular matrices and ${k=1}$. In general, the ${{\mathbb Z}^k}$ or ${{\mathbb R}^k}$ factors are there only if groups with infinite fundamental groups are encountered.

Example. ${G=\mathbb{C}\times_\alpha{\mathbb R}}$ where ${{\mathbb R}}$ acts on ${\mathbb{C}}$ by multiplication by ${e^{t+it}}$. Then ${L=\mathbb{C}\times_\beta{\mathbb R}^2}$ where ${{\mathbb R}^2}$ acts on ${\mathbb{C}}$ by multiplication by ${e^{t+iu}}$.

Theorem 19 (Gordon-Wilson 1985) If two triangulable groups are isometric, then they are isomorphic.

Bold conjecture. Replace isometric with quasi-isometric ?

This is widely open, even in the nilpotent case.

5.3. Asymptotic cones

Let ${(X_n)}$ be a sequence of metric spaces. Let ${\omega}$ be a nonprincipal ultrafilter on ${{\mathbb N}}$. One can define a metric space

$\displaystyle \begin{array}{rcl} X_\omega=\lim_{\omega}X_n \end{array}$

as follows. The “distance” on the ultraproduct ${\prod_{\omega}X_n}$ takes values in the ultraproduct ${\prod_\omega {\mathbb R}_+}$, which is too big.

Instead, pick a marked point ${o_n}$ in each ${X_n}$ and consider sequences ${(x_n)}$, ${x_n\in X_n}$, such that ${d(x_n,o_n)}$ stays bounded. Then identify sequences ${(x_n)}$ and ${(x'_n)}$ such that

$\displaystyle \begin{array}{rcl} \lim_{\omega}d(x_n,x'_n)=0. \end{array}$

A metric is well defined on the quotient set by

$\displaystyle \begin{array}{rcl} d((x_n),(x'_n))=\lim_{\omega}d(x_n,x'_n). \end{array}$

The resulting metric space indeed depends on the choice of the marked points ${o_n}$.

Important special cases.

1. If all ${(X_n,o_n)=(X,o)}$ are the same, then ${X}$ isometrically embeds into ${X_\omega}$ (which is usually much larger than ${X}$ unless ${X}$ is locally compact).
2. If ${(X_n,d_n,o_n)=(X,nd,o)}$, the limit is called a tangent cone at ${o}$.
3. If ${(X_n,d_n,o_n)=(X,\frac{1}{n}d,o)}$, the limit ${cone^\omega(X)}$ is called an asymptotic cone (it does not depend on the choice of ${o}$).

Examples. Start with ${Z^2}$ in the word metric for the obvious generating system. Then ${cone^\omega(X)=({\mathbb R}^2,\ell_1}$ metric).

5.4. The large scale category

Objects are metric spaces, morphisms are large scale Lipschitz maps, up to the following equivalence: identify ${f}$ and ${f'}$ if ${d(f(x),f'(x))}$ is bounded on ${X}$. Isomorphisms are quasi-isometries. Large scale Lipschitz maps induce Lipschitz maps between asymptotic maps. Therefore, ${cone^\omega}$ is a functor of the large scale category to the Lip category (metric spaces and Lipschitz maps).

Compactly generated groups carry many invariant metrics, all of which are quasi-isometric. Therefore their ${cone^\omega}$ is well-defined up to bi-Lipschitz homeomorphisms.

5.5. The sublinear category

Being quasi-isometric is very restrictive. Much of one usually does is compatible with le

Say a map ${f:X\rightarrow Y}$ between metric spaces is sublinearly Lipschitz if

$\displaystyle \begin{array}{rcl} d(f(x),f(x'))\le C\,d(x,x')+q(d(x,o)+d(x',o)), \end{array}$

for some sublinear function ${q}$. Identify maps ${f}$ and ${f'}$ when they are sublinearly close, i.e.

$\displaystyle \begin{array}{rcl} d(f(x),f'(x))=o(d(x,o)). \end{array}$

Again, ${cone^\omega}$ is a functor from the sublinear category to the Lip category.

Main example. Let ${\mathfrak{g}}$ be a nilpotent Lie algebra, let ${\mathfrak{g}^{i+1}=[\mathfrak{g},\mathfrak{g}^{i}]}$ be its lower central series. The Lie bracket

$\displaystyle \begin{array}{rcl} \mathfrak{g}^{i}\times \mathfrak{g}^{j}\rightarrow \mathfrak{g}^{i+j}\quad \textrm{induces}\quad\mathfrak{g}^{i}/\mathfrak{g}^{i+1}\times \mathfrak{g}^{j}/\mathfrak{g}^{j+1}\rightarrow \mathfrak{g}^{i+j}/\mathfrak{g}^{i+j+1}, \end{array}$

defining a Lie algebra bracket on

$\displaystyle \begin{array}{rcl} Car(\mathfrak{g})=\bigoplus_i \mathfrak{g}^{i}/\mathfrak{g}^{i+1}. \end{array}$

Choosing complementary subspaces, one can view ${Car(\mathfrak{g})}$ as a new, simpler, Lie bracket on ${\mathfrak{g}}$. Via exponential maps, this produces a new group law ${Car(G)}$ on the Lie group ${G}$ such that ${Lie(G)=\mathfrak{g}}$ (starting from dimension 5, it is usually not isomorphic to ${G}$).

Theorem 20 (Pansu, Breuillard) The identity map ${G\rightarrow Car(G)}$ is a sublinear Lipschitz equivalence.

For a Carnot group like ${Car(G)}$, asymptotic cones are easy to determine: ${Car(G)}$ admits a one parameter group of automorphisms which are homthetic relative to left-invariant sub-Riemannian metrics. Therefore ${cone^\omega(Car(G))=Car(G)}$. It follows that ${cone^\omega(G)=Car(G)}$. Using Malcev theory, this gives the asymptotic cones for all finitely generated nilpotent groups.

Question. Classify compactly generated groups up to sublinear Lipschitz equivalence ?

Possibly, all hyperbolic groups might be sublinearly Lipschitz equivalent.

5.6. Geometry of triangulable groups

For a triangulable group ${G}$, the lower central series stabilizes to ${\mathfrak{g}^{\infty}}$. Call

$\displaystyle \begin{array}{rcl} G^\infty=\exp(\mathfrak{g}^{\infty}) \end{array}$

the exponential radical of ${G}$. It is a normal subgroup.

Say ${G}$ is splittable if ${\mathfrak{g}}$ is a semi-direct product ${\mathfrak{g}=\mathfrak{g}^{\infty}\oplus\mathfrak{n}}$ where ${\mathfrak{n}}$ is nilpotent.

Theorem 21 Let ${G}$ be triangulable. There is a new group law ${G'}$ on ${G}$, that does not change the exponential radical, which is splittable and sublinearly Lipschitz equivalent. Furthermore, the action of ${N=G'/G^\infty}$ on ${G^\infty}$ is diagonalizable over ${{\mathbb R}}$.

This new law gets rid of all sublinear phenomena, including imaginary parts of eigenvalues.

Example. Let ${G}$ be the semidirect product ${{\mathbb R}^2\times{\mathbb R}^2}$ where ${{\mathbb R}^2}$ acts by matrices

$\displaystyle \begin{array}{rcl} \begin{pmatrix} e^t & u\,e^t \\ 0 & e^t \end{pmatrix} \end{array}$

The action of ${N=}$ the normal ${{\mathbb R}^2}$ is not diagonalisable. The new law replaces the matrix by

$\displaystyle \begin{array}{rcl} \begin{pmatrix} e^t & 0 \\ 0 & e^t \end{pmatrix} \end{array}$

resulting into a group quasi-isometric to ${H^3\times{\mathbb R}}$, where ${H^3}$ denotes hyperbolic 3-space. Both laws are not quasi-isometric, although they have isometric asymptotic cones ${\mathbb{T}\times{\mathbb R}}$ (${\mathbb{T}}$ is the universal ${{\mathbb R}}$-tree of degree ${2^{\aleph_0}}$).

5.7. Dimensions of asymptotic cones

Corollary 22 ${\mathrm{top. dim}(cone^\omega(G))=\mathrm{dim}(G/G^\infty)}$.

Proof. The exact sequence ${1\rightarrow G^\infty\rightarrow G\rightarrow N\rightarrow 1}$ induces a fibration

$\displaystyle \begin{array}{rcl} cone^\omega(G^\infty,d_G)\rightarrow cone^\omega(G)\rightarrow cone^\omega(N). \end{array}$

Guivarc’h-Osin: ${(G^\infty,d_G)}$ is quasi-ultrametric (ultrametric up to an additive constant), so ${cone^\omega(G^\infty,d_G)}$ is ultrametric, hence 0-dimensional. It implies (Gromov, Burillo), that ${\mathrm{top. dim}(cone^\omega(G))\leq0+\mathrm{top.dim}(cone^\omega(N))=\mathrm{dim}(N)}$, since ${N}$ is nilotent. Hence

$\displaystyle \mathrm{top. dim}(cone^\omega(G))\leq\mathrm{dim}(G/G^\infty).$

Reverse inequality is obvious if ${G}$ is splittable, since ${cone^\omega(N)}$ embeds isometrically into ${cone^\omega(G)}$. General case requires some more work.

The ${cone^\omega(G)}$ is bi-Lipschitz to a fiber product of ${cone^\omega(N)}$ and ${cone^\omega(E\times N/[N,N])}$ above ${cone^\omega(N/[N,N])}$

${cone^\omega(E\times N/[N,N])}$ can be described up to bi-Lipschitz equivalence as follows. Let ${^\omega{\mathbb R}}$ denote the ultrapower of ${{\mathbb R}}$, let

$\displaystyle \begin{array}{rcl} v:\omega{\mathbb R}\rightarrow[-\infty,+\infty],\quad v((x_n))=-\lim_{\omega}\frac{1}{n}\log|x_n|. \end{array}$

Define the Robinson field

$\displaystyle \begin{array}{rcl} {\mathbb R}^*=\{x\in\omega{\mathbb R}\,;\,v(x)>-\infty\}/\{x\,;\,v(x)=+\infty\}. \end{array}$

${v}$ is a ${{\mathbb R}}$-valued valuation on ${{\mathbb R}^*}$, with valuation ring ${\mathbb{A}=\{x\in\omega{\mathbb R}\,;\,v(x)\geq 0\}}$. Then, for every Lie group ${H}$,

$\displaystyle \begin{array}{rcl} cone^\omega(H)=H({\mathbb R}^*)/H(\mathbb{A}). \end{array}$

We use the notation of algebraic groups, but the construction extends to connected Lie groups for most fields and rings.

Corollary 23 Assume Continuum Hypothesis. Then, for all connected Lie groups ${G}$, then all ${cone^\omega(G)}$ are bi-Lipschitz to each other.

Continuum Hypothesis forces the Robinson field to be unique. The converse is true for higher rank absolutely simple groups: if Continuum Hypothesis is assumed to fail, such groups have several different asymptotic cones (Kramer-Shelah-Tent-Thomas).

5.9. Explicit examples of cones

Hyperbolic groups. If ${G}$ is nonelementary hyperbolic, ${cone^\omega(G)}$ is isometric to the universal ${{\mathbb R}}$-tree ${\mathbb{T}}$. Note that hyperbolic Lie groups are known.

Theorem 24 (Pansu, Cornulier-Tessera) A connected Lie group ${G}$ is nonelementary hyperbolic if and only if

1. either ${G=R\times({\mathbb R}\times K)}$ where ${{\mathbb R}}$ acts on ${E}$ by contracting automorphisms and ${K}$ is compact,
2. or ${G/W(G)}$ is an open subgroup in a rank 1 simple Lie group.

${SOL}$ groups. ${SOL_\lambda}$, ${\lambda>0}$, is the semidirect product ${{\mathbb R}^2\times{\mathbb R}}$ where ${{\mathbb R}}$ acts on ${{\mathbb R}^2}$ by matrices

$\displaystyle \begin{array}{rcl} \begin{pmatrix} e^t & 0 \\ 0 & e^{-\lambda t} \end{pmatrix}. \end{array}$

It is a subgroup in the product of two copies of the affine group, the kernel of a homomorphism to ${{\mathbb R}}$ which is a Buseman function. Thus ${cone^\omega(SOL_\lambda)}$ is a horosphere in ${\mathbb{T}\times\mathbb{T}}$, it does not depend on ${\lambda}$. It is 1-dimensional and not simply connected.

This works as well for semidirect products ${(N_1\times N_2)\times{\mathbb Z}}$ where ${{\mathbb Z}}$ contracts ${N_1}$ and dilates ${N_2}$. For instance, if ${N_1=\mathbb{Q}_p}$ and ${N_2={\mathbb R}}$, the group contains Baumslag-Solitar group ${BS(1,p)}$ as a lattice. Thus this group has asymptotic cones bi-Lipschitz equivalent to those of ${SOL}$ groups.

More generally, one can handle semidirect products ${{\mathbb R}^k\times{\mathbb R}^\ell}$ with a diagonal action. Asymptotic cones are ${k}$-dimensional, and not ${k}$-connected.

5.10. Questions

Which cones are ${CAT(0)}$ ? I can merely say which cones are contractible. This happens if and only if ${G=G^\infty\times N}$ and ${N}$ contains an element that contracts ${G^\infty}$.

6. Anders Karlsson: Nonpositive curvature, metric functionals and ergodic theorems

6.1. Towards metric functionals

Metric functionals generalize horofunctions.

In the category of vectorspaces, one encounters lines and linear functionals, and duality holds.

In the category of metric spaces, one encounters geodesics and metric functionals will play the role of linear functionals. There is a weak topology on them, that allows compactness.

6.2. Busemann functions

Let ${X}$ be a metric space. Let ${\gamma:{\mathbb R}_+\rightarrow X}$ be a geodesic ray. Then

$\displaystyle \begin{array}{rcl} b_\gamma(x)=\lim_{t\rightarrow\infty}d(x,\gamma(t))-d(\gamma(0),\gamma(t)) \end{array}$

exists. Indeed, by triangle inequality, ${t\mapsto d(x,\gamma(t))-t}$ is nonincreasing.

This idea arose in Riemannian geometry, both in nonnegative and nonpositive curvature. These functions appeared much earlier in Poissons’s integral formula for harmonic functions on the disk.

6.3. Horofunctions

Gromov gives a different exposition, inspired by Martin’s boundary in potential theory. Let ${X}$ be a metric space, and ${o\in X}$ some marked point. Map ${X}$ to ${C(X)}$ (continuous functions on ${X}$, equipped with the topology of uniform convergence on bounded sets) by

$\displaystyle \begin{array}{rcl} \Psi: x\mapsto d(\cdot,x)-d(o,x). \end{array}$

Consider the closure ${\bar{X}^h}$ of ${\Psi(X)}$.

Example. If ${X}$ is ${CAT(0)}$, ${\bar{X}^h}$ coincides with the visual bordification of ${X}$.

Example. For nonproper metric spaces, ${\partial^h X}$ may be empty. For instance, let ${X}$ be a bouquet of longer and longer geodesic segments.

6.4. Metric functionals

We change topology. We replace ${C(X)}$ with ${{\mathbb R}^X}$, i.e. pointwise convergence. Denote by ${\partial X=\bar{X}\setminus(X\cup \hat{X})}$, where ${\hat{X}}$

Example. Let ${X}$ be a bouquet of longer and longer geodesic segments. Then

Proposition 25 Let ${H}$ be an infinite dimensional real Hilbert space. Then the elements in ${\bar{H}}$ are parametrized by ${0 and vectors ${v}$ such that ${0<|v|\leq 1}$ or ${r=0}$, ${v=0}$.

1. ${h_x(y)=|y-x|-|x|}$, ${r=|x|}$, ${v=\frac{x}{|x|}}$.
2. ${h_{r,v}(y)=\sqrt{r(1-|v|^2)+|y-rv|^2}-r}$.
3. ${h_{\infty,v}(y)=-(y,v)}$.

A sequence ${x_n\in H}$ converges in ${\bar{H}}$ iff ${|x_n|}$ tends to ${r}$ and ${\frac{x_n}{|x_n|}}$ converges weakly to ${v}$.

Example. Let ${(e_n)}$ be an orthonormal basis of ${H}$. Then ${e_n}$ tends to ${h_{1,0}}$. However, given a sequence ${\lambda_n\leq 1}$, ${\lambda_n e_n}$ does not always converge.

6.5. Connection with other notions

From now now on, denote by ${\bar{X}=X\cup \partial X\cup X_f}$ where ${\partial X}$ is the horoboundary.

Caprace-Lytchak and Bader-Duchesne-Lecureux also use compactifications in weak topologies. The Roller boundary used by Alessandra Iozzi also pertains to the same idea.

Exercise. When ${X}$ is a normed vectorspace, which Busemann functions are linear ?

Example. Let ${X}$ be a countably infinite simplicial tree, with infinitely many branches at vertex ${o}$. As in Hilbert space, the sequence of neighbours of ${o}$ converges to a point in ${X_f}$.

6.6. Action of isometries

Of course, construction is natural under isometries.

Example. For ${{\mathbb Z}^2}$, the horoboundary consists of a square with countable sides. Translations along one factor fix two sides and translate the two others.

Proposition 26 Let ${\Gamma. Fix origin ${o\in X}$. Let ${\lambda}$ be a ${\Gamma}$-invariant probability measure on ${\bar{X}}$. Then

$\displaystyle \begin{array}{rcl} T(g)=\int_{\bar{X}}h(g^{-1}o)\,d\lambda(h) \end{array}$

is a group homomorphism ${\Gamma\rightarrow{\mathbb R}}$.

Exercise. Let ${X=Cay(\Gamma,S)}$.

1. If ${\Gamma}$ fixes a point of ${\partial X}$, then ${\Gamma}$ surjects onto ${{\mathbb Z}}$.
2. If ${\partial X}$ is countable, then ${\Gamma}$ has a finite orbit in ${\partial X}$.

Find conditions (growth ?) on ${\Gamma}$ that ensure that ${\partial X}$ is countable.

Cormac Walsh: If ${\Gamma}$ is nilpotent, there exists a finite orbit in ${\partial X}$.

6.7. Towards a metric spectral principle

Let ${E}$ be a normed complex vectorspace. Let ${A:E\rightarrow E}$ be a bounded linear operator. Then spectral radius

$\displaystyle \begin{array}{rcl} \rho(A)=\lim_{n\rightarrow\infty}|A^n|^{1/n} \end{array}$

exists. It coincides with the sup of the spectrum. In finite dimension, ${A}$ has a Jordan form with eigenvalues ordered with decreasing absolute values, whence a filtration ${V_i}$. Given a vector ${v\in E}$,

$\displaystyle \forall v\in V_i \setminus V_{i-1},\quad|A^n v|^{1/n}\rightarrow|\lambda_i|.$

This fails in general in infinite dimensions. This is related with the invariant subspace problem: does a bounded linear operator on Hilbert space have a proper closed invariant subspace ?

On the other hand, this works in certain situations. For instance, random products of matrices (Furstenberg-Kesten 1960, Oseledets 1968).

6.8. Translation length

This will be our nonlinear generalization of spectral radius. In the linear setting, for a matrix ${A}$, ${\log|A|=d(I,A)}$.

Theorem 27 (Karlsson 2001) Let ${f:X\rightarrow X}$ be a 1-Lip map. Define

$\displaystyle \tau=\lim_{n\rightarrow\infty} \frac{1}{n}d(o,f^n(o)).$

Then there exists ${h\in\bar{X}}$ such that ${h(f^k(o))\leq -\tau k}$ for all ${k}$, and

$\displaystyle \begin{array}{rcl} \lim -\frac{1}{n}h(f^n(o))=\tau. \end{array}$

Only the second statement is really original. I think of it as a weak Jordan decomposition. With Sebastien Gouezel, we have recently been able to extend it to random compositions of 1-Lipschitz maps.

Proof. 1-Lip and triangle inequality imply subadditivity.

Fix sequence ${\epsilon_i}$ decreasing to 0. Set

$\displaystyle \begin{array}{rcl} b_i(n)=d(o,f^n o)-(\tau-\epsilon_i)n. \end{array}$

For fixed ${i}$, this is unbounded in ${n}$, so pick subsequence ${n_i}$ such that

$\displaystyle \begin{array}{rcl} \forall m

Then

$\displaystyle \begin{array}{rcl} h_{f^no}(f^ko)&=&d(f^ko,f^{n_i}o)-d(o,f^{n_i}o)\\ &\le&b_i(n-k)+(\tau-\epsilon_i)(n_i-k)-b_i n_i -(\tau-\epsilon_i)n_i\\ &\leq&-(\tau-\epsilon_i)k. \end{array}$

By compactness, there is a limit point

6.9. Applications

\subsubsection{Mean ergodic theorem}

Here, ${X=H}$ is Hilbert space. Let ${U}$ be a linear operator of norm ${\le 1}$. Let ${v\in H}$. Consider ${f:H\rightarrow H}$, ${f(w)=Uw+v}$, so that

$\displaystyle \begin{array}{rcl} f^n(0)=\sum_{k=0}^{n-1} U^k v. \end{array}$

Then either ${\tau=0}$, and ${\frac{1}{n}\sum_{k=0}^{n-1} U^k v}$ tends to 0, or ${\tau>0}$. Our Theorem provides a metric (linear in this case) functional ${h}$ such that ${-\frac{1}{n}h(f^no)}$ tends to ${\tau}$. This implies that ${\frac{1}{n}\sum_{k=0}^{n-1} U^k v}$ tends to ${\tau y}$, where ${h=(\cdot,y)}$, and ${|y|=1}$. Indeed,

$\displaystyle \begin{array}{rcl} |\frac{1}{n}\sum_{k=0}^{n-1} U^k v-\tau y|^2&=&\frac{1}{n^2}|\sum_{k=0}^{n-1} U^k v|^2-\frac{2}{n}(\sum_{k=0}^{n-1} U^k v,\tau y)+\tau^2|y|^2\\ &=&\frac{1}{n^2}|f^n(0)|^2-\frac{2}{n}\tau h(f^n(0))+\tau^2|y|^2\\ &\rightarrow&\tau^2-2\tau^2+\tau^2=0. \end{array}$

\subsubsection{Hyperbolic metric spaces}

If ${f^no}$ is unbounded, it should converge to a point ${\xi}$ in the Gromov boundary. If ${f}$ is an isometry, ${f(\xi)=\xi}$.

\subsubsection{Several complex variables}

To a complex space, a Kobayashi pseudo-metric is associated. It is a true distance only when ${X}$ contains no rational curves. Holomorphic maps between complex spaces give rise to 1-Lipschitz maps. This goes back to Schwarz-Pick’s Lemma.

Theorem 28 (Wolff, Denjoy 1926) Given a holomorphic function ${f:D\rightarrow D}$. Then either there is a fixed point in ${D}$, or iterates converge to a boundary point.

In this case, ${\tau=\inf_{z\in D}d(z,f(z))}$.

6.10. Surface homeomorphisms

This appears in a manuscript of Thurston in 1976 (appeared in Bull. Amer. Math. Soc. in the 1980’s). Thurston calls this a spectral theorem, or Jordan normal form.

Let ${\Sigma}$ be a closed hyperbolic surface. For a closed curve ${\alpha}$, denote by ${\ell(\alpha)}$ the length of the closed geodesic freely homotopic to ${\alpha}$.

Theorem 29 Given a homeomorphism ${f:\Sigma\rightarrow\Sigma}$, there exist numbers ${\lambda_1\ge\cdots\ge\lambda_s}$ such that for every simple closed curve ${\alpha}$ on ${\Sigma}$,

$\displaystyle \begin{array}{rcl} \exists i\quad\textrm{such that}\quad \ell(f^n \alpha)^{1/n}\rightarrow \lambda_i. \end{array}$

Let ${X}$ denote Teichmüller space. In analogy with the norm of operators, Thurston defines an asymetric metric on ${X}$ as follows.

$\displaystyle \begin{array}{rcl} d_{Th}(x,y)=\log \sup_\alpha \frac{\ell_y(\alpha)}{\ell_x(\alpha)}. \end{array}$

\subsubsection{Back to Hilbert space}

Let ${G}$ denote invertible linear operators on Hilbert space ${H}$. Let ${Sym}$ denote symmetric ones. Let ${Pos}$ denote the positive ones. ${G}$ acts on ${Pos}$, and there is a ${G}$-invariant Finsler metric, defined by following norm at point ${p}$,

$\displaystyle \begin{array}{rcl} \forall X\in Sym,\quad |X|_p=|p^{-1/2}Xp^{1/2}|. \end{array}$

Theorem 30 (Carach-Porta-Recht 1993) ${Pos}$ is complete and Busemann with respect to Finsler geodesics (those which satisfy the ODE for geodesics).

Busemann property is equivalent to Segal’s inequality

$\displaystyle \begin{array}{rcl} |e^{X+Y}|\leq|e^{X/2}e^Ye^{X/2}. \end{array}$

The Finsler exponential maps coincides with the operator exponential, therefore ${d(1,p)=|\log p|}$. Thus if ${p_n=g^n.1=g^n(g^*)^n}$, then

$\displaystyle \begin{array}{rcl} \lim \frac{1}{n}|\log p_n|=\tau. \end{array}$

A similar argument shows that there exists a unit norm linear functional ${F}$ on bounded operators such that

$\displaystyle \begin{array}{rcl} \lim \frac{1}{n}F(\log p_n)=\tau. \end{array}$

6.11. Ergodic theorems

This began with Bernoulli’s law of large numbers. Let ${X_i}$ be independent, identically distributed integrable random variables. Then

$\displaystyle \begin{array}{rcl} \frac{1}{n}(X_1+\ldots+X_n)\rightarrow \mathop{\mathbb E}(X_1). \end{array}$

Replace real valued variables with ${G}$-valued ones, ${G}$ a group, and multiply instead of adding. This gives rise to random walks. Next replace a group with the semi-group of 1-Lipschitz maps of a metric space ${X}$. Replace independence by the following setting

Theorem 31 (Karlsson-Ledrappier 2006 for isometries, Gouezel-Karlsson 2015) Let ${T:\Omega\rightarrow\Omega}$ be a probability measure preserving transformation. Let ${\phi:\Omega\rightarrow 1-Lip(X)}$. Assume that for all ${x\in X}$, ${d(\phi(x),x)}$ is integrable. Set

$\displaystyle \begin{array}{rcl} u_n(\omega)=\phi(\omega)\circ\phi(T\omega)\circ\cdots\circ\phi(T^{n-1}\omega). \end{array}$

There is a random metric functional ${h=h(\omega)}$ such that almost surely,

$\displaystyle \begin{array}{rcl} \lim-\frac{1}{n}h(u_nx)=\tau=\lim\frac{1}{n}d(x,u_nx). \end{array}$

The main step in the proof is the following rather hard lemma.

Lemma 32 (Gouezel) Let

$\displaystyle \begin{array}{rcl} a(n,\omega)=d(n,u_n(\omega)x). \end{array}$

Almost surely, there exists a subsequence ${n_i}$ and numbers ${\delta_m}$ tending to 0 such that for all ${m,

$\displaystyle \begin{array}{rcl} |a(n_i,\omega)-a(n_i-m,T^n\omega)-m\tau|\leq m\delta_m. \end{array}$

\subsubsection{Random walks on groups}

Let ${\Gamma}$ be a finitely generated group of subexponential growth, with no homomorphisms to ${{\mathbb Z}}$. For any random walk on ${\Gamma}$, distance to the origin growth sublinearly.

\subsubsection{Random walks in ${CAT(0)}$ spaces}

Let ${X}$ be a ${CAT(0)}$ metric space. Our theorem implies that there exists a unique random geodesic ray ${\gamma}$ from ${o}$ such that the random walk stays sublinearly close to ${\gamma}$,

$\displaystyle \begin{array}{rcl} \frac{1}{n}d(u_no,o,\gamma(\tau n)\rightarrow 0. \end{array}$

When applied to ${X=Pos_n}$, we get Oseledets’ theorem.

1. Oseledets’ theorem has some infinite dimensional version: in the 1980’s, Ruelle stated a version for operators of the form ${1+}$ compact. Possibly, our nonlinear version might have such an extension and apply to PDE’s for instance.

2. Further potential applications ? Furstenberg’s work on random walks plus Mostow’s rigidity theorem inspired Margulis work on superrigidity. The starting point of Margulis’ arguments was Oseledets’ theorem.

7. Alessandra Iozzi: Bounded cohomology, boundary maps, and the Roller boundary

Joint work with Indira Chatterji and Talia Fernos.

7.1. The result

Let ${X}$ be a finite dimensional ${CAT(0)}$ cube complex. For each ${n\ge 2}$, there is a cohomology class, the median class

$\displaystyle \begin{array}{rcl} m_n\in H_b^2(Aut(X),\mathcal{E}^n), \end{array}$

where ${\mathcal{E}^n}$ is a Banach space to be defined below. Let ${\Gamma}$ be a group and ${\rho:\Gamma\rightarrow Aut(X)}$ be a homomorphism.

Theorem 33 (Chatterji-Fernos-Iozzi) If the action ${\rho}$ is nonelementary, then ${\rho^*m_n\not=0}$.

If ${\rho}$ is elementary (i.e. there is a finite orbit in ${\partial X}$), either ${\rho^*m_n=0}$ or there is a finite index subgroup ${\Gamma'<\Gamma}$ and a ${Gamma'}$-invariant ${CAT(0)}$ subcomplex ${X'}$ of lower dimension, on which the action is nonelementary, and ${(\rho^*m_n)_{|\Gamma'}=(\rho_{|\Gamma'})^*m_n\not=0}$.

Similar results arise in Hamenstädt, Bestvina-Bromberg-Fujiwara, Hull-Osin. Thus we see that bounded cohomology of ${\Gamma}$ does not vanish. Furthermore, the obtained cohomology class is rather handy.

Corollary 34 Let ${X}$ be an irreducible finite dimensional ${CAT(0)}$ cube complex. Let ${\Gamma}$ be an irreducible lattice in the product ${G}$ of (at least two) locally compact groups. Let ${\rho:\Gamma\rightarrow Aut(X)}$ be an essential and nonelementary action. Then ${\rho}$ extends continuously to ${G}$, factoring through one of the factors.

Corollary 35 Let ${\Gamma}$ be an irreducible lattice in a higher rank semisimple Lie group, then any action of ${\Gamma}$ on a finite dimensional ${CAT(0)}$ cube complex has a fixed point.

This requires Caprace’s description of stabilizers in ${Aut(X)}$ of finite subsets of ${\partial X}$.

7.2. Roller boundary

Cube complexes are spaces with walls. Indeed, each cube has mediating hyperplanes, which propagate into subsets called hyperplanes. Each hyperplane is a ${CAT(0)}$ cube complex on its own right. It separate ${X}$ into two half-spaces.

On the set of half-spaces, consider the following family of utrafilters ${\alpha}$. ${\alpha}$ selects one of the two half-spaces for each hyperplane. If ${h\in\alpha}$ and ${h\subset h'}$, then ${h'\in\alpha}$. We neglect principal ultrafilters (consisting of all hyperplanes containing a fixed one). The others constitute the Roller boundary ${\partial X}$. Ultrafilters in which no descending chain has a minimal element are called nonterminating. They form a proper subset ${\partial_{NT}X}$ of ${\partial X}$.

Nevo-Sageev: if ${X}$ is locally finite and has a cocompact isometry group, then ${\partial_{NT}X\not=\emptyset}$. If ${\Gamma}$ is a cocompact isometry group with a nonelementary action on ${X}$, then its closure is a minimal strongly proximal ${\Gamma}$-space.

Theorem 36 (Chatterji-Fernos-Iozzi) Let ${X}$ be a finite dimensional ${CAT(0)}$ cube complex, with a nonelementary action of ${\Gamma}$. Let ${(B,\theta)}$ be a strong ${\Gamma}$-boundary (i.e. ${\Gamma}$ acts on ${B}$ amenably and doubly ergodically with coefficients). Then there is a ${\Gamma}$-equivariant measurable map ${B\rightarrow \partial_{NT}X}$.

According to Nicolas Monod, bounded cohomology classes translate into functions on a strong boundary (no quotient any more). Combined with Monod’s theory, the above Theorem allows to express ${\rho^*m_n}$ as a rather explicit function on ${\partial_{NT}X}$ and check that it does not vanish.

7.3. Proof

Furstenberg’s lemma provides us with a map ${\phi:B\rightarrow \mathcal{M}(\bar{X})}$, ${\bar{X}=X\cup\partial X}$. A measure on ${\bar{X}}$ that gives different measures two half-spaces with the same boundary hyperplane defines an ultrafilter, and thus a point in ${\partial X}$. Let us show that this happens almost everywhere along ${\phi(B)}$. When ${X}$ is a tree, hyperplanes are points. If things go bad for some measure ${\mu}$, the bad points form a connected set where every vertex has valency at most 2, therefore a point, a segment, a ray or a line. We get an equivariant map ${B\rightarrow X}$ or to similar spaces. Only the last case (lines) is compatible with double ergodicity. Double ergodicity implies that any two lines intersect or any two lines do not intersect….

8. Stéphane Lamy: On the Cremona group acting on infinite hyperbolic space

I will develop analogies between the Cremona group and ${SL(2,{\mathbb Z})}$ on one hand and mapping class group on the other hand.

8.1. Basics

The Cremona group ${Bir(P^2)}$ is the group of birational transformations of projective plane. Elliptics of ${PGL(3)}$ or ${PGL(2)\times PGL(2)}$ can be considered as elliptic elements of ${Bir}$. Jonquieres maps

$\displaystyle \begin{array}{rcl} (x,y)\mapsto (x,\frac{a(x)y+b(x)}{c(x)y+d(x)}) \end{array}$

can be considered as parabolic elements of ${Bir}$.

${SL(2 ,{\mathbb Z})}$ embeds into ${Bir}$ as follows,

$\displaystyle \begin{array}{rcl} \begin{pmatrix} a & b \\ c & d \end{pmatrix}\mapsto ((x,y)\mapsto (x^a y^b,x^c y^d)). \end{array}$

8.2. Action on infinite dimensional hyperbolic space

In analogy with the mapping class group action on Teichmüller space, one introduces an action of ${Bir}$ on an infinite dimensional hyperbolic space ${\mathbb{H}^{\infty}}$.

In general, a birational map is undefined at finitely many points. Any birational map in 2 dimensions is a composition of blow-ups (Zariski). On the space of divisors, an intersection form is defined. It is an integer valued quadratic form. For instance, if ${E}$ is the exceptional divisor arising from a blow up, ${E\cdot E=-1}$. On ${P^2}$, the space of divisors is 1-dimensional (generated by lines), with positive intersection form ${L\cdot L=1}$. Blowing up ${n}$ points produces an ${n+1}$-dimensional space of divisors, with an intersection form of signature ${(1,n)}$. Blow-up maps induce isometric embeddings of quadratic spaces. Let ${\mathbb{H}^{\infty}_c}$ be the unit sphere in the inductive limit of these quadratic spaces and maps. It is not complete (it consists of vectors with only finitely nonzero entries), so take its ${\ell^2}$ completion, ${\mathbb{H}^{\infty}}$.

Compositions of blow-ups act isometrically on ${\mathbb{H}^{\infty}}$.

8.3. Dynamical degree

This is the analgue of spectral radius (for ${SL(2,{\mathbb Z})}$) or stretch factor (for pseudo-Anosov surface homeomorphisms).

The naive degree of a birational map is not conjugacy invariant, but the limit

$\displaystyle \begin{array}{rcl} \lambda(g)=\lim_{n\rightarrow\infty}(\mathrm{deg}(g^n))^{1/n} \end{array}$

is. It is always ${\geq 1}$.

Example. If ${g(x,y)=(y,y^n+x)}$, then ${\lambda(g)=n}$.

If ${h(x,y)=(x^a y^b,x^c y^d)}$, then ${\lambda(h)}$ is the spectral radius of the matrix.

Theorem 37 (Fabre-Diller, Gizatullin) If ${g\in Bir}$ and ${\lambda(g)=1}$, then

1. either the sequence ${(\mathrm{deg}(g^n))}$ is bounded, and there exists ${n\in{\mathbb N}}$ such that ${g^n}$ in an automorphism.
2. either ${(\mathrm{deg}(g^n))\sim c\,n}$, and ${g}$ is Jonquières,
3. or ${(\mathrm{deg}(g^n))\sim c\,n^2}$, and ${g}$ preserves an elliptic fibration.

8.4. Tits alternative

Theorem 38 (Cantat, 2010) If ${G}$ is a finitely generated subgroup of ${Bir}$, then either ${G}$ contains a free group, or ${G}$ is virtually solvable.

This uses essentially the action on ${\mathbb{H}^{\infty}}$.

Question. Over ${\mathbb{C}}$, is it necessary to assume that ${G}$ is finitely generated ?

8.5. Non simplicity

Theorem 39 (Cantat-Lamy, 2013, Lonjou 2015 for arbitrary fields) For general ${g\in Bir}$ and ${n}$ large enough, the normal subgroup generated by ${g^n}$ is a proper subgroup.

This uses ideas from hyperbolic groups.

8.6. What next ?

In 1992, Wright introduced an action of ${Bir}$ on a 2-dimensional simplicial complex. Is this complex hyperbolic ?

What happens in dimension 3 ?

9. Jean Lécureux: Non-linearity of groups acting on exotic affine buildings

Joint work with Uri Bader et Pierre-Emmanuel Caprace.

10. ${\tilde{A}_2}$-buildings

These are 2-dimensional simplicial complexes whose links are 1-dimensional finite spherical buildings. There is Bruhat-Tits one, associated to ${SL(3,k)}$, ${k}$ a discrete valuation field, but there are many others. Especially, some of them have cocompact lattices (Cartwright-Mantero-Steger-Zappa). These lattices have Kazhdan’s property T (Pansu, Zuk), they act amenably on their visual boundary (Robertson-Steger).

Theorem 40 Let ${\Gamma}$ act properly dscontinuously, cocompactly on an ${\tilde{A}_2}$-building ${X}$ which is not a Bruhat-Tits building. Then ${\Gamma}$ is not linear: any homomorphism to ${GL(n,K)}$ (any field) has finite image.

The proof is inspired by Margulis superrigidity.

10.1. Reduction to a local field

A trick due to Jacques Tits (in his proof of Tits alternative) allows to transform an infinite representation in ${GL(n,K)}$ into an unbounded homomorphism to ${GL(n,k)}$, ${k}$ local. Property T allows to assume that the Zariski closure is semisimple.

10.2. Boundary

It is a spherical building, in fact a projective plane. Therefore, singular points split into two colors, which we call lines and points.

Two asymptotic rays ultimately belong to the same Weyl chamber, where they are parallel. This defines a pseudo-distance on the set ${X_\xi}$ of rays in a boundary point ${\xi}$. This set is a segment (for regular points) and a regular tree (for singular points).

For opposite boundary points (i.e. joined by a line), there is a canonical isometry ${X_{\xi'}\rightarrow X_{\xi}}$.

10.3. Projectivity group

Consider a chain ${\xi_0,\xi_1,\ldots,\xi_4=\xi_0}$ of boundary points, each of which is opposite to the previous one. Compose canonical isometries and get an automorphism of the tree ${T=X_{\xi_0}}$. The group generated by such maps is called the projectivity group ${P}$ of ${X}$.

Fact. ${P}$ is 2-transitive on ${\partial T}$.

Theorem 41 Assume that ${X}$ is not a Bruhat-Tits building. Then ${P}$ is not linear (no faithful continuous representation on any ${GL(d,K)}$).

This follows from the work of several people on projective planes, going back to the 1970’s.

So we aim at producing a faithful representation of ${P}$ from one of ${\Gamma}$.

Let ${\Gamma}$ be a countable group, with a Zariski dense unbounded representation into some ${G(k)}$, ${G}$ simple algebraic group. Let ${Y}$ be a measure space with a measure-class preserving, ergodic action of ${\Gamma}$. Bader-Furman show that if another group ${H}$ acts on ${Y}$, commuting with the action of ${\Gamma}$, then there is a natural representation ${H\rightarrow GL(d,k)}$.

We need to show that this representation is faithful. Naturality will help, as well as

10.5. Ergodicity of the geodesic flow

Let ${\mathcal{G}}$ denote the space of isometric maps ${{\mathbb R}\rightarrow X}$. ${{\mathbb R}}$ acts by precomposition. The geodesic flow is the ${{\mathbb R}}$-action on ${\mathcal{G}/\Gamma}$. This does not quite fit as a space ${Y}$, because of the decomposition into regular and singular geodesics, and the occurrence of parallel lines, which we would like to identify. So we introduce

$\displaystyle \begin{array}{rcl} \mathcal{S}=\{\gamma:{\mathbb R}\times T\rightarrow X\}. \end{array}$

The quotient space ${{\mathbb R}\setminus\mathcal{S}}$ has commuting actions of ${\Gamma}$ and ${Aut(T)}$. One must put a measure on it in order that the ${{\mathbb R}}$-action on ${\mathcal{S}/\Gamma}$ be ergodic.

Look at the classical (Bruhat-Tits) example. ${\mathcal{S}=G/S}$ where ${S}$ is the diagonal subgroup. Ergodicity follows from Howe-Moore.

Here, ergodicity will arise from the Hopf argument.

11. Anne Parreau: Introduction to real Euclidean buildings

${{\mathbb R}}$-buildings are to buildings what ${{\mathbb R}}$-trees are to trees: buildings which may branch everywhere.

Bibliography : Tits 1986, Kleiner-Leeb 1993, Parreau 2000, Rousseau 2009, Kramer 2012, Bennett-Schwer-Struyve 2013.

11.1. Apartments

The model apartment ${\mathbb{A}}$ is a finite dimensional real vectorspace equipped with a finite linear reflection group ${W}$. Basic example is

$\displaystyle \mathbb{A}={\mathbb R}^n/{\mathbb R}(1,\ldots,1)=\{\alpha\in{\mathbb R}^n\,;\,\sum\alpha_i=0\}, \quad W=Sym_n,$

(called apartment of type ${A_{n-1}}$). The hyperplanes fixed by reflections of ${W}$ are called walls. They split ${\mathbb{A}}$ into polyhedral cones called Weyl chambers. For instance,

$\displaystyle \begin{array}{rcl} \mathbb{C}=\{\alpha_1>\alpha_2>\cdots>\alpha_n\}. \end{array}$

Roots are linear functionals ${\phi_{ij}:\alpha\mapsto \alpha_i-\alpha_j}$. Those which vanish on walls bounding ${\mathbb{C}}$ (i.e. ${j=i+1}$) are called simple roots. Since the closure ${\bar{\mathbb{C}}}$ is a fundamental domain for ${W}$, there is a projection ${\Theta:\mathbb{A}\rightarrow\bar{\mathbb{C}}}$.

By extension, all translates of walls, Weyl chambers will be called walls, Weyl chambers as well.

An affine model adds the data of a subgroup ${T<\mathbb{A}}$ of translations. The affine reflection group is ${W_{aff}=W\times T}$.

11.2. Buildings

An atlas modelled on ${(\mathbb{A},W_{aff})}$ is a collection of injections ${\mathbb{A}\rightarrow X}$ into a set ${X}$, such that

1. Invariance under precomposition by ${W_{aff}}$.
2. Transition maps are in ${W_{aff}}$.

The apartments, Weyl chambers,… of ${X}$ are the images of injections, of Weyl chambers… of ${\mathbb{A}}$. A sub-chamber of a Weyl chamber ${C}$ means a Weyl chamber contained in ${C}$. The germ of a Weyl chamber is a neighborhood of the base point.

The atlas is called an ${{\mathbb R}}$-building provided the following two extra properties hold.

1. Any two germs of Weyl chambers in ${X}$ lie in a common apartment.
2. Any two Weyl chambers admit sub-chambers that lie in a common apartment.

11.3. Examples

1. Model apartment themselves.
2. Real trees with extendible geodesics. There, ${\mathbb{A}={\mathbb R}}$ with ${W=\pm 1}$. The atlas consists of all geodesics.
3. Products of ${{\mathbb R}}$-buildings are ${{\mathbb R}}$-buildings.
4. Bruhat-Tits buildings associated to reductive algebraic groups over ultrametric fields.
5. Asymptotic cones of symmetric spaces are ${{\mathbb R}}$-buildings (Kleiner-Leeb).

11.4. The space of ultrametric norms

I will describe the Bruhat-Tits buildings associated to ${SL(n,\mathbb{K})}$.

Let ${\mathbb{K}}$ be a field equipped with an absolute value satisfying the ultrametric triangle inequality

$\displaystyle \begin{array}{rcl} |x+y|\le\max\{|x|,|y|\}. \end{array}$

For instance, the ${p}$-adic absolute value on ${\mathbb{Q}}$. It takes a discrete set of values. We are also interested in nondiscrete absolute values. Here is an example. Fix an arbitrary additive subgroup ${\Lambda<{\mathbb R}}$. ${\Lambda}$-nomials are finite sums ${P(X)=\sum_{\lambda\in\Lambda}a_\lambda X^\lambda}$. Set

$\displaystyle \begin{array}{rcl} v(P)=\min Supp(a),\quad |P|=e^{-v(P)}. \end{array}$

This is an ultrametric absolute value on the field ${\mathbb{K} =k(X^\lambda;\,\lambda\in\Lambda)}$ of ${\Lambda}$-nomials.

Let ${V}$ be a finite dimensional ${\mathbb{K}}$-vectorspace. A norm on ${V}$ is a nonnegative function ${\eta}$ on ${V}$ such that

1. ${\eta(av)=|a|\eta(v)}$.
2. ${\eta(v)=0\Leftrightarrow v=0}$.
3. ${\eta(u+v)\le\max\{\eta(u),\eta(v)\}}$.

Say ${\eta}$ is splittable if there exists a basis ${(e_i)}$ of ${V}$ and numbers ${\alpha_i}$ such that

$\displaystyle \begin{array}{rcl} \eta(\sum x_i e_i)=\max e^{-\alpha_i}|x_i|. \end{array}$

Set ${X=\{}$splittable norms${\}/}$homotheties. We shall equip it with the structure of a building.

Remark. For certain fields (maximally complete fields), all norms are splittable.

11.5. Apartments

More generally, say that a norm ${\eta}$ splits over a decomposition ${V=v_1\oplus\cdots\oplus V_k}$ if it is the max of norms on summands.

Fix a decomposition ${V=v_1\oplus\cdots\oplus V_n}$ with ${\mathrm{dim}(V_i)=1}$ (call this a frame). Fix a norm ${|\cdot|_i}$ on each ${V_i}$. Let ${\mathbb{A}}$ denote the model apartment of type ${A_{n-1}}$. The map

$\displaystyle \begin{array}{rcl} f:\mathbb{A}\rightarrow X,\quad \alpha\mapsto \max e^{-\alpha_i}|\cdot|_i \end{array}$

will be an apartment in ${X}$.

Let ${\Lambda=-\log|\mathbb{K}|}$ be the group of values of the absolute value on ${\mathbb{K}}$. Let ${T=\Lambda^n/{\mathbb R}(1,\ldots,1)}$ be the corresponding group of translations of ${\mathbb{A}}$.

Fact. In this way, we get a building structure on ${X}$, with an action of ${GL(V)}$.

Examples.

• The stabilizer of the point ${\eta(x_1,\ldots,x_n)=\max\{|x_i|\}}$ is ${GL(n,\mathcal{O})}$, where ${\mathcal{O}}$ is the ring of numbers of absolute value ${\leq 1}$.
• Diagonal matrices stabilize the apartment corresponding to the standard frame, they act on it by translations.
• For ${g\in GL(V)}$, ${g=\mathrm{diag}(e^{-\alpha_1},\ldots,e^{-\alpha_n}}$, the fixed point set of ${g}$ is

$\displaystyle \begin{array}{rcl} F=\bigcap_{ij}\{\lambda\in\mathbb{A}\,;\,\lambda_i-\lambda_j\leq\alpha_i-\alpha_j\}. \end{array}$

It follows that, more generally, the intersection of two apartments is a Weyl-convex subset (i.e. an intersection of half-spaces bounded by walls). This is sufficient to verify all axioms of buildings.

11.6. The ${CAT(0)}$ metric

Since changes of charts are isometries, Euclidean metrics on apartments piece together into a well defined two-point function ${d}$ on ${X}$. Triangle inequality and ${CAT(0)}$ property need be proved.

11.7. The retraction

A building can be folded onto a single apartment.

Proposition 42 Let ${X}$ be a building, ${A\subset X}$ an apartment. Fix a point ${o\in A}$. There is a unique distance nonincreasing retraction ${r:X\rightarrow A}$ such that ${r^{-1}(x)=\{x\}}$.

Indeed, fix a germ ${C\subset A}$ of Weyl chamber at ${x}$. Given ${y\in X}$, there is an apartment ${A'}$ containing ${y}$ and ${C}$, and a unique isometry ${A'\rightarrow A}$ fixing ${C}$. It maps ${y}$ to ${r(y)}$.

By construction, ${r}$ maps germs of Weyl chambers to germs of Weyl chambers.

11.8. The ${\mathbb{C}}$-distance

It is a refinement of ${d}$, which takes its values in the closure of the model Weyl chamber ${\bar{\mathbb{C}}}$. Given ${x}$ and ${y\in X}$, pick an apartment ${f:\mathbb{A}\rightarrow X}$ containing both and set

$\displaystyle \begin{array}{rcl} d^{\mathbb{C}}(x,y)=\Theta(f^{-1}(y)-f^{-1}(x)). \end{array}$

The model apartment ${\mathbb{A}}$ has an order, where ${\mathbb{A}_{\geq 0}}$ is the cone dual to ${\mathbb{C}}$. Note that the folding map ${\Theta}$ is subadditive:

$\displaystyle \begin{array}{rcl} \Theta(u+v)\leq_{\mathbb{A}}\Theta(u)+\Theta(v). \end{array}$

This implies

Theorem 43 (Triangle inequality for ${d^{\mathbb{C}}}$, Lidskii 1950)

$\displaystyle \begin{array}{rcl} d^{\mathbb{C}}(x,z)\leq_{\mathbb{A}}d^{\mathbb{C}}(x,y)+d^{\mathbb{C}}(y,z). \end{array}$

Equality holds if and only if ${x}$ and ${y}$ belong to opposite Weyl chambers at ${y}$.

11.9. Finsler metrics

Any ${W}$-invariant norm ${N}$ on ${\mathbb{A}}$ gives rise to a metric ${d_N}$ on ${X}$ (this follows from the triangle inequality for ${d^{\mathbb{C}}}$ (and the fact that ${N}$ is nondecreasing on ${\bar{\mathbb{C}}}$).

One can speak of ${\mathbb{C}}$-geodesics: maps of a totally ordered set to ${X}$ which satisfy equality in the triangle inequality for ${d^{\mathbb{C}}}$. It means that simple root coordinates are nondecreasing. Alternatively, ${\mathbb{C}}$-geodesics coincide with the geodesics of the Finsler metric which is linear on ${\mathbb{C}}$.

11.10. Weak convexity

A subset ${Y\subset X}$ is weakly convex if for every ${x,y\in Y}$, at least one ${\mathbb{C}}$-geodesic from ${x}$ to ${y}$ is contained in ${Y}$.

This allows to speak of weak convex cocompactness for isometric group actions on ${X}$. Interesting examples exist: actions of surface groups on ${X(\mathbb{K}^3}$ arising from cubic differentials. To describe them, one passes via projective plane ${P^2(\mathbb{K})}$, which is the visual boundary of ${X}$ in its ${CAT(0)}$ metric.

11.11. Visual boundary

The building is equipped with its ${CAT(0)}$ metric. We mean the set of equivalence classes of geodesic rays, equipped with Tits’ angle metric.

Proposition 44 Let ${X}$ be a building modelled on ${(\mathbb{A},W)}$. The visual boundary ${\partial X}$ is a spherical building modelled on ${(\partial\mathbb{A},W)}$. Its apartments are visual boundaries of apartments of ${X}$ (tiled spheres), its Weyl chambers are boundaries of Weyl chambers of ${X}$.

Example. If ${\mathbb{A}}$ is of type ${A_{n-1}}$, any corresponding spherical building is the incidence pattern of some ${n-1}$-dimensional projective space.

For ${X(\mathbb{K}^3)}$, the space of splittable norms on ${\mathbb{K}^3}$, ${\mathbb{A}}$ has two types of walls. Type 1 correspond to lines in ${\mathbb{K}^3}$, i.e. points in ${P^1(\mathbb{K})}$ (call this type “point”), type 2 to planes in ${\mathbb{K}^3}$, i.e. lines in ${P^1(\mathbb{K})}$ (call this type “line”). A wall of type 1 is a geodesic ray, therefore it contributes a point in the visual boundary. Therefore, singular points of the boundary fall into two disjoint subsets which are copies of ${P^1(\mathbb{K})}$. Regular points fall into interiors of Weyl chambers. A closed Weyl chamber at infinity is an arc of circle (a sixth of a circle) bounded by two singular points, one of each type. A “point” and a “line” bound a Weyl chamber if and only if they are incident in ${P^1(\mathbb{K})}$.

11.12. Cross-ratios

They are defined on the transverse spaces ${X_\xi}$ (see Caprace and Lecureux’s lectures). I describe them only in the example ${X(\mathbb{K}^3)}$.

If ${\xi}$ is a singular boundary point, ${X_\xi}$ is a tree. Its ideal boundary identifies with ${P^1(\mathbb{K})}$. Therefore 4 points of ${X_\xi}$ have an algebraic cross-ratio ${b\in\mathbb{K}}$. They also have a geometric cross-ratio: in the tree, project ${p_2}$ and ${p_4}$ onto the line joining ${p_1}$ to${p_3}$ to ${q_2}$ and ${q_4}$, and set ${c=d(q_2,q_4)}$. Then

$\displaystyle \begin{array}{rcl} c=\log|b|. \end{array}$

12. Bernhard Leeb: Finsler bordifications of symmetric spaces

Joint work with Misha Kapovich and Joan Porti.

To understand the geometry of a symmetric space, or the dynamics of a (infinite covolume) group of isometries, it turns out that a Finsler viewpoint is useful. We learned this when we found a higher rank version of Morse Lemma: regular quasigeodesics are approximated by Finsler geodesics. I will not develop Morse Lemma, but discuss boundaries instead.

12.1. Horoboundary

Let us focus on the horoboundary construction. Embed an arbitrary metric space ${Y}$ in ${C(Y)}$ mod additive constants. If ${Y}$ is proper, one gets a compactification. For a ${CAT(0)}$ space, one gets the visual compactification.

Assume that ${X}$ is a symmetric space in its Riemannian metric. In the boundary, each Weyl chamber is a cross section for the ${G}$ action. Away from singular directions, boundary is a product Weyl chamber times Furstenberg boundary. I.e. many orbits have the same stabilizer.

In order to get rid of this redundancy, we shall switch to a Finsler metric. We pick a polyhedral Finsler metric on a reference maximal flat which is invariant under the Weyl group, and move it around with ${G}$ action. The simplest example consists of picking an affine hyperplane that cuts a compact neighborhood of the vertex in a Weyl chamber.

What horofunction boundary does one get ? It turns out to be independant of the choice of hyperplane.

Theorem 45

1. The ${G}$ action on the Finsler compactification has finitely many orbits, corresponding to faces of the reference spherical Weyl chamber. The smallest one (it is in the closure of any other orbit) is ${G/B}$, the Furstenberg boundary, the largest is ${X}$ itself.
2. The stratification by orbits is a manifold with corners structure.
3. The Finsler compactification is homeomorphic to a ball. There is a (non-canonical) ${K}$-equivariant homeo with the unit ball of the dual Finsler metric.
4. It coincides with the maximal Satake compactification of ${X}$, known to be real analytic.

12.2. Application to discrete subgroups

Let ${\Gamma be a discrete, weakly uniformly regular subgroup. We would like to compactify ${\Gamma\setminus X}$ by attaching quotients of domains of proper discontinuity at infinity. We need to understand the action of ${\Gamma}$ at infinity.

The most chaotic part of the action, the limit set, is defined to live in the Furstenberg boundary. Remove a thickening of the limit set: this is the union, over all points ${\sigma}$ of the limit set, of all Finsler boundary points ${h}$ such that ${\sigma}$ is contained in ${\{h\le const.\}}$. Denote by ${\Omega}$ its complement.

Theorem 46 Let ${\Gamma be a discrete, ${\tau_{mod}}$-regular subgroup.

1. The ${\Gamma}$ action on ${X\cup \Omega}$ is properly continuous. ${(X\cup \Omega)/\Gamma}$ is an orbifold with corners.
2. This bordification is a compactification if and only if ${\Gamma}$ is RCA (regular, conical and antipodal).

RCA generalizes one of the equivalent definitions of convex cocompactness in rank 1. It implies that ${\Gamma}$ is Gromov hyperbolic.

• Antipodal: any two facets in the limit set are opposite.
• Conical (see Albuquerque): every point in limit set is the limit of a sequence of points from a single orbit that stay a bounded distance away from some Weyl chamber.

12.3. Application to convergence actions

Let ${\Gamma be ${\tau_{mod}}$-RCA. Then ${\Gamma}$ is Gromov hyperbolic, and limit set ${\Lambda}$ is equivariantly homeomorphic to ideal boundary.

In the thickening of the limit set, collapse each neighborhood of ${\sigma}$ to ${\sigma}$. This leads to a quotient space

$\displaystyle \begin{array}{rcl} \Sigma=\Omega\cup\Lambda \end{array}$

Then the ${\Gamma}$ action on ${\Sigma}$ is a convergence action. We shows that ${\Gamma}$ is cocompact on ${\Omega}$. This answers a question of Peter Haissinski.

Remark. A very recent post of Gueritaud, Guichard, Kassel and Wienhard addresses the question of compactifying orbifolds. Their main results seem to be covered by ours.

12.4. Proof of Theorem 1

First we work within one maximal flat. ${G}$ is replaced with the affine Weyl group.

Riemannian picture. Let a sequence ${x_n}$ tend to infinity along a ray. Resulting horofunction is linear. So horoboundary is visual boundary. Weyl group acts only via its linear part.

Finsler picture. Say dimension is 2, reflection group is ${A_2}$. Two cases.

1. Regular convergence: the sequence moves away from the walls (distance to the walls tends to infinity). Balls converge to half-spaces. Convergence to a unique Finsler boundary point.
2. Singular convergence: the sequence stays a bounded distance away from some wall. Balls converge to the intersection of 2 half spaces. One Finsler boundary point per line parallel to the wall.

It follows that compactification is a hexagon. Translations act nontrivially on it: the vertices are fixed, ${{\mathbb R}^2}$ acts on each side via a different quotient ${{\mathbb R}^2/R}$, where ${R}$ is a singular direction.

13. Alexander Lytchak: Minimal disks and ${CAT(0)}$ spaces

Joint work with Stefan Wenger.

13.1. The result

As an upshot of our work on minimal disks, we get a new characterization of ${CAT(0)}$ spaces, at least in the proper case.

Theorem 47 Let ${X}$ be a proper geodesic space. Then ${X}$ is ${CAT(0)}$ if and only if it satisfies the Euclidean filling inequality: every loop of length ${L}$ bounds a disk of area ${\frac{L^2}{4\pi}}$.

Question. What if ${X}$ is not proper ?

13.2. Previous results

One direction due to Yuri Reshetnyak. For the opposite direction,

• Reshetnyak if ${X}$ is a ${C^2}$ Riemannian disk.
• Busemann, Santalo, Holmes-Thompson if ${X}$ is a normed plane, with Hausdorff area.
• Wenger: if filling function is ${<\frac{L^2}{4\pi}}$, then ${X}$ is a tree.

My impression is that the proof is much easier for Riemannian manifolds or simplicial complexes. The new feature is to handle general metric spaces.

13.3. Outline of proof

The main idea goes back to Gauss (in sooth cases) and Petrunin-Stadler: a minimal disk in a ${CAT(0)}$ space is again ${CAT(0)}$.

Assume some triangle is not thin. Some work is needed to reduce to a Jordan triangle ${T}$. Fill it with a minimizing disk. Thanks to Gauss, one may forget the rest of ${X}$. One would like to reduce to Reshetnyak’s result. The Busemann, Santalo, Holmes-Thompson theorem suggests that tangent planes must be Euclidean. We now that the parametrization ${u:D\rightarrow T}$ is conformal and locally Lipschitz. So we have formulae for the areas of domains ${V}$ and the lengths of (almost every) curve ${\gamma}$,

$\displaystyle \begin{array}{rcl} \mathrm{Area}(u(V))=\int_{V}f^2, \quad \mathrm{length}(u(\gamma))=\int_{\gamma} f, \end{array}$

for some ${L^2}$ function ${f}$.

Let ${B_r}$ denote concentric circles centered at some point of ${D}$. Then isopermietric inequality

$\displaystyle \begin{array}{rcl} \frac{1}{4\pi}(\int_{\partial B_r}f)^2 \ge \int_{B_r}f^2. \end{array}$

This is equivalent to the differential inequality ${\Delta\log f \geq 0}$ (Beckenbach-Rado and Reshetnyak already used this). If ${f}$ where smooth, this would mean that Riemannian metric ${f^2 g_0}$ is nonpositively curved.

13.4. Pitfalls

There are pitfalls in the above train of thought.

Example. Pick a segment in unit disk, decide it to be slightly shorter. Then ${f=1}$ almost everywhere. Nevertheless, the resulting metric space is not ${CAT(0)}$.

Example. Take an arbitrary Jordan curve in the plane, declare it to be a geodesic triangle. This can be done without changing the metric in the interior. Again, ${f=1}$ a.e.

In the first example, the isoperimetric inequality is violated for loops which contain the slit. In the second, by loops containing a piece of the boundary.

13.5. Escaping pitfalls

Let ${Y}$ be the disk equipped with the measurable Riemannian metric ${f^2 g_0}$. The analytical arguments go through and show that ${Y}$ is nonpositively curved and thus ${CAT(0)}$.

Introduce metric on disk defined by ${d(z_1,z_2)=}$ infimal length of images of rectifiable curved joining ${z_1}$ to ${z_2}$. Get a new metric space ${Z}$ with a map ${P:Y\rightarrow Z}$. One need show that ${P}$ is isometric. Map ${P}$ preserves areas and lengths of a.e. curve, but, as preceding examples show, this is not sufficient. Isoperimetric inequality must be used again.

14. Andres Navas: Barycenters on Busemann spaces

14.1. Centers

If ABC is a Euclidean triangle, and all angles are acute, the circumcenter (center of the smallest ball containing ABC) belongs to the triangle.

The barycenter (point that minimizes sum of squares of distances to A, B and C) also belongs to the triangle.

The circumcenter makes sense in complete ${CAT(0)}$ metric spaces (goes back to Chebyshev). The barycenter as well (goes back to Cartan). In fact, a barycenter is associated with any probability measure (under support restrictions).

Both operations are somewhat contracting: circumcenter with respect to diameter, barycenter with respect to Wasserstein distance ${W_1}$.

I investigate wether this generalizes to metric spaces satfisfying weaker convexity properties.

14.2. Busemann spaces

Definition 48 A metric space is Busemann if it is geodesic, complete, and if in any geodesic triangle, the distance between midpoints of two sides is at most a half of the third side.

Let ${X}$ be Busemann. Define barycenter of two points as the midpoint. For 3 points, consider the triangle of midpoints, and iterate. This converges. For 4 points, consider the 4 partial barycenters (omit a point), and iterate. And so on.

The resulting barycenter satisfies

$\displaystyle \begin{array}{rcl} d(bar_n(x_1,\ldots,x_n),bar_n(y_1,\ldots,y_n))\leq \frac{1}{n}\sum d(x_i,y_i). \end{array}$

What about measures ? For combinations of Dirac measures, one is tempted to use previous construction. This does not work. Indeed, ${(x_1,x_2)}$ and ${(x_1,x_1,x_2,x_2)}$ would represent the same measure, but have different barycenters, as exemples in trees show. Nevertheless, this not a bad idea, provided one takes many points.

Lemma 49 (Elementary but nontrivial)

$\displaystyle \begin{array}{rcl} \lim bar(x_1,\ldots,x_n,x_1,\ldots,x_n,\ldots,x_1,\ldots,x_n) \textrm{ exists.} \end{array}$

We take this limit as a definition for ${bar(\frac{1}{n}(\delta_{x_1},\ldots,\delta_{x_n}))}$.

Above inequality still holds. Note that the right hand side, once infimized under permutations of ${y_i}$‘s, is Wasserstein distance. Therefore, by density, one can define

$\displaystyle \begin{array}{rcl} bar:\mathcal{P}^1(X)\rightarrow X. \end{array}$

It is a 1-Lipschitz map.

14.3. Applications

Theorem 50 Compact group actions on Busemann spaces have fixed points.

Exercise. If a group acts on a Busemann space and preserves a compact set, then there is a fixed point.

Warning. There are isometric actions on Busemann spaces with bounded orbits but without fixed points. For instance, let ${\Gamma}$ be an arbitrary countably infinite group. Let ${X=L_0^1(\Gamma)}$ be the affine space of functions whose sum equals 1.

Question. Is there a MCG-invariant Busemann metric on Teichmüller space ?

Theorem 51 Let ${T}$ act on ${\Omega}$, preserving a probability measure. Let ${f:\Omega\rightarrow X}$ be a random variable with values in a Busemann metric space. Then

$\displaystyle \begin{array}{rcl} bar(\frac{1}{n}(\delta_{T f}+\cdots+\delta_{T^n f}) \end{array}$

converges.

14.4. Questions

Metastability: define a new barycenter of 3 points as the barycenter of the measure they define. It is presumably different. Iterate! Does the procedure converge ?

Compare to earlier works by Es Sahib-Heinich 1999, Billera-Holmes-Vogtmann 2001.

15. Pierre Py: Actions of ${PO(n,1)}$ on infinite dimensional symmetric spaces

joint work with Thomas Delzant and Nicolas Monod.

15.1. The examples

This has been known since the 1970’s to representations theorists. Sally 1967, 1970, Johnson-Wallach 1977.

Let ${H}$ be a separable Hilbert space. Fix a Hilbert basis ${(e_i)}$ and define the quadratic form

$\displaystyle \begin{array}{rcl} \forall x=\sim x_i e_i,\quad B(x,x)=x_1^2+\cdots+x_p^2 -\sum_{p+1}^{\infty}x_i^2. \end{array}$

The corresponding Grassmannian

$\displaystyle X(p,\infty)=\{p-\mathrm{dim\,subspaces\,}V\,;\,B_{|V\times V}\mathrm{\,is\,positive\,definite}\}$

is an infinite dimensional Riemannian manifold, whose study was suggested by Gromov, see Duchesne’s thesis.

Let ${s}$ denote the boundary of hyperbolic ${n}$-space. We describe representations of ${G=O(n,1)}$ on ${L^2(S)}$ parametrized by ${s\in\mathbb{C}}$.

$\displaystyle \begin{array}{rcl} \pi_s(g)(f)=f\circ g^{-1}\,|Jac(g^{-1})|^{\frac{1}{2}+s}. \end{array}$

If ${s\in i{\mathbb R}}$, ${\pi_s}$ is unitary.

Theorem 52 Assume ${s>0}$. There exists an intertwining operator

$\displaystyle \begin{array}{rcl} A_s:L^2(S)\rightarrow L^2(S) \quad\textrm{ such that }A_s\circ\pi_s(g)=\pi_{-s} (g)\circ A_s \quad \forall g\in G=O(n,1). \end{array}$

Furthermore, ${A_s}$ is ${K}$-equivariant.

Therefore, the sesquilinear form

$\displaystyle \begin{array}{rcl} B_s(f_1,f_2)=\int f_1 \overline{A_s(f_2)} \end{array}$

is ${\pi_S}$-invariant.

More about ${A_s}$: It preserves the decomposition of ${L^2(S)}$ into ${K}$-irreducibles,

$\displaystyle \begin{array}{rcl} L^2(S)=\bigoplus_{\ell=0}^{\infty}H_\ell, \end{array}$

and is scalar on each of them, ${{A_s}_{|H_{\ell}}=\lambda_{\ell}(s)}$, where

$\displaystyle \begin{array}{rcl} \lambda_{\ell}(s)=\prod_{j=0}^{\ell-1}\frac{j+\frac{n-1}{2}-(n-1)s}{j+\frac{n-1}{2}+(n-1)s}. \end{array}$

When some ${\lambda_\ell(s)}$ vanishes, ${\pi_s}$ is not irreducible. This does not happen if ${s\in(0,\frac{1}{2})}$. If fact, all ${\lambda_\ell>0}$ in this interval, so ${B_s}$ is positive definite.

If instead ${s\in( \frac{1}{2},\frac{1}{2}+\frac{1}{n-1})}$, ${B_s}$ has one minus sign, leading to an action on ${X(1,\infty)}$. And so on…

15.2. Other Lie groups

For ${SU(n,1)}$ or ${Sp(n,1)}$, ${B_s}$ also exists, but sign jumps from ${(0,\infty)}$ to ${(\infty,\infty)}$, so no finite rank examples.

Theorem 53 (Duchesne) Let ${G}$ be a simple Lie group distinct from ${SO(n,1)}$ and ${SU(n,1)}$. Let ${\Gamma be a cocompact lattice. For every irreducible action of ${\Gamma}$ on ${X(p,\infty)}$, there is a ${G}$-equivariant, totally geodesic map.

Question. What about ${PU(n,1)}$ ?

15.3. Rigidity

Say an action on ${X(p,\infty)}$ is geometrically Zariski dense if there is no fixed point at infinity and non invariant closed totally geodesic manifold. Then ${\pi_s}$ is geometrically Zariski dense for ${s\in( \frac{1}{2},\frac{1}{2}+\frac{1}{n-1}}$. What happens for larger ${s}$ is unclear.

Question. Are ${\pi_s}$ the only irreducible actions on ${X(p,\infty)}$ ?

Theorem 54 (Monod-Py) If ${n\geq 5}$ and ${2, then there is no irreducible action of ${PO(n,1)}$ on ${X(p,\infty)}$.

15.4. Proof

Given an action, ${K=SO(n)}$ has a fixed point, hence a representation with an invariant ${p}$-dimensional vectorspace ${V}$. If ${n\geq 5}$, ${SO(n)}$ has no irreducible linear representations of dimension ${\leq n}$ but for the trivial one.

It is a general fact that in any irreducible unitary representation of ${G}$, ${K}$-invariants have dimension 1 (2 for orthogonal representations). We prove an analogous result with ${U(H)}$ replaced with ${U(p,\infty)}$

15.5. Infinite dimensional hyperbolic space

Theorem 55 (Monod-Py) Any irreducible action of ${PO(n,1)}$ on ${X(1,\infty)}$ belongs to the ${\pi_s}$ family.

This uses geometric arguments (fixed points of elliptics…) an a bit of Fourier analysis.

Theorem 56 (Monod-Py) Consider a ${\pi_s}$ action of ${PO(n,1)}$ on ${X(1,\infty)}$. Let ${C_s}$ denote the closed convex hull of the ${G}$-orbit of the ${K}$-fixed point. Then ${C_s}$ is locally compact, ${G}$ is cocompact on it. Its isometry group is precisely ${G}$. Distinct ${s}$ provides nonhomothetic sets.

This illustrates a theorem of Caprace-Monod. They show that geodesically complete ${CAT(0)}$ space with a cocompact action of a simple Lie group has to be the symmetric space. Of course, ${C_s}$ is not geodesically complete.

Question. What happens as ${s}$ tends to ${\frac{1}{2}}$ ?

16. Boundary dinner

Present: Karlsson, Lederle, Leeb, Magnot, Pansu, Parreau, Pozzetti.

16.1. Karlsson’s bordification

Anders Karlsson has explained more examples. In particular, why it coincides with Roller’s bordification for nonlocally finite simplicial trees.

16.2. Compactifications of symmetric spaces and buildings

Anne Parreau has explained her idea of Busemann compactification using the ${\mathbb{C}}$-valued refined distance. This does not involve any choice of metric. The result is homeomorphic to the maximal Satake compactification, as in Bernhard Leeb’s talk.

Anne has also generalized the ${\mathbb{C}}$-valued refined distance to other settings, like Riemann surfaces equipped with holomorphic cubic differentials, this leads to canonical (equivariant) maps of such surfaces into ${\tilde{A}_2}$-buildings. Such differentials arise in higher Teichmüller theory, in connection with representations of surface groups in ${SL(3,{\mathbb R})}$. Pierre Pansu wonders wether a refined distance could be defined for certain ${CAT(0)}$ cube complexes.

16.3. Asymptotic cones of infinite dimensional symmetric spaces

Let ${X(p,q)}$ be the symmetric space of ${SO(p,q)}$ (as in Py’s talk). If ${p=1}$, all ${X(1,\infty)}$ has isometric asymptotic cones, the universal tree ${\mathbb{T}}$ (see Cornulier’s lectures). Is it still true for ${X(1,\infty)}$ ? Cornulier says yes, since it is defined from a separable Hilbert space. And dealing with a nonseparable Hilbert space would not change this.

If ${p>1}$, Maria-Beatrice Pozzetti suggests that the isometry group of the asymptotic cone of ${X(p,q)}$ be exactly ${SO(p,q,{\mathbb R}^*)}$, where ${{\mathbb R}^*}$ is the Robinson field (see Cornulier’s lectures). This should follow from a theorem by Tits. If so, different values of ${q}$ would result in nonisomorphic fields, hence nonisometric cones. Would asymptotic cones of ${X(p,\infty)}$ be distinct from all these ? Maybe Kleiner and Leeb’s result (homeomorphisms between cones arise from isometries between symmetric spaces) can be adapted.

## Notes of Samuel Lelièvre’s Orsay lecture

Groupes aléatoires

avec Moon Duchin, Kasia Jankiewicz, Shelby Kilmer, John Mackay, Andrew Sánchez. Résultat d’un cluster d’undergraduates qui a eu lieu l’an dernier à Tufts, Boston (12 participants, 6 semaines).

1. Le modèle à densité de Gromov

Dans la sphère de rayon ${\ell}$ ${S_\ell}$ du groupe libre ${F_m}$, on tire uniformément et indépendamment au hasard ${n}$ éléments. On considère le sous-groupe distingué ${N}$ qu’ils engendrent, et le groupe quotient ${G=F_m/N}$.

Obtient on des groupes non isomorphes ? Pas clair, il ne s’agit pas du tirage au hasard d’une classe d’isomorphisme de groupe de présentation ${(m,n)}$. On va voir que dans certains régimes, le groupe obtenu est en général trivial !

Pourquoi la sphère ? Ca aide beaucoup. De toutes fa\c cons, dans la boule, la plupart des éléments sont au bord.

Terminologie. On appelle densité du tirage le réel ${d}$ tel que

$\displaystyle \begin{array}{rcl} n=|S_{d\ell}|. \end{array}$

Autrement dit, on tire non pas une fraction ${d}$, mais une puissance ${d}$ du nombre total d’éléments.

Etant donnée une fonction ${n\mapsto N(\ell)}$, on dit qu’une propriété des groupes est asymptotiquement presque s\^ ure si la probabilité qu’elle ait lieu tend vers 1 lorsque, à ${m}$ fixé et ${n=N(\ell)}$, loorsque ${\ell}$ tend vers l’infini.

Comment choisir ${\ell\mapsto N(\ell)}$ ? Il semble que fixer la densité, i.e. prendre ${n=(2m-1)^{d\ell}}$, est un bon choix.

Theorem 1 (Gromov 1993)

1. Si ${d>\frac{1}{2}}$, alors asymptotiquement presque s\^ urement ${G}$ a au plus 2 éléments.
2. Si ${d<\frac{1}{2}}$, alors asymptotiquement presque s\^ urement, ${G}$ est infini, hyperbolique, sans torsion, de dimension 2, et contient des groupes de surfaces.

Il y a d’autres effets de seuil connus,

1. à ${d=\frac{1}{5}}$ propriété de Dehn,
2. à ${d=\frac{1}{5}}$ propriété ${C'(\frac{1}{6})}$.
3. entre ${d=\frac{1}{5}}$ et ${\frac{1}{3}}$, propriété (T) de Kazhdan.

2. Résultats

Theorem 2 (DJKLMS 2015) On s’intéresse à la densité convergeant vers ${\frac{1}{2}}$, ${d(\ell)=\frac{1}{2}-f(\ell)}$.

1. Si ${f(\ell)\leq \frac{\log\ell}{4\ell}-\frac{\log\log\ell}{\ell}}$, alors asymptotiquement presque s\^ urement ${G}$ a au plus 2 éléments.
2. Si ${f(\ell)\geq 10^5 \frac{(\log\ell)^{1/3}}{\ell^{1/3}}}$, alors asymptotiquement presque s\^ urement, ${G}$ est infini, hyperbolique,.

3. Démonstrations

3.1. C\^oté trivial

On s’est appuyés sur des notes de Gady Kozma. Il s’agit d’augmenter la densité effective. Si deux relateurs ont une branche commune longue et des parties restantes courtes, leur différence est un petit relateur. On voit ${G}$ comme un quotient d’un groupe aléatoire avec longueur ${\ell}$ plus petite, et densité plus grande que ${\frac{1}{2}}$, c’est gagné.

Principe des tiroirs probabiliste : si ${\ell\rightarrow\infty}$ et ${n\rightarrow\infty}$ avec ${n=o(\sqrt{\ell})}$, alors asymptotiquement presque s\^ urement, il y a une coïncidence quand on range ${\ell}$ objets dans ${n}$ tiroirs.

On analyse l’influence des lettres les unes sur les autres dans des mots aléatoires. On pose ${\mu=\frac{1}{2m-1}}$,

$\displaystyle \begin{array}{rcl} S_n=\sum_{k=0}^{n-1}(-\mu)^k \rightarrow \frac{1}{1+\mu}. \end{array}$

On écrit un mot aléatoire ${x_0x_1\ldots x_n\ldots}$. Pour ${n}$ pair,

$\displaystyle \begin{array}{rcl} \mathop{\mathbb P}(x_n=x_0)=\mu S_{n-1},\quad \mathop{\mathbb P}(x_n=y\not=x_0)=\mu S_n. \end{array}$

Pour ${n}$ impair,

$\displaystyle \begin{array}{rcl} \mathop{\mathbb P}(x_n=x_0^{-1})=\mu S_{n-1},\quad \mathop{\mathbb P}(x_n=y\not=x_0^{-1})=\mu S_n. \end{array}$

Par conséquent, toutes ces probabilités tendent vers ${\frac{\mu}{1+\mu}=\frac{1}{2m}}$.

Proposition 3 S’il existe une fonction ${\ell\mapsto k(\ell)\leq \ell}$ telle que

1. ${k-2\ell f(\ell)\rightarrow\infty}$,
2. ${\frac{\ell-2}{(2k+2)(2m-1)^{2k}}\rightarrow\infty}$,

alors asymptotiquement presque s\^ urement, ${|G|\leq 2}$.

En effet, avec le principe des tiroirs probabiliste et la première condition, on trouve un mot réduit ${w}$ de longueur ${2k}$ dans ${F_m}$ tel que ${w=1}$ dans ${G}$. On l’utilise pour réduire les autres relateurs. Pour cela, on observe que la queue (de ${k+1}$ à ${\ell}$) d’un mot tiré au hasard est un mot de longueur ${\ell-k}$ tiré au hasard. Le PTP donne deux relateurs aléatoires dont les queues coincident mais qui diffèrent à la ${k}$-ème lettre, on considère leur différence, qui est de longueur ${2k}$. On note ${R_w}$ les mots du tirage qui commencent par ${w}$. Pour ${x,y,z}$ des lettres, ${R_{xz}}$ et ${R_{yz}}$ sont typiquement disjoints et non vides, cela donne une partition en ${2m(2m-1)}$ sous ensembles…

3.2. C\^oté infini hyperbolique

On suit le livre de Yann Ollivier. Il s’agit d’estimer la probabilité qu’il existe un diagramme de van Kampen de taille ${\leq K}$ qui viole une inégalité isopérimétrique quadratique avec petite constante. Pour contr\^oler les effets de dépendance, Ollivier compte des diagrammes abstraits (cellulations du plan) et estime la probabilité qu’un diagramme abstrait de taille ${\leq K}$ soit réalisable par l’ensemble de relateurs tiré au hasard (i.e. qu’on puisse coller des étiquettes aux arêtes de sorte que les mots qu’on lit sur les bords des faces sont des conjugués cycliques des relateurs tirés). On utilise sans changement son estimation de probabilité de réalisation, on n’a besoin de retravailler que son décompte de diagrammes abstraits.

## Notes of David Hume’s Orsay lecture

Expanders and separation

How different can expanders be ?

1. Expanders

Definition 1 An ${\epsilon}$-expander is a sequence of finite graphs ${G_n}$ where – each ${G_n}$ has Cheeger constant ${>\epsilon}$.

A ${(d,\epsilon)}$-expander is a sequence of finite graphs ${G_n}$ where – each ${G_n}$ has maximal degree ${< d}$, – each ${G_n}$ has Cheeger constant ${>\epsilon}$.

Margulis: Cayley graphs of finite quotients of ${Sl(3,{\mathbb Z})}$ (or of any propertyT residually finite group) are an expander.

Lubotzky-Philipps-Sarnak, Lubotzky pursued the finite group line.

Wigderson : zig-zag construction of expanders.

2. Connection to Topology

Borel conjecture : given two closed aspherical manifolds, any homotopy equivalence is homotopic to a homeomorphism.

This is hard. An important related (philosophically weaker) question is the Novikov conjecture, which has partials solutions.

Yu : if a finitely generated group ${G}$ coarsely embeds in Hilbert space, then Novikov conjecture holds for all closed manifolds with fundamental group ${G}$.

Since expanders do not coarsely embed into Hilbert space, Gromov asked wether there exist finitely generated groups that coarsely contain expanders.

Gromov (followed by Coulon and Arzhantseva-Delzant) provided a slightly weaker construction. This was made more precise by

Osajda : there exist expander families with ${C'(1/6)}$ small cancellation labellings.

Idea : graphical small cancellation. Pick finite graphs ${G_n}$ with oriented edges, labelled by a finite set S. Define

$\displaystyle G_I=.$

Then graphical small cancellation theory gives sufficient conditions in order that the disjoint union of ${G_n}$ embeds isometrically in ${G}$.

Theorem 2 There exists a continuum of ${(d,\epsilon)}$-expanders ${G_r}$, ${r}$ real number, such that ${G_r}$ does not coarsely embed into ${G_s}$ unless ${r=s}$.

Theorem 3 There exists a continuum of finitely generated groups ${G_r}$, ${r}$ real number, such that ${G_r}$ does not coarsely embed into ${G_s}$ unless ${r=s}$.

3. Separation

To distinguish expanders, we use separation.

Definition 4 (Benjamini-Schramm-Timar) ${G}$ finite graph. The cut-size of ${G}$ is the smallest ${k}$ such that there exists a subset ${A}$ of vertices of size ${k}$ such that every connected component of the complement of ${A}$ in ${G}$ contains at most half of the vertices of ${G}$.

For an infinite graph ${X}$, the separation profile ${sep_X}$ is the function

$\displaystyle sep_X(n)= \max \{\textrm{cut-size of a subgraph of size } < n\}.$

Separation behaves rather differently from Cheeger constant.

Example 1 (Benjamini-Schramm-Timar) For bounded geometry graphs, let ${r : X\rightarrow Y}$ be Lipschitz and fibers have bounded size, then ${sep_X < C sep_Y + C}$.

Proposition 5 (Benjamini-Schramm-Timar)

$\displaystyle \begin{array}{rcl} sep_{{\mathbb Z}^k} &=& n^{k-1/k} .\\ sep_{H^k} &=& \begin{cases} \log n & \text{ if }k=2, \\ n^{k-2/k-1}& \text{ if }k\geq 3. \end{cases}\\ sep_{F_k} &=& 1.\\ sep_{F_2\times F_2} &=&\frac{n}{\log n}. \end{array}$

Theorem 6 ${X}$ infinite graph. Then ${sep_X}$ is not sublinear ${\Leftrightarrow X}$ contains an ${\epsilon}$-expander. If furthermore ${X}$ has bounded degree, then ${sep_X}$ is not sublinear ${\Leftrightarrow X}$ contains a ${(d,\epsilon)}$-expander.

4. Proof

4.1. Characterization of expanders

(i) If G is a finite graph with Cheeger constant ${h > \epsilon}$, then ${cut(G) > n \epsilon/4}$. Indeed, let ${C}$ be a cut set for ${G}$. A greedy search provides a collection ${D}$ of components of ${G\setminus C}$ with size between ${n/4}$ and ${n/2}$. Its boundary is contained in ${C}$.

(ii) Conversely, let ${G}$ be a finite graph of size ${n}$. There is a subgraph ${G'}$ of size ${\geq n/2}$ such that ${h(G')\geq cut(G)/2n}$.

4.2. Different separation profiles

To construct expanders with different separation profiles that can be embedded into groups, pick a (d,${\epsilon}$)-expander (${G_n}$) such that –

1. girth ${g(G_{n+1}) > 2 |G_n|}$, –
2. ${|G_n| > 3 |G_{n-1}|}$.

Observe that if ${M}$, ${N}$ are infinite subsets of integers, with ${M\setminus N}$ infinite, then ${sep_{G_M}}$ is not bounded above by ${sep_{G_N}}$. Indeed, below girth, separation profile is dramatically low.

When embedded into groups, the receiving groups have a separation profile governed by the graph at certain scales, and are essentially hyperbolic at others. But hyperbolic implies polynomial separation profile.

## Notes of Daniel Galicer’s lecture

The minimal distortion needed to embed a binary tree into ${\ell^p}$

I will cover theorems very similar to those in the previous talk, but the methods will be different, with a more combinatorial flavour.

1. Ramsey theory

1.1. Friends and strangers problem

How many people should be in a party in order to ensure that at least 3 of them know each other or at least three of them were strangers ?

View the ${n}$ participants as a complete graph some of whose edges are coloured. We want a triangle painted in one single colour.

The answer is ${n=6}$.

1.2. Ramsey’s theorem

Theorem 1 (Ramsey) Given ${r}$, ${s\in{\mathbb N}}$, there exists ${R(r,s)}$ such that if ${n\geq R(r,s)}$, in any graph with ${n}$ vertices, there exists either an ${r}$-clique (complete subgraph) or an ${s}$-independent set (no edges between them).

We just saw that ${R(3,3)=6}$. It is known that ${R(4,4)=18}$.

\indent{If aliens would threaten Earth of war unless we can give them the value of ${R(5,5)}$, we should put all our efforts to find it. If they would ask for ${R(6,6)}$, we should better get ready for war (Erdos).}

So do not think that these numbers are easy to compute. Finding rough estimates on them is a theory in itself.

2. Ramsey and embeddings

2.1. Embedding trees into ${p}$-convex Banach spaces

See Li’s lecture for the definition of ${p}$-convexity. Hilbert space is 2-convex. ${L^p}$ is ${p}$-convex if ${p\geq 2}$ and 2-convex if ${1.

Theorem 2 Let ${X}$ be a ${p}$-convex Banach space. Let ${B_n}$ be the complete rooted binary tree of depth ${n}$. For any embedding ${f:B_n\rightarrow X}$,

$\displaystyle \begin{array}{rcl} dist(f)\geq\Omega((\log n)^{1/p}). \end{array}$

2.2. Hanner’s inequality

If ${p>2}$,

$\displaystyle \begin{array}{rcl} (|x|+|y|)^p +||x|-|y||^p \geq |x+y|^p+|x-y|^p\geq 2(|x|^p+|y|^p). \end{array}$

If ${p<2}$, the inequalities are reversed.

2.3. Forks

A 4-tuple of points ${\{x_0,x_1,x_2,x'_2\}}$ is a ${\delta}$-fork if ${\{x_0,x_1,x_2\}}$ and ${\{x_0,x_1,x'_2\}}$ are ${(1+\delta)}$-isomorphic to ${\{0,1,2\}}$ (mapping ${x_1}$ to 1).

Lemma 3 In a ${p}$-convex Banach space, ${p\geq 2}$, every ${\delta}$-fork satisfies

$\displaystyle \begin{array}{rcl} |x_2-x'_2|=|x_0-x_1|O(\delta^{1/p}). \end{array}$

Proof. One can assume that ${x_0=0}$, ${|x_1|=1}$. Let

$\displaystyle \begin{array}{rcl} z:=x_1+\frac{x_2-x_1}{|x_2-x_1|}. \end{array}$

Then ${|z-x_1|=1}$, ${|z-x_2|\leq 2\delta}$. Apply ${p}$-convexity inequality to unit vectors ${x=x_1}$ and ${y=z-x_1}$. Their midpoint is ${\frac{z}{2}}$, and ${|\frac{z}{2}|\geq 1-\delta}$. Therefore ${|x-y|\leq C\,\delta^{1/p}}$. Furthermore,

$\displaystyle \begin{array}{rcl} |x_2-2x_1|\leq |x_2-z|+|z-2x_1|=O(\delta^{1/p}),\quad |x'_2-2x_1|=O(\delta^{1/p}), \end{array}$

thus ${|x_2-x'_2|=O(\delta^{1/p})}$.

2.4. Combinatorics in rooted trees

Let ${T}$ be a rooted tree. Let ${SP(T)}$ denote the set of pairs ${(x,y)}$ of vertices with ${x}$ on the path from the root to ${y}$.

Denote by ${T_{k,h}}$ denote the complete ${k}$-ary tree of depth ${h}$.

Lemma 4 Let ${k\geq r^{(h+1)^2}}$. Colour ${SP(T_{k,h})}$ in ${r}$ colours. Then there is a copy ${T'}$ of ${B_h}$ such that the colour of any pair ${(x,y)\in SP(T')}$ only depends on the levels of ${x}$ and ${y}$.

Claim. It the leaves of ${T_{k,h}}$ are coloured by ${r'}$ colours and ${k>r'}$, then there is a copy of ${B_h}$ inside ${T_{k,h}}$ with monochromatic leaves.

This is proven by induction on ${h}$. Below the root, there are ${k}$ trees isomorphic to ${T_{k,h-1}}$. By induction, in each of them, there is a binary subtree ${B_{h-1}}$ with monocoloured leaves. Since ${k>r'}$, two of these binary subtrees have the same leaf colour. Connect them via the root, this yields a binary subtree ${B_h}$ with monocoloured leaves.

Back to the proof of the Lemma. We label each leaf ${z}$ of ${T_{k,h}}$ by a vector whose components are the colours of the ${\frac{h(h+1)}{2}}$ successive pairs ${(x,y)}$ along the path from ${o}$ to ${z}$. So the number of labels is ${r'< r^{(h+1)^2}\leq k}$. According to the Claim, there is a binary subtree ${B_h}$ with monocoloured leaves, meaning that colours of pairs depend only on their depths.

2.5. Matousek’s proof of the Theorem

We shall use the following easy facts.

Lemma 5 ${T_{k,h}}$ embeds into ${B_n}$ for ${n=2h[\log_2 k]}$.

Lemma 6 (Path embedding lemma) Given ${\alpha}$, ${\beta\in(0,1)}$, there exists ${C}$ such that every distance non decreasing map

$\displaystyle \begin{array}{rcl} f:\{0,1,\ldots,h\}\rightarrow M \end{array}$

in some metric space ${M}$, with ${h\geq 2^{CK^{\alpha}}}$, ${K=Lip(f)}$, there exists an arithmetic progression

$\displaystyle \begin{array}{rcl} Z=\{x,x+a,x+2a\} \end{array}$

such that the restriction of ${f}$ to ${Z}$ is ${(1+\epsilon)}$-isometric, for ${\epsilon=\epsilon(\alpha,\beta,a)}$.

Let ${f:B_n\rightarrow X}$ be a distance non decreasing map, of distorsion ${K=(\log n)^{1/p}}$. We shall look for forks in the image. We start with a complete ${k}$-ary subtree ${T_{k,h}}$ of ${B_n}$. We colour the elements of ${SP(T_{k,h})}$ according to the distorsion of ${f}$: the colour is the integer part of

$\displaystyle \begin{array}{rcl} \lfloor\frac{K^p}{\beta}\frac{|f(x)-f(y)|}{d_{T_{k,h}}(x,y)}\rfloor, \end{array}$

${\beta}$ a small constant.

Thanks to our Ramsey type Lemma, we find a complete binary subtree ${B_h}$ in ${T_{k,h}}$ with nodes coloured by their depth only. Along each path from root to a leaf, we find an arithmetic progression ${\{y_0,y_1,y_2\}}$ of step ${a}$ along which ${f}$ is nearly isometric. Let ${y'_2}$ be a node situated at the same depth as ${y_2}$ and at distance ${a}$ from ${y_1}$. The colour=depth property implies that, as far as distances between images under ${f}$ are concerned, ${\{y_0,y_1,y'_2\}}$ behaves learly as ${\{y_0,y_1,y_2\}}$ does, i.e. ${f(\{y_0,y_1,y_2,y'_2\})}$ is a fork. Therefore ${d(f(y_2),f(y'_2))}$ is small, whereas ${d(y_2,y'_2)=2a}$.

## Notes of Sean Li’s lecture

Lower bounding distortion via metric invariants

Let ${X}$ and ${Y}$ be metric spaces. Recall that ${c_Y(X)}$ is the infimum of bi-Lipschtz distortions of maps ${X\rightarrow Y}$. Typically, ${Y}$ is a Banach space and ${X}$ is combinatorial (finite metric space). When ${Y=\ell_2}$, one denotes by

$\displaystyle \begin{array}{rcl} c_2(n)=\sup\{c_{\ell_2}(X)\,;\,X\textrm{ an }n\textrm{-point metric space}\}. \end{array}$

Theorem 1 (Enflo 1970) ${c_2(n)}$ tends to infinity as ${n}$ tends to infinity.

Enflo gave a lower bound of ${\sqrt{\log n}}$. This is not sharp. The sharp bound is ${\log n}$. The upper bound is due to Bourgain (1986) and the lower bound to London-Linial-Rabinovich (1994) – they used expanders.

1. Enflo’s lower bound for the Hamming cube

Enflo gave a sharp distortion lower bound, but for a suboptimal example, the Hamming cube ${D_n=\{0,1\}^n}$ with ${\ell_1}$ metric (= graph metric on 1-skeleton of cube).

Proposition 2 (Enflo) For any ${f:D_n\rightarrow\ell_2}$,

$\displaystyle \begin{array}{rcl} \sum_{diagonals\,xy}|f(x)-f(y)|^2\leq \sum_{edges\,uv}|f(u)-f(v)|^2. \end{array}$

Let us begin with a

Lemma 3 (Short diagonals) In ${\ell_2}$, for any quadrilateral, the sum of squares of diagonals is less than the sum of squares of sides.

$\displaystyle \begin{array}{rcl} |x-z|^2+|y-w|^2\leq|x-y|^2+|y-z|^2+|z-w|^2+|w-x|^2. \end{array}$

Proof. By summing over coordinates, it suffices to prove this for points on a line. There, it boils down to triangle inequality.

1.1. Proof of Proposition

By induction on ${n}$. Edges of ${D_n}$ split into ${E_0}$ and ${E_1}$ (edges of 2 opposite faces) and ${E'}$ (edges joining these faces). Let ${diag_0}$ and ${diag_1}$ be the diagonals of opposite faces (there are not diagonals of ${D_n}$). Induction hypothesis gives

$\displaystyle \begin{array}{rcl} \sum_{diag_0}|f(x)-f(y)|^2&\leq &\sum_{E_0}|f(u)-f(v)|^2,\\ \sum_{diag_1}|f(x)-f(y)|^2&\leq &\sum_{E_1}|f(u)-f(v)|^2. \end{array}$

For each diagonal ${uv\in diag_0}$, there is a unique (parallel) ${u'v'\in diag_1}$ such that ${uv'}$ and ${u'v}$ are diagonals of ${D_n}$. Then ${uu'}$ and ${vv'\in E'}$. The short diagonals lemma gives

$\displaystyle \begin{array}{rcl} |f(u)-f(v')|^2+|f(u')-f(v)|^2&\leq&|f(u)-f(v)|^2+|f(v)-f(v')|^2\\ &&+|f(v')-f(u')|^2+|f(u')-f(u)|^2. \end{array}$

Summing up such inequalities, on the left hand side, each diagonal of ${D_n}$ appears once. On the right hand side, each edge in ${E'}$ and each element of ${diag_0}$ and ${diag_1}$ occurs once. With induction hypothesis, we are done.

1.2. Proof of the sharp bound for Hamming cube

Assume ${f:D_n\rightarrow\ell_2}$ is distance non decreasing and ${D}$-Lipschitz. There are ${2^{n-1}}$ diagonals and ${n.2^{n-1}}$ edges. Diagonals in ${D_n}$ have length ${n}$ and edges have length 1, their images have lengths ${\leq D}$, therefore

$\displaystyle \begin{array}{rcl} 2^{n-1}n^2\leq n.2^{n-1}D^2, \end{array}$

hence ${D\geq\sqrt{n}}$.

1.3. Enflo type

The crucial property of ${\ell_2}$ used is expressed in the Proposition. We could have had a multiplicative constant in the Proposition, this would not have spoilt the result. This motivates the following

Definition 4 Let ${p>1}$. Say that metric space ${X}$ has Enflo type ${p}$ if there exists ${T>0}$ such that for all ${n}$ and all maps ${f:D_n\rightarrow X}$,

$\displaystyle \begin{array}{rcl} \sum_{diagonals\,xy}|f(x)-f(y)|^p\leq T\sum_{edges\,uv}|f(u)-f(v)|^p. \end{array}$

The best ${T:=T_p(X)}$ is called the Enflo type ${p}$ constant of ${X}$.

By construction, if ${X}$ has Enflo type ${p}$,

$\displaystyle \begin{array}{rcl} c_X(D_n)\geq T_p(X)^{-1}n^{1-\frac{1}{p}}. \end{array}$

Enflo type is a bi-Lispchitz invariant. The Enflo type ${p}$ constant is covariant under Lipschitz maps: if there exists a Lipschitz map ${f:X\rightarrow Y}$, then

$\displaystyle \begin{array}{rcl} T_p(X)\leq dist(f) T_p(Y). \end{array}$

This gives lower bounds by

$\displaystyle \begin{array}{rcl} dist(f)\geq\frac{T_p(X)}{T_p(Y)}. \end{array}$

2. Banach spaces

Now I give some background on Banach spaces that will lead us to more invariants.

Say an infinite dimensional Banach space ${X}$ has Rademacher type ${p}$ if for all ${n}$ and all point ${x_1,\ldots,x_n\in X}$,

$\displaystyle \begin{array}{rcl} \mathop{\mathbb E}_\epsilon(|\sum_{i=1}^n\epsilon_i x_i|^2)\leq T^p\sum_{i=1}^n|x_i|^p, \end{array}$

where ${\epsilon_i}$ are independent uniform ${\pm 1}$-valued random variables.

This notion was introduced independently by Jorgensen and by Maurey-Pisier. Note that ${p}$ has not be ${\leq 2}$ (the real line does not have Rademacher type ${p}$ for ${p>2}$). Rademacher type is Enflo type when restricting maps ${f:D_n\rightarrow X}$ to linear maps. Therefore Enflo type ${p}$ implies Rademacher type ${p}$. A weak converse holds, as the following surprising theorem states.

Theorem 5 (Pisier) For Banach spaces, Rademacher type ${p>1}$ ${\Rightarrow}$ Enflo type ${p'}$ for all ${p'.

Open question. Does Rademacher type ${p>1}$ ${\Rightarrow}$ Enflo type ${p}$ ? Known to hold for certain subclasses of Banach spaces. E.g. UMD spaces (Naor-Schechtman).

2.2. Finite representability

Locality. Rademacher type ${p}$ depends only on the geometry of finite dimensional linear subspaces. Such properties are called local.

Definition 6 Say a Banach space ${X}$ is (crudely) finitely representable in ${Y}$ if there exists ${K\geq }$ such that for every ${n}$-dimensional subspace ${Z\subset X}$, there exists a subspace ${Z'\subset Y}$ with Banach-Mazur distance from ${Z}$ at most ${K}$.

By definition, local properties are preserved by finite representability. The following theorem suggests that local properties may be expressed as bi-Lipschitz metric invariants.

Theorem 7 (Ribe) If two infinite dimensional Banach spaces ${X}$ and ${Y}$ are uniformly homeomorphic, then they are mutually finitely representable.

Suggests has to be taken seriously. It merely gives a guideline, only a small number of invariants are understood from this point of view. We shall give an example below.

2.3. ${p}$-convexity

Let ${p>1}$. Say a Banach space is ${p}$-convex if it is uniformly convex with a modulus of convexity ${\epsilon^p}$.

Proposition 8 ${X}$ is ${p}$-convex iff there exists ${K>0}$ such that for all ${x}$, ${y\in X}$,

$\displaystyle \begin{array}{rcl} |x|^p+|y|^p \geq 2|\frac{x-y}{2}|^p+2|\frac{x+y}{2K}|^p. \end{array}$

Example. ${L^p}$ is ${p}$-convex for ${p\geq 2}$, with bounded constants. For ${p<2}$, ${L^p}$ is 2-convex with 2-convexity constant ${O(\frac{1}{\sqrt{p-1}})}$.

Theorem 9 (Pisier) 1. Every uniformly convex Banach space can be renormed to be ${p}$-convex for some ${p>1}$.

2. Uniform convexity is a local property.

3. Markov type

If ${(X_t)_{t\in{\mathbb Z}}}$ is a Markov chain on some set ${\Omega}$, denote by ${(\tilde{X}_t(k))_{t\in{\mathbb Z}}}$ the Markov chain that coincides with ${X_t}$ until ${t\leq k}$, and then continues an independent life.

Say a metric space ${X}$ is Markov ${p}$-convex if there exists ${\pi>0}$ such that for every Markov chain on a set ${\Omega}$ and every map ${f:\Omega\rightarrow X}$,

$\displaystyle \begin{array}{rcl} \sum_{k=0}^{\infty}\sum_{t\in{\mathbb Z}}\frac{1}{2^{kp}}\mathop{\mathbb E}(d(f(X_t),f(\tilde{X}_t(t-2^k)))^p)\leq \pi^p\sum_{t\in{\mathbb Z}}\mathop{\mathbb E}(d(f(X_t),f(X_{t-1}))^p). \end{array}$

The best ${\pi}$ is called the Markov ${p}$-convexity constant ${\pi_p(X)}$ of ${X}$.

3.1. Trees

Let us start with an example which is not Markov convex. Let ${B_n}$ be the complete rooted binary tree of depth ${n}$. Let ${X_t}$ be the standard downward random walk on ${B_n}$ with probability ${1/2}$, stopped at leaves.

Proposition 10 There is a ${C(p)}$ such that the Markov ${p}$ convexity constant has to be

$\displaystyle \pi_p(B_n)\geq C\,(\log n)^{1/p}.$

It follows that the infinite rooted binary tree ${B_\infty}$ is not Markov convex. We see that trees are really extreme for Markov convexity: they allow maximal branching for Markov chains.

Proof. The right hand side (or denominator in ${\pi}$) equals ${n}$, since the distance ${d(X_t,X_{t+1})=1}$ by construction. To estimate the right hand side (or numerator in ${\pi}$), assume that ${0\leq t-2^k\leq t\leq n}$ and consider the independent chains ${(X_t)}$ and ${(\tilde{X}_t(t-2^k))}$ over ${2^k}$ time steps. With probability ${1/2}$, at time ${t-2^k}$, they jump to different nodes. If this happens, later on, they walk down the tree along different branches, therefore ${d(X_t,\tilde{X}_t(t-2^k))}$ increases by 2 at each step, leading to

$\displaystyle \begin{array}{rcl} d(X_t,\tilde{X}_t(t-2^k))\geq 2.2^k, \end{array}$

and thus

$\displaystyle \begin{array}{rcl} \mathop{\mathbb E}(d(X_t,\tilde{X}_t(t-2^k))^p)\geq \frac{1}{2}(2.2^k)^p\sim 2^{kp}. \end{array}$

Summing over ${t}$ from ${2^k}$ to ${n}$ gives

$\displaystyle \begin{array}{rcl} \sum_{t=2^k}^{n}\mathop{\mathbb E}(d(X_t,\tilde{X}_t(t-2^k))^p)\geq (n-2^k)2^{kp}. \end{array}$

Summing over ${k}$ from 0 to ${\log_2 n}$ gives

$\displaystyle \begin{array}{rcl} \sum_{k=0}^{\log n}\sum_{t=2^k}^{n}\mathop{\mathbb E}(d(X_t,\tilde{X}_t(t-2^k))^p)\geq n\log n, \end{array}$

whence ${\pi^p\geq\log n}$.

Theorem 11 If metric space ${X}$ is Markov ${p}$-convex, then

$\displaystyle \begin{array}{rcl} c_X(B_n)\geq C(\log n)^{1/p}. \end{array}$

Indeed, the Markov ${p}$-convexity constant is Lipschitz covariant.

3.2. The case of Banach spaces

Theorem 12 An infinite dimensional Banach space can be renormed to be ${p}$-convex iff it is Markov ${p}$-convex.

I prove only one implication. Let ${X}$ be a ${p}$-convex Banach space, i.e.

$\displaystyle \begin{array}{rcl} |x|^p+|y|^p \geq 2|\frac{x-y}{2}|^p+2|\frac{x+y}{2K}|^p. \end{array}$

One first proves a consequence, the fork inequality:

$\displaystyle \begin{array}{rcl} \frac{1}{2^{p-1}}(|x-w|^p+|x-z|^p)+\frac{1}{4^{p-1}K}|z-w|^p \leq 2|y-x|^p+|y-w|^p+|z-y|^p. \end{array}$

Note that only the tip of the fork (${z}$ and ${w}$) involves the ${p}$-convexity constant, the other coefficients take specific values.

Apply the fork inequality to the images by ${f}$ of ${X_{t-2^{k+1}}}$, ${X_{t-2^{k}}}$, ${X_t}$, ${\tilde{X}_t(t-2^k)}$ and take expectation. By independence,…

3.3. Question

Does one have a metric invariant for uniform smoothness ?

## Notes of Adriane Kaichouh’s lecture nr 2

1. Super-reflexivity

1.1. Uniform convexity

A Banach space ${E}$ is uniformly convex if the midpoint of two points on the unit sphere which are sufficiently far apart is deep inside the ball.

Example. Hilbert spaces, ${L^p}$ spaces with ${1 are uniformly convex.

Uniform convexity carries to ${\ell^2}$ direct sums ${\ell^2(X,E)}$.

1.2. Super-reflexivity

Uniform convexity is not stable under renorming.

Definition 1 A Banach space is super-reflexive if it admits an equivalent norm which is uniformly convex.

Here is an equivalent definition.

Theorem 2 (Enflo) A Banach space is super-reflexive iff every one of its ultrapowers is reflexive.

In other words, every space which is finitely representable in it is reflexive.

Recall that, given an ultrafilter ${\mathcal{U}}$ on an index set ${I}$, the ultraproduct ${\prod_{\mathcal{U}}E_i}$ of a family of Banach spaces ${(E_i)_{i\in I}}$ is the quotient of ${\ell^{\infty}(I,E_i)}$ by the subspace ${c_0(I,E_i)}$ of sequences that tend to 0 along ${\mathcal{U}}$. This is a Banach space. If all ${E_i=E}$, the ultraproduct is called an ultrapower.

2. Pestov’s theorem

Here is Pestov’s main theorem.

Theorem 3 (Pestov) Let ${X}$ be a metric space. Let ${G}$ be a locally finite group acting on ${X}$ by isometries. Assume that ${X}$ admits a coarse/uniform embedding into a Banach space ${E}$. Then there is a coarse/uniform embedding ${\psi}$ from ${X}$ to some ultrapower of some ${\ell^2(\mathcal{U},E)}$, and the action of ${G}$ on ${\psi(X)}$ extends to an action by affine isometries on the affine span of ${\psi(X)}$.

2.1. Proof

Let ${\phi}$ be the initial coarse embedding. Start with a finite subgroup ${F}$ of ${G}$. Define

$\displaystyle \begin{array}{rcl} \psi_F : X\rightarrow \ell^2(F,E),\quad \psi_F(x)(f)=\frac{1}{|F|}\phi(f^{-1}x). \end{array}$

Then ${\psi_F}$ has the same expansion and compression moduli as ${\phi}$. ${F}$ acts on ${\ell^2(F,E)}$ via left translations, and ${\psi_F}$ is equivariant for this action.

Next we glue all maps ${\psi_F}$ together. Let ${I}$ denote the set of all finite subgroups of ${G}$. Because ${G}$ is locally finite, there exists an ultrafilter ${\mathcal{U}}$ on ${I}$ such that for all ${F\in I}$, the set

$\displaystyle \begin{array}{rcl} \{H\in I\,;\,F\subset H\} \end{array}$

is in ${\mathcal{U}}$. Indeed, the intersection of two such subsets

$\displaystyle \begin{array}{rcl} \{H\in I\,;\,F\subset H\}\cap\{H\in I\,;\,F\subset H\}=\{H\in I\,;\,F\subset H\}=\{H\in I\,;\,F\subset H\} \end{array}$

and the subgroup ${±langle F_1,F_2\rangle}$ generated by two finite subgroups is finite again. So we are dealing with a filter basis.

Fix an origin ${o\in X}$. Let ${V}$ be the ultraproduct over ${\mathcal{U}}$ of spaces ${\ell^2(F,E)}$, based at ${\psi_F(o)}$ instead of 0 (this makes ${\ell^2(F,E)}$ an affine space instead of a vectorspace, and so much for the ultraproduct). Define ${\psi(x)}$ as the sequence ${(\psi_F(x))_{F\in I}}$. It is well defined and has the same expansion and compression moduli as ${\phi}$.

Given ${g\in G}$, observe that ${\mathcal{U}}$-almost surely,

$\displaystyle \begin{array}{rcl} g\psi(x)=\psi(gx) \end{array}$

is well defined, and this provides an isometry of ${\psi(X)}$. Isometries of subsets of Hilbert spaces extend to affine hulls, so the ${G}$ action extends.

As defined, ${V}$ is not quite an ultrapower. For every ${F\in I}$, ${\ell^2(F,E)}$ embeds non canonically into ${\ell^2(G,E)}$, so

2.2. Application to uniform embeddability of Urysohn space

Let ${\phi:U\rightarrow E}$ be a uniform embedding. Let ${G}$ be a dense locally finite subgroup of ${Isom(U)}$. Pestov’s theorem implies that there is a uniform embedding of ${U}$ to a reflexive Banach space ${V}$, equivariant with respect to an affine isometric action of ${G}$. ${\Psi}$ is a homeo onto ${\psi(U)}$. The topologies of pointwise convergence induced on ${G}$ by the two actions of ${\psi(U)}$ coincide, so, by density, the action on the affine span ${S}$ of ${\psi(U)}$ extends to ${Isom(U)}$. I.e. we get a continuous group homomorphism

$\displaystyle \begin{array}{rcl} Isom(U)\rightarrow Aff(S)=O(S)\times S. \end{array}$

Since there is an injective equivariant map, it is injective.

Now we use the fact that ${Isom(U)}$ contains a subgroup ${H}$ isomorphic to ${Homeo^*([0,1])}$. So we get a linear isometric representation of ${H}$ on a reflexive Banach space. According to Megerelishvili’s theorem, this must be the trivial representation. So ${H}$ acts faithfully by translations. Contradiction, since ${H}$ is non abelian.