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 unitary representation which almost admits invariant vectors. Haagerup property is equivalent to a-T-menability for -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 and .

It is equivalent to existence of a proper action on a median metric space (Chatterji-Drutu-Haglund). Examples are spaces with (measured) walls. For , 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 . 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 -quasiconformal maps. In case space satisfies a quadratic isoperimetric inequality, their Hölder exponent (or 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 , asymptotic cones are trees, thus space is hyperbolic. If constant is , 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 is the group of homeos of the boundary of a regular tree T which are piecewise tree automorphisms.

Caprace-de Mets: let be a profinite group. Let be the group of isomorphisms between open subgroups of (identify two suchmaps if they coincide on some open subgroup). Take for an arbitrary compact open subgroup of . Then .

Kapoudjian: is simple and compactly generated.

Bader-Caprace-Gelander-Mozes: does not have lattices. has infinite asymptotic dimension (it contains for all ).

** 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 and fix a bijection of each link of vertex of T with . Then any in and any vertex, one has a permutation of . Fix a subgroup of permutations of .

Burger-Mozes: .

Bader-Caprace-Gelander-Mozes: .

is a closed subgroup of , but is not. The action of on is not proper.

Definition 1Given subgroups , define

Theorem 2is compactly generated, and its asymptotic dimension equals 1 (it is thus much smaller than ). Furthermore, also appears as a group of commensurators.Assume that has index 2 in , and that is generated by its point stabilizers. Then has a simple subgroup of index 8.

Assume that F acts transitively on . Then has Haagerup property. Assume further that for every point , the stabilizer is essential in , and . Then has no lattices.

**Example**: , actig on the projective line .

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

Examples of groups with lattices are obtained when is discrete and cocompact ( simply transitive).

**4. Pierre-Emmanuel Caprace: From amenability to buildings **

Joint work with Nicolas Monod. We are interested in amenable groups, i.e. pairs where is a proper space and an amenable group acting isometrically and cocompactly on .

** 4.1. An ancient subject **

Theorem 3 (Adams-Ballmann 1998 (Avez 1970 for manifolds))If acts properly discontinuously, then is a round sphere. Furthermore, is virtually . If, in addition, is geodesically complete, then is Euclidean -space.

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

** 4.2. Motivation **

Let be a closed manifold. Assume carries a locally metric. Does 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 (Davis-Januskiewicz 1991, Davis-Januskiewicz-Lafont 2013 for dimension 4).

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

Proposition 4Let be a connected, locally Lie group. Then is solvable. In fact, more generally, if a connected Lie group acts freely and cocompactly on a a proper space , then is solvable.

This is proved by induction on the dimension of the solvable radical. The initialization step deals with semi-simple groups. having no compact subgroups, it must be a product of copies of , so its center is infinite. One can assume that acts minimally (no proper invariant closee convex subset). Then any nontrivial element of has constant displacement function, so splits a Euclidean factor on which acts cocompactly. This happens only if , so 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 spaces. I note that dimension will play a role.

** 4.3. Geometric dimension **

I will use Kleiner’s geometric dimension.

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

Fact: geometric dimension does not increase when passing from a space to its asymptotic cones, and then to its ideal boundary. The converse does not hold.

** 4.4. Main result **

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

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

Theorem 7 (Caprace-Cornulier-Monod-Tessera)Let be a propeer geodesic hyperbolic space. Assume that isometries of do not have a common fixed point on the ideal boundary of . If admits a cocompact action of an amenable group. Then, modulo a compact normal subgroup, 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 .

Corollary 8In main Theorem, if is geodesically complete, then 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 is a proper geodesically complete -space and is an irreducible spherical building of dimension , then 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 be an amenable locally compact group, acting continuously on a proper space , then either stabilizes a flat in 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 10A group is an AB group if it satisfies the conclusion of the above theorem, i.e. any action on a proper space stabilizes a flat or fixes a boundary point.

**Fact**. If is a proper space with a cocompact isometry group, then a closed subgroup is amenable if and only if it is an AB group. **Warning**. This equivalence fails in general. For instance, is an AB group but it is not amenable. Thompson’s group 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 be a discrete group acting properly cocompactly on a proper space . Then the fixed point set is contained in the maximal spherical joint factor of . If action on is minimal, this factor is the boundary of the maximal flat factor of .

**Proof**. We provide a proof that illustrate ideas that are extendable to the non discrete case. Let be a fixed point, a ray converging to . By cocompactness, there is a radial sequence for , i.e. a sequence such that stays bounded. The sequence subconverges to a line with endpoints and . Conjugate the picture with . Since fixes , stays bounded. Hence stays bounded in , i.e. is constant (up to extracting), . must fix for large enough. Since is finitely generated, every element of fixes some . stabilizes the union of all lines with endpoints , therefore is isometric to , contradicting minimality.

** 4.7. Proof of Adams-Ballmann’s first theorem **

Assuming Adams-Ballmann’s second theorem. Replace with a minimal invariant subspace . has a canonical splitting . acts on without fixed points on . Adams-Ballmann’s second theorem implies that stabilizes a flat in . By minimality, is a point.

Arguments in the proof of the proposition lead to the following

Lemma 12Let be a proper space. Let be a closed subgroup which is cocompact on , and fixes a point . Then

is nonempty and is transitive on it.

** 4.8. Geometric Levi Decomposition **

Theorem 13 (Geometric Levi Decomposition)Let be a proper space. Let be a closed subgroup which is cocompact on , and fixes a point . Pick . Then

- where is the set of whose displacement tends to zero along rays converging to .
- is normal in and is compact.
- is cocompact on the set of lines with endpoints and .
- is amenable. In fact, it is compactible, i.e. there exists a compact subgroup and a sequence that conjugates every element of into a bounded sequence all of whose accumulation points are in .

In other words, as soon as a group of isometries fixes a point at infinity , two new actions arise, the Busemann character (an action on ), and the action on the *transverse space* , the quotient set of rays converging to , which inherits a quotient metric. The kernel of the action on is . A homomorphism to is produced like in the proof of the Proposition above, by taking coherently (using an ultrafilter) limits of subsequences of .

Corollary 14Assume further that is unimodular. Then is again in the maximal spherical factor of .

Indeed, compactibility and unimodularity force to be compact, and ultimately is a single point.

** 4.9. Spherical buildings **

Recall that our goal is to prove that the boundary of a 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 be a -space. Assume that

- .
- contains a pair of antipodal points, i.e. .
- Every pair of antipodal points lie in a common -sphere.
- There is a special point with a compact neighborhood.
Then is a spherical building.

The special point can be in fact any regular point.

**Examples**.

- If , 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 space has geometric dimension , then there is -dimensional flat in . Leeb proved that every -sphere isometrically embedded in bounds a -flat.

In particular, antipodal points contained in a sphere in 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 (see Adams-Ballmann’s theorem) is our candidate. For this, we need that the fixed point set be not too large (eg contain

Proposition 16Let be a proper space, cocompact. Then

- is contained in some -sphere, .
- Its radius is at most , unless has a nonempty spherical factor.
- If, in addition, is amenable, then radius is at most , unless is a sphere.

** 4.11. Proof of (1) **

By induction on .

If , i.e. is hyperbolic, there is at most 1 fixed point (otherwise, would stabilize a point or a line of ).

Let . Pick a fixed point . We know that it has at least one opposite point . According to the geometric Levi decomposition, acts cocompactly on the union of parallel lines . Its boundary is the spherical join of the pair and .

**Claim**. .

Indeed, given , by cocompactness, one finds a point in from where the angle of equals the Tits distance, thus getting a flat sector. Again by cocompactness, one produces a flat half-plane whose boundary contains , which is contained in .

Since , conclude by induction (requires more work to produce a -sphere…).

** 4.12. Main step **

It consists in proving that , and that every interior point is“regular” in the sense that is a round sphere. (It is likely that this statement holds for all amenable groups having boundary fixed points. We are able to prove this only if is totally disconnected.)

**Consequence**. Pick such an interior point . Then is transitive on the set of -spheres containing .

Indeed, aconsider antipodes of in such spheres. An element that maps an antipode to another must send whole sphere to whole sphere, since it fixes a -dimensional subset of these spheres.

** 4.13. Tri-spheres **

Again, is an interior point of the fixed point set. One shows that two -spheres through form a tri-sphere.

Now we use the assumption that the full isometry group has no common fixed point to show that the -orbit of intersects each half-sphere of the tri-sphere. This allows to produce a reflection fixing a given -sphere . Such reflections suffice to show that -translates of cover .

**5. Yves Cornulier: Large scale geometry of Lie groups **

** 5.1. Facts about Lie groups **

For finitely generated groups, polycyclic solvable and all subgroups are finitely generated. Nilpotent groups are automatically polycyclic.

Example: Baumslag-Solitar group is finitely generated, solvable but not polycyclic.

Tits alternative: A finitely generated subgroup of 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 if and only if it is virtually polycyclic.

A virtually solvable subgroup of is contained and cocompact in a closed subgroup with finitely many connected components (beware that need not be the Zariski closure). This is proved by reduction to the nilpotent case. For groups with unipotent Zariski closure , Malcev theory shows that is cocompact in . In general, pass to derived subgroups and . is abelian, is cocompact in ,

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: is diffeomorphic to , and is diffeomorphic to .

Nevertheless, it is which contains the large scale geometry.

Theorem 18Every connected Lie group has a closed cocompact subgroup of the form where is solvable and connected.For every solvable connected Lie group , there exist proper and cocompact homomorphisms

- where is solvable connected,
- where is triangulable (i.e. a subgroup of upper triangular matrices).
Therefore, we have the following diagram of proper and cocompact homomorphisms

**Example**. . Then upper triangular matrices and . In general, the or factors are there only if groups with infinite fundamental groups are encountered.

**Example**. where acts on by multiplication by . Then where acts on by multiplication by .

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 be a sequence of metric spaces. Let be a nonprincipal ultrafilter on . One can define a metric space

as follows. The “distance” on the ultraproduct takes values in the ultraproduct , which is too big.

Instead, pick a marked point in each and consider sequences , , such that stays bounded. Then identify sequences and such that

A metric is well defined on the quotient set by

The resulting metric space indeed depends on the choice of the marked points .

**Important special cases**.

- If all are the same, then isometrically embeds into (which is usually much larger than unless is locally compact).
- If , the limit is called a tangent cone at .
- If , the limit is called an
*asymptotic cone*(it does not depend on the choice of ).

**Examples**. Start with in the word metric for the obvious generating system. Then metric).

** 5.4. The large scale category **

Objects are metric spaces, morphisms are large scale Lipschitz maps, up to the following equivalence: identify and if is bounded on . Isomorphisms are quasi-isometries. Large scale Lipschitz maps induce Lipschitz maps between asymptotic maps. Therefore, 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 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 between metric spaces is *sublinearly Lipschitz* if

for some sublinear function . Identify maps and when they are *sublinearly close*, i.e.

Again, is a functor from the sublinear category to the Lip category.

**Main example**. Let be a nilpotent Lie algebra, let be its lower central series. The Lie bracket

defining a Lie algebra bracket on

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

Theorem 20 (Pansu, Breuillard)The identity map is a sublinear Lipschitz equivalence.

For a Carnot group like , asymptotic cones are easy to determine: admits a one parameter group of automorphisms which are homthetic relative to left-invariant sub-Riemannian metrics. Therefore . It follows that . 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 , the lower central series stabilizes to . Call

the *exponential radical* of . It is a normal subgroup.

Say is *splittable* if is a semi-direct product where is nilpotent.

Theorem 21Let be triangulable. There is a new group law on , that does not change the exponential radical, which is splittable and sublinearly Lipschitz equivalent. Furthermore, the action of on is diagonalizable over .

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

**Example**. Let be the semidirect product where acts by matrices

The action of the normal is not diagonalisable. The new law replaces the matrix by

resulting into a group quasi-isometric to , where denotes hyperbolic 3-space. Both laws are not quasi-isometric, although they have isometric asymptotic cones ( is the universal -tree of degree ).

** 5.7. Dimensions of asymptotic cones **

Corollary 22.

**Proof**. The exact sequence induces a fibration

Guivarc’h-Osin: is quasi-ultrametric (ultrametric up to an additive constant), so is ultrametric, hence 0-dimensional. It implies (Gromov, Burillo), that , since is nilotent. Hence

Reverse inequality is obvious if is splittable, since embeds isometrically into . General case requires some more work.

** 5.8. More about asymptotic cones **

The is bi-Lipschitz to a fiber product of and above

can be described up to bi-Lipschitz equivalence as follows. Let denote the ultrapower of , let

Define the *Robinson field*

is a -valued valuation on , with valuation ring . Then, for every Lie group ,

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

Corollary 23Assume Continuum Hypothesis. Then, for all connected Lie groups , then all 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 is nonelementary hyperbolic, is isometric to the universal -tree . Note that hyperbolic Lie groups are known.

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

- either where acts on by contracting automorphisms and is compact,
- or is an open subgroup in a rank 1 simple Lie group.

** groups**. , , is the semidirect product where acts on by matrices

It is a subgroup in the product of two copies of the affine group, the kernel of a homomorphism to which is a Buseman function. Thus is a horosphere in , it does not depend on . It is 1-dimensional and not simply connected.

This works as well for semidirect products where contracts and dilates . For instance, if and , the group contains Baumslag-Solitar group as a lattice. Thus this group has asymptotic cones bi-Lipschitz equivalent to those of groups.

More generally, one can handle semidirect products with a diagonal action. Asymptotic cones are -dimensional, and not -connected.

** 5.10. Questions **

Which cones are ? I can merely say which cones are contractible. This happens if and only if and contains an element that contracts .

**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 be a metric space. Let be a geodesic ray. Then

exists. Indeed, by triangle inequality, 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 be a metric space, and some marked point. Map to (continuous functions on , equipped with the topology of uniform convergence on bounded sets) by

Consider the closure of .

**Example**. If is , coincides with the visual bordification of .

**Example**. For nonproper metric spaces, may be empty. For instance, let be a bouquet of longer and longer geodesic segments.

** 6.4. Metric functionals **

We change topology. We replace with , i.e. pointwise convergence. Denote by , where

**Example**. Let be a bouquet of longer and longer geodesic segments. Then

Proposition 25Let be an infinite dimensional real Hilbert space. Then the elements in are parametrized by and vectors such that or , .

- , , .
- .
- .
A sequence converges in iff tends to and converges weakly to .

**Example**. Let be an orthonormal basis of . Then tends to . However, given a sequence , does not always converge.

** 6.5. Connection with other notions **

From now now on, denote by where 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 is a normed vectorspace, which Busemann functions are linear ?

**Example**. Let be a countably infinite simplicial tree, with infinitely many branches at vertex . As in Hilbert space, the sequence of neighbours of converges to a point in .

** 6.6. Action of isometries **

Of course, construction is natural under isometries.

**Example**. For , the horoboundary consists of a square with countable sides. Translations along one factor fix two sides and translate the two others.

Proposition 26Let . Fix origin . Let be a -invariant probability measure on . Then

is a group homomorphism .

**Exercise**. Let .

- If fixes a point of , then surjects onto .
- If is countable, then has a finite orbit in .

Find conditions (growth ?) on that ensure that is countable.

Cormac Walsh: If is nilpotent, there exists a finite orbit in .

** 6.7. Towards a metric spectral principle **

Let be a normed complex vectorspace. Let be a bounded linear operator. Then *spectral radius*

exists. It coincides with the sup of the spectrum. In finite dimension, has a Jordan form with eigenvalues ordered with decreasing absolute values, whence a filtration . Given a vector ,

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 , .

Theorem 27 (Karlsson 2001)Let be a 1-Lip map. Define

Then there exists such that for all , and

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 decreasing to 0. Set

For fixed , this is unbounded in , so pick subsequence such that

Then

By compactness, there is a limit point

** 6.9. Applications **

\subsubsection{Mean ergodic theorem}

Here, is Hilbert space. Let be a linear operator of norm . Let . Consider , , so that

Then either , and tends to 0, or . Our Theorem provides a metric (linear in this case) functional such that tends to . This implies that tends to , where , and . Indeed,

\subsubsection{Hyperbolic metric spaces}

If is unbounded, it should converge to a point in the Gromov boundary. If is an isometry, .

\subsubsection{Several complex variables}

To a complex space, a Kobayashi pseudo-metric is associated. It is a true distance only when 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 . Then either there is a fixed point in , or iterates converge to a boundary point.

In this case, .

** 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 be a closed hyperbolic surface. For a closed curve , denote by the length of the closed geodesic freely homotopic to .

Theorem 29Given a homeomorphism , there exist numbers such that for every simple closed curve on ,

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

\subsubsection{Back to Hilbert space}

Let denote invertible linear operators on Hilbert space . Let denote symmetric ones. Let denote the positive ones. acts on , and there is a -invariant Finsler metric, defined by following norm at point ,

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

Busemann property is equivalent to *Segal’s inequality*

The Finsler exponential maps coincides with the operator exponential, therefore . Thus if , then

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

** 6.11. Ergodic theorems **

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

Replace real valued variables with -valued ones, 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 . Replace independence by the following setting

Theorem 31 (Karlsson-Ledrappier 2006 for isometries, Gouezel-Karlsson 2015)Let be a probability measure preserving transformation. Let . Assume that for all , is integrable. Set

There is a random metric functional such that almost surely,

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

Lemma 32 (Gouezel)Let

Almost surely, there exists a subsequence and numbers tending to 0 such that for all ,

\subsubsection{Random walks on groups}

Let be a finitely generated group of subexponential growth, with no homomorphisms to . For any random walk on , distance to the origin growth sublinearly.

\subsubsection{Random walks in spaces}

Let be a metric space. Our theorem implies that there exists a unique random geodesic ray from such that the random walk stays sublinearly close to ,

When applied to , we get Oseledets’ theorem.

** 6.12. Comments **

1. Oseledets’ theorem has some infinite dimensional version: in the 1980’s, Ruelle stated a version for operators of the form 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 be a finite dimensional cube complex. For each , there is a cohomology class, the *median class*

where is a Banach space to be defined below. Let be a group and be a homomorphism.

Theorem 33 (Chatterji-Fernos-Iozzi)If the action is nonelementary, then .If is elementary (i.e. there is a finite orbit in ), either or there is a finite index subgroup and a -invariant subcomplex of lower dimension, on which the action is nonelementary, and .

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

Corollary 34Let be an irreducible finite dimensional cube complex. Let be an irreducible lattice in the product of (at least two) locally compact groups. Let be an essential and nonelementary action. Then extends continuously to , factoring through one of the factors.

Corollary 35Let be an irreducible lattice in a higher rank semisimple Lie group, then any action of on a finite dimensional cube complex has a fixed point.

This requires Caprace’s description of stabilizers in of finite subsets of .

** 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 cube complex on its own right. It separate into two half-spaces.

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

Nevo-Sageev: if is locally finite and has a cocompact isometry group, then . If is a cocompact isometry group with a nonelementary action on , then its closure is a minimal strongly proximal -space.

Theorem 36 (Chatterji-Fernos-Iozzi)Let be a finite dimensional cube complex, with a nonelementary action of . Let be a strong -boundary (i.e. acts on amenably and doubly ergodically with coefficients). Then there is a -equivariant measurable map .

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 as a rather explicit function on and check that it does not vanish.

** 7.3. Proof **

Furstenberg’s lemma provides us with a map , . A measure on that gives different measures two half-spaces with the same boundary hyperplane defines an ultrafilter, and thus a point in . Let us show that this happens almost everywhere along . When is a tree, hyperplanes are points. If things go bad for some measure , 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 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 on one hand and mapping class group on the other hand.

** 8.1. Basics **

The Cremona group is the group of birational transformations of projective plane. Elliptics of or can be considered as elliptic elements of . Jonquieres maps

can be considered as parabolic elements of .

embeds into as follows,

** 8.2. Action on infinite dimensional hyperbolic space **

In analogy with the mapping class group action on Teichmüller space, one introduces an action of on an infinite dimensional hyperbolic space .

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 is the exceptional divisor arising from a blow up, . On , the space of divisors is 1-dimensional (generated by lines), with positive intersection form . Blowing up points produces an -dimensional space of divisors, with an intersection form of signature . Blow-up maps induce isometric embeddings of quadratic spaces. Let 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 completion, .

Compositions of blow-ups act isometrically on .

** 8.3. Dynamical degree **

This is the analgue of spectral radius (for ) or stretch factor (for pseudo-Anosov surface homeomorphisms).

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

is. It is always .

**Example**. If , then .

If , then is the spectral radius of the matrix.

Theorem 37 (Fabre-Diller, Gizatullin)If and , then

- either the sequence is bounded, and there exists such that in an automorphism.
- either , and is Jonquières,
- or , and preserves an elliptic fibration.

** 8.4. Tits alternative **

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

This uses essentially the action on .

**Question**. Over , is it necessary to assume that is finitely generated ?

** 8.5. Non simplicity **

Theorem 39 (Cantat-Lamy, 2013, Lonjou 2015 for arbitrary fields)For general and large enough, the normal subgroup generated by is a proper subgroup.

This uses ideas from hyperbolic groups.

** 8.6. What next ? **

In 1992, Wright introduced an action of 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. -buildings **

These are 2-dimensional simplicial complexes whose links are 1-dimensional finite spherical buildings. There is Bruhat-Tits one, associated to , 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 40Let act properly dscontinuously, cocompactly on an -building which is not a Bruhat-Tits building. Then is not linear: any homomorphism to (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 into an unbounded homomorphism to , 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 of rays in a boundary point . 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 .

** 10.3. Projectivity group **

Consider a chain of boundary points, each of which is opposite to the previous one. Compose canonical isometries and get an automorphism of the tree . The group generated by such maps is called the *projectivity group* of .

**Fact**. is 2-transitive on .

Theorem 41Assume that is not a Bruhat-Tits building. Then is not linear (no faithful continuous representation on any ).

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 from one of .

** 10.4. Bader-Furman’s Gate theory **

Let be a countable group, with a Zariski dense unbounded representation into some , simple algebraic group. Let be a measure space with a measure-class preserving, ergodic action of . Bader-Furman show that if another group acts on , commuting with the action of , then there is a natural representation .

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

** 10.5. Ergodicity of the geodesic flow **

Let denote the space of isometric maps . acts by precomposition. The geodesic flow is the -action on . This does not quite fit as a space , because of the decomposition into regular and singular geodesics, and the occurrence of parallel lines, which we would like to identify. So we introduce

The quotient space has commuting actions of and . One must put a measure on it in order that the -action on be ergodic.

Look at the classical (Bruhat-Tits) example. where 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 **

-buildings are to buildings what -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* is a finite dimensional real vectorspace equipped with a finite linear reflection group . Basic example is

(called apartment of type ). The hyperplanes fixed by reflections of are called *walls*. They split into polyhedral cones called *Weyl chambers*. For instance,

*Roots* are linear functionals . Those which vanish on walls bounding (i.e. ) are called *simple roots*. Since the closure is a fundamental domain for , there is a projection .

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 of translations. The affine reflection group is .

** 11.2. Buildings **

An atlas modelled on is a collection of injections into a set , such that

- Invariance under precomposition by .
- Transition maps are in .

The apartments, Weyl chambers,… of are the images of injections, of Weyl chambers… of . A sub-chamber of a Weyl chamber means a Weyl chamber contained in . The germ of a Weyl chamber is a neighborhood of the base point.

The atlas is called an -building provided the following two extra properties hold.

- Any two germs of Weyl chambers in lie in a common apartment.
- Any two Weyl chambers admit sub-chambers that lie in a common apartment.

** 11.3. Examples **

- Model apartment themselves.
- Real trees with extendible geodesics. There, with . The atlas consists of all geodesics.
- Products of -buildings are -buildings.
- Bruhat-Tits buildings associated to reductive algebraic groups over ultrametric fields.
- Asymptotic cones of symmetric spaces are -buildings (Kleiner-Leeb).

** 11.4. The space of ultrametric norms **

I will describe the Bruhat-Tits buildings associated to .

Let be a field equipped with an absolute value satisfying the ultrametric triangle inequality

For instance, the -adic absolute value on . It takes a discrete set of values. We are also interested in nondiscrete absolute values. Here is an example. Fix an arbitrary additive subgroup . -nomials are finite sums . Set

This is an ultrametric absolute value on the field of -nomials.

Let be a finite dimensional -vectorspace. A norm on is a nonnegative function on such that

- .
- .
- .

Say is *splittable* if there exists a basis of and numbers such that

Set splittable normshomotheties. 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 *splits* over a decomposition if it is the max of norms on summands.

Fix a decomposition with (call this a *frame*). Fix a norm on each . Let denote the model apartment of type . The map

will be an apartment in .

Let be the group of values of the absolute value on . Let be the corresponding group of translations of .

**Fact**. In this way, we get a building structure on , with an action of .

**Examples**.

- The stabilizer of the point is , where is the ring of numbers of absolute value .
- Diagonal matrices stabilize the apartment corresponding to the standard frame, they act on it by translations.
- For , , the fixed point set of is

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 metric **

Since changes of charts are isometries, Euclidean metrics on apartments piece together into a well defined two-point function on . Triangle inequality and property need be proved.

** 11.7. The retraction **

A building can be folded onto a single apartment.

Proposition 42Let be a building, an apartment. Fix a point . There is a unique distance nonincreasing retraction such that .

Indeed, fix a germ of Weyl chamber at . Given , there is an apartment containing and , and a unique isometry fixing . It maps to .

By construction, maps germs of Weyl chambers to germs of Weyl chambers.

** 11.8. The -distance **

It is a refinement of , which takes its values in the closure of the model Weyl chamber . Given and , pick an apartment containing both and set

The model apartment has an order, where is the cone dual to . Note that the folding map is subadditive:

This implies

Theorem 43 (Triangle inequality for , Lidskii 1950)

Equality holds if and only if and belong to opposite Weyl chambers at .

** 11.9. Finsler metrics **

Any -invariant norm on gives rise to a metric on (this follows from the triangle inequality for (and the fact that is nondecreasing on ).

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

** 11.10. Weak convexity **

A subset is *weakly convex* if for every , at least one -geodesic from to is contained in .

This allows to speak of weak convex cocompactness for isometric group actions on . Interesting examples exist: actions of surface groups on arising from cubic differentials. To describe them, one passes via projective plane , which is the visual boundary of in its metric.

** 11.11. Visual boundary **

The building is equipped with its metric. We mean the set of equivalence classes of geodesic rays, equipped with Tits’ angle metric.

Proposition 44Let be a building modelled on . The visual boundary is a spherical building modelled on . Its apartments are visual boundaries of apartments of (tiled spheres), its Weyl chambers are boundaries of Weyl chambers of .

**Example**. If is of type , any corresponding spherical building is the incidence pattern of some -dimensional projective space.

For , the space of splittable norms on , has two types of walls. Type 1 correspond to lines in , i.e. points in (call this type “point”), type 2 to planes in , i.e. lines in (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 . 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 .

** 11.12. Cross-ratios **

They are defined on the *transverse spaces* (see Caprace and Lecureux’s lectures). I describe them only in the example .

If is a singular boundary point, is a tree. Its ideal boundary identifies with . Therefore 4 points of have an algebraic cross-ratio . They also have a geometric cross-ratio: in the tree, project and onto the line joining to to and , and set . Then

**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 in mod additive constants. If is proper, one gets a compactification. For a space, one gets the visual compactification.

Assume that is a symmetric space in its Riemannian metric. In the boundary, each Weyl chamber is a cross section for the 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 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

- The 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 , the Furstenberg boundary, the largest is itself.
- The stratification by orbits is a manifold with corners structure.
- The Finsler compactification is homeomorphic to a ball. There is a (non-canonical) -equivariant homeo with the unit ball of the dual Finsler metric.
- It coincides with the maximal Satake compactification of , known to be real analytic.

** 12.2. Application to discrete subgroups **

Let be a discrete, weakly uniformly regular subgroup. We would like to compactify by attaching quotients of domains of proper discontinuity at infinity. We need to understand the action of 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 of the limit set, of all Finsler boundary points such that is contained in . Denote by its complement.

Theorem 46Let be a discrete, -regular subgroup.

- The action on is properly continuous. is an orbifold with corners.
- This bordification is a compactification if and only if is RCA (regular, conical and antipodal).

RCA generalizes one of the equivalent definitions of convex cocompactness in rank 1. It implies that 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 be -RCA. Then is Gromov hyperbolic, and limit set is equivariantly homeomorphic to ideal boundary.

In the thickening of the limit set, collapse each neighborhood of to . This leads to a quotient space

Then the action on is a convergence action. We shows that is cocompact on . 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. is replaced with the affine Weyl group.

Riemannian picture. Let a sequence 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 . Two cases.

- 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.
- 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, acts on each side via a different quotient , where is a singular direction.

**13. Alexander Lytchak: Minimal disks and 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 spaces, at least in the proper case.

Theorem 47Let be a proper geodesic space. Then is if and only if it satisfies the Euclidean filling inequality: every loop of length bounds a disk of area .

**Question**. What if is not proper ?

** 13.2. Previous results **

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

- Reshetnyak if is a Riemannian disk.
- Busemann, Santalo, Holmes-Thompson if is a normed plane, with Hausdorff area.
- Wenger: if filling function is , then 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 space is again .

Assume some triangle is not thin. Some work is needed to reduce to a Jordan triangle . Fill it with a minimizing disk. Thanks to Gauss, one may forget the rest of . 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 is conformal and locally Lipschitz. So we have formulae for the areas of domains and the lengths of (almost every) curve ,

for some function .

Let denote concentric circles centered at some point of . Then isopermietric inequality

This is equivalent to the differential inequality (Beckenbach-Rado and Reshetnyak already used this). If where smooth, this would mean that Riemannian metric 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 almost everywhere. Nevertheless, the resulting metric space is not .

**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, 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 be the disk equipped with the measurable Riemannian metric . The analytical arguments go through and show that is nonpositively curved and thus .

Introduce metric on disk defined by infimal length of images of rectifiable curved joining to . Get a new metric space with a map . One need show that is isometric. Map 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 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 .

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

** 14.2. Busemann spaces **

Definition 48A 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 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

What about measures ? For combinations of Dirac measures, one is tempted to use previous construction. This does not work. Indeed, and 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)

We take this limit as a definition for .

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

It is a 1-Lipschitz map.

** 14.3. Applications **

Theorem 50Compact 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 be an arbitrary countably infinite group. Let be the affine space of functions whose sum equals 1.

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

Theorem 51Let act on , preserving a probability measure. Let be a random variable with values in a Busemann metric space. Then

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 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 be a separable Hilbert space. Fix a Hilbert basis and define the quadratic form

The corresponding Grassmannian

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

Let denote the boundary of hyperbolic -space. We describe representations of on parametrized by .

If , is unitary.

Theorem 52Assume . There exists an intertwining operator

Furthermore, is -equivariant.

Therefore, the sesquilinear form

is -invariant.

More about : It preserves the decomposition of into -irreducibles,

and is scalar on each of them, , where

When some vanishes, is not irreducible. This does not happen if . If fact, all in this interval, so is positive definite.

If instead , has one minus sign, leading to an action on . And so on…

** 15.2. Other Lie groups **

For or , also exists, but sign jumps from to , so no finite rank examples.

Theorem 53 (Duchesne)Let be a simple Lie group distinct from and . Let be a cocompact lattice. For every irreducible action of on , there is a -equivariant, totally geodesic map.

**Question**. What about ?

** 15.3. Rigidity **

Say an action on is geometrically Zariski dense if there is no fixed point at infinity and non invariant closed totally geodesic manifold. Then is geometrically Zariski dense for . What happens for larger is unclear.

**Question**. Are the only irreducible actions on ?

Theorem 54 (Monod-Py)If and , then there is no irreducible action of on .

** 15.4. Proof **

Given an action, has a fixed point, hence a representation with an invariant -dimensional vectorspace . If , has no irreducible linear representations of dimension but for the trivial one.

It is a general fact that in any irreducible unitary representation of , -invariants have dimension 1 (2 for orthogonal representations). We prove an analogous result with replaced with

** 15.5. Infinite dimensional hyperbolic space **

Theorem 55 (Monod-Py)Any irreducible action of on belongs to the family.

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

Theorem 56 (Monod-Py)Consider a action of on . Let denote the closed convex hull of the -orbit of the -fixed point. Then is locally compact, is cocompact on it. Its isometry group is precisely . Distinct provides nonhomothetic sets.

This illustrates a theorem of Caprace-Monod. They show that geodesically complete space with a cocompact action of a simple Lie group has to be the symmetric space. Of course, is not geodesically complete.

**Question**. What happens as tends to ?

**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 -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 -valued refined distance to other settings, like Riemann surfaces equipped with holomorphic cubic differentials, this leads to canonical (equivariant) maps of such surfaces into -buildings. Such differentials arise in higher Teichmüller theory, in connection with representations of surface groups in . Pierre Pansu wonders wether a refined distance could be defined for certain cube complexes.

** 16.3. Asymptotic cones of infinite dimensional symmetric spaces **

Let be the symmetric space of (as in Py’s talk). If , all has isometric asymptotic cones, the universal tree (see Cornulier’s lectures). Is it still true for ? Cornulier says yes, since it is defined from a separable Hilbert space. And dealing with a nonseparable Hilbert space would not change this.

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