** Existence of conic Kaehler-Einstein metrics **

Joint work with Feng Wang, Zhejiang university.

A log-Fano manifold is the date of a compact Kaehler manifold , a divisor with normal crossings such that the line bundle

is positive.

A metric is a conic Kaehler-Einstein metric if it is smooth Kaehler in and for every point where is defined by in some coordinates, is equivalent (between two multiplicative constants) to the model cone metric

Say that is conic Kaehler-Einstein if

**1. Necessary conditions **

Berman 2016: If admits a conic KE metric with , then is log-K-stable.

Log-K-stability is defined as follows.

A special degeneration of is a 1-parameter family of log-pairs, consisting of

- A normal log-pair with a -equivariant map ,
- is an equivariant -ample -line bundle.
- is isomorphic to for every .

There is a natural compactification of that maps to . Defined number

If the central fiber is a log-Fano variety embedded in by , then can be interpreted as a Futaki invariant.

Say that is log-K-semistable if for any special degeneration has . Say that is log-K-stable if for any special degeneration has and equality holds only for the trivial degeneration .

**2. The result **

Theorem 1If is log-K-stable, the there exists a conic KE metric with .

Many special cases were known, as consequences of existence of KE metrics on smooth closed manifolds. For instance when is a multiple of .

**3. Motivation **

We are interested in -Fano varieties . Assume admits a resolution such that , . For small enough , define

If there exists a KE metric on , then is a degenerate conic KE metric on with conic angles along . We expect that there exist conic KE metrics on with , which Gromov-Hausdorff converge to as .

We think that we are now able to prove the following. *If is a K-stable -Fano variety. Then it admits a generalized KE metric in the above sense*.

**4. Proof **

Many steps are similar to the smooth case. Pick a large integer such that has a smooth divisor . We use a continuity method, solving

. The set of such that a solution exists is easily shown to be non-empty (it contains 0) and open. Is it closed? The key point is a estimate. It follows from a “partial -estimate” and log-K-stability. In turn, this follows from an -estimate and compactness a la Cheeger-Colding-Tian.

** 4.1. Smoothing conical KE metrics **

Say that has a K-approximation if there exist Kaehler metrics in the same cohomology class such that

- uniformly on and smoothly outside ,
- ,
- in Gromov-Hausdorff topology.

We show that if and if for all ,

for some , then has a K-approximation where .

We solve a modified equation with an extra term involving ‘s. For this, we use the variational approach by Boucksom-Eyssidieux-Guedj-Zeriahi and results of Darwan-Robinstein, Guenancia-Paun.

** 4.2. Extend B. Wang-Tian’s results to conic case **

**5. Work in progress **

To handle -Fano varieties, we need to extend Cheeger-Colding to conic cases.

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** Geometry and analysis of waves in manifolds with boundary **

The wave-front is a subset of the cotangent bundle, whose projection is the singular support. In all dimensions, in Euclidean space, it travels at constant speed along straight lines (Fermat,…, Hormander).

In general Riemannian manifolds without boundary, it travels along geodesics as long as time stays less than the injectivity radius (Duistermaat-Hormander).

We impose Dirichlet boundary conditions. Then transverse waves reflect according to Snell’s law of reflection (Chazarain). What about tangencies? Assume obstacle is convex. Do waves propagate in the shadow?

Melrose-Taylor 1975: if the boundary is , no smooth singularities in the shadow region. However, analytic singularities occur.

Inside strictly convex domains, waves reflect a large number of times. The wave shrinks in size between two reflections, it refocusses, therefore its maximum increases. Caustics appear, together with swallowtail and cusp singularities.

In the non-convex case, especially if infinite order tangencies occur, one does not even know what the continuation of a ray should be (Taylor 1976).

**1. Dispersive estimates **

It is a measurement of the decay of amplitude of waves due to spreading out while energy is conserved.

In , after a high frequency cut-off around frequency , the maximum amplitude decays like . Indeed, the wave is concentrated in an annulus of width . The same holds in Riemannian manifolds without boundary.

In the presence of boundary, propagation of singularities has brought results in the 1980’s. Later on, people have tried a reduction to the boundary-less case with a Lipschitz metric: this requires no assumptions on the boundary, but ignores reflection and its refocussing effect.

** 1.1. Within convex domains **

Theorem 1 (Ivanovici-Lascar-Lebeau-Planchon 2017)For strictly convex domains, dispersion is in

This follows from a detailed description of the wave-front, including swallow-tails. It takes into account infinitely many reflections. It is sharp.

** 1.2. Outside convex obstacles **

The Poisson spot. This is a place where diffracted light waves interfere. It is in the shadow area, but much more light concentrates there. This was confirmed experimentally by Arago, following a debate launched by Fresnel who did not believe in the wave description of light. It should exist if one believes in Fermat’s principle that light rays follow geodesics, including those which creep along the boundary surface (Keller’s conjecture). In 1994, HargĂ© and Lebeau proved that, when light creeps along the bounday, it decays like .

Theorem 2 (Ivanovici-Lebeau 2017)For strictly convex obstacles,

- if , dispersion estimates hold like in ,
- if , they fail at the Poisson spot.

The reason is that a -dimensional surface lits the Poisson point.

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** Asymptotic expansions of holonomy **

Joint with Pierre Pansu.

**1. Motivation **

Given a connection on a principal bundle , holonomy along a based loop of is an element of resulting from lifting horizontally to . We look for an expression such that is a good approximation of holonomy when is short,

We want that be simpler to compute than holonomy, and be related to curvature.

Hatton-Choset: motion of a snake with two joins. , . Experimentalists have been led to choose the Coulomb gauge, and for the integral over a disk spanning of curvature expressed in Coulomb gauge.

In this practical example, motions are tangent to a sub-bundle of the tangent bundle of . Hence our interest in expansions which are particularly efficient on such curves. We call this setting *sub-Riemannian*.

Sub-Riemannian curvature is not easy to define. The obvious approach of using adapted connections on the tangent bundle is not illuminating.

**2. Results **

- Asymptotic, gauge-free formula in Euclidean space.
- Riemannian case not that different.
- Sub-Riemannian case suggests a notion of curvature.
- For certain sub-Riemannian structures,

** 2.1. Euclidean case **

Dilations define radial fillings of loops. Use radial gauge (frame is parallel along rays through the origin). They turn out to be optimal. Using radial gauge, integrate curvature over radial filling. This defines

Say a differential form has weight if dilates are . Use radial gauge to define weight of forms on .

Theorem 1If the curvature has weight , thenFurthermore, one can expand in termes of Taylor’s expansion of curvature.

Since curvature has weight at least 2, one gets a 4-th order approximation.

** 2.2. Sub-Riemannian case **

The flat sub-Riemannian case corresponds to Carnot groups, i.e. a Lie group whose Lie algebra has a gradation

and is generated by . **Example**: Heisenberg group.

Fix a norm on . Left translates of define a sub-Riemannian metric, for which dilations on are homothetic.

According to Le Donne, sub-Riemannian Carnot groups are characterized by being the only locally compact homogeneous geodesic metric spaces with homothetic homeos.

Carnot groups come with a left-invariant horizontal basis, we pick a connection on the tangent bundle which makes it parallel. It has torsion. We combine it with the principal bundle connection to define iterated covariant derivatives of curvature. We organize them according to weights adapted to the Lie algebra grading. The above theorem extends.

** 2.3. Horizontal holonomy **

Since we are interested only in holonomy along horizontal loops, we have the freedon to change the connection outside the horizontal subbundle.

Chitour-Grong-Jean-Kokkonen: using this freedom, there are choices which minimize the curvature in the sense that as many components as possible vanish identically. This tends to increase the weight of curvature.

**Example**: on 3-dimensional Heisenberg group, the preferred connection has curvature which vanishes on the horizontal distribution, hence has weight instead of 2. Above Theorem provides a 6-th order expansion, whose terms can be computed algebraically.

More generally, on free -step nilpotent Lie groups, the curvature of a preferred connection has order at least , whence a -th order expansion whose terms are linear in curvature (in fact, in the preferred curvature).

We expect to use it to refine the Euclidean expansion.

**3. Question **

What does this give in case of the two-joints snake? Requires to push computations further.

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** The borders of Outer Space **

Joint work with Kai-Uwe Bux and Peter Smillie.

**1. Duality groups **

I am interested in Poincare duality. For a group, assume is a smooth -manifold, then

Bieri-Eckmann observed that is suffices that acts freely cocompactly on a contractible space whose compactly supported cohomology vanishes in all degrees but , and is torsion free. Then is a duality group.

If the action is merely proper and cocompact, is a virtual duality group. Borel-Serre used this for lattices. Bestvina-Feighn used this to show that $latex {Out(F_n) is a virtual duality group. Mapping class groups also act on a contractible space.

To achieve cocompactness, Borel-Serre added to the symmetric space copies of Euclidean space forming a rational Euclidean building. The resulting bordification of the quotient is a manifold, with boundary homotopy equivalent to a wedge of spheres (Solomon-Tits). Instead, Grayson constructed an invariant cocompact subset of symmetric space.

Grayson’s work was used by Bartels-Lueck-Reich-Ruping to prove Farrel-Jones for }&fg=000000$Sl(n,{\mathbb Z})$latex {.

For }&fg=000000$Out(F_n)J_n\subset X_n$latex {. It is much easier and gives more information on the boundary.

\section{Construction}

}&fg=000000$X_nJ_n$ is obtained by chopping off some of their corners.

A core graph is a subgraph such that, when one shrinks it, one gets out of Outer Space. Only corner facing core graphs need be chopped off.

The boundary appears as a union of contractible walls (every intersection of walls is contractible).

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** Topology of ends of nonpositively curved manifolds **

Joint work with T. Nguyen Pham.

I am interested in complete Riemannian manifolds with curvature in , and finite volume.

**Example**. Product of two hyperbolic surfaces. The end is homeomorphic to , with some extra structure: is made of two pieces.

More generally, for locally symmetric spaces of noncompact type, lifts of ends are homeomorphic to , with a wedge of spheres. This description goes back to Borel-Serre.

**1. Thick-thin decomposition **

Gromov-Schroeder: assume there are no arbitrarily small geodesic loops. Then the thin part is homeomorphic to , with a closed manifold.

The condition is necessary. Gromov gives an example of a nonpositively curved infinite type graph manifold of finite volume.

Theorem 1 (Avramidi-Nguyen Pham)Under the same assumptions, any map of a polyhedron to the thin part of the universal cover can be homotoped within the thin part into a map to an -dimensional complex, .

**Consequences**:

- If , each component of the thin part is aspherical and has locally free fundamental group.
- for all .
- .

**2. Proof **

Maximizing the angle under which two visual boundary points are seen gives Tits distance, and the corresponding path metric .

In the universal cover, the thin part is the set of points moved less than away by some deck transformation . Isometries are either hyperbolic (minimal displacement is achieved) or parabolic (infimal displacement is 0). Parabolic isometries have a nonempty fix-point set at infinity. At each point , the subgroup generated by isometries moving no more than is virtually nilpotent, hence virtually has a common fixed point at infinity. This allows to define a discontinuous projection to infinity. The point is to show that the image has dimension .

** 2.1. Busemann simplices **

If and are Busmeann functions, need not be a Busemann function again, but on each sphere, there is a unique point where it achieves its minimum, and tis point depends in a Lipschitz manner on . This defines an arc in Tits boundary, hence simplices . We claim that

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** Free-by-cyclic groups and trees **

Joint work with S. Dowdall and I. Kapovich.

The Bieri-Neumann-Strebel invariant is an open subset of , it is the set of such that is surjective on . Here, is the torsion free abelian cover of and is an equivariant map representing .

If is free-by-cyclic, one can refine

Geoghegan-Mihalik-Sapir-Wise show that for every , is locally free and there exists an outer automorphism and a finitely generated subgroup such that . In particular, if , then one can take .

From now on, we assume that is atoroidal and fully irreducible. Then is hyperbolic, and there exists an expanding irreducible train track representative (Bestvina-Handel). Let be the mapping torus. It carries the suspension of , which is a one-sided flow (action of semi-group ). The representative of integral cohomology class factors to a map . Let be the subset of cohomology classes such that the representative can be chosen to be increasing along the flow. Then

Theorem 1

- is a component of . It is a rational polyhedral cone.
- For , inverse images of points are cross-sections of the flow. The first return map is an expanding irreducible train track representative of , with .

Stretch factors form a nice function on .

Theorem 2 (Algom-Kfir-Hironaka-Rafi)There exists an -analytic, convex function such that for all such that for al and ,

- .
- .
- If , then .

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** Action dimension and Cohomology **

Joint work with Giang Le and Mike Davis.

**1. Action dimension **

This is the minimal dimension of contractible manifolds which admit a proper -action. The geometric dimension replaces manifolds with complexes.

** 1.1. Examples **

If is of type , then . This comes from embedding complexes into . would be easy. is Stallings’ theorem, using a suitable model of .

Bestvina-Feighn: For lattices in semi-simle Lie groups, is the dimension of the symmetric space.

Desputovic: Teichmuller space.

** 1.2. Our favourite examples **

Today, we focus on graph products of fundamental groups of closed aspherical manifolds and complements of hyperplane arrangements. We are concerned with lower bounds: when can one reduce from the obvious dimension?

The first class (circles) includes RAAG, covered by Avramidi-Davis-Okun-Shreve.

** 1.3. Motivation from -cohomology **

Let denote the -Betti numbers of the universal covering.

**Singer conjecture**: If is a closed aspherical manifold of dimension , then vanish if .

This suggests

**Action dimension conjecture**. If , then .

Okun and I have shown that both conjectures are in fact equivalent.

**2. Graph products **

Let be a flag complex with vertex set . The graph product of a family of groups over is the quotient of the free product of by the normal subgroup generated by , when is an edge of .

**Examples**. If all , we get RAAG. If all are finite cyclic, we get RACG.

Theorem 1Let be a -dimensional flag complex, let be the corresponding graph product of fundamental groups of closed aspherical -manifolds. Then

- If , then .
- If , then .

** 2.1. Constructing aspherical manifolds **

The only way to make new aspherical manifolds is to glue aspherical manifolds with boundary along codimension 0 submanifolds of their boundaries. For instance, Salvetti complexes, made of tori, do not work. We replace tori with tori interval.

In general, we glue together products of , which is dimensional, which is sharp in some cases, as we show next. The fact that has vanishing homology allows to decrease dimension.

** 2.2. Obstructions to actions **

Bestvina-Kapovitch-Kleiner coarsify van Kampen’s obstruction to embedding complexes into . This lives in (configuration of pairs of points).

Theorem 2 (Bestvina-Kapovitch-Kleiner)Let be or hyperbolic, let with . Then

**Example**. If , contains , hence (in fact, ).

For graph products of closed aspherical manifolds, we construct a complex, denoted by , in . It is a join of -spheres based on .

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** The dynamics of classifying geometric structures **

**1. Marked geometric structures **

Moduli spaces of geometric structures do not all behave like the moduli space of Riemann surfaces: in general, it is not a well behaved space, it is a quotient by a group action with interesting dynamics.

Lie and Klein (1872), Ehresmann (1936) suggest to study -structures on manifolds . Experience shows that it is useful to introduce a deformation space of *marked *-structures, on which the *mapping class group* acts. A marking is the data of a -manifold and a diffeomorphism .

In some cases (e.g. hyperbolic structures on surfaces), this action is properly discontinuous, resulting in a quotient space which is a manifold mere singularities. In general, it is not.

** 1.1. Example: complete affine surfaces **

All Euclidean structures on the 2-torus are affinelu isomorphic. Other affine structures, discovered by Kuiper, are obtained from the polynomial diffeomorphism

Indeed, change of charts turn out to be affine.

The mapping class group acts ergodically on the deformation space (Moore 1966).

**2. Moduli spaces of representations **

Let be a closed surface, . Let be a simple Lie group. Connected components of are indexed by .

With Forni, we try to use Teichmuller dynamics, and replace the difficult action by a simpler action. This is defined on the unit tangent bundle of Teichmuller space .

Let . This is a bundle over . Let be its unit tangent bundle.

Theorem 1 (Forni-Goldman)For compact, the Teichmuller flow is strongly mixing on .

Each element of defines a character function, hence a Hamiltonian flow. Dehn twists suffice to generate the ring of functions, hence

** 2.1. An example: compact surfaces of Euler characteristic **

There are 4 of them, all have . was determined as early as 1889. It is isomorphic (as a complex manifold) to .

The function is invariant under (Nielsen). Level sets have invariant symplectic structures. Interesting involutions arise as deck transformations of branched double coverings given by coordinate projections to .

Level sets for values in contain a component corresponding to unitary representations, on which the action is ergodic.

The case of the once-punctured Klein bottle is particularly interesting. The action does not extend to projective space.

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** Extending group actions on metric spaces **

Joint work with David Hume and C. Abbott.

**Question**. Let be groups. Given an isometric action of on a metric space , does it extend to an action on a (possibly different) metric space ?

**1. Extensions of actions **

What to be mean by extension? We have in mind induction of representations.

Let act on and . Say that a map is *coarsely equivariant* if for every , is bounded on .

Definition 1Say an action of group on is an extension of the action of subgroup on is there exists a coarsely -equivariant quasi-isometric embedding .

Definition 2We say that the extension problem (EP) for is solvable if every action of on a metric space extends to an action of .

** 1.1. Examples **

This is rather flexible.

- If has bounded orbits, the trivial action of is an extension.
- If is a retract of (i.e. there exists a homomorphism which is the identity on ), then every actions of extends.
- Fix finite generating systems of and . Assume is undistorted in . Then the action of on its Cayley graph extends to the action of on its Cayley graph.
- An example where (EP) is not solvable. Let . Then every action of on a metric space has bounded orbits (Cornulier). If , no action of with unbounded orbits can extend.
- A converse of (3) holds: if is finitely generated and (EP) is solvable for then is finitely generated and undistorted in . Whence many examples where (EP) is not solvable. Furthermore, if is finitely generated and elementarily amenable, then (EP) is solvable for all implies that is virtually abelian.
- Let be a free group and where exchanges generators. Then translation action of on with one generator acting trivially cannot extend to . Indeed, one generator of has bounded orbits, the other does not, but both are conjugate in .

** 1.2. Hyperbolic embeddings **

The following definition appears in Dahmani-Guirardel-Osin. Let be a subset such that generates . Let be the metric on induced by the embedding of (as vertex set of complete graph ) into with edges of removed. Say that is *hyperbolically embedded* in if

- is hyperbolic,
- is proper.

For instance,

- is not hyperbolically embedded into , but it is into .
- Observe that there exists a finite subset such that is hyperbolically embedded into iff is hyperbolic relative to .
- If is pseudo-Anosov, then there exists a virtually cyclic subgroup containing which is hyperbolically embedded in .

** 1.3. Acylindrically hyperbolic groups **

This class contains , , finitely presented groups of deficiency (argument uses -Betti numbers).

Theorem 3 (Dahmani-Guirardel-Osin)If is acylindrically hyperbolic, then it contains hyperbolically embedded subgroups of the form finite for all .

**2. Results **

Theorem 4Let be hyperbolically embedded. Then (EP) is solvable for . Moreover, every action of on a hyperbolic metric space extends to a action of on a hyperbolic metric space.

Corollary 5Let be a hyperbolic group, and .

- If is virtually cyclic, then (EP) is solvable for .
- If is quasi-convex and almost malnormal ( for all ), then (EP) is solvable for .
- Conversely, if (EP) for is solvable, then is quasi-convex.

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** Hyperbolic groups whose boundary is a Sierpinski -space **

Joint work with Bena Tshishiku.

**1. Sierpinski -space **

Start with an -dimensional sphere. Remove a dense family of balls with disjoint interiors. Get . Up to homeo, balls need not be round. One merely needs that their diameters tend to 0.

Any homeo of permutes the distinguished *peripheral spheres*.

**Examples**.

- Free groups have ideal boundary .
- If is a compact negatively curved -manifold with nonempty totally geodesic boundary, then .
- Let be a nonuniform lattice of isometries of . Then is cocompact on the complement of a union of horospheres, hence .

** 1.1. Cannon conjecture **

What properties of the group follow from specifying the topology of the boundary ? This is what Cannon’s conjecture is about: if , must be a cocompact lattice in ?

Here is a topological variant of Cannonc’s conjecture.

Theorem 1 (Bartels-Lueck-Weinberger)If is torsion-free hyperbolic, and , and , then there exists a unique closed aspherical -manifold with .

**2. Result **

Theorem 2If is torsion-free hyperbolic, and , and , then there exists a unique aspherical -manifold with nonempty boundary with . Moreover, every boundary component of corresponds to a quasi-convex subgroup of .

**3. Proof **

** 3.1. Step 1 **

Kapovitch-Kleiner: is a relative -group, relative to the collection of stablizers of peripheral spheres.

** 3.2. Step 2 **

Realize as , where is a finite relative complex, relative to a finite subcomplex . We use the Rips complex for but the Bartles-Lueck-Weinberger complexes for parabolic subgroups .

** 3.3. Surgery theory **

Browder-Novikov-Sullivan-Wall surgery theory provides obstructions to finding a manifold homotopy equivalent to . They belong to the space that appears in the algebraic surgery exact sequence

A similar exact sequence appears in 4-periodic surgery exact sequence, with replaced with a very similar (and with ). They have the same homotopy groups and differ only in their 0-spaces

There is a long exact sequence

It turns out that . Furthermore, thanks to the (L-theoretic) Farrell-Jones isomorphism conjecture (which holds for hyperbolic groups, Bartels-Lueck-Reich), . Hence , the obstruction vanishes, so there exists a homology manifold model for .

** 3.4. groups **

Bartels-Lueck-Reich cover groups. In the relative case (replace spheres with Sierpinski spaces), much of the argument carries over, but the first step.

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