## Notes of Frédéric Bourgeois’ lecture

Derived category of sheaves

1. Generalities on categories

The categories we shall encounter are

1. Sets.
2. Top, Diff.
3. Abelian Groups.
4. ${k}$-modules.
5. Sh${(M)}$, sheaves on ${M}$.

Examples of functors that we shall encounter are

1. Forgetful functor to Sets.
2. ${Hom(X,\cdot)}$.
3. ${\otimes^n}$
4. ${Hom_k(\cdot,M)}$.
5. Sections of a sheaf ${\Gamma}$: OpenSets${(M)\rightarrow}$ Abelian Groups.

There is a notion of morphism of functors. A functor is representable if it is ${Hom(X,\cdot)}$ for some object ${X}$.

An additive category is a category such that

1. There is an object ${0}$ such that ${Hom(0,0)=0}$.
2. All ${Hom(X,Y)}$ are abelian groups, and composition is bilinear.
3. There is a direct sum, i.e. for all ${X_1}$, ${X_2}$, an object ${Z}$ with arrows ${i_j \in Hom(X_j,Z)}$ such that ${Hom(X_j,Y)=Hom(Z,Y)\circ i_j}$ for ${j=1,2}$.
4. There is a direct product, i.e. for all ${Y_1}$, ${Y_2}$, an object ${W}$ with arrows ${k_j \in Hom(W,Y_j)}$ such that ${Hom(X,Y_j)=k_j \circ Hom(X,W)}$ for ${j=1,2}$ and all ${X}$.

Ab, ${k}$-Mod, Sh${(M)}$ are additive.

A functor ${F}$ is additive if the map ${Hom(X,Y)\rightarrow Hom(F(X),F(Y))}$ is a group homomorphism.

1.2. Complexes

If ${\mathcal{C}}$ is an additive category, the category of complexes ${C(\mathcal{C})}$ is additive as well. We denote by ${T^n}$ the shift functor that re-numbers spaces in a complex. A homotopy between chain maps ${f,g:X\rightarrow Y}$ is a morphism ${h:X\rightarrow Y[-1]}$ such that ${f-g=hd+dh}$. The intermediate category ${\mathcal{K}(\mathcal{C}))}$ is the additive category with objects complexes and morphisms homotopy classes of chain maps.

This category is merely an intermediate step.

1.3. Abelian categories

Homology cannot be defined for general additive categories, since we do not have kernels and images. For ${f\in Hom(X,Y)}$, the kernel functor ${ker(f):\mathcal{C}^0\rightarrow }$ Sets, is defined as follows :

$\displaystyle \begin{array}{rcl} ker(f)(W)=\{g\in Hom(W,X)\,;\,f\circ g=0\}. \end{array}$

If this functor is representable, this provides us with an object, also denoted by ${ker(f)}$, such that ${ker(f)(W)=Hom(W,ker(f))}$.

Similarly, given ${f\in Hom(X,Y)}$, there is a cokernel functor ${coker(f):\mathcal{C}\rightarrow}$ Sets. If this functor is representable, this provides us with an object, also denoted by ${coker(f)}$.

Remark 1 ${Z\mapsto Hom(Z,Y)/f\circ Hom(Z,X)}$ is a functor, but it is not representable, usually. So this is not the right definition for cokernel.

Indeed, in Ab, let ${f:{\mathbb Z}\rightarrow{\mathbb Z}}$ is multiplication by ${n}$, and ${Z={\mathbb Z}_n}$. Then ${Hom({\mathbb Z}_n,{\mathbb Z})=0}$ whereas ${Hom({\mathbb Z}_n,{\mathbb Z}_n)\not=0}$.

Definition 1 In an abelian category,

1. For all ${f\in Hom(X,Y)}$, ${ker(f)}$ and ${coker(f)}$ exist.
2. Let ${c:Y\rightarrow coker(f)}$ and ${k:ker(f)\rightarrow X}$ be the natural maps. Then ${coker(k)}$ and ${ker(c)}$ are canonically isomorphic.

In fact, ${f}$ has a canonical factorization ${X\rightarrow I\rightarrow Y}$ where ${I=coker(k)=ker(c)}$.

1.4. Homology

If ${\mathcal{C}}$ is an abelian category, given a complex ${X}$, for all degrees ${n}$, the differential ${d^{n-1}:X^{n-1}\rightarrow X^n}$ factors as ${a^n : X^{n-1}\rightarrow ker(d^n)}$ and ${d^{n}:X^{n}\rightarrow X^{n+1}}$ factors as ${b^{n-1} : coker(d^{n-1})\rightarrow X^{n+1}}$. One defines the homology of ${X}$ as ${H(X)^n=\ker(a^n)=coker(b^{n-1})}$.

A quasi-isomorphism between complexes is a morphism which induces and isomorphism on homology. We want to think of quasi-isomorphisms as as good as isomorphisms.

Example 1 Consider the following two complexes.

$\displaystyle \begin{array}{rcl} 0\rightarrow{\mathbb Z}\stackrel{\times 2}{\rightarrow}{\mathbb Z}\rightarrow 0,\\ 0\rightarrow 0\rightarrow {\mathbb Z}/2{\mathbb Z}\rightarrow 0, \end{array}$

There is an obvious chain map between them. This is a quasiisomorphism, but there is no homotopy from it to 0.

1.5. Derived category

The derived category of an abelian category ${\mathcal{C}}$ has for objects complexes and for morphisms a localization of chain maps mod homotopy, which we now define as the solution of a universal problem.

Definition 2 The derived category of an abelian category ${\mathcal{C}}$ is the unique category such that there exists a functor ${Q:K(\mathcal{C})\rightarrow D(\mathcal{C})}$ such that

1. ${Q(}$quasiisomorphism${)=}$isomorphism.
2. For any category ${\mathcal{D}}$, any functor ${F:K(\mathcal{C})\rightarrow\mathcal{D}}$ such that

$\displaystyle F(quasiisomorphism)=isomorphism$

factorizes through ${Q}$.

Concretely, ${D(\mathcal{C})}$ is constructed as follows. Let ${S}$ be the collection of quasiisomorphisms in ${K(\mathcal{C})}$. Define

$\displaystyle \begin{array}{rcl} Hom_{D(\mathcal{C})}(X,Y)=\{(s,f)\in (Hom_{K(\mathcal{C})}(Z,X)\cap S)\times Hom_{K(\mathcal{C})}(Z,Y)\}/\sim \end{array}$

where the equivalence relation ${(s,f)\sim(t,g)}$ roughly means that ${f\circ s^{-1}=g\circ t^{-1}}$, and precisely means that there exist quasiisomorphisms ${s,v}$ such that

$\displaystyle \begin{array}{rcl} s\circ u=t\circ v,\quad g\circ v=f\circ u. \end{array}$

The functor ${Q}$ is defined as identity on objects (i.e. complexes) and on morphisms ${f}$, by ${Q(f)=(id,f)}$.

1.6. Localizing systems

The proof of the statement included in the definition of the derived category is not straightforward.

For the definition to work, one needs the following properties of ${S}$ (say ${S}$ is a localizing system) :

1. Identity belongs to ${S}$.
2. ${S}$ is stable under composition (whenever defined).
3. Given ${f\in Hom(X,Y)}$ and a quasiisomorphism ${t\in Hom(Z,Y)}$, there exists a complex ${W}$, ${g\in Hom(W,Z)}$ and a quasiisomorphism ${s\in Hom(W,X)}$ such that ${t\circ g=f\circ s}$.
4. For all ${f,g\in Hom(X,Y)}$, existence of ${t\in S}$ such that ${t\circ f=t\circ g}$ implies existence of ${s\in S}$ such that ${f\circ s=g\circ s}$.

Proposition 3 The collection ${S}$ of quasiisomorphisms is a localizing system.

For the proof, we shall use the notion of mapping cone. Given ${f\in Hom_{C(\mathcal{C})}(X,Y)}$, define the mapping cone ${M(f)}$ as ${M(f)=X[1]\oplus Y}$ with differential ${\begin{pmatrix} d & 0 \\ f & d \end{pmatrix}}$. There is an exact sequence ${X\stackrel{f}{\rightarrow}Y\rightarrow M(f)\rightarrow X[1]}$.

Now we can prove property 3 of localizing systems. Given ${t}$ and ${f}$, let ${j=j'\circ f}$ where ${j':Y\rightarrow M(g)}$ arises in the mapping cone exact sequence for ${t}$. Let ${W=M(j)[-1]}$ and ${s:M(j)[-1]\rightarrow X}$ arise in the mapping cone exact sequence for ${j}$. Then ${s}$ is a quasiisomorphism, there is a natural map ${g:M(j)[-1]\rightarrow Z}$ making the diagram commute.

Property 4 is proven in a similar way.

1.7. Derived functors

Any functor ${F}$ from ${C(\mathcal{C})}$ to itself induces a functor ${K(F)}$ from ${K(\mathcal{C})}$ to itself, but ${K(F)(}$quasiisomorphisms${)\not=}$quasiisomorphism in general (counterexample ?).

Say a functor ${F}$ if it maps short exact sequences to short exact sequences. Say ${F}$ is left exact if it maps short exact sequences to short sequences which are exact on the left and middle.

Proposition 4 An exact functor maps quasiisomorphisms to quasiisomorphisms, and so induces a functor ${D(F)}$ on derived categories.

Indeed, ${f}$ is a quasiisomorphism iff ${M(f)}$ is acyclic. For an exact functor, ${X}$ acyclic implies ${F(X)}$ acyclic (split ${X}$ into short exact sequences using kernels and images).

So life with exact functors is easy. Unfortunately, the functors we are interested in are not exact. For instance, ${Hom_k(\cdot,M)}$ is only left exact. ${\otimes^n}$ is right exact. The section functor for a sheaf is left exact.

Definition 5 The derived functor of an additive left exact functor from ${\mathcal{C}}$ to ${\mathcal{D}}$ is a pair

1. of an exact functor ${RF}$ from ${D^+(\mathcal{C})}$ to ${D^+(\mathcal{D})}$,
2. and a morphism of functors ${\epsilon_F}$ from ${Q_{\mathcal{D}}\circ K^+(F)}$ to ${RF\circ Q_{\mathcal{C}}}$,

such that for any exact functor ${G}$ from ${D^+(\mathcal{C})}$ to ${D^+(\mathcal{D})}$ and morphism of functors ${\epsilon}$ from ${Q_{\mathcal{D}}\circ K^+(F)}$ to ${G\circ Q_{\mathcal{C}}}$, there exists a unique morphism of functors ${\eta}$ from ${RF}$ to ${G}$ such that

$\displaystyle \begin{array}{rcl} \eta\circ Q_{\mathcal{C}}\circ \epsilon_F=\epsilon. \end{array}$

Looks like a mess.

1.8. Strategy for computing derived functors.

Select a subclass of objects from ${K(\mathcal{C})}$ (typically, resolutions). If functor ${F}$ is nice on subclass, work componentwise. If subclass is large enough, its localization is equivalent to the derived category ${D(\mathcal{C})}$.

## Notes of Davide Barilari’s lecture Nr 2

Heat diffusion for generic Riemannian and sub-Riemannian manifolds

With Boscain, Charlot, Jendrej, Neel.

1. Motivation

Understand interplay between analysis (heat) and geometry (distance, geodesics, curvature). In particular, asymptotics of ${p_t(x,y)}$ when ${y}$ lies in the cutlocus of ${x}$.

1.1. Example : the round 2-sphere

If ${x}$ and ${y}$ are antipodal,

$\displaystyle \begin{array}{rcl} p_t(x,y)\sim\frac{1}{t^{3/2}}e^{-d^2(x,y)/4t}, \end{array}$

where the exponent ${3/2}$ replaces the usual ${1=2/2}$.

1.2. Surfaces of revolution

For ellipsoids of revolution close to round, the cutlocus is a segment. This holds more generally for surfaces of revolution symmetric w.r.t. the equator.

Theorem 1 Assume further a non degeneracy condition (conjugate locus of ${x}$ does not osculate the cutlocus at its endpoint ${y}$). Then

$\displaystyle \begin{array}{rcl} p_t(x,y)\sim\frac{1}{t^{5/4}}e^{-d^2(x,y)/4t}. \end{array}$

In non generic cases, exponents like ${\frac{n}{2}-\frac{r}{3}}$ arise for all integer ${r\geq 3}$.

2. Sub-Riemannian cut and conjugate loci

In the sub-Riemannian case, distance${^2}$ is not smooth at ${x}$, it is smooth on an open and dense subset ${\Sigma(x)}$, the complement of the cut and conjugate loci (Agrachev). One can define an exponential map on the cotangent space at ${x}$, whence the notion of conjugate points.

3. General results

The Laplacian is defined once a smooth volume is defined.

Theorem 2 (Léandre) Assume further a non degeneracy condition (conjugate locus of ${x}$ does not osculate the cutlocus at its endpoint ${y}$). Then

$\displaystyle \begin{array}{rcl} \lim_{t\rightarrow 0}4t\log p_t(x,y)-d^2(x,y). \end{array}$

Theorem 3 (Bénarous) If ${y\notin\Sigma(x)}$,

$\displaystyle \begin{array}{rcl} p_t(x,y)\sim\frac{1}{t^{n/2}}e^{-d^2(x,y)/4t}. \end{array}$

Along the diagonal,

$\displaystyle \begin{array}{rcl} p_t(x,x)\sim\frac{1}{t^{Q/2}}, \end{array}$

where ${Q}$ is the homogeneous dimension à ${x}$.

Theorem 4 (Barilar, Boscain, Neel) If ${y\in\Sigma(x)}$ and every optimal geodesic joining ${x}$ to ${y}$ is strongly normal. Then

$\displaystyle \begin{array}{rcl} \frac{1}{t^{n/2}}e^{-d^2(x,y)/4t}\leq p_t(x,y)\leq\frac{1}{t^{n-1/2}}e^{-d^2(x,y)/4t}. \end{array}$

If ${y}$ is conjugate to ${x}$ or order ${r}$ (rank of differential of exponential map is ${n-r}$) along every optimal geodesic from ${x}$ to ${y}$, then

$\displaystyle \begin{array}{rcl} p_t(x,x)\sim\frac{1}{t^{(n-r)/2}}, \end{array}$

where ${Q}$ is the homogeneous dimension à ${x}$.

3.1. Proof

Use semi-group property and apply Bénarous’ estimate at ${t/2}$. Then

${y}$ is conjugate to ${x}$ along ${\gamma}$ ${\Leftrightarrow}$ ${Hess (d^2)}$ at midpoint is degenerate.

Furthermore, if there exist coordinates such that

$\displaystyle \begin{array}{rcl} h_{x,y}(z)=\frac{1}{4}d^2(x,y)+z_1^{m_1}+\cdots+z_k^{m_k}+o(|z|^{m_k}), \end{array}$

we get an expansion whose leading term in ${t}$ is ${t^{-n+\sum\frac{1}{2m_j}}}$.

This applies when kernel of Hessian is 1-dimensional (Gromoll-Meyer), and in particular in the Heisenberg group.

4. Generic results

4.1. Exponential map as a Lagrangian map

A map ${\pi:E\rightarrow M}$, ${E}$ symplectic, is Lagrangian if the fibers are Lagrangian. Exponential map does, since at each point it is a composition of projection with an hamiltonian flow.

Theorem 5 (Arnold’s school) Classification of generic Lagrangian singularities (i.e. Lagrangian maps up to symplectic coordinate changes of the domain and coordinate changes of the range) in dimensions ${\leq 5}$.

For instance, the A3 singularity in dimension 2 is the usual fronce.

4.2. Generic Riemannian results

Theorem 6 In dimension ${\leq 5}$, fix a point. For a generic Riemannian metric, the singularities of the exponential map are generic Lagrangian singularities.

Theorem 7 In dimension ${\leq 5}$, fix a point. For a generic Riemannian metric, the singularities of the exponential map are of type ${A_3}$ or ${A_5}$. ${A_3}$ arise only in dimension ${\geq 2}$, ${A_5}$ only in dimension ${\geq 4}$.

Corollary 8 Fof generic metrics in dimension ${n\leq 5}$, heat kernel asymptotics have exponent ${\frac{n}{2}+\frac{1}{4}}$ in case of an ${A_3}$ singularity, and ${\frac{n}{2}+\frac{1}{6}}$ in case of an ${A_5}$ singularity.

4.3. Generic Sub-Riemannian results

Stick to 3-dimensional contact structures. A generic points, the cut locus is a surface made of two opposite horned triangles.

Theorem 9 If no optimal geodesic from ${x}$ to ${y}$ is conjugate, then exponent is ${3/2}$.

If at least one optimal geodesic from ${x}$ to ${y}$ is conjugate, then genericly, exponent is ${7/4}$.

## Notes of Andrei Agrachev’s lecture

Some models of constant curvature in sub-Riemannian geometry

This is a reaction on Luca Rizzi’s talk last time (october 22nd).

1. GO spaces

V. Berestovskii has classified a class of sub-Riemannian manifolds which I now describe.

Definition 1 Say a Sub-Riemannian manifold is GO if every geodesic is an orbit of a one-parameter group of isometries.

This happens for a rather large class of Riemannian homogeneous spaces. Taking limits of such Riemannian homogeneous spaces provides examples of GO sub-Riemannian homogeneous spaces. Berestovskii has found examples which do not arise in this way.

Why are we interested in GO spaces ? Luca’s comparison theorem involves a notion of curvature (asociated to a point and geodesic) which is pretty complicated to compute. In the GO case, such curvature is constant along each geodesic.

1.1. Examples

Tangent cones of GO spaces are GO. Many step 2 Carnot groups are GO (not the most interesting, since curvature vanishes for Carnot groups).

Here is a less trivial example discovered by Berestovskii. Let ${M}$ be a Lie group with a bi-invariant pseudo-Riemannian metric (e.g. ${M}$ is semi-simple). Fix a left-invariant distribution generated by ${\Delta=\mathfrak{k}^{\bot}}$, where ${\mathfrak{k}\subset\mathfrak{m}}$ is a subalgebra. If ${M}$ is simple, ${\Delta}$ is bracket-generating. Assume that metric is definite positive on ${\Delta}$. Then the corresponding sub-Riemannian metric is GO.

Question. Are these examples all 2-step or not ?

1.2. Proof that many 2-step Carnot groups are GO

Let ${X_1,\ldots,X_k}$ be a basis of the distribution ${\Delta}$. Each ${X_i}$ defines a function ${u_i(p,q)=p\cdot X_i(q)}$ on ${T^*M}$. Let

$\displaystyle \begin{array}{rcl} h=\frac{1}{2}\sum u_i^2. \end{array}$

denote the Hamiltonian. Set

$\displaystyle \begin{array}{rcl} u_{ij}=\{u_i,u_j\}=p\cdot[X_i,X_j] \end{array}$

be there Poisson brackets. Then ${u_{ij}}$‘s are constant along the motion. Indeed, there derivative is a Poisson bracket, which involves a third order Lie bracket, which vanishes, by assumption. The equations for geodesics are

$\displaystyle \begin{array}{rcl} \dot{q}=\sum u_i X_i,\quad \dot{u}_i=\sum u_{ji}u_j, \end{array}$

This can be readily integrated : ${u=e^{tA}v}$, then the horizontal projection

$\displaystyle \begin{array}{rcl} q^{hor}=\int \sum e^{tA}v_i X_i. \end{array}$

The matrix ${A=(u_{ji})_{ij}}$ is skew-symmetric. Therefore it is a good candidate for an isometric automorphism, provided it lifts to an isomorphism, which happens often.

Question. Classify 2-step Carnot groups which are GO.

1.3. Proof that examples attached to simple Lie groups are GO

For any ${a\in M}$ and ${b\in K}$, ${x\mapsto axb}$ is an isometry. On computes that every geodesic has the form ${t\mapsto e^{at}e^{-ut}}$ where ${a\in \mathfrak{m}}$, ${a=u+v}$ where ${v\in\mathfrak{k}}$ and ${u\in\Delta}$. So it is an orbit of a one-parameter group.

2. Structure of isometries

Capogna-Le Donne : sub-Riemannian isometries are smooth, they are determined by their derivative, they form compact Lie groups (on compact manifolds).

Question : is an isometry determined by its restriction to the distribution at one point ? Related question : on a Carnot group, is every isometry affine, i.e. translation times automorphism ?

2.1. Discussion

Breuillard : doesn’t this follow from Capogna and Le Donne’s results ?

Pansu : generic 2-step Carnot groups with dimension vectors ${(n,p)}$ such that ${3\leq p\leq \frac{n(n-1)}{2}-3}$ have no graded automorphisms but dilations, so no isometries fixing a point. Those cannot be GO.

## Notes of Patrick Massot’s lecture

Microsupport of sheaves in symplectic topology

1. Goal of the working seminar

Use sheaves to prove a weak form of one of Arnold’s conjectures.

If ${\phi}$ is a hamiltonian isotopy in ${T^* N}$, then ${N\cap \phi_t(N)\not=\infty}$ for all ${t}$.

Proved by Hofer, and reproved by Laudenbach-Sikorav. The sheaf-theoretic proof is due to Tamarkin, Guillermou, Kashiwara-Shapira.

Sources : Claude Viterbo’s lecture notes.

A more remote goal is Stéphane Guillermou’s 2012 preprint on quantification of Lagrangian submanifolds.

2. The bundle of contact elements

2.1. Definition

${M}$ a manifold. ${\mathcal{C}M=\{}$ cooriented hyperplanes in ${TM\}}$. It carries a tautological contact structure ${\xi_H=\pi_*^{-1}(H)}$.

If ${Z\subset M}$ is a closed submanifold, let

$\displaystyle \begin{array}{rcl} \mathcal{N}Z=\{H\,;\,H\supset TZ\}. \end{array}$

If ${Z}$ has codimension 1 and is cooriented, one can define the subset ${\mathcal{N}_+ Z\subset\mathcal{N}Z}$ of positive elements. These are Legendrian submanifolds.

2.2. Lifting

If ${L\subset T^* N}$ is exact Legendrian, i.e. ${\lambda_{|L}=df}$, then ${L}$ lifts to

$\displaystyle \begin{array}{rcl} \hat{L}=\{(q,-f(p,q)),[p,1])\,;\,(q,p)\in L\}, \end{array}$

which is Legendrian in ${\mathcal{C}(M\times{\mathbb R})}$.

Exact hamiltonian isotopies of ${\mathcal{C}M}$ lift to ${\mathcal{C}(M\times{\mathbb R})}$ as well.

The symplectization of ${(V,\xi)}$ is

$\displaystyle \begin{array}{rcl} S(V,\xi)=\{(q,p)\in T^*V\,;\,\mathrm{ker}(p)=\xi_q\}. \end{array}$

Contact isotopies correspond to ${{\mathbb R}_+}$-invariant functions on ${S(V,\xi)}$ in the following manner : ${\phi_t}$ correspond to ${H_t(q,p)=p\circ(T_q\phi_t)^{-1}}$.

Special case of contact elements: ${S(\mathcal{C}(M),\xi_{can})=T^*M\setminus 0_M}$.

The lift ${\hat{\phi}_t}$ of ${\phi_t}$ arises from the Hamiltonian

$\displaystyle \begin{array}{rcl} \hat{H}_t((n,s,p_n,p_s))=p_s H_t(n,\frac{p_n}{p_s}). \end{array}$

If ${\phi_t}$ has compact support, then ${\hat{\phi}_t}$ exists on the whole of ${\mathcal{C}(N\times{\mathbb R})}$.

2.3. Walls

If ${\psi:M\rightarrow{\mathbb R}}$ has no critical point, then it defines a wall in ${\mathcal{C}M}$,

$\displaystyle \begin{array}{rcl} W_\psi =\{(q,\mathrm{ker}(d_q\psi)\}\subset\mathcal{C}M, \end{array}$

which is foliated by ${\mathcal{N}\psi^{-1}(*)}$.

2.4. Contact version of the disjunction conjecture

Let ${(\phi_t)}$ be a hamiltonian isotopy with compact support. Then intersection points ${\phi_t(0_N)\cap 0_N}$ correspond to intersection points of ${\phi_t(\mathcal{L})\cap W_\psi}$, for ${\psi(n,s)=s}$.

So from now on we shall work in the contact setting only.

3. Microsupport and intersections

3.1. Strategy

There is a category whose objects generalize submanifolds of ${M}$ and local systems: this is the derived category of sheaves on ${M}$. A submanifold ${Z\subset M}$ corresponds to the constant sheaf along ${Z}$, ${k_Z}$.

If ${U}$ is a codimension 0 submanifold with smooth cooriented boundary, there is also a corresponding ${k_U}$.

Any object in this category has a support ${supp(\mathcal{F})\subset M}$. It also has a microsupport ${\mathcal{N}\mathcal{F}}$ which is a closed subset of ${\mathcal{C}M}$.

To any subset ${A\subset M}$, there is an associated cohomological object ${R\Gamma(A,\mathcal{F})}$. For instance, for the constant sheaf, ${R\Gamma(A,k_M)}$ contains the same amount of information as ${H^*(A,k)}$ when ${k}$ is a field.

Proposition 1 (Morse Lemma) Let ${\psi:M\rightarrow{\mathbb R}}$ be proper on the support of ${\psi}$. Fix ${a. Assume that for all ${x}$ such that ${a\leq \psi(x),

• either ${d\psi(x)=0}$ and ${x}$ is not in the support of ${\mathcal{F}}$,
• or ${[d\phi(x)]\notin\mathcal{NF}}$,

then

$\displaystyle \begin{array}{rcl} R\Gamma(\{\psi

Example 1 For the constant sheaf, ${\mathcal{NF}}$ is empty, this is nothing but the usual Morse Lemma.

3.2. Main Quantization Theorem

Theorem 2 (Guillermou-Kashiwara-Shapira) Suppose ${\mathcal{F}_0}$ has compact support in ${M}$. and ${\Phi}$ is a contact isotopy in ${\mathcal{C}M}$. Then there exists a family ${(\mathcal{F}_t)}$ in the category such that

$\displaystyle \begin{array}{rcl} \Phi_t(\mathcal{NF}_0)=\mathcal{NF}_t, \quad\textrm{and}\quad R\Gamma(M,\mathcal{F}_t)\simeq R\Gamma(M,\mathcal{F}_0). \end{array}$

Corollary 3 If ${\psi:M\rightarrow{\mathbb R}}$ has no critical points, ${\mathcal{F}_0}$ has compact support and ${R\Gamma(M,\mathcal{F}_0)\not=0}$, then for all ${t}$,

$\displaystyle \begin{array}{rcl} \phi_t(\mathcal{NF}_0)\cap W_\psi\not=\emptyset. \end{array}$

This clearly implies the form of Arnold’s conjecture we are aiming at. Indeed, to the 0-section ${0_N}$, there corresponds ${\mathcal{N}k_U}$ where ${U=\{(n,s)\,;\,s\leq 0\}}$. Use function ${\psi(n,s)=s}$. Then ${R\Gamma(M,k_U)\simeq H^*(N,{\mathbb R})}$.

3.3. Proof of corollary

From Theorem 1 and Morse Lemma.

The Theorem provides us with the family ${\mathcal{F}_t}$ of objects (quantification of the transported Lagrangians). By contradiction, assume that for some ${t}$, ${\phi_t(\mathcal{NF}_0)=\mathcal{NF}_t}$ does not intersect the wall ${W_\psi}$. Then Morse Lemma applies to ${\mathcal{F}_t}$ with ${a}$ negative enough in order that ${\{\psi does not intersect the support of ${\mathcal{F}_t}$, and ${b}$ large enough so that ${\{\psi. Then cohomology ${R\Gamma(M,\mathcal{F}_t)\simeq R\Gamma(M,\mathcal{F}_0)}$. This contradicts the fact that ${R\Gamma(\{\psi.

4. Quantizing isotopies

The above Theorem 1 follows from a more abstract theorem, which quantizes isotopies. The quantization can be applied to ${\mathcal{F}_0}$ to get the family ${\mathcal{F}_t}$. Next goal is to quantize ${\Phi}$, getting ${\mathcal{K}_\phi}$, such that

$\displaystyle \begin{array}{rcl} \mathcal{K}_\phi(\mathcal{F}_0)=\mathcal{F}_t. \end{array}$

Before we get into thatn we need more geometry.

4.1. Functoriality in contact tautology

Let ${f:M\rightarrow N}$ be an arbitrary (smooth) map between manifolds. It pushes forward arbitrary subsets ${L\subset\mathcal{C}M}$ in the following way,

$\displaystyle \begin{array}{rcl} f_*(L)=\{(m,[p_m])\in\mathcal{C}M\,;\,\exists (n,[p_n])\in L,\,m=f(n), \,p_m\circ T_n f=p_n\}. \end{array}$

Dually, if ${L'\subset \mathcal{C}M}$, get ${f^*L'\subset \mathcal{C}N}$.

Example 2 If ${f}$ is a diffeo, this amounts to the usual lift of ${f}$ to ${\mathcal{C}M}$ and ${\mathcal{C}N}$.

Example 3 If ${f:E\rightarrow M}$ is a submersion, ${W\subset E}$ is a cooriented hypersurface, in generic situations, ${\mathcal{L}=f_*(\mathcal{N}_+ W)}$ is a Legendrian submanifold in ${\mathcal{C}M}$, and ${W}$ is called a generating hypersurface for ${\mathcal{L}}$.

This covers much more submanifolds in ${\mathcal{C}M}$ than you might expect, see forthcoming talk by Ferrand.

4.2. Contact correspondances

Let

$\displaystyle \begin{array}{rcl} \mathcal{C}(N,M)=(\mathcal{C}(N\times M),\xi=\mathrm{ker}(-\lambda_N+\lambda_M)). \end{array}$

Note the sign in the contact structure. This is contactomorphic with ${\mathcal{C}(N\times M)}$.

Definition 4 A contact correspondance between ${\mathcal{C}(N)}$ and ${\mathcal{C}(M)}$ is a Legendrian submanifold in ${\mathcal{C}(N,M)}$.

Switching ${M}$ and ${N}$ produces a contactomorphism which map a correspondance to the dual correspondance, by definition.

Example 4 Given ${f:N\rightarrow M}$, let ${f_*=\{(n,m,[p_n,p_m])\,;\,m=f(n),\,p_m\circ T_n f=p_m\}}$ is a contact correspondance.

4.3. Graphs

Example 5 The graph of a contactomorphism ${\phi:M\rightarrow M}$

$\displaystyle \begin{array}{rcl} \Gamma_\phi=\{(x,y,[px,py]),(y,[p_y])=\phi(x,[p_x])\}. \end{array}$

defines a correspondance.

More generally, a contact isotopy ${\Phi}$ with Hamiltonian ${H_t}$ has a Legendrian graph

$\displaystyle \begin{array}{rcl} \Gamma_\Phi \subset\mathcal{C}(M,M\times I), \end{array}$

defined by

$\displaystyle \begin{array}{rcl} \Gamma_\Phi =\{(q,q',t,[p_q,p_{q'},p_t])\,;\,\phi_t(q,[p_q])=(q',[p_{q'}]),\,p_t=H_t(q,p_t)\}. \end{array}$

By quantizing a Hamiltonian isotopy ${\Phi=(\phi_t)_{t\in I}}$, we mean an object in our category corresponding to ${\Gamma_\Phi}$.

4.4. More abstract quantization

Theorem 5 (Guillermou-Kashiwara-Shapira) For any contact isotopy ${\Phi}$ in ${\mathcal{C}M}$, the is an object ${\mathcal{K}_\Phi}$ of our category on ${M\times M\times I}$ such that

$\displaystyle \begin{array}{rcl} (\mathcal{NK}_\Phi)^a=\Gamma_\Phi. \end{array}$

4.5. Microsupport functoriality

We need some more notation in order to explain how Theorem 2 implies Theorem 1.

A map ${f:N\rightarrow M}$ induces a functor ${RF_! : Db N\rightarrow D^b M}$, related to the cohomology of fibers. If ${f}$ is proper on the support of ${\mathcal{F}}$, then

$\displaystyle \begin{array}{rcl} \mathcal{N}(Rf_! \mathcal{F})\subset f_*(\mathcal{NF}). \end{array}$

Definition 6 A submanifold ${L\subset \mathcal{C}M}$ is non characteristic for a map ${f:N\rightarrow M}$ if

$\displaystyle \begin{array}{rcl} f_*(pr_M^*(L))=\emptyset \subset \mathcal{C}(N,M). \end{array}$

There is also a functor ${f^{-1}}$ in the reverse direction. If ${\mathcal{NF}}$ is non characteristic for ${f}$, then ${\mathcal{N}(f^{-1}(\mathcal{F}))\subset f^{*}(\mathcal{NF})}$.

4.6. Theorem 2 implies Theorem 1

${\mathcal{K}_{\Phi}}$ lives on ${M\times M\times I}$. It can be used to push forward ${\mathcal{F}_0}$, the resulting object ${\mathcal{F}=\mathcal{K}_\Phi(\mathcal{F}_0)}$ is defined on ${M\times I}$. Freezing at ${M\times\{t\}}$ yields ${\mathcal{F}_t =j_t^{-1}\mathcal{F}}$. Since ${\mathcal{NF}}$ is non characteristic for ${j_t}$,

$\displaystyle \begin{array}{rcl} \mathcal{NF}_t\subset j_t^*(\mathcal{NF})\subset \Gamma_{\phi_t}. \end{array}$

It fact, equality holds but we shall not need it. Cohomology does not change because ${R_{\pi_{!}}\mathcal{F}}$ is locally constant on ${I}$. This is related to the fact that its microsupport ${\mathcal{N}(R_{\pi_{!}}\mathcal{F})}$ is empty. Indeed, this is contained in the projection of the microsupport of ${\mathcal{F}=\mathcal{K}_\Phi(\mathcal{F}_0)}$. ${\mathcal{NF}}$ is contained in the composition of the correspondance ${\Gamma_\Phi}$ with ${\mathcal{NF}_0}$. This is empty because ${\mathcal{F}}$ is non characteristic for injections ${j_t}$.

## Notes of dario Prandi’s lecture

Complexity of control affine systems

1. Definitions of complexities

A control system takes the form

$\displaystyle \begin{array}{rcl} \dot{q}=f(q,u), \end{array}$

where ${q\in M}$ some manifold and ${u\in \mathcal{U}}$ a set of admissible controls. A cost function ${J:\mathcal{U}\rightarrow[0,+\infty]}$ is given.

Problem.

1. Find a path that minimizes ${J}$. In general, it is not admissible.
2. Approximate it by admissible trajectories.

1.1. Definition

Complexity of a path ${\Gamma}$ is a function of the precision ${\epsilon}$ of approximation,

$\displaystyle \begin{array}{rcl} \sigma(\Gamma,\epsilon)=\phi(\epsilon)\inf_{u\in A_\epsilon}J(u). \end{array}$

The set of approximating controls can be defined via interpolation,

$\displaystyle \begin{array}{rcl} A_\epsilon=\{u\,;\,q_u(t_i)\in\Gamma,\,J(u_{|[t_i,t_{i+1}]})<\epsilon\}. \end{array}$

Then ${\phi(\epsilon)=\frac{1}{\epsilon}}$ is the suitable rate.

If the cost depends on the parametrization, one may want to replace ${A_\epsilon}$ with time interpolation,

$\displaystyle \begin{array}{rcl} A^t_\delta = \{u\,;\,q_u(t_i)=\gamma(t_i),\,|t_i-t_{i+1}|<\delta\}. \end{array}$

Tubular interpolation complexity arises from

$\displaystyle \begin{array}{rcl} A^a_\epsilon=\{u\,;\,q_u\subset \mathrm{Tube}(\Gamma,\epsilon)\}. \end{array}$

The parametrized version of this is

$\displaystyle \begin{array}{rcl} A^n_\delta = \{u\,;\,q_u(t)\in B(\gamma(t),\epsilon)\}. \end{array}$

Each class ${A^?_\epsilon}$ defines a corresponding complexity. Parametrized ones tend to 0 when ${\delta}$ tends to 0, unparametrized ones tend to infinity when ${\epsilon}$ tends to 0.

2. Results

2.1. Sub-Riemannian geometry

Sub-Riemannian geometry deals with ${f(q,u)}$ which is linear in ${u}$,

$\displaystyle \begin{array}{rcl} \dot{q}=\sum_{i=1}^m i f_i(q), \end{array}$

and cost

$\displaystyle \begin{array}{rcl} J(u)=\int\sqrt{\sum u_i^2}\,ds. \end{array}$

Let ${\Delta(q)}$ be the distribution spanned by the vector-fields ${f_i(q)}$. Let ${\Delta^2(q)}$ the distribution generated by brackets of sections of ${\Delta}$, and so on. We assume that Hörmander’s condition ${\Delta^r=TM}$ is satisfied, as well as equiregularity: ranks of ${\Delta^p}$ are constant.

Theorem 1 Let ${\Gamma}$ be a path in ${M}$ such that ${T\Gamma\subset \Delta^k \setminus \Delta^{k-1}}$. Then

$\displaystyle \begin{array}{rcl} \sigma_c(\Gamma,\epsilon)\sim\sigma_a(\Gamma,\epsilon)\sim \sigma_n(\Gamma,\epsilon)\sim\frac{1}{\epsilon^k}. \end{array}$

The time interpolation complexity has the same behaviour,

$\displaystyle \begin{array}{rcl} \sigma_t(\gamma,\delta)\sim\delta^{1/k}. \end{array}$

2.2. Control affine systems

Control affine systems are of the form

$\displaystyle \begin{array}{rcl} \dot{q}=f_0 (q)+\sum_{i=1}^m i f_i(q). \end{array}$

Again, cost is

$\displaystyle \begin{array}{rcl} J(u)=\int\sqrt{\sum u_i^2}\,ds. \end{array}$

We assume the strong Hörmander condition, i.e. ${f_1,\ldots,f_m}$ alone satisfy Hörmander condition. ${\mathcal{U}}$ consists of all ${L^1}$ controls on sub-intervals of ${[0,\tau]}$.

Theorem 2 Assume that ${f_0 \in \Delta^s \setminus \Delta^{s-1}}$, ${s>1}$. Let ${\Gamma}$ be a path in ${M}$ such that ${T\Gamma\subset \Delta^k \setminus \Delta^{k-1}}$. Then, for ${\tau}$ small enough,

$\displaystyle \begin{array}{rcl} \sigma_c(\Gamma,\epsilon)\sim\sigma_a(\Gamma,\epsilon)\sim \frac{1}{\epsilon^k}. \end{array}$

For time interpolation, assume further that ${\dot{\gamma}\not= f_0(\gamma)}$ mod ${\Delta^{s-1}}$. Then

$\displaystyle \begin{array}{rcl} \sigma_t(\gamma,\delta)\sim\delta^{1/\max\{k,s\}}, \quad \sigma_n(\gamma,\epsilon)\sim \frac{1}{\epsilon^{\max\{k,s\}}}. \end{array}$

When ${\tau}$ is large enough, certain curves can be approximated more easily, and our results break down. Extreme case : drift ${f_0}$ is a recurrent vector field. Then minimum cost is zero.

3. Proofs

Both theorems require variants of the ball-box theorem. Assumptions can probably be weakened.

For application to mechanics, i.e. second order

4. Questions

Agrachev: Beware that approximation may be impossible in general. The end point map may not have an open image, the minimum cost between two points may be discontinuous, depending on the cost function. If ${L^1}$ norm is replaced with ${L^p}$ norm, approximability may depend on the number of brackets needed to fill in space.

## Notes of Victor Chepoi’s july 2013 lecture

Graphes de bases de matroides

On va établir un mécanisme analogue à la caractérisation des complexes cubiques ${CAT(0)}$ : il suffit de vérifier une condition locale (sur les liens) et la simple connexité.

Curiosité : Maurer, Isbell, Nash ont eu le meme directeur de thèse, Albert Tucker.

1. Matroides

1.1. Trois définitions équivalentes d’un matroide

Un matroide, c’est un complexe simplicial fini qui satisfait l’axiome d’échange : Si ${I}$ et ${I'}$ sont des simplexes, avec ${|I'|>|I|}$, alors il existe ${e\in I'\setminus I}$ tel que ${I\cup\{e\}}$ est un simplexe.

Exemple : les ensembles de vecteurs linéairement indépendants.

Je vais utiliser une autre axiomatique, celle des bases. Une base d’un matroide, c’est un simplexe maximal. L’axiome d’échange entraine que toutes les bases ont le meme cardinal. Donc, dans cette version de la définition, on se donne un ensemble de bases, qui satisfait

Si ${B}$,${B'}$ sont deux bases, alors pour tout ${e\in B'\setminus B}$, il existe ${e'\in B\setminus B'}$ tel que ${B\setminus\{e\}\cup \{e'\}}$ est une base.

Troisième définition possible : les circuits. Les circuits sont les ensembles dépendants minimaux. Ils sont sujets à 3 axiomes

1. ${\emptyset}$ n’est pas un circuit.
2. minimalité.
3. pour toute arete commune à deux circuits, il existe un circuit contenu dans la réunion qui évite cette arete.

1.2. Le graphe des bases

Les sommets correspondent aux bases. Une arete relie deux bases ${B}$ et ${B'}$ dès que ${|B\Delta B'|=2}$.

C’est le 1-squelette du polyèdre du matroide, enveloppe convexe des vecteurs colonnes de la matrice d’incidence élément/base.

Conjecture (Mihail, Sudan). Les graphes des bases des matroides forment un expanseur, avec constante de Cheeger égale à 1 : si l’ensemble des sommets est partitionné en ${S}$ et ${S'}$, alors

$\displaystyle \begin{array}{rcl} E(S,S')\geq \min\{|S|,|S'|\}. \end{array}$

1.3. Caractérisation des graphes des bases

Question (Maurer). Quels graphes apparaissent comme graphes des bases de matroides ?

Voici les conditions nécessaires recensées par Maurer en 1973.

1. Les graphes de bases sont connexes (plus, ils se plongent dans des hypersimplexes).
2. Condition d’intervalles : si ${d(B,B')=2}$, alors ${B}$ et ${B'}$ appartiennent à un carré, une pyramide à base carrée ou un octaèdre.
3. Condition du lien : soit ${B}$ une base. Les bases à distance 1 de ${B}$ forment le graphe d’incidence des aretes d’un graphe biparti.
4. Condition du positionnement : soit ${B}$ une base, on s’intéresse à la fonction distance à ${B}$, et à ses restrictions possibles à un carré. Il y a 3 configurations possibles : constante, prend 2 valeurs (constante sur deux cotés), prend 3 valeurs (constante sur exactement une diagonale).

Theorem 1 (Maurer 1973) ${G}$ est le graphe des bases d’un matroide si et seulement il satisfait ces 4 propriétés.

Maurer était jeune et optimiste, il a conjecturé que certaines conditions ne sont pas indispensables. Dès 1977, un contre exemple a été trouvé à sa conjecture initiale.

Proposition 2 Soit ${G}$ un graphe. On note ${X(G)}$ le 2-complexe obtenu en remplissant les triangles et les carrés dans ${G}$. Si ${G}$ est le graphe des bases d’un matroide, alors ${X(G)}$ est simplement connexe.

Maurer a conjecturé que cette propriété pouvait remplacer l’une ou plusieurs des 4 précédentes.

2. Résultats

2.1. Caractérisation locale/globale des graphes des bases

Theorem 3 (Chalopin, Chepoi, Osajda) Pour un graphe ${G}$, les conditions suivantes sont équivalentes.

1. ${G}$ est le graphe des bases d’un matroide.
2. ${X(G)}$ est simplement connexe et toute boule de rayon 3 de ${G}$ est comme une boule de rayon 3 d’un matroide.
3. ${X(G)}$ est simplement connexe, ${G}$ satisfait la condition d’intervalles, la condition de positionnement locale (positionnement pour les carrés situés à distance ${\leq 2}$), et ${G}$ possède un sommet de degré fini.

2.2. Preuve

Le plus intéressant, c’est ${3.\Rightarrow 2.}$. On s’appuie sur le théorème de Maurer : il faut vérifier la condition de lien et la condition de positionnement global. On le fait par récurrence sur les niveaux de la distance à un sommet, mais en reconstruisant les couches. La construction s’apparente à celle du revetement universel.

2.3. Questions

La condition de simple connexité est elle indispensable dans le théorème ? Oui. Sinon, il y a ce contre exemple de 1977.

## Notes of Peter Haissinsky’s july 2013 lecture

Groupes hyperboliques de bord planaire

Il s’agit d’un survol destiné à souligner les difficultés rencontrées dans l’étude de la conjecture de Cannon.

1. La conjecture de Cannon

1.1. Groupes kleinéens

Un groupe kleinéen ${G}$ est un sous-groupe discret d’isométries de l’espace hyperbolique de dimension 3. ${G}$ agit par homographie sur la sphère à l’infini, laquelle se décompose en un fermé invariant, l’ensemble limite ${\Lambda}$ (adhérence commune à toutes les orbites) et son complémentaire ${\Omega}$, sur lequel l’action est proprement discontinue.

On dit que ${G}$ est convexe-cocompact si l’action sur l’enveloppe convexe de l’ensemble limite est cocompacte. Dans ce cas, ${M_G=G\setminus(H^3 \cup \Omega)}$ est une variété de dimension 3 compacte à bord.

Theorem 1 (Thurston-Perelman) Si ${M}$ est une variété orientable, irréductible, compacte à bord, de dimension 3, de groupe fondamental infini mais ne contenant aucun sous-groupe isomorphe à ${{\mathbb Z}\oplus{\mathbb Z}}$, alors il existe un groupe kleinéen convexe-cocompact ${G}$ tel que ${M}$ est difféomorphe à ${M_G}$.

L’objectif est de sortir du cadre des 3-variétés.

1.2. Propriété de convergence uniforme

Soit ${Z}$ un espace compact métrisable. Une action de groupe sur ${Z}$ est de convergence uniforme si par homéomorphismes et l’action diagonale sur les triplets de points disctincts est proprement discontinue et cocompacte.

La classe des groupes possédant une action de convergence uniforme coincide avec la classe des groupes hyperboliques au sens de Gromov (l’action étant l’action sur le bord à l’infini).

Par conséquent, l’action d’un groupe sur le bord, du seul point de vue topologique, contient déjà beaucoup d’informations. Cela motive la question suivante.

Conjecture. Soit ${G}$ un groupe hyperbolique dont le bord est planaire. Alors ${G}$ est virtuellement isomorphe à un groupe kleinéen convexe cocompact.

Cela contient deux questions classiques.

Conjecture (Cannon). Soit ${G}$ un groupe hyperbolique dont le bord est une 2-sphère. Alors ${G}$ est virtuellement isomorphe à un groupe kleinéen cocompact.

Conjecture (Kapovitch-Kleiner). Soit ${G}$ un groupe hyperbolique dont le bord est homéomorphe au tapis de Sierpinsky. Alors ${G}$ est virtuellement isomorphe à un groupe kleinéen convexe cocompact.

Kapovitch et Kleiner on ramené leur conjecture à celle de Cannon.

Haïssinsky a montré que

1. si la conjecture de Kapovitch-Kleiner est vraie,
2. si ${G}$ n’a pas de 2-torsion,
3. et si le bord de ${G}$ est planaire sans etre toute la sphère,

alors ${G}$ est virtuellement kleinéen.

2. Approche analytique

2.1. Stratégie

On fixe un système générateur fini. On place une boule de rayon 1 à chacun des points d’une orbite dans ${{\mathbb R}^3}$ et on considère son ombre portée sur le bord. Soit ${\mathcal{S}_n}$ la collection des ombres correspondant aux éléments de ${G}$ de longueur ${\leq n}$.

Son graphe d’incidence est une triangulation de degré borné de la 2-sphère. Il y a un empilement de cercles associé (Koebe) : il a le meme graphe d’incidence. Soit ${\phi_n}$ l’application qui envoie chaque centre d’ombre de ${\mathcal{S}_n}$ sur le centre du cercle associé.

But. Montrer que les ${\phi_n}$ sont uniformément équicontinues et que toute limite est un homéomorphisme.

Si on y arrive, on a gagné.

2.2. Modules de courbes

D’après Cannon, il s’agit de controler des modules d’anneaux, à définir proprement. Commencons par la définition classique, dans un espace métrique mesuré ${Z}$.

Definition 2 Soit ${\Gamma}$ une famille de courbes, ${p\geq 1}$. Son module ${mod_p(\Gamma)}$ est la borne inférieure, sur toutes les fonctions boréliennes ${\rho}$ dont l’intégrale sur chaque courbe ${\gamma}$ de la famille est ${\geq 1}$, de ${\int_Z\rho^p}$.

Example 1 ${Z={\mathbb C}}$, ${p=2}$, ${\Gamma=}$ les courbes qui traversent d’un bord à l’autre d’un anneau ${A}$. Le nombre obtenu est l’unique invariant des anneaux à bijection conforme près. Si ${\Gamma=}$ les lacets qui font le tour du trou, le nombre obtenu est l’inverse du précédent.

On n’a pas de distance canonique, mais, à la place, le recouvrement ouvert ${\mathcal{S}_n}$. En remplacant chaque intégrale par une somme, on obtient un nombre ${mod_p(\Gamma,\mathcal{S}_n)}$.

L’application ${\phi_n}$ envoie un anneau sur la réunion des cercles centrés aux points de ${\phi_n(A)}$, kégèrement augmentés pour garnir les trous entre cercles. Pour ${n}$ assez grand, c’est à nouveau un anneau.

Proposition 3

$\displaystyle \begin{array}{rcl} mod_2(A,\mathcal{S}_n)\sim mod_2(\phi_n(A)). \end{array}$

Theorem 4 Soit ${G}$ un groupe hyperbolique dont le bord est homéomorphe à la 2-sphère. Supposons établi que pour tout anneau ${A}$ du bord, il existe ${m>0}$ tel que pour tout ${n}$ assez grand,

$\displaystyle \begin{array}{rcl} mod_2(A,\mathcal{S}_n) \geq m, \end{array}$

Alors ${G}$ est virtuellement kleinéen.

En fait, il suffit de minorer les modules d’un nombre fini d’anneaux.

Corollary 5 (Bourdon-Kleiner) Soit ${G}$ un groupe hyperbolique dont le bord est homéomorphe à la 2-sphère. Supposons établi qu’il existe ${\delta>0}$ tel que les modules ${mod_2(\Gamma_\delta,\mathcal{S}_n)}$ restent bornés, où ${\Gamma_\delta}$ est la famille des courbes de diamètre ${\geq \delta}$. Alors ${G}$ est virtuellement kleinéen.

Il en résulte (mais c’était déjà connu) que tout groupe de Coxeter hyperbolique dont le bord est homéomorphe à la 2-sphère est virtuellement kleinéen.

3. La jauge conforme

La jauge conforme de ${G}$ est l’ensemble des distances Ahlfors-régulières sur le bord de ${G}$ pour lesquelles l’action de ${G}$ est uniformément quasi-Mobius.

Uniformément quasi-Mobius signifie que les birapports sont préservés à un homéomorphisme de ${{\mathbb R}_+}$ près.

Cette famille de distances est non vide (Gromov, Coornaert), deux distances de la jauge sont quasi-Mobius.

Definition 6 La dimension conforme Ahlfors-régulière de ${G}$, ${dim_{AR}(G)}$, est la borne inférieure des dimensions de Hausdorff des métriques de la jauge.

C’est un invariant topologique de l’action de ${G}$ sur son bord.

Theorem 7 (Bonk-Kleiner) Soit ${G}$ un groupe hyperbolique dont le bord est homéomorphe à la 2-sphère. Si la dimension conforme Ahlfors-régulière est atteinte, alors ${G}$ est virtuellement kleinéen.

Ils montrent qu’une métrique minimisante est Loewner, ce qui donne une borne inférieure sur le module, qui, à son tour, minore ${mod_Q(A,\mathcal{S}_n)\leq mod_2(A,\mathcal{S}_n)}$. Ce n’est pas exactement comme cela qu’ils procèdent, je mélange leur méthode et celle de Cannon et al.

4. Difficultés restantes

4.1. Instabilité des modules analytiques

On sait montrer qu’il existe une famille de courbes ${\Gamma}$ de ${mod_Q(\Gamma)>0}$ si et seulement si ${Q=dim_{AR}(G)}$. Autrement dit, si la dimension conforme Ahlfors-régulière n’est pas atteinte, on ne peut rien tirer des modules analytiques.

Espoir : les modules combinatoires

Theorem 8 (Keith-Kleiner, Carrasco)

$\displaystyle \begin{array}{rcl} dim_{AR}(G)=\inf\{Q\geq 2\,;\, \lim_{n\rightarrow\infty}mod_Q(\Gamma_\delta,\mathcal{S}_n)=0\}. \end{array}$

4.2. Faiblesse des modules combinatoires

On ne sait pas établir la sous-additivité dénombrable. On aimerait savoir que si des familles de courbes ${\Gamma_k}$ satisfont ${\lim_{n\rightarrow\infty}mod_Q(\Gamma_k,\mathcal{S}_n)=0}$, alors

$\displaystyle \begin{array}{rcl} \lim_{n\rightarrow\infty}mod_Q(\bigcup_{k}\Gamma_k,\mathcal{S}_n)=0. \end{array}$

4.3. Contre exemples

Il existe des distances sur ${{\mathbb C}}$ qui sont ${Q}$-Ahlfors-régulières, mais telles que, pour tout anneau ${A}$, ${mod_2(A,\mathcal{S}_n)}$ tend vers 0. Il s’agit des paillassons de Rickman.

5. Approche de Markovic

Markovic : construire une action sur un complexe cubique ${CAT(0)}$. utiliser les sous-groupes de surfaces.

Calegari-Walker : inspiré par les groupes aléatoires. Ne marche que lorsque les relateurs sont longs.