## Pansu’s Lille slides

Slides of Pansu’s 2 lectures on Lp cohomology and large scale conformal maps.

lille_15oct14_beamer

## Notes of Romain Tessera’s Lille lecture nr 2

1. Consequence

Corollary 1 Let ${G}$ be a real Lie group. Then ${G}$ is hyperbolic iff it acts isometrically and transitively on a negatively curved Riemannian manifold. Let ${G}$ be a ${p}$-adic Lie group. Then ${G}$ is hyperbolic iff it acts isometrically and transitively on a regular tree.

2. Proof of classification theorem

On shows that unless ${B}$ is Heintze, reduced ${\ell^p}$ cohomology vanishes.

Splits into two cases, whether ${B}$ is unimodular or not.

I will illustrate the argument withe the groups ${Sol(\lambda_1,\lambda_2)}$, semi-direct product of ${{\mathbb R}^2}$ by ${{\mathbb Z}}$ acting via a diagonal matrix with eigenvalues ${\lambda_1,\lambda_2}$.

2.1. Case ${B}$ is non unimodular

Let ${T}$ be the generator of the ${{\mathbb Z}}$ factor. Let ${\Delta}$ be the modular function.

I prove that ${\bar{H}^1_p(Sol(\lambda_1,\lambda_2))=0}$ if ${\lambda_1+\lambda_2\not=0}$.

Lemma 2 Let ${u\in D^p(G)}$. There exists ${u_\infty\in D^p(G)}$ such that ${u-u_\infty\in\ell^p}$. Conversely, if ${u_\infty}$ is constant, cohomology class is 0.

For that example,

$\displaystyle \begin{array}{rcl} txt^{-1}=\begin{pmatrix} e^{\lambda_1}&0\\ 0&e^{-\lambda_2} \end{pmatrix}. \end{array}$

Let ${W=\{g\,;\, t^{-n}wt^n}$ is bounded independantly of ${n\}}$. This a subgroup of the form ${W=(0\times {\mathbb R})\times{\mathbb Z}}$. I show that function ${u}$ is left ${W}$-invariant. Thus ${u}$ cannot have finite energy, unless ${u=0}$.

2.2. Case ${B}$ is unimodular

E.g. ${G=SOL}$, ${\lambda_1+\lambda_2=0}$.

We show that any ${u\in D^p}$ can be approximated by ${L^p}$ functions. We need good F\o lner sequences. Let

$\displaystyle \begin{array}{rcl} F_n=\{(x,y,m)\,;\,|x|\leq e^{2n},\,|y|\leq e^{2n},\,|m|\leq n\}. \end{array}$

One checks that

$\displaystyle \begin{array}{rcl} \frac{|sF_n\Delta F_n|}{|F_n|}\leq \frac{C}{n},\quad\textrm{and}\quad F_n\subset B(100n). \end{array}$

We shall approximate ${u}$ with

$\displaystyle \begin{array}{rcl} u_n(g)=\frac{1}{|F_n|}\int_{F_n}(u(gh)-u(h))\,dh. \end{array}$

${u_n}$ is an average of left translates of expressions ${\rho_h(u)-u}$, so it is in ${\ell^p}$.

The error term ${v_n=u_n-u}$ is the average of ${u}$ over ${F_n}$. Since ${F_n}$ is nearly left invariant, ${v_n}$ is nearly right-invariant, meaning that ${\|\rho_s(v_n)-v_n\|_p}$ tends to 0. This means that ${v_n}$ tends to 0 in ${D^p(G)}$.

2.3. Final step

Let ${G}$ be a locally compact compactly generated group. Let ${S\subset G}$ be open. Let ${G}$ act in a mixing manner on an infinite measure space ${(X,\mu)}$ (e.g. left action of ${G}$ on itself). Then for ever 1-cocycle ${b}$ for the representation ${L^p(X,\mu)}$,

$\displaystyle \begin{array}{rcl} \|b(g)\|_p=o(|g|_S). \end{array}$

## Notes of François Le Maître’s lecture

Full groups in the locally compact measure preserving setting

1. Orbit equivalence

Let ${G}$ be a locally compact secound countable group, ${(X,\mu)}$ a standard probability measure space, on which ${G}$ acts preserving ${\mu}$. Let ${R_G}$ denote the corresponding equivalence relation.

1.1. A general lemma

Lemma 1 Let ${G}$ and ${H}$ act on ${X}$. The following are equivalent.

1. There is a full measure subset on which the equivalence relations ${R_G}$ and ${R_H}$ restrict to the same equivalence relation.
2. For all ${g\in G}$,

$\displaystyle \begin{array}{rcl} \mu(\{x\in X\,;\,gx\in Hx\})=1. \end{array}$

Proof: set ${A=\{x\in X\,;\, \textrm{for a.e. }g\in G,\,gx\in Hx\}}$.

1.2. Full groups

Definition 2 Say two actions ${G}$, ${H}$ on ${(X,\mu)}$ are orbit equivalent if there exists a subset ${A}$ of full measure and an automorphism (measure preserving bijection mod null sets) of ${(X,\mu)}$ that maps ${R_G}$ to ${R_G}$ in restriction to ${A}$.

Definition 3 Given an action of ${G}$ on ${(X,\mu)}$, the associated full group ${[R_G]}$ is defined by

$\displaystyle \begin{array}{rcl} [R_G]=\{T\in Aut(X,\mu)\,;\,\forall x\in X,\,T(x)\in Gx\}. \end{array}$

Orbit equivalence boils down to conjugacy of full groups. This follows from the Lemma above.

Proposition 4 Let ${G}$ and ${H}$ act on ${(X,\mu)}$. Let ${S\in Aut(X,\mu)}$. The following are equivalent.

1. ${S}$ is an orbit equivalence between the actions of ${G}$ and ${H}$.
2. ${S[R_G]S^{-1}=[R_H]}$.

2. From isomorphism to conjugacy

2.1. Dye’s reconstruction theorem

In 1959, Dye defined full groups a sfollows: a group of automorphisms of ${X,\mu)}$ is a full group if it is stable under cutting and pasting. This amounts, given a countable partition of ${X}$ and elements of ${G}$ that move them around in order to obtain again a partition of ${X}$, to put these elements together to define a new automorphism.

Theorem 5 (Dye’s reconstruction theorem, 1963) Let ${G_1}$, ${G_2}$ be ergodic full groups. Suppose ${\psi:G_1\rightarrow G_2}$ is an abstract isomorphism. Then ${\psi}$ coincides with conjugation by some automorphism.

Corollary 6 ${Aut(X,\mu)}$ itself has no outer automorphisms.

Theorem 7 (Carderi-Le Maitre) Let ${G}$ and ${H}$ act ergodically on ${(X,\mu)}$. Then ${[R_G]}$ and ${[R_H]}$ are abstractly isomorphic iff the actions are orbit equivalent.

3. More properties of full groups

3.1. The countable discrete case

If ${G}$ is countable discrete, ${[R_G]}$ has a topology (the uniform metric) that turns it into a Polish space (separable, metrizable). Here are a few known fact.

Theorem 8 (Giordano-Pestov 2005) Let ${G}$ be countable discrete and act freely on ${(X,\mu)}$. The following are equivalent.

1. ${G}$ is amenable.
2. ${[R_G]}$ is amenable.

Definition 9 Let ${G}$ be a topological group. Its topological rank ${t(G)}$ is the minimum number of elements needed to generate a dense subgroup.

Example. ${t({\mathbb R}^n)=n+1}$.

Theorem 10 Let ${G}$ be countable discrete and act ergodically. Then

$\displaystyle \begin{array}{rcl} t([R_G])=\lfloor Cost(R_G)\rfloor +1. \end{array}$

For instance, for a free action of the free group ${\mathbb{F}_n}$, ${t([R_{\mathbb{F}_n}])=n+1}$.

3.2. The locally compact second countable case

We use the topology of convergence in measure. If ${Y}$ is a Polish space, pick a metric on ${Y}$ and equip the space of measurable maps ${X\rightarrow Y}$ with the distance ${d(f,g)=\int_X d(f(x),g(x))\,d\mu(x)}$. The resulting topology does not depend on the choice of metric on ${Y}$.

Theorem 11 (Carderi-Le Maitre) With this topology, ${[R_G]}$ is a Polish group.

Giordano-Pestov’s result generalizes.

Theorem 12 (Carderi-Le Maitre) Let ${G}$ be locally compact, second countable and unimodular. Assume that ${G}$ acts freely on ${(X,\mu)}$. The following are equivalent.

1. ${G}$ is amenable.
2. ${[R_G]}$ is amenable.

Theorem 13 (Carderi-Le Maitre) Let ${G}$ be non discrete and act ergodically. Then

$\displaystyle \begin{array}{rcl} t([R_G])=2. \end{array}$

In fact, a dense ${G_\delta}$ of pairs of elements generate a dense subgroup.

Proof is inspired by the work of Kyed-Vaas on ${\ell^2}$ Betti numbers of locally compact groups. Especially, their notion of discrete section is useful, it more or less allows to reduce to the case ${G=S^1\times\Gamma}$, ${X=S^1\times Y}$, where ${\Gamma}$ is countable discrete acting on ${Y}$ ergodically, and ${S^1}$ acts on itself by translation. Cost 2 comes from the ${S^1}$ factor.

## Notes of Henrik Petersen’s lecture

Quasiisometries of nilpotent groups

Joint work (in progress) with David Kyed. The paper is not fully written, so be careful.

1. Nilpotent groups

Let ${\Gamma}$ be finitely generated torsion free nilpotent. Mal’cev’s theorem states that there exists a unique connected Lie group ${G=\Gamma\otimes{\mathbb R}}$ where ${\Gamma}$ sits as a discrete cocompact subgroup.

This is not hard for the free step ${d}$ nilpotent group, the general case follows. See Baumslag’s notes.

Fact. Any two cocompact lattices in a Lie group are quasiisometric. Therefore nilpotent group having the same Mal’cev completion are quasiisometric.

Question. Is the converse true ?

2. Earlier results

In 1989, Pansu showed that quasiisometric nilpotent groups have the same graded Lie algebras. Given a Lie algebra ${\mathfrak{g}}$, let ${\mathfrak{g}_{[i]}}$ denote the descending central series. Then

$\displaystyle \begin{array}{rcl} gr(\mathfrak{g})=\bigoplus_i \mathfrak{g}_{[i]}/\mathfrak{g}_{[i+1]}, \end{array}$

with the induced Lie bracket.

In 2002, Shalom showed that quasiisometric nilpotent groups have the same usual Betti numbers. Apart from degree one, this does not follow from the previous result.

2.1. Sketch of Shalom’s argument

Two groups ${\Gamma}$ and ${\Lambda}$ are quasiisometric iff they are uniformly measure equivalent.

Definition 1 ${\Gamma}$ and ${\Lambda}$ are uniformly measure equivalent if they have commuting measure preserving actions on some measure space ${\Omega}$, and both action are co-finite. Furthermore, pick a fundamental domain ${X}$ with resulting cocycle ${\omega:\Gamma\times X\rightarrow \Lambda}$; one requires that for fixed ${\gamma\in\Gamma}$, ${\omega(\gamma,\cdot):X\rightarrow \Lambda}$ is bounded.

In this case, cohomology of the module ${L^2(X)}$ can be transferred from ${\Gamma}$ to ${\Lambda}$.

Next, Shalom uses Property ${H_T}$ (see below). This implies that

$\displaystyle H^n(\Gamma,{\mathbb R})\simeq H^n(\Gamma,L^2(X))\simeq H^n(\Lambda,L^2(X))\simeq H^n(\Lambda,{\mathbb R}).$

3. Cohomology and higher order cohomology

3.1. Cohomology

Definition 2 A continuous ${G}$-module is relatively injective if for every exact sequence

$\displaystyle \begin{array}{rcl} 0\rightarrow A\rightarrow B \end{array}$

of continuous ${G}$-modules, which admits a continuous linear section ${\sigma}$ (when this is the case, we speak of a strengthened morphism), also admits a ${G}$-equivariant continuous linear section.

Proposition 3 For every ${G}$-module ${E}$, the module ${C(G,E)}$ of continuous equivariant maps ${G\rightarrow E}$ is relatively injective.

A strengthened resolution of ${E}$ follows:

$\displaystyle \begin{array}{rcl} 0\rightarrow E\rightarrow C(G,E)\rightarrow C(G\times G,E)\rightarrow\cdots. \end{array}$

Definition 4 Let ${E}$ be a ${G}$-module. Pick a relatively injective resolution. The continuous cohomology of ${E}$ is the cohomology of the subcomplex of ${G}$-invariant vectors of the resolution.

Note that we do not take closures.

3.2. Higher order invariants

Definition 5 For ${\xi\in E}$ and ${g\in G}$, denote by

$\displaystyle \begin{array}{rcl} \partial_g(\xi):=g\xi-\xi. \end{array}$

For instance, invariant vectors

$\displaystyle \begin{array}{rcl} E^G=\{\xi\in E\,;\,\forall g\in G,\,\partial_g \xi=0\}. \end{array}$

Definition 6 For ${\xi\in E}$ and ${g\in G}$, denote by

$\displaystyle \begin{array}{rcl} E^{G(d)}:=\{\xi\in E\,;\,\forall g_1,\ldots,g_d\in G,\,\partial_{g_1}\cdots\partial_{g_d}\xi=0\}. \end{array}$

The functor ${E\mapsto E^{G(d)}}$ is left-exact. As for all left-exact functors, one can define its polynomial cohomology.

Definition 7 Let ${E}$ be a ${G}$-module. Pick an arbitrary relative injective resolution

$\displaystyle \begin{array}{rcl} 0\rightarrow E\rightarrow E_0\rightarrow E_1\rightarrow\cdots. \end{array}$

Its polynomial cohomology in degree ${d}$, ${H^{n}_{(d)}(G,E)}$, is the cohomology of the resulting complex of order ${d}$ invariants ${\cdots\rightarrow E_i^{G(d)}\rightarrow\cdots}$.

3.3. Computation

As an exercise, let us compute ${C(G,{\mathbb R})^{G(2)}}$. ${f\in C(G,{\mathbb R})^{G(2)}}$ iff for all ${g_1}$ and ${g_2\in G}$,

$\displaystyle \begin{array}{rcl} f(g_1g_2)-f(g_1)-f(g_2)+f(e)=0. \end{array}$

I.e. ${f}$ is a homomorphism plus a constant.

Elements of ${C(G,{\mathbb R})^{G(3)}}$ are known as bi-characters

${C({\mathbb R}^n,{\mathbb R})^{{\mathbb R}^n(d)}=\{}$polynomials of degree ${\leq d-1\}}$. Whence the name polynomial cohomology.

3.4. Polynomial maps

More generally, Lazard (’50s), Leibman (2002) call elements of ${C(G,{\mathbb R})^{G(d)}}$ polynomial maps on ${G}$. Note that polynomials can be multiplied together.

Proposition 8

$\displaystyle \begin{array}{rcl} H^1_{(d)}(G,{\mathbb R})=Pol_d(G)/Pol_{d-1}(G). \end{array}$

Now I describe polynomials in the nilpotent Lie group case. Fix a Mal’cev basis, i.e. a linear basis ${\{X_{ij}\}}$ adapted to the descending central filtration. Denote ${g_{ij}:=\exp(X_{ij})}$. Then the map

$\displaystyle \begin{array}{rcl} (t_{ij})\mapsto g=\prod g_{ij}^{t_{ij}} \end{array}$

is a diffeomorphism. Each coordinate ${\xi_{ij}:g\mapsto t_{ij}}$ becomes a function on ${G}$, this is a polynomial map of degree ${i}$.

Theorem 9 ${G}$ nilpotent Lie group. Then ${Pol(G)}$ is generated as an algebra by functions ${\xi_{ij}}$.

Example. ${G=}$ Heisenberg group. Then

$\displaystyle \begin{array}{rcl} H^1_{(2)}(G,{\mathbb R})=span\langle \xi_x^2,\xi_y^2,\xi_x\xi_y,\xi_z\rangle. \end{array}$

4. Main result

Theorem 10 (Kyed-Petersen) Let ${G}$, ${H}$ be nilpotent Lie groups. Assume they are uniformly measure equivalent. Then for all ${n}$ and ${d}$,

$\displaystyle \begin{array}{rcl} H^n_{(d)}(G,{\mathbb R})\simeq H^n_{(d)}(H,{\mathbb R}). \end{array}$

Let ${m:G\times G\rightarrow G}$ be the group multiplication. One can show that it induces ${m^*:Pol(G)\rightarrow Pol(G\times G)}$ which is a complete invariant of ${G}$. Our strategy is to express ${m^*}$ in terms of polynomial cohomology. In this way, we hope to be able to prove that quasiisometric nilpotent Lie groups must be isomorphic.

5. Proof

Recall that uniform measure equivalence means there exists a measure space ${(\Omega,\mu)}$ with a ${G\times H}$ action, isomorphisms to ${G\times Y}$ and ${G\times X}$ respectively, with ${X}$ and ${Y}$ compact.

Theorem 11 (Reciprocity theorem, inspired by Monod-Shalom)

$\displaystyle H^n_{(d)}(G,L^2_{loc}(\Omega, E)^{H(d')})\simeq H^n_{(d')}(H,L^2_{loc}(\Omega, E)^{G(d)}).$

Then Property ${H_T}$ leads to

$\displaystyle \begin{array}{rcl} H^n_{(d)}(G,{\mathbb R})\simeq H^n_{(d)}(G,L^2(X))\simeq H^n(H,L^2(Y,Pol_{d-1}(G))). \end{array}$

which relates to the higher cohomology of ${Pol_{d-1}(H)}$. Then recursion.

## Notes of Damien Gaboriau’s Lille lecture nr 3

Lück’s approximation theorem

For residually finite groups, ${\ell^2}$-Betti numbers can be obtained as limits of ordinary Betti numbers.

Theorem 1 (Lück) Let ${\Gamma_n}$ be decreasing finte index normal subgroups of ${\Gamma}$. Assume that

$\displaystyle \begin{array}{rcl} \bigcap_n \Gamma_n=\{e\}. \end{array}$

Assume that ${\Gamma}$ acts freely cocompactly on some simplicial complex ${L}$. Then, for all ${d}$,

$\displaystyle \begin{array}{rcl} \lim_{n\rightarrow\infty}\frac{1}{[\Gamma:\Gamma_n]}Betti_d(\Gamma_n\setminus L)=\beta_d(L,\Gamma). \end{array}$

1. Preparation: counting roots

Let ${M(T)=\prod(T-\lambda_i)}$ be a monic integral polynomial.

$\displaystyle \begin{array}{rcl} |\{\lambda_i\,;\,|\lambda_i|\in)0,\lambda)\}|\leq degree(M)\frac{\log D}{-\log\lambda}, \end{array}$

where ${D=\max|\lambda_i|}$.

Indeed, the product of nonzero roots equals the first nonzero coefficient of ${M}$, whose absolute value is ${\geq 1}$.

2. Proof

Write ${\beta_i(L,\Gamma)=\sum_{\sigma\subset D}\langle p_{ker}\sigma,\sigma\rangle}$, where ${p_{ker}}$ is the orthogonal projector from ${\ell^2}$ cochains to ${\ell^2}$ harmonic cochains. The usual Betti number has a similar expression, except that ${\ell^2}$ is replaced with ${\Gamma_n}$-invariant.

Since it involves only a fixed neighborhood of each simplex, the Laplacian does not distinguish ${\ell^2}$ from ${\Gamma_n}$-invariant cochains. This holds as well for every polynomial ${Q(\Delta)}$ in the Laplacian. Therefore, for ${n}$ large enough,

$\displaystyle \begin{array}{rcl} \frac{1}{[\Gamma:\Gamma_n]}Trace(Q(\Delta_n))=Trace_{\Gamma_n\setminus\Gamma}(Q(\Delta_n))=Trace_\Gamma(\Delta). \end{array}$

Let ${I\subset{\mathbb R}}$ be an interval. Let ${E_n(I)}$ denote the sum of the eigenspaces of ${\Delta_n}$ relative to eigenvalues belonging to ${I}$. The orthogonal projection onto ${E_n(I)}$ is denoted by ${\Pi_n(I)}$. Let

$\displaystyle \begin{array}{rcl} f_n(I)=\frac{1}{[\Gamma:\Gamma_n]}dim(E_n(I)). \end{array}$

Let ${E(I)}$, ${\Pi(I)}$ and ${f(I)}$ be the corresponding objects in ${\ell^2}$. ${f}$ is increasing and

$\displaystyle \begin{array}{rcl} \lim_{\lambda\rightarrow 0}f((0,\lambda))=0. \end{array}$

Our discussion of monic polynomials shows that

$\displaystyle \begin{array}{rcl} f_n((0,\lambda))\leq \alpha_d \frac{\log D}{-\log\lambda} \end{array}$

where ${D}$ is an upper bound on ${\|\Delta_n\|}$. This does not depend on ${n}$, and ${\alpha_d}$ is the number of orbits of ${d}$-simplices in ${L}$.

The idea is to approximate the indicator function of an interval by polynomials. Forevery ${\eta>0}$, there exists a polynomial ${Q_\eta}$ such that

$\displaystyle \begin{array}{rcl} 1_{[0,\eta)}\leq Q_\eta\leq (1+\eta)1_{[0,2\eta)}+\eta 1_{[0,T]}. \end{array}$

The same inequality holds for operators and for traces,

$\displaystyle \begin{array}{rcl} f([0,\eta))\leq Trace_\Gamma Q_\eta(\Delta)\leq(1+\eta)f([0,2\eta))+\eta \alpha_d. \end{array}$

This shows that ${Trace_\Gamma Q_\eta(\Delta)}$ tends to ${f(0)=\beta_d}$ as ${\eta}$ tends to 0.

Similarly,

$\displaystyle \begin{array}{rcl} f_n([0,\eta))\leq Trace_{\Gamma_n\setminus\Gamma} Q_\eta(\Delta_n)\leq(1+\eta)f_n([0,2\eta))+\eta \alpha_d. \end{array}$

Thanks to the monic estimate, ${f_n((0,\eta))}$ tends to 0 as ${\eta}$ tends to 0 uniformly in ${n}$. Therefore ${Trace_{\Gamma_n\setminus\Gamma} Q_\eta(\Delta_n)}$ converges uniformly to ${Betti_d(\Gamma_n\setminus L)}$ as ${\eta}$ tends to 0. This proves that ${Betti_d(\Gamma_n\setminus L)}$ tends to ${\beta_d}$.

3. Generalization

Theorem 2 (Bergeron-Gaboriau 2004) Let ${\Gamma_n}$ be decreasing finite index subgroups of ${\Gamma}$. Then

$\displaystyle \begin{array}{rcl} \lim_{n\rightarrow\infty}\frac{1}{[\Gamma:\Gamma_n]}Betti_d(\Gamma_n\setminus L)=\beta_d(\mathcal{L},\Gamma), \end{array}$

where ${\mathcal{L}}$ is a compact laminated space with free ${\Gamma}$ action.

$\displaystyle \mathcal{L}=\Gamma\setminus(\partial T)\times L,$

where ${T}$ is the rooted tree associated to the sequence ${\Gamma_n}$. Here, we use the extension of ${\beta_d}$ to equivalence relations, an elaboration on Connes’ work for foliations due to Gaboriau.

In case ${\Gamma}$ action on ${\partial L}$ is free and ${L}$ is contractible, ${\beta_d(\mathcal{L},\Gamma)=\beta_d(\Gamma)}$.

Connection with Lécureux’s talk. Each ${\Gamma_n}$ defines an (atomic) IRS. The limiting ${\ell^2}$ Betti number depends only on the limiting IRS, which is the IRS associated to the ${\Gamma}$ action on ${\partial T}$.

## Notes of Gaboriau’s Lille lecture nr 2

1. Formulae

Free products of infinite groups: ${\beta_n}$ add up for ${n\geq 2}$, ${\beta_1}$ add up plus one.

Free products with amalgamation: idem (${\beta_n}$ add up…) provided amalgamated subgroup is amenable.

2. Betti numbers of amenable groups

Following Cheeger and Gromov, I show that ${\ell^2}$ Betti numbers vanish for infinite amenable groups. I make the simplifying assumption that group ${\Gamma}$ acts freely and cocompactly on a contractible simplicial complex ${L}$.

The result obviously follows from

Lemma 1 The forgetful map to ordinary cohomology

$\displaystyle \begin{array}{rcl} \bar{H}^n_{(2)}(L)\rightarrow H^n(L,{\mathbb R}) \end{array}$

is injective.

We study the ${\Gamma}$-dimension of the kernel ${K}$ of the forgetful map. By definition,

$\displaystyle \begin{array}{rcl} dim_{\Gamma}(K)=\sum_{\sigma\subset D}\langle p_K\sigma,\sigma\rangle. \end{array}$

Let ${D\subset L}$ be a fundamental domain. Let ${F_j}$ be an increasing F\o lner sequence of finite subsets of ${\Gamma}$. Let ${X_j=F_j D}$. Then

$\displaystyle \begin{array}{rcl} |\{g\in F_j\,;\, gD\cap X_j\not=\emptyset\}|=o(|F_j|). \end{array}$

The ${\Gamma}$-dimension can be rewritten

$\displaystyle \begin{array}{rcl} dim_{\Gamma}(K)=\frac{1}{|F_j|}\sum_{\sigma\in F_j D}\langle p_K\sigma,\sigma\rangle. \end{array}$

Let ${p_j}$ denote restriction of cochains to ${X_j}$. I claim that

$\displaystyle \begin{array}{rcl} dim_{\Gamma}(K)\leq\frac{1}{|F_j|}\,dim_{{\mathbb R}}(p_j(K)). \end{array}$

Indeed, the composition ${p_j\circ p_K}$ is a contraction, so ${trace(p_j\circ p_K)\leq dim_{\mathbb R}(im(p_j\circ p_K))}$. Thus the ${F_j}$ block of the matrix of ${p_K}$, ${p_j\circ p_K\circ p_j}$, satisfies

$\displaystyle \begin{array}{rcl} trace(p_j\circ p_K\circ p_j)\leq dim_{\mathbb R}(im(p_j\circ p_K)). \end{array}$

Now I claim that

$\displaystyle \begin{array}{rcl} codim_{\mathbb R}(K\cap ker(p_j))\leq |\partial X_j|. \end{array}$

Finally, for ${h\in K}$, ${h=\delta b}$

$\displaystyle \begin{array}{rcl} \|p_j(h)\|^2=\langle p_j h,p_j h\rangle=\langle h,p_j h\rangle=[h,p_j h]=[\delta b,p_j]=[b,\partial p_j h]. \end{array}$

If the support of ${h}$ does not meet ${\partial X_j}$, then ${\partial p_j h=p_j \partial h}$, and thus ${[b,\partial p_j h]=0}$ and ${p_j h=0}$. Thus ${K\cap ker(p_j)}$ contains all the kernels of linear froms “evaluation on simplices of ${\partial X_j}$”. This yield the codimension estimate.

Since ${codim(K\cap ker(p_j))=dim(p_j(K))}$, we see that

$\displaystyle \begin{array}{rcl} dim_{\Gamma}(K)\leq\frac{|\partial F_j|}{|F_j|} \end{array}$

which tends to 0, so ${dim_{\Gamma}(K)=0}$ and the Lemma is proved.

3. Euler-Poincaré characteristic

Theorem 2 (Atiyah)

$\displaystyle \begin{array}{rcl} \chi(\Gamma\setminus L)=\sum_i (-1)^i b_i(L,\Gamma)=\sum_i (-1)^i\beta_i(L,\Gamma). \end{array}$

I will prove a stronger inequality,

3.1. Morse inequalities

Theorem 3 For all ${n\leq dim(L)}$,

$\displaystyle \begin{array}{rcl} \sum_{i=0}^n (-1)^{n-i} \alpha_i\geq\sum_{i=0}^n (-1)^{n-i}\beta_i(L,\Gamma), \end{array}$

where ${\alpha_i}$ is the number of ${k}$-simplices in ${\Gamma\setminus L}$.

Proof: usual diagram chasing.

3.2. Consequences

Corollary 4 If ${\Gamma}$ is finitely generated and infinite, with ${g}$ generators, ${\beta_1(\Gamma)\leq g-1}$.

If ${\Gamma}$ is finitely presented with ${r}$ generators, then

$\displaystyle \begin{array}{rcl} \beta_2(\Gamma)-\beta_1(\Gamma)+\beta_0(\Gamma)\leq r-g+1. \end{array}$

## Notes of Jean Lécureux’s lecture

Amenable invariant random subgroups

Joint with Bader, Duchesne, Glasner, Lazarovich.

1. IRS

Let ${G}$ be a locally compact second counable group. Let ${S(G)}$ be the topological space of closed subgroups of ${G}$: ${H_n}$ converges to ${H}$ if

• Every ${h\in H}$ is a limit of elements ${h_n\in H_n}$.
• Every converging subsequence of a sequence ${h_n\in H_n}$ converges to an element of ${H}$.

Definition 1 (Abert-Glazner-Virag) An invariant random subgroup (IRS) is a ${Ad_G}$-invariant probability measure on ${S(G)}$.

Examples.

• ${\delta_{\{e\}}}$, ${\delta_{G}}$.
• If ${N}$ is normal in ${G}$, ${\delta_N}$ is an IRS.
• Let ${\Gamma be a lattice. Then ${\Gamma\setminus G\rightarrow S(G)}$ mapsto
• Every IRS arises as follows. Let ${X}$ be a probability space with measure preserving ${G}$-action. Then ${x\mapsto Stab_G(x)}$ maps the measure to an IRS.

The space of IRS is compact. This raises the question of describing limits of IRS associated with lattices. Also, which sequences of lattices have the property that the corresponding IRS co,verge to ${\delta_{\{e\}}}$ ?

Lück’s theorem on ${\ell^2}$ Betti numbers can be interpreted in terms of convergence of IRS.

2. Amenable IRS

Definition 2 An IRS is amenable if it is supported on amenable subgroups.

The following is a generalization of a theorem of Kesten about normal subgroups.

Theorem 3 (Abert-Glasner-Virag) Let ${\mu}$ be an IRS. Then ${\mu}$ is amenable iff for ${\mu}$ a.e. subgroup ${H, the spectral radius of he simple random walk does not change when passing from ${G}$ to ${G/H}$.

2.1. Structure theorem

Theorem 4 (Bader-Duchesne-Lecureux) Every amenable IRS is supported on ${S(R_a(G))\subset S(G)}$, where ${R_a(G)}$ is the amenable radical, i.e. the largest normal amenable subgroup of ${G}$.

2.2. Proof

Can assume IRS is not supported on the space of subgroups of an any proper subgroup. Then one must show that ${G}$ is amenable. Use the fixed point property. Let ${G}$ act on a weakly compact convex set ${C}$. Assume ${C}$ is minimal (no proper compact convex invariant set). Almost every ${H}$ fixes a point in ${C}$. It suffices to prove that there is a common fixed point for a.e. ${H}$.

Baby case. ${H}$ has a unique fixed point. The map ${H\mapsto}$ fixed point of ${H}$ pushes forward ${\mu}$ to a ${G}$-invariant measure on ${C}$. Its barycenter is a fixed point in ${C}$.

General case. The fixed point set is a convex set ${C_H}$.

Lemma 5 The space ${C(E)}$ of compact convex subsets of ${E}$ is a convex cone in a a topological locally convex vector space. The subspace of convex subsets of a fixed convex set ${C}$ is convex and comapct in that space.

Indeed, compact convex sets can be added and dilated, and this corresponds to the linear structure on ${{\mathbb R}^{E^*}}$ when a convex set ${C}$ is mapped the function which to a linear form associates its max on ${C}$.

Since ${G}$ acts on ${{\mathbb R}^{E^*}}$

3. IRS in groups acting on ${CAT(0)}$ spaces

Let ${X}$ be a ${CAT(0)}$ space. Assume that either ${X}$ is proper and ${\partial X}$ is finite dimensional, or that ${X}$ has finite telescopic dimension. Assume too that ${X}$ is irreducible, and not a line.

Definition 6 Say ${G}$ acts geometrically densely on ${X}$ if there is no invariant convex subspace and no fixed point at the boundary.

Theorem 7 Assume that ${G}$ acts geometrically densely on ${X}$. Let ${\mu}$ be an IRS that does not charge the trivial subgroup. Then ${\mu}$-a.e. subgroup ${H}$ acts geometrically densely.

This is an elaboration on a result by Caprace and Monod.

3.1. Proof

Use Adams-Ballmann fixed point theorem for amenable groups acting on ${CAT(0)}$ spaces.

Let ${C_H}$ denote the fixed point set of subgroup ${H}$. Average distance to ${C_H}$ over ${\mu}$. Get concex, ${G}$-quasi-invariant function. If inf is achieved, it is on a convex subset. Otherwise, it is achieved on ${\partial X}$. Then ${[H,H]}$ stabilizes horoballs.