Groupes de Coxeter sphériques associés aux algèbres de Lie

Algèbres semisimples déployées

Soient \( F \) un corps de caractéristique 0 et \( \mathfrak{g} \) une algèbre de Lie semisimple sur \( F \). Soit \( E \) la clotûre algébrique de \( F \) ; on sit que \( \mathfrak{g} \) est déployée sur \( F \) s’il existe une sous-algèbre de Cartan \( H \) de \( \mathfrak{g} \otimes_F E \) telle que \( \dim_F(H \cap \mathfrak{g}) = \dim_E(H) \) et pour tout \( x \in H \cap \mathfrak{g} \) on a que \( \mathrm{ad}_x \) est diagonalisable sur \( F \). On peut alors appliquer la théorie vue dans le cas algébriquement clos à \( \mathfrak{g} \) sur \( F \) et on obtient le résultat suivant.

Théorème : Soit \( g \) une algèbre de Lie semisimple déployée sur \( F \). Soit \( H \) une sous-algèbre de Cartan de \( \mathfrak{g} \) et \( H^* \) son dual. Il existe un sous-ensemble \( \Phi = \Phi(\mathfrak{g}, H) \) de \( H^* \) tel que l’on ait la décomposition
\[
\mathfrak{g} = H \oplus \bigoplus_{\alpha \in \Phi} \mathfrak{g}_\alpha, \: \mathfrak{g}_\alpha = \{ x \in \mathfrak{g} : \forall h \in H, \mathrm{ad}_h(x) = \alpha(h)x \}.
\]

Groupe de Coxeter d’un système de racines

Si \( h \in H \) on a d’après la décomposition radicielle \( \mathrm{tr}(\mathrm{ad}_h) = \sum_{\alpha \in \Phi} \alpha(h)^2 \). En particulier, si \( F = {\mathbb R} \) la forme de Killing est définie positive en restriction à \( H \).

En général, pour définir un espace euclidien associé à \( \mathfrak{g} \) (et au choix de la sous-algèbre de Cartan) on utilise le lemme suivant.

Lemme : On peut choisir des générateurs \( X_\alpha \) des \( \mathfrak{g}_\alpha \) et \( H_1, \ldots, H_r \) de \( H \) tels que les crochets de Lie \( [X_\alpha, X_\beta] \) et \( [X_\alpha, H_i] \) soient des \( {\mathbb Q} \)-combinaisons linéaires des \( X_\alpha, H_i \).

La démonstration utilise la théorie des \( \mathfrak{sl}_2 \)-triplets et est donc un peu longue pour être donnée ici. On peut ausi déduire le lemme a posteriori de la classification des algèbres de Lie semsisimples scindées : il se trouve qu’elles sont toutes définissables sur \( {\mathbb Q} \).

Il suit du lemme que si on définit \( \mathfrak{g}_{\mathbb Q} \) comme le \( {\mathbb Q} \)-sous-espace de \( \mathfrak{g} \) engendré par les \( X_\alpha, H_i \) alors c’est une \( {\mathbb Q} \)-sous algèbre de Lie de \( \mathfrak{g} \) et on a \( \mathfrak{g} = F \otimes_{\mathbb Q} \mathfrak{g}_{\mathbb Q} \). La forme quadratique induite sur \( H_{\mathbb Q} \otimes_{\mathbb Q} {\mathbb R} \) par la forme de Killing \( K \) est alors définie positive. On définit alors l’espace euclidien \( V \) comme l’espace dual \( H_{\mathbb Q}^* \otimes_{\mathbb Q} {\mathbb R} \) munie de la forme duale de \( K \).

Il suit du lemme que \( \Phi \subset H_{\mathbb Q}^* \). La démonstration du lemme ci-dessus montre en fait que \( \Phi \) est un système de racines au sens axiomatique (cf. Bourbaki). La seule conséquence qui nous intéresse ici est la suivante.

Lemme : Soit \( \alpha \in \Phi \) et \( s_\alpha \) la réflexion orthogonale de \( V \) de miroir l’orthogonal de \( \alpha \). On a \( s_\alpha\Phi = \Phi \).

Soit \( W \) le sous-groupe de \( \mathrm O(V) \) engendré par les réflexions \( s_\alpha \). Par la théorie générale exposée dans les notes de Stéphane il existe un sous-ensemble \( \Delta \subset \Phi \) ayant les propriétés suivantes :

1. \( \Delta \) est une base de \( H^* \) ;
2. Si \( \alpha \in \Phi \) elle s’écrit sous la forme \( \pm \sum_{\beta \in \Delta} n_\beta \beta \) pour des \( n_\beta \in \mathbb N \) ;
3. \( (W, s_\beta) \) est un groupe de Coxeter.

Un tel \( \Delta \) est appelé une base du système de racines \( \Phi \). On notera \( \Phi^+ \) l’ensemble des racines positives pour \( \Delta \), c’est-à-dire qui s’écrivent \( \sum_{\beta \in \Delta} n_\beta \beta \), \( n_\beta \mathfrak{g}e 0 \).

Exemple : \( \mathfrak{g} = \mathfrak{sl}_3(F) \)

Dans ce cas on prend comme sous-algèbre de Cartan :
\[
H = \left\{ \begin{pmatrix} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c \end{pmatrix} : a, b, c \in F, a + b + c = 0 \right\}.
\]
et on note \( L_i \) la forme linéaire sur \( \mathfrak{gl}_3(F) \) défine par \( L_i(a_{ij}) = a_{ii} \). Alors \( \Phi = \{ \alpha_{ij} = L_i – L_j : i\not= j \} \) est le système de racines de \( \mathfrak{sl}_3 \) : le sous-espace \( (\mathfrak{sl}_3)_{\alpha_{ij}} \) est donné par les matrices \( (a_{kl}) \) avec \( a_{kl} = 0 \) si \( (k, l) \not= (i, j) \).

Une base de \( \Phi \) est donnée par exemple par \( \{L_1 – L_2, L_2 – L_3\} \). Le groupe de Coxeter est de type \( \mathrm A_3 \), en particulier isomorphe au groupe symétrique \( S_3 \).

Appartement d’un groupe algébrique semisimple

Soit \( \mathbf{G} \) un groupe algébrique connexe sur \( F \) dont l’algèbre de Lie est \( \mathfrak{g} \). Il existe un unique sous-\( F \)-groupe connexe \( \mathbf T \le \mathbf G \) tangent à \( H \). On note \( X(\mathbf T) = \hom_F(\mathbf T, F^\times) \) le groupe des caractères de \( \mathbf T \), qui est un groupe abélien libre de rang \( \dim(\mathbf T) \). On note \( \mathbf N = N_{\mathbf G}(\mathbf T) \) le normalisateur dans \( \mathbf{G} \) de \( \mathbf T \) (qui est un \( F \)-sous-groupe) ; on remarque que comme \( \mathbf{G} \) est connexe \( \mathbf T \) est égal à son propre centralisateur.

Lemme : On a des isomorphismes naturels \( e : V \cong X^*(\mathbf T) \otimes_{\mathbb Z} {\mathbb R}\) et \( f : W \cong \mathbf N(F)/\mathbf T(F) \). De plus \( e(w\xi) = f(w)e(\xi) \) pour touts \( \xi \in V \) et \( w \in W \).

BN-paires et construction de l’immeuble

Sous-groupe de Borel

Soit \( \mathfrak{n} = \bigoplus_{\alpha \in \Phi^+} \mathfrak{g}_\alpha \). Alors \( \mathfrak{n} \) est une sous-algèbre de Lie de \( \mathfrak{g} \) d’après la règle \( [\mathfrak{g}_\alpha, \mathfrak{g}_\beta] \subset \mathfrak{g}_{_alpha+\beta} \), puisque \( \Phi^+ \) est stable par addition. Il existe un unique sous-\( F \)-groupe connexe \( \mathbf U \le \mathbf{G} \) tangent à \( \mathfrak{n} \), et comme \( \mathfrak{n} \) est normalisée par \( H \), le sous-groupe \( \mathbf B = \mathbf T\mathbf U \) est le sous-\( F \)-groupe connexe tangent à \( H + \mathfrak{n} \). On l’appelle sous-groupe de Borel de \( \mathbf G \).

On peut montrer « à la main » qu’il existe un immeuble \( X \) dont les appartements sont des complexes de Coxeter pour \( V \) et sur lequel \( \mathbf G(F) \) agit en prolongeant l’action de \( \mathbf N(F) \) sur \( V \). Les chambres de cet immeuble sont en bijection avec l’ensemble \( \mathbf G(F) / \mathbf B(F) \), sur lequel on peut construire une relation d’adjacence ad hoc. Dans la suite on va plutôt expliquer une machinerie formalisant cette construction purement en termes de théorie des groupes.

BN-paires

Définition : Soit \( G \) un groupe et \( B, N \) des sous-groupes de \( G \) ; on note \( T = N \cap B \), on suppose que \( T \triangleleft N \) et on pose \( W = N/T \). On dit alors que \( (B, N) \) est une BN-paire si \( G = \langle B, N \rangle \), et il existe une famille génératrice \( S \) de \( W \) vérifiant les deux conditions suivantes :

1. Pour touts \( s \in S, w \in W \) on a \( sBw \subset BwB \cup BswB \) ;
2. Pour tout \( s \in S \) on a \( sBs^{-1} \not\subset B \).

Il suit de cette définition que \( (W, S) \) est un groupe de Coxeter (cf. Abramenko–Brown, Proposition 6.40) ; dans les cadres où on l’appliquera ceci sera déjà connu. Le lien avec les immeubles est donné par le résultat suivant (loc. cit., Theorem 6.56).

Théorème : Avec les notations ci-dessus il existe un immeuble épais dont les appartements sont isomorphes au complexe de Coxeter de \( (W, S) \), l’ensemble des chambres est \( G/B \) et les appartements sont les orbites des classes à gauche \( gT, g \in G \) sur cet ensemble. En particulier \( G \) agit par automorphismes sur cet immeuble, et l’action est transitive sur les chambres.

Les stabilisateurs des simplexes de l’immeuble sont les sous-groupes paraboliques de \( G \) associés à la BN-paire. Pour les décrire on commence par définir un sous-groupe parabolique standard comme suit : c’est un sous-groupe de la forme \( \langle T, B \rangle \) où \( T \subset S \). Ces derniers sont les stabilisateurs des faces de la chambre fondamentale (correspondant à la classe triviale \( 1B \in G/B) \). Un sous-groupe est donc parabolique s’il est conjugué par un élément de \( G \) à un sous-groupe parabolique standard.

BN-paire d’un groupe algébrique

L’ingrédient qui nous manque encore pour la construction de l’immeuble d’un \( F \)-groupe déployé \( \mathbf G \) est le suivant.

Lemme : Si \( \mathbf N, \mathbf B \) sont les \( F \)-sous-groupes de \( \mathbf G \) définis plus haut alors \( (\mathbf B(F), \mathbf N(F)) \) est une BN-paire.

De plus le groupe de Coxeter associé est bien le groupe de Weyl \( W = \mathbf N(F) / \mathbf T(F) \). On a donc bien un immeuble dont les appartements sont \( V \) et sur lequel \( \mathbf G(F) \) agit.

Dans ce cadre la définition ci-dessus de sous-groupe parabolique standard correspond à la définition classique : ce sont les groupes \( \mathbf P(F) \) où \( \mathbf B \le \mathbf P \le \mathbf G \) est un \( F \)-sous-groupe. Ils sont en bijection avec les sous-ensembles de la base \( \Delta \) ; si \( \Theta \subset \Delta \) le sous-groupe associé est tangent à la sous-algèbre
\[
H \oplus \mathfrak{n} \oplus \bigoplus_{\theta \in \Theta} \mathfrak{g}_{-\theta}.
\]

Exemple : \( \mathrm{SL}_3 \)

Si \( \mathbf G = \mathrm{SL}_3 \) son algèbre de Lie est \( \mathfrak{sl}_3 \). On a alors
\[
\mathbf T(F) = \left\{ \begin{pmatrix} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c \end{pmatrix} : abc = 1 \right\}, \, \mathbf B(F) = \left\{ \begin{pmatrix} a & x & y \\ 0 & b & z \\ 0 & 0 & c \end{pmatrix} : abc = 1 \right\}
\]
et \( W \) est le sous-groupe des matrices de permutation. Les sous-groupes paraboliques standard sont \( \mathbf G, \mathbf B \) et les conjugués de
\[
\mathbf P(F) = \left\{ \begin{pmatrix} a & b & x \\ c & d & y \\ 0 & 0 & e \end{pmatrix} : (ad – bc)e = 1 \right\}
\]
(qui correspond à la racine \( L_1 – L_2 \)).

On peut donner dans ce cas une interprétation géométrique de l’immeuble \( I \) associé à \( \mathbf G(F) \) : \( \mathbf G(F)/\mathbf B(F) \) est l’ensemble des drapeaux de \( F^3 \), qui représentent donc les 1-simplexes de \( I \). Les sommets adjacents à un drapeau \( 0 \subset D \subset P \subset F^3 \) sont la droite \( D \) et le plan \( P \) ; deux chambres sont donc adjacentes si elles sont représentées par des drapeaux ayant une droite ou un plan en commun.

Cet immeuble est donc un graphe, de valence \( |F| + 1 \) (le cardinal de la droite projective sur \( F \). Dans le cas où \( F = \mathbb F_2 \) (a priori non traité par les arguments ci-dessus, mais on peut les adapter) il est représenté par l’image suivante.

Formalisme des algèbres de Lie

Algèbres de Lie

Définition : Une algèbre de Lie sur \( F \) est un \( F \)-espace vectoriel \( L \) muni d’un crochet de Lie \( [\cdot, \cdot] \) qui est une application bilinéaire \( L \times L \to L \) qui est antisymétrique et vérifie la relation de Jacobi :
\[
[x, [y, z]] + [y, [z, x]] + [z, [x, y]] = 0
\]
pour touts \( x, y, z \in L \).

On définit de manière évidente les sous-algèbres de Lie et les morphismes. Une représentation d’une algèbre de Lie \( L \) est un morphisme \( L \to \mathfrak{gl}(V) \) pour un espace vectoriel \( V \) (que l’on supposera toujours de dimension fini dans la suite).

Exemples :

• Soit \( V \) un espace vectoriel, alors \( \mathfrak{gl}(V) = \mathrm{End}(V) \) muni du crochet \( [a, b] = ab – ba \) est une algèbre de Lie.

  Le sous-espace \( \mathfrak{sl}(V) = \{ x \in \mathfrak{gl}(V) : \mathrm{tr}(x) = 0 \} \) est une sous-algèbre de Lie.

  Si \( \dim(V) = n \) le choix d’une base de \( V \) détermine un isomorphisme \( \mathfrak{gl}(V) \cong \mathfrak{gl}_n(F) \) où \( \mathfrak{gl}_n(F) \) est l’algèbre de Lie des matrices \( n \times n \) munies du crochet évident ; de même \( \mathfrak{sl}(V) \cong \mathfrak{sl}_n(F) \) où \( \mathfrak{sl}_n(F) \) est la sous-algèbre des matrices de trace nulle.

• D’autres sous-algèbres importantes de \( \mathfrak{gl}_n(F) \) sont \( \mathfrak{t}_n(F) \), formée des matrices triangulaires supérieures, et \( \mathfrak{n}_n(F) \), formée des matrices strictement triangulaires supérieres.

Représentation adjointe, centre, idéaux

Soit \( A \) une \( F \)-algèbre (pas forcément associative). Une dérivation de \( A \) est un endomorphisme \( \delta \in \mathrm{End}_F(A) \) qui satisfait la règle de Leibniz :
\[
\delta(ab) = a\delta(b) + \delta(a)b
\]
pour touts \( a, b \in A \). L’ensemble des dérivations est une sous-algèbre de Lie de \( \mathfrak{gl}(V) \).

Si \( L \) est une algèbre de Lie et \( x \in L \) on note \( \mathrm{ad}_x \) l’endomorphisme linéaire de \( L \) défini par \( \mathrm{ad}_x(y) = [x, y] \). L’application \( \mathrm{ad} : L \to \mathfrak{gl}(L) \) , \( x \mapsto \mathrm{ad}_x \) est un morphisme d’algèbres de Lie. On l’appelle la représentation adjointe de \( L \). On vérifie en que \( \mathrm{ad}(L) \subset \mathrm{Der}(L) \).

Le centre \( Z(L) \) de \( L \) est par définition :
\[
Z(L) = \ker(\mathrm{ad}) = \{ x \in L : \forall y \in L, [x, y] = 0 \}.
\]
On dit que \( L \) est abélienne si \( L = Z(L) \).

En général le centre est un idéal de \( L \), c’est-à-dire un sous-espace \( I \le L \) tel que \( [x, y] \in I \) pour touts \( y \in I \) et \( x \in L \) (en particulier c’est une sous-algèbre de Lie). Un autre exemple d’idéal est l’algèbre dérivée de \( L \) définie par :
\[
[L, L] = \{ [x, y] : x, y \in L\}.
\]
On dit qu’une algèbre de Lie est simple si ses seuls idéaux sont elle-même et le sous-espace nul. Par exemple \( \mathfrak{sl}(V) \) est simple (la démonstration est la même que celle de la simplicité du groupe \( \mathrm{SL}(V) \)).

Algèbres résolubles et nilpotentes

La série dérivée \( L^{(0)}, L^{(1)}, \ldots \) de \( L \) est définie par récurrence comme suit :
\[
L^{(0)} = L, L^{(i+1)} = [L^{(i)}, L^{(i)}].
\]
On dit que \( L \) est résoluble si \( L^{(i)} = 0 \) pour \( i \) assez grand. Par exemple, l’algèbre \( \mathfrak{t}_n(F) \) des matrices triangulaires supérieures est résolubles : en effet le \( i \)-ème terme de sa série dérivée est contenu dans les \( (a_{kl} \) telles que \( l \le k + i \Rightarrow a_{kl} = 0 \) donc on a \( L^{(n)} = 0 \).

On a les propriétés de stabilité suivante pour cette notion (les démonstrations sont immédiates d’après les définitions).

Proposition :

1. Si \( L \) est résoluble alors toute sous-algèbre ou image de \( M \) est résoluble.
2. Si \( I \) est un idéal résoluble de \( L \) et \( L/I \) est aussi résoluble alors \( L \) elle-même doit être résoluble.
3. Si \( I, J \) sont des idéaux résolubles de \( L \) alors \( I + J \) aussi.

Il suit de la propriété 3. ci-dessus qu’une algèbre de Lie (de dimension finie) \( L \) contient un unique idéal résoluble maximal. Ce dernier est appelé radical résoluble de \( L \) et noté \( \mathrm{Rad}(L) \).

Définition : On dit que \( L \) est semisimple si l’une des conditions équivalentes suivantes est satisfaite :

• On a \( \mathrm{Rad}(L) = 0 \) ;
• Il n’existe pas d’idéal abélien \( I \subset L \) ;

La série centrale \( L^0, L^1, \ldots \) de \( L \) est définie par :
\[
L^0 = L, L^{i+1} = [L, L^i].
\]
On dit que \( L \) est nilpotente si \( L^i = 0 \) pour \( i \) assez grand. Par exemple l’algèbre \( \mathfrak{n}_n(F) \) est nilpotente par le même argument que celui utilisé pour démontrer que \( \mathfrak{t}_n \) est résoluble (noter que \( \mathfrak{t}_n(F) \) elle-même n’est pas nipotente, vu que \( [\mathfrak{t}_n, \mathfrak{t}_n^{(i)}] = \mathfrak{t}_n^{(i)} \) pour \( i \ge 1 \)).

La nilpotence est stable par passage aux sous-algèbres et aux images. On a de plus les propriétés importantes suivantes.

Proposition :

1. \( L \) est nilpotente si et seulement si \( L/Z(L) \) est nilpotente.
2. Si \( L \) est nilpotente alors\( Z(L) \not= 0 \).

Théorèmes fondamentaux

Théorème d’Engel

Si \( L \) est une algèbre de Lie nilpotente et \( x_0, \ldots, x_i \in L \) on a \( \mathrm{ad}_{x_i} \cdots \mathrm{ad}_{x_1} \in \mathrm{ad}(L^{i}) \) et cet élément est donc nul pour \( i \) assez grand. En particulier il existe un \( n \) tel que \( (\mathrm{ad}_x)^n = 0 \) pour tout \( x \in L \). Autrement dit tous les éléments d’une algèbre nilpotente sont nilpotents (au sens usuel) dans la représentation adjointe. Le théorème d’Engel est une réciproque de cet énoncé.

Théorème (Engel) : Soit \( L \) une algèbre de Lie. Si \( \mathrm{ad}_x \) est un endomorphisme nilpotent de \( L \) pour tout \( x \in L \) alors \( L \) est nilpotente.

Démonstration

\( L \) est nilpotente si et seulement si \( \mathrm{ad}(L) \) l’est. On obtient alors cet énoncé par récurrence sur \( \dim(L) \), comme conséquence du lemme d’algèbre linéaire suivant : si \( L \) est une sous-algèbre de Lie de \( \mathfrak{gl}(V) \) telle que tout \( x \in L \) est un endomorphisme nilpotent alors il existe un vecteur \( v \in V \) non-nul tel que \( xv = 0 \) pour tout \( x \in L \).

Théorème de Lie

Théorème (Lie) : On suppose que \( F \) est algébriquement clos. Soit \( L \subset \mathfrak{gl}(V) \) une sous-algèbre de Lie résoluble. Il existe un \( v \in V \), \( v \not= 0 \) tel que \( xv \in Fv \) pour tout \( x \in L \).

Démonstration

On obtient cet énoncé par récurrence sur \( \dim(L) \) ; il est évidemment vrai pour \( \dim(L) = 0, 1 \). Le point de départ de la récurrence est le lemme suivant.

Lemme 1 : Il existe un idéal \( I \subset L \) de codimension 1.

Par l’hypothèse de récurrence il existe un \( v \in V \setminus 0 \) tel que \( xv \in Fv \) pour tout \( x \in I \). Soit \( \lambda \) la forme linéaire sur \( I \) telle que \( xv = \lambda(x)v \) pour \( x \in I \). On définit un sous-espace
\[
W = \bigcap_{x \in I} \ker(x – \lambda(x))
\]
qui est non-nul puisque \( v \in W \). On a alors

Lemme 2 : \( LW \subset W \)

On peut alors conclure de la manière suivante : on écrit \( L = Fz + I \) (pour n’importe quel \( z \in L \setminus I \)) et on obtient le vecteur désiré en prenant n’importe quel vecteur propre de \( z \) dans le sous-espace stable \( W \) (c’est ici que l’hypothèse sur \( F \) est utilisée).

Démonstration des lemmes

Le lemme 2 est une conséquence à peu près immédiate de ce que \( I \) est un idéal. Le lemme 1 se démontre comme suit : l’algèbre \( L^a = L/[L, L] \) est abélienne et non-nulle. On choisit un sous-espace \( J \subset L^a \) de codimension 1 ; il suit immédiatement que \( J + [L, L] \) est un idéal de codimension 1 dans \( L \).

Conséquences

Le théorème de Lie a les corollaires immédiats suivants :

1. Si \( L \) est une sous-algèbre résoluble de \( \mathfrak{gl}(V) \) alors il existe un drapeau de \( V \) (c’est-à-dire une suite de sous-espaces \( 0 = V_0 \subset V_1 \subset \cdots \subset V_n = V \) avec \( \dim(V_{i+1}/V_i) = 1 \)) stabilisé par \( L \) ; autrement dit \( L \) est une sous-algèbre d’un conjugué de \( \mathfrak{t}_n \). (Noter que le théorème d’Engel implique un énoncé similaire pour les sous-algèbres nilpotentes.)
2. \( L \) est résoluble si et seulement s’il existe des sous-algèbres \( L_0 = 0 \subset \cdots \subset L_n = L \) telles que \( \dim(L_{i+1}/L_i) = 1 \) et \( L_i \) est un idéal de \( L_{i+1} \).
3. \( L \) est résoluble si et seulement si son algèbre dérivée \( [L, L] \) est nilpotente.

Noter que le dernier point est valide même si \( F \) n’est pas algébriquement clos.

Critère de Cartan

Il suit de la caractérisation traciale des endomorphismes nilpotents, du théorème d’Engel et du dernier critère de résolubilité ci-dessus que si \( L \subset \mathfrak{gl}(V) \) vérifie que \( \mathrm{tr}(xy) = 0 \) pour tout \( x \in [L, L] \) et tout \( y \in L \) alors elle est résoluble. On obtient ainsi le critère de résolubilité suivant.

Théorème (Cartan) : Soit \( L \) une algèbre de Lie. Si \( \mathrm{tr}(\mathrm{ad}_x\mathrm{ad}_y) = 0 \) pour touts \( x \in [L,L] \) et \( y \in L \) alors \( L \) est résoluble.

Structure des algèbres de Lie semisimples

Forme de Killing

Définition : La forme de Killing d’une algèbre de Lie \( L \) est la forme bilinéaire symétrique \( K = K_L \) sur \( L \) donnée par :
\[
K(x, y) = \mathrm{tr}(\mathrm{ad}_x\mathrm{ad}_y).
\]

Les propriétés suivantes sont immédiates :

1. On a \( K([x, y], z) = K(x, [y, z]) \) (invariance de \( K \)).
2. Le noyau \( S = \ker(K) = \{ x \in L : K(\cdot, x) = 0\} \) est un idéal de \( L \).
3. Si \( I \subset L \) est un idéal alors \( K_I = K_L|_I \).

Le résultat utile pour la suite sur la forme de Killing est alors le théorème suivant.

Théorème : \( L \) est semi-simple si et seulement si \( K \) est non-dégénérée.

Démonstration

L’idéal \( S = \ker(K) \) est résoluble (ceci suit directement du critère de Cartan) et si \( L \) est semisimple on a donc \( S \subset \mathrm{Rad}(L) = 0 \) donc \( K \) est non-dégénérée.

Réciproquement, si \( S = 0 \) alors \( L \) ne contient pas d’idéal abélien non-nul (un tel idéal est contenu dans \( S \)). Il suit que \( L \) ne contient pas non plus d’idéal résoluble non-nul (le dernier terme de la série dérivée d’un tel idéal serait un idéal abélien non-nul de \( L \)), et donc que \( \mathrm{Rad}(L) = 0 \).

Décomposition en algèbres simples

Dans toute la suite on suppose que \( L \) est semisimple.

Théorème : Il existe des idéaux simples \( L_1, \ldots, L_n \) de \( L \), uniques à permutation près, tels que l’on ait la décomposition
\[
L = L_1 \oplus \cdots \oplus L_n.
\]

Démonstration

Ceci suit d’une récurrence sur la dimension mise en place comme suit : si \( L \) n’est pas simple elle contient un idéal \( 0 \not= I \not= L \), qui doit lui aussi être semisimple (sinon \( \mathrm{Rad}(I) \) serait un idéal résoluble de \( L \)). Son orthogonal \( I^* \) pour \( K \) est alors un idéal, et on a \( L = I \oplus I^* \) car \( K \) et \( K|_I = K_I \) sont non-dégénérées. On conclut en appliquant l’hypothèse de récurrence à \( I \) et \( I^* \).

Conséquences

Le théorème de décomposition a les corollaires suivants, que l’on peut aussi déduire directement de sa démonstration.

1. On a \( [L, L] = L \).
2. Les idéaux et quotients de \( L \) sont aussi semisimples.

Représentation adjointe des algèbres semisimples

La représentation adjointe \( \mathrm{ad} : L \to \mathrm{Der}(L) \) de l’algèbre de Lie semisimple \( L \) a les propriétés suivantes :

1. \( \mathrm{ad} \) est fidèle ;
2. \( \mathrm{ad}(L) = \mathrm{Der}(L) \) ;

La première suit immédiatement du fait que \( \ker(\mathrm{ad}) \) est un idéal abélien. La seconde se démontre comme suit : si \( \delta \in \mathrm{Der}(L) \) et \( x \in L \) on a \( \mathrm{ad}_{\delta(x)} = [\delta, \mathrm{ad}_x] \) et il suit que \( \mathrm{ad}(L) \) est un idéal de \( \mathrm{Der}(L) \). Soit \( J \) son orthogonal pour la forme de Killing \( K_{\mathrm{Der}(L)} \) ; alors \( J \cap \mathrm{ad}(L) = 0 \) vu que \( K_I = K_{\mathrm{Der}(L)}|_I \) est non-dégénérée, et par inégalité sur les dimensions \( \mathrm{Der}(L) = J \oplus \mathrm{ad}(L) \). Il suit aussi que \( \mathrm{ad}_{\delta(x)} = [\delta, \mathrm{ad}_x] \in J \cap \mathrm{ad}(L) \) est nul pour touts \( x \in L \) et \( \delta \in J \). Comme \( \mathrm{ad} \) est injective il suit que pour tout \( \delta \in J \) on a \( \delta(x) = 0 \) pour tout \( x \in L \), c’est-à-dire \( \delta = 0 \). On conclut que \( J = 0 \) et donc que \( \mathrm{ad}(L) = \mathrm{Der}(L) \).

Décomposition de Jordan

Proposition : Pour tout \( x \in L \) il existe une paire \( (x_n, x_s) \in L \times L \) telle que \( \mathrm{ad}_{x_n} \) est nilpotent, \( \mathrm{ad}_{x_s} \) est semisimple, \( [x_n, x_s] = 0 \) et \( x = x_s + x_n] \).

Pour démontrer ceci on admet le fait suivant : si \( x \) est une endomorphisme linéaire de \( L \) on note \( x_n \) sa partie nilpotente et \( x_s \) sa partie semisimple (diagonalisable) données par le théorème de réduction de Jordan ; on a alors :

Si \( x \in \mathrm{Der}(L) \) alors \( x_n, x_s \in \mathrm{Der}(L) \).

Le résultat suit alors immédiatement du fait (démontré ci-dessus) que \( \mathrm{ad} \) est un isomorphisme \( L \to \mathrm{Der}(L) \).

La décomposition de Jordan est fonctorielle, c’est-à-dire que pour tout morphisme \( \rho : L \to \mathfrak{gl}(V) \) (dont l’image est forcément contenue dans \( \mathfrak{sl}(V) \)) et tout \( x \in L \) on a \( \rho(x)_n = \rho(x_n) \).

Décomposition radicielle

Soit \( H \subset L \) une sous-algèbre abélienne dont tous les éléments sont semisimples (c-à-d \( x = x_s \) pour tout \( x \in H \)), et maximale pour ces propriétés. (On peut démontrer sans difficulté que demander que \( H \) soit abélienne est superflu.) Une telle sous-algèbre est appelée sous-algèbre de Cartan de \( L \).

Il existe une base de \( L \) qui diagonalise simultanément tous les éléments de \( H \) ; autrement dit il existe un sous-ensemble fini \( \Phi \) du dual \( L^* \) tel que
\[
L = H \oplus \bigoplus_{\alpha \in \Phi} L_\alpha, \: L_\alpha = \{ x \in L : \forall h \in H, \mathrm{ad}_h(x) = \alpha(h)x \}.
\]
Cette décomposition est appelée décomposition radicielle de \( L \) et \( \Phi \) est appelé un système de racines de \( L \). Tout ceci dépend du choix de \( H \) mais on peut démontrer que toutes les sous-algèbres de Cartan sont conjuguées l’une à l’autre et il en va de même pour les décompositions radicielles et systèmes de racines.

On a les propriétés importantes suivantes :

1. \( [L_\alpha, L_\beta] \subset L_{\alpha+\beta} \) (qui est nul si \( \alpha + \beta \not= 0, \not\in \Phi \) et \( H \) si \( \alpha + \beta = 0 \)).
2. Pour tout \( \alpha \in \Phi \) le sous-espace \( L_\alpha \) est \( \mathrm{ad} \)-nilpotent.
3. Si \( \alpha \not= \beta \) alors \( L_\alpha \) est Killing-orthogonal à \( L_\beta \).

Immeubles

Contemplons la définition suivante d’immeuble :

Définition Un immeuble est un complexe simplicial \(\Delta\) obtenu comme union de sous-complexes \(\Sigma\) (les appartements) satisfaisant les axiomes suivants:

1. Chaque appartement \(\Sigma\) est un complexe de Coxeter.
2. Pour tout couple de simplexes \(A, B \in \Delta\), il existe un appartement \(\Sigma\) contenant les deux.
3. Si \(\Sigma\) et \(\Sigma’\) sont deux appartements contenant des simplexes \(A\) et \(B\), alors il existe un isomorphisme \(\Sigma \to \Sigma’\) fixant \(A\) et \(B\) point par point.

Dans ces exposés on va introduire la notion de complexe de Coxeter, qui sont des complexes simpliciaux basiques qui serviront à contruire les immeubles.

Les axiomes ci-dessus disent qu’un immeuble est en un certain sens un objet très symétrique, on verra plus tard leurs conséquences. Noter cependant qu’il est possible qu’un immeuble n’admette aucun automorphisme non trivial, comme dans l’exemple suivant :

Exemple Soit \(\Delta\) un arbre simplicial, dont tous les sommets sont de valences \(\ge 3\) deux à deux distinctes. Alors \(\Delta\) est un immeuble, dont les appartements sont les droites simpliciales plongées dans \(\Delta\).
On verra que ce sont des complexes de Coxeter de type \(\tilde A_1\).

Groupes et matrices de Coxeter

Un groupe de Coxeter \(W\) est un groupe engendré par un ensemble \(S\) (disons fini, de cardinal \(n\)) d’involutions, et qui admet une présentation par générateurs et relations très simple :
\[
W = \left\langle S \mid (s,t)^{m(s,t)} = 1\right\rangle
\]
Ici la matrice \(M = (m(s,t))_{1 \le s,t \le n}\) est symétrique à coefficients entiers positifs, avec des \(1\) sur la diagonale est des entiers \(\ge 2\) partout ailleurs. On admet aussi la valeur \(m(s,t) = \infty\). On dit que \(M\) est la matrice de Coxeter encodant \(W\). Si on veut souligner le choix de \(S\), on parle de système de Coxeter \((W,S)\).

Exemple

1. Le groupe \(S_3\) des symétries d’un triangle équilatéral, engendré par les transpositions \((12)\) et \((23)\) dont le produit est d’ordre 3.
2. Le groupe infini engendré par les symétries par rapport aux côtés d’un
   triangle équilatéral, dont les produits deux à deux sont d’ordre 3.
3. Le groupe \(\mathrm{PGL}_2({\mathbb Z})\) est engendré par les matrices
   \[
   s_1 = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \quad
   s_2 = \begin{pmatrix} -1 & 1 \\ 0 & 1 \end{pmatrix}, \quad
   s_3 = \begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}.
   \]
   et les produits \(s_1s_2\), \(s_1s_3\), \(s_2s_3\) sont d’ordre respectif \(3,2,\infty\).

Graphes de Coxeter

La donnée d’une telle matrice \(n \times n\) est encodée par un graphe de Coxeter \(\Gamma\), avec \(n\) sommets et une arête avec étiquette \(m(s,t)\) pour chaque \(m(s,t) \ge 3\). Il est d’usage de ne pas noter les étiquettes 3 sur les dessins. On dit que la matrice de Coxeter \(M\) est irréductible (ou indécomposable) si le graphe \(\Gamma\) associé est connexe.

La matrice de Gram d’un graphe de Coxeter est la matrice symétrique \(A\) avec coefficients \(-\cos \frac{\pi}{m(s,t)}\) (à interpréter comme \(-1\) dans le cas \(m(s,t) = \infty\)). Pour les calculs il est souvent plus aisé de considérer \(2A\) (notamment pour que les \(-\cos \frac{\pi}{3}\) deviennent des \(-1\)…)

Si \(\Gamma\) est un graphe de Coxeter, on notera \(\det (\Gamma)\) le déterminant \(\det (2A)\), où \(A\) est la matrice de Gram associée.

Exemple
Groupe \(W\) d’ordre 48 des isométries de \({\mathbb R}^3\) préservant un cube:

• \( M = \left(\begin{array}{ccc} 1 & 3 & 2 \\ 3 & 1 & 4 \\ 2 & 4 & 1 \end{array}\right)\)
• \(\Gamma = \)
• \( 2A = \left(\begin{array}{ccc} 2 & -1 & 0 \\ -1 & 2 & -\sqrt 2 \\ 0 & -\sqrt 2 & 2 \end{array}\right) \)
• \(\det (\Gamma) = 2 \).

Représentation canonique

Soit \((W,S)\) un système de Coxeter, \(V\) l’espace vectoriel \(V = \oplus_{s \in S} {\mathbb R} e_s\), et \(B(.,.)\) la forme bilinéaire définie par
\[
B(e_s,e_t) = – \cos \frac{\pi}{m(s,t)}.
\]
On fait agir \(W\) sur \(V\) en posant, pour chaque \(s \in S\)
\[\sigma_s(v) = v – 2 B(e_s,v) e_s,\]
ou autrement dit \(\sigma_s\) est la (une) réflexion orthogonale de plan
\(e_s^\perp\) par rapport à la forme \(B\).
Par acquis de conscience, on peut vérifier que pour tout \(v,v’ \in V\), on a :
\[\begin{array}{rl}
B(\sigma_s (v), \sigma_s (v’) ) &= B(v-2B(e_s,v)e_s, v’-2B(e_s,v’)e_s) \\
&= B(v,v’) – 2 B(e_s,v) B(e_s,v’) – 2 B(e_s,v’)B(v,e_s) +4 B(e_s,v)B(e_s,v’)\cdot 1
\\
&= B(v,v’).
\end{array}\]
On obtient donc un morphisme de \(W\) vers le groupe orthogonal \(\mathrm O(V)\) pour la forme \(B\).

Lemme
Soit \(W = \langle S; (st)^{m(s,t)} = 1 \rangle\) un groupe de Coxeter. Alors chaque \(s \in S\) est d’ordre 2 dans \(W\), et chaque produit \(st\) est d’ordre \(m(s,t)\).

Démonstration :
On a un morphisme signature \(W \to \{\pm 1\}\) qui envoie chaque \(s \in S\) sur \(-1\), donc les \(s\) sont d’ordre 2 dans \(W\). Si \(s\neq t \in S\), alors \(\sigma_s\) et \(\sigma_t\) préserve le plan \(\mathrm{Vect}(e_s, e_t)\), et comme la restriction de \(B\) est de matrice définie positive (on traite d’abord le cas \(m = m(s,t)\) fini)
\[
\begin{pmatrix}
1 & -\cos \frac{\pi}{m} \\ -\cos \frac{\pi}{m} & 1
\end{pmatrix},
\]
le groupe \(\langle \sigma_s, \sigma_t \rangle\) est diédral d’ordre \(m\), car l’angle entre les deux vecteurs \(e_s\) et \(e_t\) est \(\pi – \frac{\pi}{m}\). Cela montre que \(st\) est d’ordre \(m\) dans \(W\).

Représentation duale

On part de l’action de \(W\) sur \(V’ = \oplus_{s \in
S} {\mathbb R} e_s\), par
\[s'(f) = f – 2 B(e_s,f) e_s\]
On note \(V\) le dual de \(V’\).
On pense aux éléments de \(V’\) comme des formes linéaires, et à ceux de \(V\) comme des vecteurs.
Pour tout \(s \in S\), il existe un unique vecteur \(v_s \in V\) tel que pour tout \(f \in V’\),
\[ f(v_s) = B(e_s,f). \]

Lemme
L’action duale de \(S\) sur \(V\) est donnée par
\[
s(v) = v-2e_s(v) v_s.
\]

Démonstration :
\[\begin{array}{rl}
\langle e_t, s(v) \rangle
&= \langle e_t, v- 2 e_s(v) v_s \rangle \\
&= e_t(v) – 2 e_t(v_s) e_s(v) \\
&= e_t(v) – 2B(e_s,e_t)e_s(v) \\
&= \langle e_t – 2B(e_s,e_t)e_s,v \rangle \\
&= \langle s'(e_t), v \rangle.
\end{array}\]

Exemple
Soit \(W\) le groupe diédral infini:

• \( M = \begin{pmatrix} 1 & \infty \\ \infty & 1 \end{pmatrix} \)
• \(\Gamma = \)
• \(A = \begin{pmatrix} 1 & -1 \\ -1 & 1 \end{pmatrix} \)
• \( \det (\Gamma) = 0 \).

On considère l’espace vectoriel \(V’ = {\mathbb R} e_s \oplus {\mathbb R} e_t\), et \(V\) son dual. On définit comme précédemment deux réflexions \(s’\) et \(t’\) sur \(V’\):
\[\begin{array}{c}
s'(f) &= f – 2B(e_s,f)e_s, \\
t'(f) &= f – 2B(e_t,f)e_t.
\end{array}\]
Observer que \(s'(e_s) = -e_s\), \(t'(e_t) = -e_t\) et \(s’,t’\) fixent toutes deux point par point la droite engendrée par \(e_s + e_t\), ce qui donne :
\[\begin{array}{cc}
s'(e_t) = 2e_s + e_t, && t'(e_s) = e_s + 2e_t.
\end{array}\]

Notons \(s,t\) les réflexions linéaires sur \(V\) induites par \(s’,t’\), alors en notant \((x,y)\) les coordonnées sur \(V\) telles que
\[\begin{array}{cc}
e_s(x,y) = x, && e_t(x,y) = -x + y,
\end{array}\]
on obtient
\[\begin{array}{cc}
s (x,y) = (-x,y), && t(x,y) = (-x + 2y, y).
\end{array}\]
En effet :
\[\begin{array}{ccccccccccccccccc}
\langle e_s, s(s,y) \rangle &&=&&\langle e_s, (-x,y) \rangle&& = &&-x &&=&& \langle -e_s, (x,y) \rangle &&=&& \langle s'(e_s), (x,y) \rangle, \\
\langle e_t, s(s,y) \rangle &&=&& \langle e_t, (-x,y) \rangle&& = &&x + y &&=&& \langle 2e_s+e_t, (x,y) \rangle &&= &&\langle s'(e_t), (x,y) \rangle, \\
\langle e_s, t(x,y) \rangle &&=&&\langle e_s, (-x+2y,y) \rangle&& = &&-x+2y &&=&& \langle e_s+2e_t, (x,y) \rangle &&=&& \langle t'(e_s), (x,y) \rangle, \\
\langle e_t, t(x,y) \rangle &&=&&\langle e_t, (-x+2y,y) \rangle&& = &&x-y &&=&& \langle -e_t, (x,y) \rangle &&=&& \langle t'(e_t), (x,y) \rangle.
\end{array}\]

Le cas \(W\) fini

Théorème
Soit \((W,S)\) un système de Coxeter, et \(B\) la forme quadratique associée. Le groupe \(W\) est fini si et seulement si la forme \(B\) est définie positive.

Démonstration :
On peut se ramener à \((W,S)\) irréductible, en travaillant sur chaque bloc de la matrice de \(B\).

Si \(B\) est définie positive, \(W\) s’identifie à un groupe discret du groupe
compact \(O(V)\), et est donc fini.

Réciproquement si \(W\) est fini, par moyennisation du produit scalaire standard
sur \(V\) il préserve un produit scalaire. On a donc deux formes bilinéaires invariantes, \(B\) et un produit scalaire \(\langle \cdot, \cdot \rangle\). On veut montrer que \(B = c\, \langle \cdot, \cdot \rangle\) pour une constante \(c > 0\). Tout d’abord \(B\) est non dégénérée, car sinon \(\ker B\) serait un sous-espace stable sans supplémentaire stable. Tout sous-espace stable devrait être dans le noyau de \(B\), donc la représentation de \(W\) est irréductible. Le centralisateur de \(W\) est réduit aux matrices scalaires. Cela implique qu’il existe une unique forme bilinéaire symétrique invariante (à un scalaire près), et donc \(B\) est un multiple (positif, puisque admet des 1 sur la diagonale) du produit scalaire invariant.

Classification

Graphe (semi)-définis positifs

On dit qu’un graphe de Coxeter est défini positif (resp. semi-défini positif) si la matrice de Gram \(A\) associée l’est (ou de façon équivalente, \(2A\)).

On dit qu’un graphe de Coxeter \(\Gamma\) contient un graphe de Coxeter \(\Gamma’\) si \(\Gamma’\) s’obtient à partir de \(\Gamma\) en supprimant des arêtes et/ou en diminuant des poids.
On obtient aussi une relation d’ordre partiel sur les graphes de Coxeter, notée \(\Gamma’ \prec \Gamma\) pour indiquer une inclusion stricte.

Exemple

Lemme
Une matrice symétrique est définie positive si et seulement si tous ses mineurs principaux sont strictement positifs.

Variante: si le déterminant est nul est que tous les autres mineurs principaux sont strictement positifs, alors la matrice est semi-définie positive.

Démonstration :
Si \(A\) est symétrique définie positive alors ses valeurs propres sont réelles strictement positives, et donc \(\det A > 0\). De plus la restriction de \(q_A\) à tout sous-espace est encore définie positive, ce qui donne la positivité des mineurs en restreignant au sous-espace engendré par \(e_1, \dots, e_k\) pour chaque \(k\).

La réciproque est claire en dimension 1. Supposons maintenant la réciproque vraie en dimension \(n-1\), et montrons la en dimension \(n\).
Écrivons \({\mathbb R}^n = {\mathbb R}^{n-1} \oplus {\mathbb R}\), alors la restriction de \(q_A\) au facteur \({\mathbb R}^{n-1}\) est définie positive par hypothèse de récurrence.
De plus \(\det A >0\), donc \(q_A\) est non dégénéré, donc \(({\mathbb R}^{n-1})^\perp\) est une droite \(\mathrm{Vect} (e)\), et on a une somme directe orthogonale \({\mathbb R}^n = {\mathbb R}^{n-1} \oplus \mathrm{Vect} (e)\).
Notons \(A’\) la matrice de la restriction de \(q_A\) à \({\mathbb R}^{n-1}\) (dans une base quelconque), alors \(\det A\) et \(\det A’ \cdot q_A(e)\) ont même signe, et donc \(q_A(e) > 0\) et \(A\) est définie positive.

Preuve de la variante :
à nouveau on écrit \({\mathbb R}^n = {\mathbb R}^{n-1} \oplus {\mathbb R}\), et la restriction de \(q_A\) au
facteur \({\mathbb R}^{n-1}\) est définie positive par ce qui précède.
Soit \(e\) un vecteur isotrope pour \(q_A\), alors on a une somme directe orthogonale \({\mathbb R}^n = {\mathbb R}^{n-1} \oplus \mathrm{Vect} (e)\), et \(q_A\) est positive comme somme directe d’une définie positive sur \({\mathbb R}^{n-1}\) et de la forme triviale sur \({\mathbb R}\).

Une matrice est indécomposable si aucune permutation des vecteurs de
la base canonique ne la rend (non trivialement) diagonale par blocs.

Lemme
Soit \(A\) une matrice symétrique semi-définie positive indécomposable, avec tous les coefficients non diagonaux \(a_{ij}\) négatifs ou nuls. Alors le cone isotrope est ou bien réduit à \(\{0\}\), ou bien égal à une droite. De plus la plus petite valeur propre de \(A\) a multiplicité \(1\), et il existe un vecteur propre associé dont toutes les coordonnées sont strictement positives.

Démonstration :
Comme \(A\) est semi-définie le cône isotrope est égal au noyau.
Soit \(x \neq 0\) dans le cône isotrope, et \(z\) le vecteur obtenu en prenant la
valeur absolue des coordonnées de \(x\). L’hypothèse \(a_{ij} \le 0, i\neq j,\)
implique
\[ 0 \le z^t A z \le x^t A x = 0 \]
donc \(z\) est aussi dans le cône isotrope.
Pour tout \(i\), on a \(\sum a_{ij} z_j = 0\).
Soit \(J\) l’ensemble des indices \(j\) avec \(z_j > 0\), et \(I\) son complément, correspondant à l’ensemble des indices \(i\) avec \(z_i = 0\).
On obtient que \(a_{ij} = 0\) pour tout \(i \in I, j \in J\), et donc \(I\) est vide par indécomposabilité de \(A\).
Ainsi le noyau de \(A\) est \(\{0\}\) ou une droite (tout sous-espace de dimension \(\ge 2\) contient un vecteur non nul avec au moins une coordonnée nulle), et la conclusion s’obtient en appliquant ce fait à la matrice symétrique semi-définie positive \(A – \lambda \mathrm{Id}\), avec \(\lambda\) la plus petite valeur propre.

Proposition
Soit \(\Gamma’ \prec \Gamma\) deux graphes de Coxeter distincts.
Si \(\Gamma\) est connexe et semi-défini positif, alors \(\Gamma’\) (qui n’est
pas forcément connexe) est défini positif.

Démonstration :
On numérote les sommets de \(\Gamma’\) de \(1\) à \(k\), puis ceux restant de \(\Gamma\) de \(k+1\) à \(n\).
Soit \(A’\) la matrice de \(\Gamma’\), c’est une matrice \(k \times k\) avec pour tous \(1 \le i,j \le k\)
\[
a_{ij}’ = – \cos \frac{\pi}{m_{ij}’} \ge – \cos \frac{\pi}{m_{ij}} = a_{ij}.
\]
Par l’absurde, supposons que \(x= (x_1, \dots, x_k) \neq 0\) vérifie \(x^t A’ x \le 0\).
Alors en appliquant \(A\) au vecteur \((|x_1|, \dots, |x_k|, 0, \dots, 0)\) on obtient
\[
0 \le \sum a_{ij} |x_i||x_j| \le \sum a_{ij}’ |x_i||x_j| \le \sum a_{ij}’ x_ix_j \le 0.
\]
Donc les inégalités sont des égalités.
Par le lemme, l’égalité \(0 = \sum a_{ij} |x_i||x_j|\) implique \(k = n\), et \(|x_i| > 0\) pour tout \(i\). Alors l’égalité \(\sum a_{ij} |x_i||x_j| = \sum a_{ij}’ |x_i||x_j|\) implique \(a_{ij} = a_{ij}’\) pour tous \(i,j\), contredisant que \(\Gamma’\) est un sous-graphe propre.

Théorème

1. Les graphes de la figure 1 sont la liste complète des graphes de Coxeter connexes défini positifs.
2. En ajoutant les graphes de la figure 2 on obtient la liste complète des graphes de Coxeter connexes semi-défini positifs.

Remarque
Les restrictions sur \(n\) sont pour éviter les redondances. En particulier on a les coïncidences suivantes :
\[\begin{array}{cccccc}
A_2 = I_2(3); &&
B_2 = I_2(4); &&
H_2 = I_2(5); &&
G_2 = I_2(6).
\end{array}\]

Preuve du théorème : cas de 1 ou 2 sommets

Les calculs sont directs. Pour \(\Gamma\) de type \(A_1\), on a \(2A = (2)\) et \(\det \Gamma = 2\). Pour \(\Gamma\) de type \(I_2(m)\) avec \(m \ge 3\), on a
\[\begin{array}{c}
2A =
\begin{pmatrix}
2 & -2\cos \frac{\pi}{m} \\
-2\cos \frac{\pi}{m} & 2
\end{pmatrix},
&&
\det \Gamma = 4\sin^2 \frac{\pi}{m}.
\end{array}\]
On obtient en particulier le tableau de valeurs

Remarque
Calcul de \(\cos \frac{\pi}{5}\) et \(\sin^2 \frac{\pi}{5}\).

D’abord
\[\cos 2\theta = {\mathbb R}e (e^{2\theta}) = {\mathbb R}e ((e^\theta)^2) = \cos^2 \theta – \sin^2 \theta = 2\cos^2 \theta -1.\]
Posons \(a = \cos \frac{\pi}{5}\) et \(b = \cos \frac{2\pi}{5}\). Remarquons que \(-a = \cos \frac{4\pi}{5}\), et donc
\[\begin{array}{cc}
b = 2a^2 -1, && -a = 2b^2 -1.
\end{array}\]
Par soustraction, \(a+ b = 2(a+b)(a-b)\), et donc \(a -b = \frac12\).
En remplaçant, on trouve que \(a\) est la racine positive de
\[4a^2 -2a -1 = 0\]
et finalement
\[\begin{array}{cc}
\cos \frac{\pi}{5} = \frac{1+\sqrt 5}{4},
&&
\sin^2 \frac{\pi}{5} = \frac{5-\sqrt 5}{8}.
\end{array}\]

Les graphes de la figure 1 sont définis positifs

Pour chacun des graphes de la figure (sauf \(D_4\), voir plus bas), on peut numéroter les \(n\) sommets de façon à ce que pour chaque \(i = 1, \dots, n\), le sous-graphe \(\Gamma_i\) induit par les \(i\) premiers sommets soit connexe (et donc aussi dans la liste), et que la dernière arête rajoutée soit de poids 3 entre les sommets \(n-1\) et \(n\).
Notons \(d_i = \det \Gamma_i\).
Par hypothèse les \(d_i\) sont les mineurs principaux de la matrice \(2A\), qui est de la forme
\[
2A =
\begin{pmatrix}
* & \dots & * & * & 0 \\
\vdots & \ddots & \vdots & \vdots & \vdots \\
* & \dots & * & * & 0 \\
* & \dots & * & 2 & -1 \\
0 & \dots & 0 & -1 & 2
\end{pmatrix}
\]
En développant suivant la dernière ligne on obtient la relation
\[
d_n = 2d_{n-1} – d_{n-2}.
\]
Par récurrence, on obtient la table suivante qui calcule les déterminants de tous les graphes de la figure (et aussi de \(D_2 = A_1 \times A_1\) qui est non connexe, mais qui avec \(D_3 = A_3\) intervient dans le calcul de \(D_4\)).
En particulier les mineurs \(d_i\) sont tous strictement positifs, et on conclut par le critère de Sylvester.

Précisément on utilise les suites suivantes pour \(d_{n-2}, d_{n-1}, d_n\):

• \(A_{n-2} \prec A_{n-1} \prec A_n\) initialisée avec \(\det A_1 = 2\), \(\det
  I_2(3) = 3\);
• \(B_{n-2} \prec B_{n-1} \prec B_n\) initialisée avec \(\det A_1 = 2\), \(\det
  I_2(4) = 2\);
• \(D_{n-2} \prec D_{n-1} \prec D_n\) initialisée avec \(\det A_1 \times A_1 =
  4\), \(\det A_3 = 4\);
• \(A_4 \prec D_5 \prec E_6\);
• \(D_5 \prec E_6 \prec E_7\);
• \(E_6 \prec E_7 \prec E_8\);
• \(A_2 \prec B_3 \prec F_4\);
• \(A_1 \prec I_2(5) \prec H_3\), noter que \(3 – \sqrt 5 \simeq 0.76 > 0\);
• \(I_2(5) \prec H_3 \prec H_4\), noter que \(\frac{7 – 3\sqrt 5}{2} \simeq
  0.15 > 0\).

Les graphes de la figure 2 sont semi-définis positifs

Tous d’abord on observe que l’on peut réaliser chacun des graphes \(\Gamma\) comme une suite croissante de graphes connexes dont tous les éléments sauf le dernier sont dans la liste, et tel que la dernière arête ajoutée est de poids \(3\) ou \(4\) entre les sommets \(n-1\) et \(n\). Par la variante du critère de Sylvester, il suffit de montrer que \(\det \Gamma = 0\).

On l’a déjà vu pour \(\tilde A_2\), et pour \(\tilde A_n\), \(n \ge 3\), il suffit de constater que chaque ligne comporte un \(2\), deux \(-1\) et des \(0\), ainsi la somme des colonnes est nulle.

Pour les graphes \(\tilde B_n\) et \(\tilde C_n\), comme précédemment on utilise une relation de récurrence entre les mineurs \(d_{n-2}, d_{n-1}\) et \(d_n\), mais cette fois le poids de la dernière arête ajoutée étant 4 cette relation prend la forme
\[
d_n = 2 (d_{n-1} – d_{n-2}).
\]
obtenue en développant le déterminant de
\[
2A =
\begin{pmatrix}
* & \dots & * & * & 0 \\
\vdots & \ddots & \vdots & \vdots & \vdots \\
* & \dots & * & * & 0 \\
* & \dots & * & 2 & – \sqrt 2 \\
0 & \dots & 0 & -\sqrt 2 & 2
\end{pmatrix}
\]

On trouve \(\det \Gamma = 0\) à l’aide des suites

• \(D_{n-1} \prec D_n \prec \tilde B_n\);
• \(B_{n-1} \prec B_n \prec \tilde C_n\).

Pour les graphes restants, on peut supposer la dernière arête de poids 3 et utiliser la relation de récurrence
\[
d_n = 2 d_{n-1} – d_{n-2}.
\]

Précisément, on peut utiliser les suites

• \(A_1 \times A_1 \times A_1 \prec D_4 \prec \tilde D_4\);
• \(A_1 \times D_{n-2} \prec D_n \prec \tilde D_n\) pour \(n \ge 5\);
• \(A_5 \prec E_6 \prec \tilde E_6\);
• \(D_6 \prec E_7 \prec \tilde E_7\);
• \(E_7 \prec E_8 \prec \tilde E_8\);
• \(A_3 \prec B_4 \prec \tilde F_4\);
• \(A_1 \prec G_2 \prec \tilde G_2\).

Figure 1 : graphes de Coxeter définis positifs

Figure 2 : graphes de Coxeter semi-définis positifs

Tout graphe semi-défini positif est dans une des deux listes

Supposons que \(\Gamma\) est un graphe connexe semi-défini positif à \(n\) sommets
et poids maximal \(m\), qui n’apparaisse pas dans une des deux listes.

1. Le nombre \(n\) de sommets est au moins 3, puisqu’on a traité exhaustivement
   les cas de 1 ou 2 sommets.
2. \(m\) est fini et \(m \le 5\), puisque \(\Gamma\) ne contient pas le graphe
   \(\tilde A_1\) ni le graphe \(\tilde G_2\).
3. Le cas \(m = 5\) est impossible, car \(\Gamma\) ne contenant pas \(\tilde B_n\)
   n’admet donc pas de point de branchement, et les deux graphes
   • >\(Z_4 =\)
   • >\(Z_5 =\)

   sont de déterminant \(<0\), ce que l'on voit par exemple en appliquant la formule
   \[
   d_n = 2 d_{n-1} – d_{n-2}.
   \]
   aux suites

   • \(A_2 \to H_3 \to Z_4\);
   • \(H_3 \to H_4 \to Z_5\).
4. Supposons maintenant \(m =4\).
   \(\Gamma\) ne contient pas \(\tilde C_n\), donc une seule arête admet le poids \(4\),
   et comme il ne contient pas \(\tilde B_n\), il n’a pas de point de branchement.
   Les deux arêtes extrèmes sont de poids 3, sinon \(\Gamma = B_n\).
   Comme \(\Gamma \neq F_4\) et \(\Gamma\) ne contient pas \(\tilde F_4\), cela épuise
   toutes les possibilités avec \(m = 4\).
5. Finalement supposons \(m=3\).
   \(\Gamma\) doit admettre un point de branchement, sinon \(\Gamma = A_n\).
   \(\Gamma\) ne contient pas \(\tilde D_4\), donc admet seulement des point de
   branchement triples.
   \(\Gamma\) ne contient pas \(\tilde D_n\), donc admet exactement un point de
   branchement triple, avec branche de longueur \(2 \le a \le b \le c\).
   \(\Gamma\) ne contient pas \(\tilde E_6\), donc \(a = 2\).
   Comme \(\Gamma \neq D_n\), on a \(b \ge 3\), mais comme \(\Gamma\) ne contient pas
   \(\tilde E_7\), on a \(b = 3\).
   Finalement \(3 \le c \le 5\) puisque \(\Gamma\) ne contient pas \(\tilde E_8\), mais
   les cas \(c = 3, 4 , 5\) correspondraient à \(\Gamma = E_6, E_7, E_8\), ce qui est
   exclu.

Réalisation

Groupes de réflexions finis

Soit \(V\) un espace vectoriel euclidien, \(H \subset V\) un hyperplan, \(\alpha \in V\) un vecteur.
On note \(s_H\) la réflexion orthogonale associé à \(H\), et \(s_\alpha\) la réflexion orthogonale associée à \(\alpha^\perp\).
Explicitement:
\[
s_\alpha(x) = x – 2\frac{\langle \alpha,x \rangle}{\langle \alpha, \alpha \rangle} \alpha.
\]

Un groupe de réflexions fini est un groupe fini \(W \subset \mathrm O(V)\) engendré par des réflexions orthogonales.
Le groupe \(W\) est dit essentiel si l’origine est le seul point fixe (ou autrement dit, \(\bigcap_{s_H \in W} H = \{ 0 \}\)).

Un ensemble de vecteur \(\Phi = \{\alpha\}\) est un système de racine (généralisé) si les réflexions \(s_\alpha\), \(\alpha \in \Phi\) engendre un groupe fini \(W\), et que \(\Phi\) est invariant par \(W\).

On suppose toujours un tel système de racine réduit, au sens où \(\alpha, \beta \in \Phi\) colinéaires implique \(\alpha = \pm \beta\).

Un système de racine est dit cristallographique si pour tout \(\alpha, \beta \in \Phi\), le coefficient \(2\frac{\langle \alpha,x \rangle}{\langle \alpha, \alpha \rangle}\) est entier.
Cela implique que le groupe additif engendré par \(\Phi\) est un réseau de \(V\).

On donne des réalisations combinatoires et/ou géométriques des groupes de Coxeter des diagrammes du tableau.

• \(A_n\) correspond au groupe symétrique, pour le système de générateurs \((i \, i+1)\).
  Géométriquement, c’est le groupe des isométries préservant un simplexe régulier dans l’espace euclidien de dimension \(n\).

• \(B_n\) correspond au groupe des permutations signées, pour le système de générateurs \((i \, i+1)\) plus le « flip » de \(n\).
  Géométriquement, c’est le groupe des isométries préservant un hypercube dans l’espace euclidien de dimension \(n\) (\(n\) paires de faces opposées).

• \(D_n\) est le groupe des permutations signées avec un nombre pair de \(-\).
  Géométriquement, c’est le groupe des isométries préservant un demi-hypercube dans l’espace euclidien de dimension \(n\)

• \(H_3\) est le groupe de l’icosaèdre (ou dodécaèdre), \(H_4\) le groupe du \(120\)-cellules (ou \(600\)-cellules), et \(F_4\) le groupe du \(24\)-cellules.

Complexes de coxeter

Groupe de réflexion fini

Soit \(W\) un groupe de réflexions fini, c’est à dire \(W \subseteq \mathrm O_n({\mathbb R})\) est
un sous-groupe fini du groupe orthogonal standard engendré par des réflexions
orthogonales.
On suppose \(W\) irréductible (pas de sous-espace propre invariant) et essentiel
(pas de point fixe global à part l’origine).
Soit \(H_i\) l’ensemble des hyperplans correspondant à des réflexions
orthogonales \(s_i \in W\) (pas seulement les générateurs), et \(\ell_i\) des
formes linéaires définissant les \(H_i\).
On appelle chambre une composante connexe de \({\mathbb R}^n \setminus \bigcup H_i\).
En particulier le choix des \(\ell_i\) conduit à un choix de chambre fondamentale
\[
C = \{ v \in {\mathbb R}^n \mid \ell_i(v) > 0 \text{ pour tout } i\}
\]
Deux chambres qui ont une face de codimension 1 commune (correspondant à traverser l’un des \(H_i\)) sont dites adjacentes.
Une suite de chambres \((C_j)_{0\le j \le r}\) est une gallerie de longueur \(r\) si \(C_j\) est adjacente à \(C_{j+1}\) pour tout \(j\).
La gallerie est dite minimale s’il n’y a pas de gallerie de longueur \(< r\) entre \(C_0\) et \(C_r\).
Le fait suivant me semble géométriquement clair:

Deux chambres sont toujours connectées par au moins une gallerie, et une gallerie est minimale si et seulement si elle ne traverse aucun hyperplan plus d’une fois.

Le lemme suivant semble être implicite dans \cite[p. 36]{AB}:

Lemme
Le groupe \(W\) agit transitivement sur les chambres.

Démonstration :
Le fait qu’il existe une action est claire, il s’agit de montrer que toute chambre \(D\) est dans l’orbite de la chambre fondamentale \(C\).
On procède par récurrence sur la distance \(r\) entre \(C\) et \(D\) (\(r\) est le
nombre d’hyperplans séparant ces deux chambres).
Si \(r = 1\), on envoie \(D\) sur \(C\) par la réflexion d’hyperplan séparant \(D\) et \(C\).
Si \(r > 1\), on considère une gallerie \(C_0 = C, C_1, \dots, C_r = D\), on prend
\(s_1\) la réflexion échangeant \(C\) et \(C_1\), et on applique l’hypothèse de
récurrence à \(C = s_1 C_1\) et \(s_1 D\).

Proposition
Soit \(\{H_s\}\) l’ensemble des hyperplans d’appui de \(C\), \(S\)
l’ensemble des réflexions orthogonales \(s \in W\) associées, et \(e_s\) le vecteur
unitaire normal à \(H_s\) et pointant vers le demi-espace contenant \(C\).
Alors:

1. Pour chaque \(s,t \in S\), on a \(\langle e_s, e_t \rangle = – \cos \frac{\pi}{m(s,t)} \le 0\) où
   \(m(s,t)\ge2\) est l’ordre de \(st\).
2. La chambre \(C\) est simpliciale, ce qui revient à dire que \(S\) est de cardinal
   \(n\);
3. \(S\) engendre \(W\);
4. \(\bar C\) est un domaine fondamental pour l’action de \(W\) sur
   \({\mathbb R}^n\);
5. \(W\) agit simplement transitivement sur les chambres.

Démonstration :
1. C’est juste la remarque que le groupe \(\langle s,t \rangle\) agissant sur \((H_s \cap
H_t)^\perp = {\mathbb R} e_s \oplus
{\mathbb R} e_t\) est le groupe diédral d’ordre \(m(s,t)\).

2. L’ensemble des hyperplans d’appui est de cardinal \(r \ge n\) sinon l’action neserait pas essentielle. Par l’absurde supposons \(r > n\). Donc les \(\ell_s = \langle e_s, \cdot \rangle\) forment une famille liée, et il existe une relation linéaire nulle non-triviale dont tous les coefficients sont positifs (sinon on écrit une égalité entre combinaisons linéaires non nulles à coefficients positifs, et comme les coefficients non diagonaux sont \(\le 0\), le produit scalaire est \(\le 0\), mais aussi \(\ge 0\) puisque c’est le produit scalaire d’un vecteur avec lui-même, donc \( =0\), absurde). Une telle combinaison est incompatible avec la définition de \(C\) comme quadrant positif.

3. Dans le lemme on a en fait montré que le groupe \(\langle S \rangle\) agit transitivement sur les chambres. Toute réflexion \(s_i\) dans \(W\) correspondant à un mur d’au moins une chambre \(D\), en utilisant le fait qu’il existe \(w \in \langle S \rangle\) tel que \(w D = C\), on obtient \(w s_i w^{-1} \in S\), et donc également \(s_i \in \langle S \rangle\). Comme les \(s_i\) engendrent \(W\), on obtient \(W = \langle S \rangle\).

4. Soit \(w \in W\) et \(x,y \in \bar C\) tel que \(wx = y\). On veut montrer \(x = y\). On écrit \(w = s_1 \dots s_r\) sous forme réduite, et on procède par récurrence sur la longueur \(r\) de \(w\). Si \(r = 0\), \(w = \mathrm{Id}\) et c’est fini. Sinon, on remarque que \(wC\) et \(C\) sont de part et d’autre de l’hyperplan
\(H_{s_1}\) (par minimalité de \(r\), et par la propriété « deletion » qui est assez claire géométriquement). Donc \(wx = y \in H_{s_1}\). Mais alors en appliquant \(s_1\) on trouve \(s_2 \dots s_r x = s_1 y = y\), et on conclut par hypothèse de récurrence.

5. Juste la remarque que dans la preuve précédente, si \(x = y \in C\), alors le seul \(w \in W\) fixant \(x\) est l’identité.

La décomposition simpliciale de la sphère \(S^{n-1} \subset {\mathbb R}^n\) est appelée le complexe de Coxeter (« sphérique ») associé au groupe de réflexion fini \(W\).

Une autre propriété importante est le

Théorème
Avec les notations de la proposition, \((W,S)\) est un système de Coxeter.

La preuve, combinatoire, est liée au problème du mot, est ne semble pas plus facile dans le cas fini que dans le cas général.

Groups of automorphisms of rooted trees

Let \( T \) be a \( d \)-regular rooted tree (each vertex has \( d \) children). We assume that \( T \) is embedded in the plane, equivalently the children of each vertex are totally ordered. Let \( \mathrm{Aut}_f(T) \) be the group of finitary isomorphisms of \( T \), i.e. those automorphisms that are order-preserving on the children for all but finitely many vertices.

Such an an automorphism is prescribed by an automorphism of a finite subtree, and extended by order-preserving isomorphisms between the components of its complement. We say that it is elementary if there exists a vertex such that the automorphism permutes its children, and acts as the identity everywhere else. Such automorphisms generate \( \mathrm{Aut}_f(T) \); equivalently \( \mathrm{Aut}_f(T) \) is isomorphic to an iterated wreath product of copies of the symmetric group \( \mathfrak S(d) \).

We say that an elementary automorphism with associated permutation \( \sigma \in \mathfrak S(d) \) is alternating if \( \sigma \) is alternating. The subgroup of \( \mathrm{Aut}_f(T) \) generated by such automorphisms is called the finitary alternating automorphism group and denoted by \( \mathrm{Alt}_f(T) \).

In the sequel we will always assume that \( d \ge 5 \). The countable group \( \Gamma = \mathrm{Alt}_f(T) \) then has the following property:

Any normal subgroup of \( \Gamma \) fixes a ball around the root (in other words it is the « stabiliser of a level » of \( T \)).

Invariant random subgroups

Examples

Let \( \overline\Gamma \) be the completion of \( \Gamma \) in \( \mathrm{Aut}(T) \). It is a compact group, and as such it has an invariant measure \( \nu \) on the boundary \( \partial T \), the pushforward of the normalised Haar measure of \( \overline\Gamma \), which is in fact the visual measure from the root. It follows that the stabiliser in \( \Gamma \) of a \( \nu \)-random point of \( \partial T \) is an IRS of \( \Gamma \).

More generally, if \( C \) is a closed subset of \( \partial T \) we can define an invariant random subgroup in \( \Gamma \) as follows: take a random \( g \in \Gamma \), and then take the (pointwise) stabiliser in \( \Gamma \) of \( gC \). Note that this IRS can be written as an « intersection » of those obtained in the previous paragraph (as \( \mathrm{Stab}_\Gamma(gC) = \bigcap_{x \in C} \mathrm{Stab}_\Gamma(gx) \)). This construction yields a priori lots of distinct invariant random subgroups, as there are plenty of \( \overline\Gamma \)-orbits on \( 2^{\partial T} \).

The main result is as follows.

Theorem: Let \( H \) be a nontrivial ergodic IRS of \( \Gamma \), and assume that almost surely \( H \) has no global fixed point on \( \partial T \). Then \( H \) is induced from a finite-index subgroup of \( \Gamma \).

For example, let \( d’ < d \) and \( T' \subset T \) a \( d' \)-regular subtree. Then the subgroup of \( \mathrm{Alt}_f(T) \) preserving \( T' \) must have a global fixed point (as its conjugacy class supports an IRS): it is possible to see by elementary means that it is in fact equal to the subgroup fixing \( T' \) pointwise.

A more descriptive result is as follows.

Theorem: Let \( H \) be a nontrivial ergodic IRS of \( \Gamma \). There exists a closed subset \( C \) of \( \partial T \) such that the distribution of the fixed subset of \( H \) is the same as that of a random translate of \( C \).

This is not a complete classification, as different IRSs of \( \Gamma \) might have the same distribution for their fixed subsets. However there is a universal construction as follows: let \( C \) be the subset given by the theorem, and \( T_C \) the subtree of \( T \) such that \( C = \partial T_C \). For each component \( s \) of \( T \setminus T_C \) choose an integer \( m_s \). Let \( L(C, (m_s)_s) \) be the subgroup defined as follows: it fixes \( T_C \), and the ball of radius \( m_s \) around the root of \( s \) for each \( s \). Then \( L(gC, g_*(m_s)_s) \) is an invariant random subgroup (where \( g \in \overline\Gamma \) is a Haar-random element) and we have the following statement.

There exists \( (m_s)_s \) such that \( H \) contains \( L(gC, g_*(m_s)_s) \) almost surely.

Schreier graphs of invariant random subgroups

Let \( \Gamma \) be a group generated by a finite subset \( S \). To a subgroup \( H \le \Gamma \) one associates the Schreier coset graph \( \mathrm{Sch}(H \backslash \Gamma, S) \): this is the graph with vertex set \( H \backslash \Gamma \), where two vertices \( Hg, Hg’ \) are joined by an edge if and only if \( g’ \in HgS \); in other words it is the quotient of the Cayley graph of \( \Gamma \) with respect to \( S \) by the action of \( H \) by left-translations.

A random subgroup in \( \Gamma \) thus yields a random rooted graph: the Schreier graph of the random subgroup, rooted at the identity coset. Let \( \mathcal G^S \) be the set of rooted \( S \)-labeled graphs (i.e. every vertex has exactly one incoming and one outcoming edge labeled by every element of \( S \)). This set has an action of the free group \( F_S \) on \( S \), where a generator moves the root to the head of the outcoming edge lebeled by it. The following result is essentially immediate as the Schreier graph \( \mathrm{Sch}(sHs^{-1} \backslash \Gamma, S) \) is the same as \( \mathrm{Sch}(H \backslash \Gamma, S) \) but rooted at the coset \( sHs^{-1}s \).

Proposition: A random subgroup is conjugacy-invariant if and only if the corresponding random rooted graph is invariant under this action.

We can reformulate this property as follows: let \( \mathcal G_2 \) be the space of doubly pointed, \( S \)-labeled graphs, up to isomorphism.

Proposition (Mass Transport Principle for unimodular graphs): Let \( \lambda \) be a Borel measure on \( \mathcal G \). It is \( F_S \)-invariant if and only if, for any positive Borel function \( f \) on \( \mathcal G_2 \) we have:
\[
(\mathrm{MTP}) \hspace{1cm} \int_{\mathcal G} \sum_{o_2 \in X} f(X, o_1, o_2) d\lambda(X, o_1) = \int_{\mathcal G} \sum_{o_1 \in X} f(X, o_1, o_2) d\lambda(X, o_2).
\]

The proof of this is not very long. Assume first that \( \lambda \) satisfies (MTP). We want to prove that \( \lambda(sE) = \lambda(E) \) for all \( s \in S \) and all Borel sets \( E \subset G \). For this we define the following function on \( \mathcal G_2 \) to which we’ll apply (MTP):
\[
f(X, v, w) =
\begin{cases}
1 & \text{ if } (X, v) \in E \text{ and } (X, v) = s\cdot (X, w) \\
0 & \text{ otherwise}
\end{cases}
\]
(in other words it indicates whether the two roots are joined by an \( s \)-labeled edge). We compute:
\[
\int_{\mathcal G} \sum_w f(X, v, w) d\lambda(X, v) = \int_{\mathcal G} 1_{(X,v) \in E} d\lambda(X, v) = \lambda(E)
\]
as there is exactly one nonzero value of \( f \) in the sum over \( w \) when \( (X, v) \in E \) and none otherwise; and similarly
\[
\int_{\mathcal G} \sum_{v \in X} f(X, v, w) d\lambda(X, w) = \lambda(sE).
\]
The mass transport principle implies that both integral are equal, hence \( \lambda(sE) = \lambda(E) \).

The converse follows from a similar computation by reducing to the case where \( f \) is supported on a set where the two roots are adjacent.

Unimodularity for unlabeled graphs

From now on we use \( \mathcal G \) (resp. \( \mathcal G_2 \)) to denote the space of pointed (resp. doubly pointed) unlabeled locally finite (not necessary regular?) graphs. We say that a random graph is unimodular if its law satisfies the mass transport principle (MTP) (which makes sense in this generality). Some easy examples are:

1. A finite fixed graph with uniformly chosen root;
2. A de-labeled Schreier graph of an invariant random subgroup in a discrete group;
3. If a graph is vertex transitive with unimodular isomorphism group then it is (rooted at an arbitrary vertex) a unimodular graph (note that this is not true without the unimodularity hypothesis on the group of isomorphisms).

The no-core principle

This is a statement which formalises the idea that a unimodular random graph looks everywhere the same, more precisely that it is impossible to distinguish a finite region (a « core ») in an infinite unimodular random graph in a measurable manner. It goes as follows.

Theorem: Let \( \lambda \) be a unimodular random graph. Let \( A \) be a Borel subset of the space \( \mathcal G \) of pointed graphs. If
\[
0 < |\{ w \in X :\: (X, w) \in A \}| < +\infty \quad (\ast)
\]
with positive probability then \( X \) is finite with positive probability.

To prove this assume that \( (\ast) \) holds for \( (X, v) \in B \) with \( \lambda(B) > 0 \). As it has an enumerable range we can assume that \( |\{ w \in X :\: (X, w) \in A \}| \) takes only the value \( 0 \in < +\infty \) on \( B \). Then applying (MTP) to the function
\[
(X, v, w) \mapsto 1_A(X,v) 1_B(X,w)
\]
we get that :
\[
\int_B \sum_{w \in X} 1_A(X,v) d\lambda(X, v) = \int_{\mathcal G} \sum_{w \in X} 1_A(X, w) d\lambda(X, v) = N.
\]
It follows that the right-hand side must be finite, but as it is equal to \( \int_{\mathcal G} 1_A(X,v) \cdot |X| d\lambda(X,v) \) this implies that \( |X| \ +\infty \) for all \( (X, v) \in B \).

As an application of the no-core principle one can prove the following classification result for unimodular random graphs.

Corollary: Let \( \lambda \) be a unimodular random graph. Then for \( \lambda \)-almost all \( (X, v) \in \mathcal G \) the graph \( X \) is either finite (zero ends) or has exactly one, two or a Cantor set of ends.

We prove first the weaker statement that \( X \) has either \( 0, 1, 2 \) or infinitely many ends. Assume to the contrary that with positive probability \( X \) has \( N \) ends, where \( N \ge 3 \) is an integer. Then there exists \( R > 0 \) such that with positive probability for \( (X, v) \) the ball \( B_X(v, R) \) of radius \( R \) in \( X \) around \( v \) separates those \( N \) ends. We define a function as follows:
\[
f(X, v) =
\begin{cases}
1 & \text{ if } X \setminus B_X(v, R) \text{ has at least } N \text{ unbounded components; } \\
0 & \text{otherwise. }
\end{cases}
\]
This yields a contradiction with the NCP: given \( X \) with \( N \) ends and \( v, w \in X \), if both \( f(X, v), f(X, w) = 1 \) then \( B_X(v, R) \cap B_X(w, R) \neq \emptyset \) (otherwise \( X \) would have \( > N \) ends). THis implies that for such \( X \) the subset \( \{v \in X :\: f(X, v) = 1\} \) is finite. But as \( X \) is infinite this contradicts the theorem above.

Unimodular random manifolds

Let \( \mathcal M^d \) be the space of isometry classes of pointed complete Riemannian manifolds (the topology on this space is not obvious but it exists and is locally compact) and \( \mathcal M_2^d \) the space of doubly pointed such manifolds.

A Borel probability measure \( \lambda \) on \( \mathcal M^d \) is called unimodular if it satisfies a mass transport principle, that is for any Borel function \( f \) on \( _mathcal M_2^d \) the followin gequality holds:
\[
\int_{\mathcal M^d} \int_M f(M, x, y) d\mathrm{vol}_M(y) d\lambda(M, x) = \int_{\mathcal M^d} \int_M f(M, x, y) d\mathrm{vol}_M(x) d\lambda(M, y).
\]

Examples

1. A finite-volume complete manifold, with the root chosen with respect to the normalised volume measure;
2. If \( \mathrm{Isom}(X) \) is transitive, then the random manifold \( (X, x_0) \) (where \( x_0 \in X \) is an arbitrarily chosen point) is unimodular if and only if the locally compact group \( \mathrm{Isom}(X) \) is unimodular;
3. Let \( X \) be a symmetric space without compact or Euclidean factors and \( G = \mathrm{Isom}(X) \). Then any torsion-free and discrete IRS in \( G \) gives a unimodular random manifold locally isometric to \( X \), and vice-versa.

Benjamini–Schramm convergence

As a space of probability measures on a locally compact space the set of nimodular random manifolds has a natural locally topology (weak-star convergence). Convergence in this space is called Benjamini–Schramm convergence (these authors first studied the corresponding notion for unimodular random graphs). This topology has many applications in Riemannian geometry, such as the following result.

Theorem (Abért–Bergeron–Biringer–Gelander): Let \( X \) be a symmetric space without Euclidean or compact factors, and assume that it is not isometric to hyperbolic 3–space \( \mathbb H^3 \). Let \( M_n \) be a sequence of finite-volume, complet \( X \)-manifolds which is convergent in the Benjamini–Schramma topology. Then the sequences of normalised Betti numbers \( b_k(M_n) / \mathrm{vol}(M_n) \) converge for every \( 0 \le k \le \dim(X) \).

Critical exponents

Let \( X \) be a Gromov-hyperbolic space and \( o \in X \). Let \( \Gamma \) be a discrete subgroup of \( \mathrm{Isom}(X) \). For any \( s \ge 0 \) we can define the Poincaré series
\[
P(\Gamma, s) = \sum_{\gamma \in \Gamma} e^{-sd(o, \gamma o)} \in [0, +\infty].
\]
The critical exponent \( \delta(\Gamma) \) of \( \Gamma \) is then defined by
\[
\delta(\Gamma) = \inf(s > 0 :\: P(\Gamma, s) < +\infty) = \sup(s \ge 0 :\: P(\Gamma, s) = +\infty).
\]
It measures the exponential growth rate of the orbits of \( \Gamma \) on \( X \), as it is easily seen to also be given by the following formula:
\[
\delta(\Gamma) = \liminf_{R \to +\infty} \left( \frac{\log|B(o, R) \cap \Gamma\cdot o|}{R} \right).
\]
It is also readily seen that it does not depend on the chosen origin \( o \).

Part of the finer asymptotics of this growth is captured by the following notion: if \( P(\Gamma, s) < +\infty \) for \( s = \delta(\Gamma) \) then \( \Gamma \) is said to be of convergence type; otherwise it is said to be of divergence type.

This is a well-studied invariant and a sample of its known properties is given by the following list:

• \( \delta(\Gamma) = 0 \) if and only if \( \Gamma \) is elementary;
• Let \( \partial X \) be the Gromov boundary of \( X \); then if \( \Gamma \) is a uniform lattice (acts cocompactly on \( X \)) we have \( \delta(\Gamma) = \dim(\partial X) \) (where \( \dim \) is the Hausdorff dimension of a metric space);
• For any discrete subgroup \( \Gamma \le \mathrm{Isom}(X) \) we have \( 0 \le \delta(\Gamma) \le \dim(\partial X) \). Moreover there are subgroups with \( \delta(\Gamma) \) arbitrarily close to zero, which may be taken to be of either convergenec or divergence type;
• \( \delta(\Gamma) \) equals the dimension of the radial, or conical, limt set of \( \Gamma \).

Main result

Viewed as a function on Chabauty space (or rather its subspace containing only discrete groups) \( \delta \) is not continuous. An important property for the sequel is the following weaker property.

The function \( \delta \) is measurable on the discrete part of the Chabauty space.

As it is obvious that \( \delta \) is conjugacy-invariant, it follows that if \( \mu \) is an ergodic IRS of \( \mathrm{Isom}(X) \) the value \( \delta(\Gamma) \) is \( \mu \)-essentially constant and we can take its value to be \( \delta(\mu) \). Similarly, being of divergence/convergence type is a Borel property and hence well-defined for ergodic IRSs.

Theorem (Gekhtman–Levit): Let \( \mu \) be an ergodic discrete IRS of \( \mathrm{Isom}(X) \). Then
\[
\delta(\mu) \ge \frac 1 2 \dim(\partial X).
\]
Moreover, if \( \mu \) is of divergence type then necessarily \( \delta(\Gamma) = \dim(\partial X) \).

The main tool in the proof is an ergodic theorem due to Nevo and Zimmer.

Consequences

A theorem of Corlette states that if \( G \) is a Lie group of rank 1 with Kazhdan’s property (T) then \( \delta(\Gamma) = \dim(X) \) if and only if \( \Gamma \) is a lattice in \( G \). Thus, a corollary of the theorem above is that

If \( G \) is as above then any ergodic IRS of \( G \) of divergence type is supported on the conjugacy class of a lattice in \( G \).

• Lecture 1: Invariant random subgroups in locally compact groups and Lie groups
• Lecture 2: Margulis Normal Subgroup and Factor Theorems (Jean Lécureux)
• Lectures 3, 4 and 5: Intermediate factor Theorem and proof of the Stuck–Zimmer Theorem (Arie Levit)

Lecture 1: Invariant random subgroups in locally compact groups and Lie groups

Invariant random subgroups

Let \( G \) be a locally compact, second countable group. The set \( {\mathrm{Sub}}_G \) of closed subgroups is endowed with the Chabauty topology: it is a metrisabe topology which can be defined sequentially as follows: a sequence \( H_n \in {\mathrm{Sub}}_G \) converges to \( H \) if:

• For every convergent sequence \( h_n \in H_n \) we have \( \lim_{n \to +\infty} h_n \in H \);
• For every \( h \in H \) there exists a sequence \( h_n \in H_n \) such that \( h = \lim_{n \to +\infty} h_n \).

See also this paper for a definition of the metric. The group \( G \) acts continuously on \( {\mathrm{Sub}}_G \) by conjugation.

Definition: An invariant random subgroup of \( G \) is a \( G \)-invariant Borel probability measure on \( {\mathrm{Sub}}_G \). We denote by \( {\mathrm{IRS}}(G) \) the set of such measures, with the topology induced from the weak-* topology on the space of Borel measures on \( {\mathrm{Sub}}_G \) of which it is a convex compact subspace.

An invariant random subgroup \( \mu \) is said to be ergodic if it is an extremal point in this convex; equivalently the action of \( G \) on \( ({\mathrm{Sub}}_G, \mu) \) is ergodic.

It is also convenient to view an IRS as a random variable taking values in \( {\mathrm{Sub}}_G \). There is also a dynamical interpretation given by the following proposition.

Proposition: Let \( G \) act on a Borel probability space \( (X, \nu) \) preserving the measure \( \nu \). Then the map \( {\mathrm{Stab}} : X \to 2^G \) associating to a point \( x \in X \) its stabliser \( {\mathrm{Stab}}_G(x) \) has its image contained in \( {\mathrm{Sub}}_G \), and the measure \( {\mathrm{Stab}}^*\nu \) is an invariant random subgroup.

Conversely, for every IRS \( \mu \) there exists a pmp action of \( G \) on a space \( (X, \nu) \) such that \( \mu = {\mathrm{Stab}}^*\nu \).

The proof that stabilisers of Borel actions of \( G \) (on countably separated spaces) are closed follows immediately from a simple construction of Varadarajan which shows that such actions are embeddable in continuous \( G \)-actions.

The proof in the other direction needs some technical arguments. Ideally an IRS \( \mu \) would arise from the conjugation action of \( G \) on \( ({\mathrm{Sub}}_G, \mu) \), but this is obviously not the case since in general, for a closed subgroup \( H \in {\mathrm{Sub}}_G \) the normaliser \( N_G(H) \) has no reason to be equal to \( H \). To remedy to this problem one uses a action on the bundle over \( {\mathrm{Sub}}_G \) with fiber \( G/H \) over \( H \), and an invariant measure arising from an invariant point process on the fibers.

Examples

1. The Dirac mass \( \delta_G \) supported on \( G \) is obviously an IRS; there is only one associated pmp action, the trivial action of \( G \) on a point.
2. The Dirac mass \( \delta_{\mathrm{Id}} \) supported on the trivial subgroup is also an IRS. Any essentially free pmp action of \( G \) has it as its stabiliser (for example the action on Poisson point processes on \( G \), but in general there are many such actions).
3. More generally, let \( N \) be any closed normal subgroup of \( G \), then the Dirac mass \( \delta_N \) is an IRS arising from any essentially free pmp action of \( G/N \).
4. Getting nonatomic examples is more involved. In this paragraph we will discuss when such examples may be supported on a single conjugacy class.
   
   Assume that \( G \) is unimodular (right- and left-invariant Haar measures coincide), then there is a \( G \)-invariant measure \( \mu_{Haar} \) on the coset space \( G/H \) if and only if \( H \) itself is unimodular as well (this measure is then unique up to scaling). For example \( H \) might be discrete, or compact. A non-example is given by the subgroup of diagonal matrices in \( \mathrm{SL}_n({\mathbb R}) \).
   
   If the measure \( \mu_{Haar} \) has finite mass (H is then called cofinite in \( G \)) then by rescaling it we may assume it is a probability measure. The action of \( G \) on \( (G/H, \mu_{Haar}) \) is then a pmp action and the corresponding IRS is supported on the conjugacy class of \( H \) in \( G \) (which is not necessarily closed but at least Borel in this case).
   
   To define this IRS as a measure consider the continuous map from \( G/H \) to \( {\mathrm{Sub}}_G \) defined by
   \[
   gH \mapsto gHg^{-1}.
   \]
   It is \( G \)-equivariant, hence the pushforward \( \mu_H \) of \( \mu_{Haar} \) by this map is then a probability measure on \( {\mathrm{Sub}}_G \) which is invariant under conjugation.
   
   Examples of cofinite subgroups are rather hard to come by in general. When such a subgroup is discrete it is called a lattice of \( G \); we will see below that in semisimple Lie groups without compact factors (where they always exist by a result of Borel–Harder) these are all cofinite subgroups. A nondiscrete example is a cocompact subgroup of automorphisms of a tree with infinite vertex stabilisers, for example \( \mathrm{PGL}_2(\mathbb Q_p) \) inside the automorphisms of its Bruhat–Tits tree.

Note that all examples above are ergodic, but none is properly ergodic, that is all are supported on a single conjugacy class. Properly ergodic IRSs do exist in some groups. The constructions above show that whenever a lcsc group \( G \) either is not simple, or has a proper nontrivial cofinite subgroup it has IRSs beyong the « trivial » ones \( \delta_G \) and \( \delta_{\mathrm{Id}} \).

There are discrete (Thompson groups) and non-discrete (see Adrien le Boudec’s talk) examples of totally disconnected groups where these are the only ergodic IRSs. There is however no known example of a nondiscrete, compactly generated group where this propert of « non nontrivial IRSs » holds. A candidate for this is the Neretin group, which we discuss in another series of lectures.

Invariant random subgroups in semisimple Lie groups and the Stuck–Zimmer Theorem

A connected Lie group \( G \) is said to be simple if it has finite center, and all proper normal subgroups are contained in its center. Equivalent conditions are the Lie algebra of \( G \) being simple, or \( G \) being the group of real points of a simple algebraic group over a subfield of \( {\mathbb R} \). An example is \( \mathrm{SL}_n({\mathbb R}) \).

A connected lie group \( G \) is said to be semisimple if there exists normal closed subgroups \( G_1 \ldots, G_r \) such that

• For each \( i \), \( G_i \) is a simple Lie group;
• If \( i \not= j \) the intersection \( G_i \cap G_j \) is finite and central in \( G \);
• \( G = G_1 \cdots G_r \).

The \( G_i \) are then uniquely determined by \( G \) and called its simple factors. The group is said to have no compact factors if none of them is compact; equivalently it does not admit a nontrivial morphism to a compact group.

A product \( G_1 \times \cdots \times G_r \) of simple Lie groups is semisimple. In general a semisimple group does not have to be a product but it maps with finite kernel to one. For example
\[
G = \mathrm{SL}_2({\mathbb R}) \times \mathrm{SL}_2({\mathbb R}) / \pm({\mathrm{Id}}, {\mathrm{Id}})
\]
is semisimple and not a product, but it admits a surjective map whose kernel is \( (-{\mathrm{Id}}, {\mathrm{Id}}) \) to the product \( \mathrm{PSL}_2({\mathbb R}) \times \mathrm{PSL}_2({\mathbb R}) \).

Let \( G \) be a semisimple Lie group. A discrete subgroup \( H \) in \( G \) is said to be irreducible if its projection to every simple factor is dense. Every simple Lie groups admits dense lattices, and a semisimple Lie group admits irreducible lattices if and only if all of its simple factors have the same absolute type. The only cofinite irreducible subgroups in Lie groups are lattices.

An IRS \( \mu \) of \( G \) is said to be irreducible if every simple factor of \( G \) acts ergodically on \( ({\mathrm{Sub}}_G, \mu) \). If \( \Gamma \) is a lattice in \( G \) then the IRS \( \mu_\Gamma \) defined above is irreducible in this sense if and only if \( \Gamma \) itself is.

The goal of the remaining lectures will be to give a proof of the following theorem.

Theorem (Stuck–Zimmer): Let \( G \) be a semisimple Lie group of higher rank, without compact factors, and which has property (T). Then any irreducible IRS in \( G \) is of the form \( \mu_\Gamma \) for \( \Gamma \) a lattice in \( G \).

A more general statement for non-necessarily irreducible IRSs also holds. The original statement of Stuck and Zimmer deals with actions rather than IRSs: any irreducible pmp action of such a group \( G \) is either essentially free, or it is essentially transitive (admits an orbit of full measure). The statement above follows immediatly since being essentially transitive means that stabilisers belong to a single conjugacy class, which has to be that of a cofinite subgroup and we saw above that all irreducible cofinite subgroups in semisimple Lie groups are lattices.

The theorem applies to all IRSs in higher rank simple Lie groups such as \( \mathrm{SL}_n({\mathbb R}), n \ge 3 \) since such groups have property (T). It is known not to hold for all rank 1 Lie groups (we will shortly describe the construction below). The remaining open cases are thus those of products of rank 1 groups without property (T). For example the following question is completely open at present.

Question: Are there irreducible IRSs in \( \mathrm{SL}_2({\mathbb R}) \times \mathrm{SL}_2({\mathbb R}) \) which are not lattices?

Induction of IRSs

Let \( G \) be a lcsc group. It is useful to be able to induce IRSs from a subgroup to \( G \). This is of course not possible in general but it is if the smaller group is a lattice or normal subgroup. More generally it is possible to induce « nested IRSs » but the definition for this is rather technical and not really more useful than the one for lattices. We will describe induction from lattices in three ways:

1. As a random variable: Let \( \Gamma \) be a lattice in \( G \) and \( \mu_0 \) an IRS of \( \Gamma \). Choose a \( \mu_0 \)-random subgroup \( \Lambda \) in \( \Gamma \), and a random conjugate \( g\Gamma g^{-1} \) of \( \Gamma \), and output the subgroup \( g\Lambda g^{-1} \).
2. As a measure: Let \( G, \Gamma, \mu_0 \) as above. The space \( G \times {\mathrm{Sub}}_\Gamma \) with the action given by \( g \cdot (h, \Lambda) = (ghg^{-1}, \Lambda) \) has a (non-finite) measure given by \( \tilde\mu = \mu_{Haar} \otimes \mu_0 \). Define the map:
   \[
   \tilde\Phi :
   \begin{array}{rl}
   G \times {\mathrm{Sub}}_\Gamma &\to {\mathrm{Sub}}_G \\
   (g, \Lambda) &\mapsto g^{-1}\Lambda g
   \end{array}
   \]
   Let \( \Gamma \) act by \( \gamma\cdot(g, \Lambda) = (\gamma g, \gamma\Lambda\gamma^{-1}) \), then \( \Phi \) is \( \Gamma \)-invariant, and the action of \( \Gamma \) is cofinite for \( \tilde\mu \). Let \( \Phi \) be the map \( \Gamma \backslash G \times {\mathrm{Sub}}_\Gamma \to {\mathrm{Sub}}_G \) and \( \mu \) the quotient measure on \( \Gamma \backslash G \times {\mathrm{Sub}}_\Gamma \), then \( \Phi_*\mu \) is an IRS of \( G \).
3. As an action: Let \( \Gamma \) act on \( X_0 \) preserving an probability measure \( \mu_0 \). Let \( Y = \{ y_g : g \in G \} \) be a fundamental domain for \( \Gamma \) in \( G \) such that \( y_g \in \Gamma g \) and let \( X = Y \times X_0 \) with the probability measure \( \mu_{Haar} \otimes \mu_0 \). Then the following action of \( G \) on \( X \) preserves \( \mu \): for \( g \in G \) there is a unique \( \gamma \in \Gamma \) such that \( g = \gamma y_g \). Let :
   \[
   g \cdot (y, x) = (y_{y_g y}, \gamma \cdot x).
   \]

Lecture 2: Margulis Normal Subgroup and Factor Theorems (Jean Lécureux)

The goal of this lecture is to give a proof of Margulis’ normal subgroup theorem assuming the Factor Theorem. The statement of the former is as follows:

Theorem (Margulis): Let \( G \) be a semisimple Lie group of higher rank, and \( \Gamma \) an irreducible lattice in \( G \). If \( N \) is a normal subgroup of \( \Gamma \) then either \( \Gamma/N \) or \( N \) is finite, and in the latter case \( N \subset Z(G) \).

We note that it is not needed to assume that \( G \) has property (T) in this statement. For groups which do have property (T) it is implied by the Stuck–Zimmer theorem (the induction process above would give a non-trivial, non-lattice IRS in \( G \) if the statement was in default).

Structure of semisimple Lie groups

In this section we describe some structure theory for parabolic subgroups of semsisimple Lie groups. We will stick to the case of \( \mathrm{SL}_n({\mathbb R}) \) for simplicity, but modulo definitions the notation can be used in the later sections for the proof of the NST without changes.

Let \( G = \mathrm{SL}_n({\mathbb R}) \). We define the following subgroups:
\[
A = \left\{ \left( \begin{array}{ccc}* & 0 & 0 \ & \ddots & 0 \\ 0 & 0 & * \end{array}\right) \in \mathrm{SL}_n({\mathbb R}) \right\}, P = \left\{ \left( \begin{array}{ccc} * & * & * \\ 0 & \ddots & \vdots \\ 0 & 0 & * \end{array}\right) \in \mathrm{SL}_n({\mathbb R}) \right\}
\]
(a maximal split torus and associated minimal parabolic subgroup). A subgroup is said to be minimal parabolic if it is conjugated in \( G \) to \( P \). Another we will use later is the opposite parabolic defined by
\[
P^- = \left\{ \left( \begin{array}{ccc}
* & 0 & 0 \\
\vdots & \ddots & 0 \\
* & * & *
\end{array}\right) \in \mathrm{SL}_n({\mathbb R}) \right\}.
\]
The unipotent radical of \( P \) is the subgroup
\[
U = \left\{ \left( \begin{array}{ccc}
1 & * & * \\
0 & \ddots & \vdots \\
0 & 0 & 1
\end{array}\right) \in \mathrm{SL}_n({\mathbb R}) \right\}
\]
and \( U^- \) is defined similarly.

An important property of semisimple groups is the Bruhat decomposition (which in this case follows immediately from the pivot algorithm to invert linear systems): let W be the group of \( n \times n \) permutation matrices, then we have :
\[
G = \bigsqcup_{w \in W} PwP
\]
This decomposition has many useful corollaries :

1. The \( P \)-orbits on the quotient \( G/P \) (the set of complete flags in \( {\mathbb R}^n \)) are in natural bijection with \( W \).
2. Let
   \[
   w_0 = \left(\begin{array}{ccc} 0 & 0 & 1 \\ 0 & \vdots & 0 \\ 1 & 0 & 0 \end{array} \right) \in W.
   \]
   Then \( Pw_0 P \) is the only open orbit of \( P \).
3. As \( w_0 P w_0 = P^- \), \( P \cap P^- = A \) and \( P^- = AU^- \) we see that there is an isomorphism of Borel spaces between \( G/P \) and \( U^- \) (note that \( G \) does a priori not act on \( U^- \)). It preserves the class of Haar measures on both sides.

A parabolic subgroup is a subgroup of \( G \) which contains a conjugate of \( P \). It follows from the Bruhat decomposition that such subgroups are conjugated to one of the subgroups:
\[
P_\Theta = \left\{ \left( \begin{array}{ccc}
g_1 & * & * \\
0 & \ddots & * \\
0 & 0 & g_r \end{array} \right) \in \mathrm{SL}_n({\mathbb R}), g_i \in \mathrm{GL}_{k_i}({\mathbb R}) \right\}
\]
where \( \Theta = (\theta_1, \ldots, \theta_r) \) with \( \theta_i = k_1 + \cdots + k_i -1 \). For example, \( P_\emptyset = P \) and \( P_{(1, \ldots, n)} = G \).

The only way we’ll use higher rank in the sequel (in the proof of the Factor Theorem) is through the following fact.

There exists a parabolic subgroup \( P < Q < G \) if and only if \( G \) is of real rank at least two.

The unipotent radical of \( P_\theta \) is the subgroup
\[
U_\Theta = \left\{ \left( \begin{array}{ccc}
\mathrm{Id}_{k_1} & * & * \\
0 & \ddots & * \\
0 & 0 & \mathrm{Id}_{k_r} \end{array} \right) \in \mathrm{SL}_n({\mathbb R}) \right\}
\]
and its Levi component is
\[
L_\Theta = \left\{ \left( \begin{array}{ccc}
g_1 & 0 & 0 \\
0 & \ddots & 0 \\
0 & 0 & g_r \end{array} \right) \in \mathrm{SL}_n({\mathbb R}), g_i \in \mathrm{GL}_{k_i}({\mathbb R}) \right\}
\]
For example \( U_\emptyset = U, L_\emptyset = A \). We then have the product decomposition
\[
P_\Theta = U_\Theta \rtimes L_\Theta.
\]

Statement of the Factor Theorem

We will use the terminology \( G \)-space to mean a Borel action of \( G \) on a standard Borel probability space \( (X, \mu) \), where \( G \) preserves the class of \( \mu \).

If \( (X, \mu) \) is a \( G \)-space a \( G \)-factor is another \( G \)-space space \( (Y, \nu) \) together with a \( G \)-equivariant Borel map \( \pi : X \to Y \) such that \( \pi_*\mu = \nu \).

If \( X \) is a Borel set the measure algebra \( \mathcal B(X) \) of \( X \) is the sigma-algebra of measurable sets, modulo the null sets.

Let \( (X, \mu) \) be a \( G \)-space. There is a bijection between \( G \)-factors of \( X \) and \( G \)-invariant subalgebras of \( \mathcal B(X) \).

The bijection is simply \( (Y, \nu, \pi) \mapsto \pi^*\mathcal B(Y) \).

For example, if \( X = G/P \) with a probability measure in the class of the Haar measure, as the action is essentially transitive all \( G \)-factors are of the form \( G/Q \) with \( Q \) a parabolic subgroup. The algebra corresponding to \( G/Q \) is the subalgebra of right-\( Q \)-invariant Borel subsets in \( \mathcal B(G/P) \).

The statement of the Factor Theorem is then as follows.

Theorem (Margulis): Let \( G \) be a semisimple Lie group of higher rank, \( P \) a minimal parabolic subgroup and \( \Gamma \) an irreducible lattice in \( G \). Then any \( \Gamma \)-factor of \( G/P \) is a \( G \)-factor.

Equivalently, any \( \Gamma \)-invariant sub-sigma-algebra of \( \mathcal B(G/P) \) is in fact \( G \)-invariant.

The Factor Theorem implies the Normal Subgroup Theorem

Let \( G, \Gamma, N \) as in the statement of the theorem. There are two steps in the proof that either \( \Gamma/N \) or \( N \) must be finite, only the second of which uses the factor theorem:

1. Prove that \( \Gamma / N \) has property (T);
2. Prove that if \( N \not{\mathrm{Sub}}set Z(G) \) then \( \Gamma/N \) is amenable.

The first step is immediate if \( G \) itself has property (T); the general case is more delicate and we won’t give the proof here.

To prove the second step we let \( \mathcal B \) be the subalgebra of \( N \)-invariant subsets in \( \mathcal B(G/P) \). It is a \( \Gamma \)-invariant subalgebra (by normality of \( N \) in \( \Gamma \)), and by the Factor Theorem it follows that it is of the form \( \mathcal B(G/P_\theta) \) for some \( \theta {\mathrm{Sub}}set \{1, \ldots, n\} \).

If \( P_\theta \not= G \) then \( N \) acts essentially trivially on \( G/P_\theta \), hence \( N {\mathrm{Sub}}set gP_\theta g^{-1} \) for almost all \( g \in G \) and it follows that
\[
N {\mathrm{Sub}}set \bigcap_{g \in G} gP_\theta g^{-1} = Z(G)
\]
if \( G \) is simple. If \( G \) is not simple the right-hand side can be a proper product of factors, by the irreducibility of \( \Gamma \) then implies that we must still have \( N {\mathrm{Sub}}set Z(G) \).

It remains to prove that \( \Gamma/N \) is amenable if \( P_\theta = G \). To do this we use the following criterion for amenability:

A group is amenable if and only if it preserves a Borel probability measure on any compact space on which it acts continuously.

Thus let \( X \) be a compact \( \Gamma/N \)-space. We use the following well-known lemma, which follows essentially immediately from the amenability of the minimal parabolic subgroup \( P \).

Lemma (Furstenberg): There exists a \( \Gamma \)-equivariant Borel map \( \varphi : G/P \to {\mathrm{Prob}}(X) \).

The Borel space \( ({\mathrm{Prob}}(X), \varphi_*\mu) \) (where \( \mu \) is again any probability measure in the class of the Haar measure) is then a \( \Gamma \)-factor of \( G/P \). By the hypothesis that \( P_\theta = G \) we have \( \mathcal B = \{ G \} \), in other words there are no proper invariant subsets in \( \mathcal B(G/P) \) which are \( N \)-invariant, hence \( \varphi_*\mu \) must be supported on a single point in \( {\mathrm{Prob}}(X) \) (otherwise the preimage of any proper subset would be a \( N \)-invariant subset). This point must be fixed by \( \Gamma \), hence it is a \( \Gamma/N \)-invariant probability measure on \( X \).

Lectures 3, 4 and 5: Intermediate factor Theorem and proof of the Stuck–Zimmer Theorem (Arie Levit)

These three final lectures are dedicated to give some details about the proof of the Stuck–Zimmer Theorem and the Factor Theorem. The main tool will be a generalisation of the latter called the Intermediate Factor Theorem. The Stuck–Zimemr theorem will be deduced from it in a manner similar to how we proved that the FT implied the NST, and then we will present Margulis’ proof of the FT and explain how to modify it to get the IFT.

A reccurent idea will be to replace lattices by more general pmp actions.

Cocycles

Let \( G \) be a lcsc group, \( X \) a \( G \)-space and \( H \) another group. A cocycle of \( (G, X) \) with values in \( H \) is a Borel function
\[
\alpha : G \times X \to H
\]
which satisfies the chain rule
\[
\alpha(gh, x) = \alpha(g, hx) \alpha(h, x).
\]
The idea behind this definition is that whenever \( Y \to X \) is a measurable map, with a group \( H \) acting measurably on \( Y \) preserving the fibers, a cocycle allows to lift the action of \( G \) to an action on the space of sections \( L^\infty(X, Y) \) by the following rule:
\[
(g \cdot f)(x) = \alpha(g, x)f(gx).
\]
This does not give an action on \( Y \), rather the fibers are « affine spaces » over \( G \).

Two cocycles \( \alpha, \beta \) are said to be cohomologous if there exists a Borel function \( \varphi : X \to X \) such that
\[
\beta(g, x) = \varphi(gx)^{-1}\alpha(g, x)\varphi(x)
\]
for almost all \( x \in X \). (The actions are the same modulo multiplication by a map which is constant along fibers).

A more fancy way of phrasing this definition would be to say that cocycles are morphisms from the groupoid \( G \times X \) to the group \( H \).

Here are some examples of cocycles.

1. If \( \pi: G \to H \) is a group morphism then
   \[
   \alpha_\pi(g, x) = \pi(g)
   \]
   defines a « constant » cocycle.
2. If \( X = G/\Gamma \) is an homogeneous space and \( \alpha \) any cocycle on \( G \times X \) then the restriction to the orbit of the identity coset
   \[
   \gamma \mapsto \alpha(\gamma, {\mathrm{Id}}\Gamma)
   \]
   is a morphism from \( \Gamma \) to \( H \). For any morphism there is a cocycle which restricts to it.

Let us prove the last claim: to construct a cocycle \( \alpha_\varphi \) restricting to a given morphism \( \varphi : \Gamma \to H \) it suffices to deal with the case of the identity morphism \( \Gamma \to \Gamma \) since the composition \( \varphi \circ \alpha_{\mathrm{Id}} \) is still a cocycle.

To deal with this case we choose a right fundamental domain \( D \) for \( \Gamma \) in \( G \); then for any \( x \in D \) there are unique \( \bar x \in D \cap \Gamma x \) and \( \gamma \in \Gamma \) such that \( g\bar x\gamma^{-1} \in D \) and we put \( \alpha(g, x) = \gamma \).

The construction in the previous paragraph is essentially the same as the one used to construct induced actions from \( \Gamma \) to \( G \) above. It is obtained by the action on the space \( (G/\Gamma) \times X_0 \) given by
\[
g \cdot (h\Gamma, x) = (gh\Gamma, \alpha(g, h\Gamma)x)
\]
where \( \alpha \) is a cocycle obtained from the identity morphism as above. This action is pmp or ergodic whenever the original action of \( \Gamma \) is and its stabilisers are conjugates to that of the latter.

Amenability of actions

Recall that a group \( G \) is amenable if and only if it admits a fixed point in any affine action on a compact convex subset of a Banach space. This is immediately seen to be the same as the characterisation given above: probability spaces are compact convex subsets, and in the other direction one obtains fixed points by taking barycenters of invariant probability measures.

Amenability of actions (in the sense of Zimmer) is a bit more technical to define. Let \( (X, m) \) be a \( G \)-space. To define it we need to consider any Banach space \( E \) with dual (in the weak-* topology) unit ball \( E_1^* \), and any cocycle
\[
\alpha: G \times X \to {\mathrm{Isom}}(E_1^*).
\]
Then for any \( \alpha \)-equivariant Borel family of subsets \( A_x:x \in X \) (this means that \( A_x: x \in B \) is Borel for any Borel subset \( B {\mathrm{Sub}}set X \), and that \( \alpha(g, x)\cdot A_x = A_{gx} \) for all \( g \in G \) and \( x \in X \)) we have a space of sections \( L^\infty(X, (A_x)_x) \) with a natural \( G \)-action given by the cocycle \( \alpha \).

Definition: The action of \( G \) on \( (X, m) \) is said to be amenable if for any data \( E, A_x \) as above there is a fixed point of \( G \) in \( L^\infty(X, (A_x)_x) \).

The fixed point given by the definition should be thought of as an invariant section of the bundle over \( X \) with fibers \( A_x \).

The following properties of amenable actions are essentially immediate.

1. A group is amenable if and only if its trivial action is amenable.
2. If \( G \) admits an action which is both pmp and amenable then \( G \) is amenable (as above we can take barycenters, this time of an invariant section of the product bundle).
3. If \( H \) is a closed amenable subgroup of \( G \) then the action of \( G \) on \( G/H \) (with the Haar measure class) is amenable (this follows from the same argument as the lemma above).
4. If \( X, Y \) are two \( G \) spaces and \( X \) is amenable then \( X \times Y \) is amenable as well (this follows from the fact that there is an isomorphism between the spaces \( L^\infty(X \times Y, (A_{x, y})) \) and \( L^\infty(X, (L^\infty(Y, A_{x, y}))) \)).

Weakly amenable actions

A cocyle \( \alpha : G \times X \to H \) is said to be orbital if for all \( g \in G, x \in X \) such that \( gx = x \) we have \( \alpha(g, x) = {\mathrm{Id}} \). Using the chain rule we see that this means that the value of \( \alpha \) at any pair \( (g, x) \) depends only on the pair of points \( (x, gx) \); so the cocycle depends only on the equivalence relation on \( X \) induced by the \( G \)-action.

An action is said to be weakly amenable if the conclusions of the definition of an amenable action hold for every orbital cocycle. An important characterisation is the following result.

A Borel action of a lcsc group on s standard space is weakly amenable if and only if the induced equivalence relation is amenable.

Other definitions are the following: an action is weakly amenable if and only if

• for almost every \( x \in X \) the stabiliser \( G_x \) acts ergodically on \( G/P \) (where \( P \) is a minimal parabolic in \( G \));
• for almost every \( x \in X \) there exists bounded harmonic functions on \( G/G_x \).

An example of an ergodic, weakly amenable but not amenable action can be constructed as follows for a group \( G \) of rank 1: let \( \Gamma \) be a lattice in \( G \) and \( N \le \Gamma \) a normal infinite, infinite index subgroup. Let \( X, m \) be a pmp \( G \)-space whose stabiliser is conjugate to \( N \) almost surely. Then this space is weakly amenable as \( N \) acts ergodically on \( G/P \) (because \( \Gamma \) does and it is normal there). But the action is pmp, hence it cannot be amenable. Note also that the action is not essentially transitive as \( G/N \) does not have a \( G \)-invariant Borel measure.

Proof of the Stuck–Zimmer theorem assuming the Nevo–Zimmer factor theorem

We recall that here \( G \) is assumed to have higher rank and property (T). Let \( (X, m) \) be an irreducible \( G \)-space with pmp action. We assume that the action is not essentially transitive and we want to prove that it must then be essentially free. To do this there are two steps, which are similar to those in the proof of the NST:

1. Prove that a consequence of property (T) is that the pmp action on \( (X, m) \) cannot be weakly amenable;
2. Using the intermediate factor theorem of Nevo–Zimmer (to be stated later) prove that a non-weakly amenable action of \( G \) must be essentially free.

Proof of Step 1

We now assume that the action of \( G \) on \( (X,m) \) is weakly amenable. To get to a contradiction we will use the two following results (we will not define amenability of a measured equivalence relation but rather use it as a black box).

An action is weakly amenable if and only if its orbital equivalence relation is amenable.

Theorem (Connes–Feldman–Weiss): Any amenable equivalence relation on a standard space is the orbit relation of an essentially free Borel action of \( {\mathbb Z} \) or \( {\mathbb R} \) (according to whether the classes are countable or not).

Together these two results imply that we get a cocycle \( \alpha: G \times X \to {\mathbb R} \) defined by setting (using freeness of the action):
\[
\alpha(g, x) = t \text{ if } gx = tx.
\]
A theorem of Zimmer then implies that \( \alpha \) is cohomologous to the trivial cocycle. This means that there is a Borel function \( \varphi : X \to {\mathbb R} \) such that
\[
\forall t \in {\mathbb R} \forall x \in X : \: \varphi(tx) – \varphi(x) = t
\]
but because of ergodicity of the \( {\mathbb R} \)-action this is not possible unless it is essentially transitive.

Proof of Step 2

Here \( X, m \) is a properly ergodic \( G \)-space which is not weakly amenable and we want to prove that it is essentially free. This will be a consequence of the intermediate factor theorem.

Theorem (Nevo–Zimmer): Let \( G \) be a semisimple Lie group of higher rank, \( P \) a minimal parabolic and assume there are \( G \)-spaces \( X, m \) and \( Y, \mu \) with measure-class preserving \( G \)-maps
\[
X \times G/P \to Y \to X.
\]
Then there exists a parabolic subgroup \( Q \supset P \) such that \( Y \cong G/Q \times X \) and the maps are conjugated to the natural projections via this isomorphism.

We note that the Margulis factor theorem is a consequence of this statement: let \( Z \) be a \( \Gamma \)-factor of \( G/P \). Then inducing the \( \Gamma \)-action on \( G/P \) to a \( G \)-action gives the product \( G/\Gamma \times G/P \), and that on \( Y \) to some \( G \)-space \( Y \) with a \( G \)-map \( Y \to G/\Gamma \). We can then apply the IFT to the sequence
\[
G/\Gamma \times G/P \to Y \to G/\Gamma
\]
and it follows that \( Z \) was of the form \( G/Q \).

Now we finish the proof of the Stuck–Zimmer theorem: the \( G \)-action on \( (X, m) \) is not weakly amenable, hence there exists a bundle \( A_x, x \in X \) with trivial stabiliser actions \( G_x \curvearrowright A_x \) and without a \( G \)-invariant section, which can be made into a \( G \)-space which we’ll denote \( E \). As \( G/P \) and hence also \( G/P \times X \) are amenable we get a \( P \)-invariant section \( s \) of \( A_x \), and then a \( G \)-map from \( G/P \times X \) to \( E \) defined by
\[
(gP, x) \mapsto (s(x, gP), x).
\]
Applying the IFT to the sequence
\[
G/P \times X \to E \to X
\]
we get that \( E \) is isomorphic (as a \( G \)-space) to \( G/Q \times X \) for \( Q \) a parabolic subgroup. We finally want to see that \( Q \not= G \): this follows from the non-amenability as \( G = Q \) would mean that there is a \( G \)-invariant section of \( E \).

By hypothesis the stabiliser action \( G_x \curvearrowright G/Q \) is trivial for almost every \( x \in X \) and it follows that
\[
G_x {\mathrm{Sub}}set \bigcap_{g \in G} gQg^{-1}
\]
hence \( G_x {\mathrm{Sub}}set Z(G) \).

Proof of the Margulis factor theorem

Recall from previous lectures that the Factor Theorem is equivalent to the following statement:

Any \( \Gamma \)-invariant subalgebra of \( \mathcal B(G/P) \) is in fact \( G \)-invariant.

We will use the topology on \( \mathcal B(G/P) \) induced by the metric
\[
d(A_1, A_2) = m(A_1 \Delta A_2)
\]
which is second-countable and complete. Moreover any subalgebra is closed. We will use notation similar to that used in previous lectures :

• \( P \) is a minimal parabolic and \( S {\mathrm{Sub}}set P \) a maximal torus;
• Let \( \Sigma(G, S) \) be a basis of positive roots; for \( \Theta {\mathrm{Sub}}set \Sigma(G, S) \) we denote by \( P_\Theta \) the associated parabolic subgroup;
• \( \bar P \) is the opposite parabolic and \( \bar V \) its unipotent radical;
• Similarly, \( \bar P_\Theta , \bar V_\Theta \) are the parabolic opposed to \( P_\Theta \) and its unipotent radical;
• we will need the decomposition
  \[
  \bar V = \bar L_\Theta \rtimes \bar V_\Theta
  \]
  where \( \bar L_\Theta \) is the intersection of \( bar V \) with the Levi component of \( P_\Theta \);

We recall that we have an identification of Borel spaces
\[
G / P_\Theta \cong \bar V_\Theta.
\]

Preliminary sketch

The main idea will be, starting from a given \( \Gamma \)-invariant Borel set in \( \mathcal B(X) \), to produce a \( G \)-equivariant family of new subsets in \( \mathcal B(X) \) to be able to prove in the end that this subalgebra must equal one of the \( G \)-invariant subalgebras of \( \mathcal B(G/P) \).

The main tool to do this will be the following procedure, adequately modified below. By a variant of the Howe–Moore theorem, the action of any nontrivial element of the maximal split torus \( S \) is ergodic on \( G/\Gamma \). This means in particular that, for almost all \( x \in G \), the subset
\[
\{ s^n x \Gamma : n \in \mathbb N \}
\]
is dense in \( G \). For such \( x \) and a \( E \in \mathcal B(X) \) it follows, modulo technical details that will be partially described below, that any limit \( E_\infty \) of the sequence
\[
s^m x E, \, m\ge 1
\]
belongs to \( \mathcal B(X) \), as does \( g E_\infty \) for any \( g \in G \).

Images of Borel sets under the \( S \)-action

We will study the following problem: given \( E \in \mathcal B(G/P) \), \( x \in G \) and \( s \in S \), how does \( s^n x E \) behave as \( n \to +\infty \)?

For this we will work with the identification \( G/P = \bar V \); if \( v \) represents \( xP \) in this identification (i.e. \( vP = xP \)) then we have that \( s^n x E \) is identified with \( s^n vE s^{-n} \). Now we may choose \( s \in S \) whose conjugation action is expanding on \( \bar V \), that is for all compact subsets \( K {\mathrm{Sub}}set V \) and any neigbourhood \( U \) of \( {\mathrm{Id}} \) in \( \bar V \) we have \( s^n K s^{-n} {\mathrm{Sub}}set U \) for large \( n \). For such an \( s \) we thus expect that :

• If \( {\mathrm{Id}} \in vE \) then \( \lim_n (s^n x E) = G/P \) in \( \mathcal B(G/P) \);
• If \( {\mathrm{Id}} \not\in vE \) then the limit is empty.

This is immediate if \( E \) is open. In general one needs to use a version of the Lebesgue density theorem to prove the following result.

Lemma: Let \( E \in \mathcal B(\bar V) \). For almost every \( v \in \bar V \) we have:
\[
\lim_{n \to +\infty} \left( s^n vE s^{-n} \right) =
\begin{cases}
\emptyset & \text{ if } {\mathrm{Id}} \not\in vE; \\
\bar V & \text{ else.} \\
\end{cases}
\]

We need to generalise this to all parabolics. For the remainder of the section we fix \( \Theta {\mathrm{Sub}}set \Sigma(G, S) \) such that \( P_\Theta \neq G \). Then we have an isomorphism
\[
\mathcal B(G / P_\Theta) \cong \mathcal B(\bar V_\Theta)
\]
and we recall that we have a decomposition \( \bar V = \bar V_\Theta \rtimes L_\Theta \).

Let \( s \in S \) such that \( s \) acts expansively on \( \bar V_\Theta \), and trivially on \( L_\Theta \). We define for E \in \mathcal B(\bar V_\Theta) \):
\[
\psi_\Theta(E) = \bar V_\Theta \cdot (E \cap L_\Theta).
\]
The same arguments used in the case \( \Theta = \Sigma(G, S) \) give the following generalisation of the lemma above.

Lemma: Let \( \Theta, s \) be as above and \( E \in \mathcal B(\bar V) \). Then for almost every \( v \in \bar V \) we have:
\[
\lim_{n \to +\infty} \left( s^n vE s^{-n} \right) = \psi_\Theta(vE).
\]

As a corollary to the lemma above and the arguments in the sketch we get the following statement:

Let \( \mathcal B {\mathrm{Sub}}set \mathcal B(\bar V) \) be a \( \Gamma \)-invariant subalgebra and \( E \in \mathcal B \). Then for all \( \Theta {\mathrm{Sub}}set \Sigma(G, S) \), \( g \in G \) and for almost every \( v \in \bar V \) we have that \( g\psi_\Theta(vE) \in \mathcal B \).

Conclusion of the proof

Let \( \mathcal B \) be a \( \Gamma \)-invariant subalgebra of \(\mathcal B(G/P) \) \( Q \) be a minimal parabolic such that \( \mathcal B(G/Q) {\mathrm{Sub}}set \mathcal B \). We want to show that \( \mathcal B(G/Q) = \mathcal B \), so we can assume in the rest of the proof that \( Q \neq P \).

Suppose that some \( E \in \mathcal B \) is not \( Q \)-invariant. As \( Q \neq P \) it is generated by the \( P_\theta \) for \( \theta \in \Sigma(G, S) \) a root which is trivial on \( Q \). Thus we may assume that \( E \) is in fact not invariant under some \( P_\theta \). Then by the conclusion to the previous section we have that \( \psi_\theta(vE) \) is either empty (if \( vE \cap L_\theta \) is) or also not in \( \mathcal B(G/Q) \) for almost all \( v \in \bar V \). As the first possibility cannot occur for almost all \( v \), neither can \( \psi_\theta(vE) \) be equal to \( \bar V \) for almost all \( v \).

To conclude, there exists a proper Borel set \( \psi_\theta(vE) \) such that \( g\psi(vE) \in \mathcal B \) for all \( g \in G \) but \( \psi_\theta(vE) \not\in \mathcal B(G/Q) \). If follows, if \( Q = P_\Theta \), that \( Q’ = P_{\Theta \setminus \theta} \) also satisfies that \( \mathcal B(G/Q’) {\mathrm{Sub}}set \mathcal B \) (the former is equal the subalgebra generated by \( \mathcal B(G/Q) \) and \( \psi_\theta(vE) \) as it is \( G \)-invariant. This contradicts the minimality of \( Q \), hence we must have had \( \mathcal B = \mathcal B(G/Q) \) and this finishes the proof.

Some ideas for the intermediate factor theorem

We have a sequence of \( G \)-factors
\[
G/P \times X \to Y \to X
\]
which yields inclusions
\[
\mathcal B(X) {\mathrm{Sub}}set \mathcal B(Y) {\mathrm{Sub}}set \mathcal B(G/P \times X)
\]
and we want to prove that there exists a parabolic \( Q \supset P \) such that \( \mathcal B(Y) = \mathcal B(G/Q \times X) \). As for the factor theorem we take a minimal \( P’ \) such that
\[
\mathcal B(X) {\mathrm{Sub}}set \mathcal B(Y) {\mathrm{Sub}}set \mathcal B(G/P’ \times X)
\]
and we want to prove that the above holds with \( Q = P’ \). If that is not the case then there exists a \( E \in \mathcal B(Y) \) such that \( E \not\in \mathcal B(G/P’ \times X \).

The proof uses a disintegration
\[
\mathcal B(Y) = \int_X \mathcal B_x, \, \mathcal B_x {\mathrm{Sub}}set \mathcal B(G/P)
\]
which is \( G \)-equivariant (that is \( \mathcal B_{gx} = g\mathcal B(x) \) for all \( g \in G \)). The set \( E \) decomposes as \( \int_X E_x \) and we have that on a non-null set \( E_x \not \in \mathcal B(G/P’) \). Then, assuming that \( P’ \not= G \), we may apply an argument similar to that used for the factor theorem in each subalgebra \( \mathcal B_x \) where \( E_x \not\in \mathcal B(G/P’) \) to conclude. To do this a further generalisation of Howe–Moore is needed, where the action of \( G \times S \) on \( G \times X \) is proven to have dense orbits, and use convergence not only in the fibers but also on the base space.

Introduction

Let \( G \) be a locally compact, second countable group. We introduced the Chabauty space \( \mathrm{Sub}_G \) of closed subgroups of \( G \) and the space \( \mathrm{IRS}(G) \) of invariant random subgroups of \( G \) in the lectures on the Nevo–Stück–Zimmer theorem. A corresponding topological notion is given by the following objects introduced by Eli Glasner and Benjamin Weiss.

Definition: A uniformly reccurent subgroup (URS) of \( G \) is a minimal \( G \)-invariant subset in \( \mathrm{Sub}_G \).

Recall that a closed \( G \)-invariant subset of a topological \[ G \)-space is called minimal if it does not contain a proper closed invariant \( G \) subset. As in the case of IRSs there are immediate examples of URSs:

• The singletons \( \{ {\mathrm{Id}} \}, \{ G \} \) are always URSs;
• More generally, if \( N \) is a closed normal subgroup of \( G \) tnen \( \{ N \} \) is an URS of \( G \).

An URS is said to be nontrivial if it is not one of the two examples in the first item above.

An example which often differs from IRSs (in the sense that it is not the support of any IRS of \( G \)) is given by cocompact subgroups: if \( P \le G \) is cocompact then the orbit \( P^G = \{ gPg^{-1} :\: g \in G \} \) is closed and hence an URS. For an example which is not the support of an URS we can take \( G \) to be semisimple a Lie group (e.g. \( G = \mathrm{SL}_2(\mathbb R) \)) and \( P \) a proper parabolic subgroup (e.g. the upper triangular matrices).

The correspondence between URSs and continuous actions on compact space is looser than in the emasurable category (as stated in this result): it is given by the following result.

Proposition (Glasner–Weiss): If \( G \) acts minimally on a compact space \( X \) then the subset
\[
\overline{ \{ \mathrm{Stab}_G(x) :\: x \in X \} }
\]
contains a unique URS.

Similarly to the case of IRSs the following question is open:

Are there any URSs in the Neretin group \( N_p \) which do not come from the action on \( \partial T_p \)?

We also note that an example of a discrete group which admits (many) IRSs but no non-trivial URSs is the group of finitary premutations of a countable set. On the contrary the Thompson group admits non non-trivial IRSs (Dudko–Medynets) but has URSs coming from the action on the boundary.

The goal of this lecture is to explain the construction of a family of non-dicrete lcsc groups which have no nontrivial URSs. They turn out to have no non-trivial IRSs as well. They are not compactly generated, and no such examples are known.

These groups will occur as subgroups of the group of piecewise affine homeomorphisms of \( {\mathbb Q}_p \). Such maps are by definition invertible maps \( g : {\mathbb Q}_p \to {\mathbb Q}_p \) such that there exists a decomposition into disjoint clopen subsets
\[
{\mathbb Q}_p = X_1 \sqcap \cdots \sqcap X_r
\]
and \( a_1, b_1, \ldots, a_r, b_r \in {\mathbb Q}_p \) such that
\[
\forall x \in X_i :\: gx = a_i x + b_i
\]
(i.e. the restriction to \( X_i \) is affine). It is clear that they form a group, which will be denoted by \( \mathrm{PL}({\mathbb Q}_p) \); it is not closed in \( \mathrm{Homeo}({\mathbb Q}_p) \) but still inherits a group topology from it.

Theorem: Let \( F_n,\, n \ge 1 \) be a family of finite groups. There exists a locally compact subgroup \( G \le \mathrm{PL}({\mathbb Q}_p) \) which contains \( U \) as an open subgroup, and has non nontrivial URSs or IRSs

Biapproximations

In this section \( G \) is a lcsc group.

Definition: A biapproximation of \( G \) is a sequence \( U_n, G_n \) of subgroups satisfying the following properties:

1. For all \( n \), \( U_n \le G_n \);
2. For all \( n \), \( G_n \) is open, \( G_n \le G_n{n+1} \) and
   \[
   G = \bigcup_{n \ge 1} G_n;
   \]
3. The subset \( \bigcup_{n \ge 1} U_n \) is relatively compact in \( G \);
4. \( U_n \) converges to the trivial subgroup \( \{{\mathrm{Id}}\} \) as \( n \to +\infty \).

For example we can have \( U_n \supset U_{n+1} \) and \( bigcap_{n \ge 1} U_n = \{{\mathrm{Id}}\} \).

Definition: Let \( U \le G \) be a compact subgroup and let \( H \le G \). The \( U \)-saturation of \( H \) in \( G \) is the subgroup of elements preserving setwise all orbits of \( H \) in \( G/U \) :
\[
[H]_U^G = \{ g :\: \forall x \in G/U, gx \in Hx \}
\]

It follows immediately from the definition that
\[
[H]_U^G = \bigcap_{x \in G/U} HxUx^{-1}
\]
which implies that

• \( [H]_U^G \) is a closed subgroup of \( G \);
• If \( U \triangleleft G \) then \( [H]_G^U = HU \).

Definition: The subgroup \( H \) is said to be \( U \)-saturated if \( [H]_U^G = H \).

Note that in case \( U \triangleleft G \) saturated subgroups are exactly subgroups containg \( U \) by the above, so that there is a bijection between the \( U \)-saturated subgroups and \( \mathrm{Sub}_{G/U} \). In general we have the following fact.

Lemma: The map \( \mathrm{Sub}_G \to \mathrm{Sub}_G \) defined by \( H \mapsto [H]_U^G \) is semi-continuous, in particular Borel.

Groups with biapproximations

Let \( U_n, G_n \) be a biapproximation of a lcsc group \( G \). Then for all \( n \) it follows from the above lemma the self-map \( \lambda_n \) of \( \mathrm{Sub}_G \) defined by
\[
\lambda_n(H) = [H \cap G_n]_{U_n}^{G_n}
\]
is semi-continuous. Let \( \mathcal F(G) \) be the space of closed subsets of \( \mathrm{Sub}_G \), with the Hausdorff topology, and define the map \( \lambda_n^* \) by
\[
\bar\lambda_n(\mathcal H) = \overline{\lambda_n(\mathcal H)}.
\]
The crucial result is then the following.

Proposition:

• The maps \( \lambda_n^* \) converge uniformly to the identity on ths space \( \mathrm{Prob}(\mathrm{Sub}_G) \) of IRS of \( G \);
• If \( \mathcal H_n \to \mathcal H \) in \( \mathcal F(G) \) then \( \bar\lambda_n(\mathcal H_n) \to \mathcal H \) as well.

An immediate consequence for later use is that the following properties are equivalent for the group \( G \):

1. \( G \) has no non-trivial IRSs;
2. If \( \mu_n \) is an IRS of \( G_n \) which is almost surely \( U_n \)-saturated then every accumulation point of \( \mu_n \) in \( \mathrm{Prob}(\mathrm{Sub}_G) \) is a convex combination of \( \delta_{\mathrm{Id}}, \delta_G \).

Application

Let \( \Gamma_p \) be the subgroup of \( \mathrm{PL}({\mathbb Q}_p) \) consisting of elements which are compactly supported (i.e. equal to the identity outside of a compact subset) piecewise affine maps whose coefficients belong to \( p^{\mathbb Z} \) (for the multiplicative part) and \( {\mathbb Z}[1/p] \) (for the additive part).

For \( n \ge 1 \) le \( \mathcal X_n \) be the union of nontrivial cosets of \( p^{-n}{\mathbb Z}_p \) in \( p^{-n-1}{\mathbb Z}_p \):
\[
\mathcal X_n = p^{-n-1}{\mathbb Z}_p \setminus p^{-n}{\mathbb Z}_p.
\]
Let \( \mathcal F = (F_m)_{m \ge 1} \) be a family of finite subgroups such that any elements of \( F_m \) is supported on \( \mathcal X_m \), and define the following subgroup: \( G_{\mathcal F} \) is the subgroup consisting of all \( g \in \mathrm{PL}({\mathbb Q}_p) \) such that:

1. \( g(p^{-N}{\mathbb Z}_p) \mathrm{Sub}set p^{-N}{\mathbb Z}_p \) for some \( N \ge 1 \);
2. \( g|_{p^{-N}{\mathbb Z}_p} \in \Gamma_p|_{p^{-N}{\mathbb Z}_p} \);
3. \( g|_{{\mathbb Q}_p \setminus p^{-N}{\mathbb Z}_p} \in \prod_{m \ge N+1} F_m \).

This subgroup contains \( \Gamma_p \) and \( \prod_{m \ge 1} F_m \); it is not closed in the compact-open topology but we give it a locally compact group topology by decreting the commensurated subgroup \( \prod_n F_n \) to be open.

This group \( G_{\mathcal F} \) admits a biaproximation by the following subgroups:

• \( G_n \) is defined by the conditions 1, 2, 3 above but with \( N = n \) fixed;
• \( U_n \) is defined by 1, 3 (with \( N = n \)) and \( g|_{p^{-n}{\mathbb Z}_p} = {\mathrm{Id}} \) for all \( g \in U_n \).

As \( U_n \triangleleft G_n \), the application of the criterion in the previous section shows that \( G_{\mathcal F} \) has nontrivial IRSs if and only if the locally finite group
\[
\bigcup_{n \ge 1} G_n/U_n \cong \Gamma_p
\]
does. As \( \Gamma_p \) is isomorphic to a Thompson group and those were proven to not have nontrivial IRSs by Dudko–Medynets we conclude that \( G_{\mathcal F} \) does not as well.

A similar reasoning gives the result for URSs.

The Neretin groups are a family of locally compact subgroups of homeomorphisms of the Cantor set.

Before defining them we will present an Archimedean analogue for them. The group \( \mathrm{PGL}_2({\mathbb R}) \) acts by homographies on the circle \( \mathbb S^1 \) (identified with the projective space \( \mathbb P^1({\mathbb R}) \)), and it is a locally compact subgroup of \( \mathrm{Homeo}(\mathbb S^1) \). This is a Lie group and it is reasonably well-understood. A more exotic group is obtained by taking those homeomorphism which are only piecewise homographies; however this group is not closed in \( \mathrm{Homeo}(\mathbb S^1) \), and there is no natural way to turn it into a locally compact group. (It has, however, many interesting discrete subgroups such as the Thompson groups and the « Frankenstein groups » defined by Monod).

In the totally discontinuous world, the group \( \mathrm{PGL}_2({\mathbb Q}_p) \) acts on \( \mathbb P^1({\mathbb Q}_p) \) (topologically a Cantor set) by homographies, and this action can be identified with the action on the boundary \( \partial T_p \) at infinity of its Bruhat–Tits tree \( T_p \) (a \( p+1 \)-regular tree). It will be more convenient to consider the entire group \( \mathrm{Isom}(T_p) \) of isometries of this tree, whose action also extends to \( \partial T_p \). The Neretin group \( N_p \) will be the subgroup of homeomorphisms of \( \partial T_p \) which are piecewise restictions of isometries. The totally discontinuous nature of \( \partial T_p \) allows a nice description of this.

Definition: Let \( F_1, F_2 \) be two (possibly empty) finite subtrees of \( T_p \) such that \( T_p \setminus F_i \) have the same number of components. If \( \phi \) is an isometry between \( T_p \setminus F_1 \) and \( T_p \setminus F_2 \) it induces an homeomorphism \( \phi_* \) of \( \partial T_p \). Then Neretin group \( N_p \) is the subgroup of \( \mathrm{Homeo}(\partial T_p) \) of all elements of this form.

Note that this subset is actually a subgroup because it is always possible to enlarge the finiet tree \( F_1 \) used in the definition, and thus to compose such elements inside the tree.

Simplicity

This is due to C. Kapoudjian. See https://arxiv.org/abs/1502.00991 for a nice account.

Topology of the Neretin groups

The Neretin group is not closed in the group \( \mathrm{Homeo}(\partial T_p) \): it is not hard to construct sequences of finite subtrees, and associated elements of \( N_p \) which converge to an homeomorphism which does not belong to \( N_p \). The size of the subtrees must go to infinity for this to happen, and it turns out that it is possible to make the Neretin group into a locally compact group, essentially by stating that such sequences must diverge.

Formally, this is done using the following fact:

If a topological group \( H \) is a commensurated subgroup in an abstract group \( G \) then there is a unique group topology on \( G \) such that \( H \) is an open subgroup.

Recall that a subgroup \( H \le G \) is said to be commensurated if for all \( g \in G \) the intersection \( H \cap gHg^{-1} \) has finite index in \( H \). This is the case for any compact-open subgroup of \( \mathrm{Isom}(T_p) \) in \( N_p \): indeed, if \( K \) is such a subgroup and \( g \in N_p \), letting \( F \) be a finite subtree outside of which \( g \) acts by homeomorphisms, and \( K_F \) the subgroup of \( K \) fixing \( F \) pointwise we have that \( K_F \) is compact open, hence of finite index in \( K \). On the other hand \( g K_F g^{-1} \) is still contained in \( \mathrm{Isom}(T_p) \), and as it follows from the previous sentence that it is compact-open its interection with \( K \) must have finite index in both, so that in particular \( K \cap gKg^{-1} \) which contains it has finite index in \( K \).

As the subgroup \( K \) is compact and open in \( N_p \) for this topology it follows that \( N_p \) is locally compact. It is second countable as the quotient \( N_p / K \) is countable.

In this topology a sequence \( g_n \in N_p \) converges if and only if there exists a pair of finite subtrees \( F \subset T_p \) such that \( g_n \) is induced from an isomorphism \( T_p \setminus F_1 \to T_p \setminus F_2 \), with a fixed map between the sets of components for \( n \) large enough and the isomorphism between components converging (in the topology of maps between rooted trees).

Absence of lattices

The goal of this section is to explain the proof of the following theorem (https://arxiv.org/abs/1008.2911).

Theorem (Bader–Caprace–Gelander–Mozes): The Neretin group has no lattices.

The proof uses the following subgroup: let \( e \) be an edge in \( T_p \) and let \( B_n \) be the ball of radius \( n \) around \( e \). Let \( O_n \) be the subgroup of elements of \( N_p \) induced by isometries of \( T_p \setminus B_n \). Then \( O_n \subset O_{n+1} \) and we can form the subgroup
\[
O = \bigcup_{n \ge 1} O_n
\]
of \( N_p \). An hyperbolic isometry of \( T_p \) cannot belong to \( O \) it is a proper subgroup of \( N_p \). Each \( O_n \) is open and compact (the stabiliser in \( isom(T_p) \) of \( e \) is a finite-index subgroup), so that:

\( O \) is an open subgroup of \( N_p \).

The main result above then follows from the following theorem.

Theorem: The group \( O \) has no lattices.

We will explain how Bader–Caprace–Gelander–Mozes rule out cocompact lattices; the general case follows the same lines but is quite a bit more involved. In what follows \( \Gamma \) is a cocompact lattice in \( O \).

First we note that for any \( n \) the action of \( O_n \) on \( \partial B_n \) gives a surjective morphism
\[
\pi_n :\: O_n \to \mathfrak S(k_n)
\]
where \( k_n = |\partial B_n| = p^n \). Let \( \Gamma_{O_n} = \Gamma \cap O_n \), a finite group, and \( \Gamma_n = \pi_n(\Gamma_{O_n}) \). As \( \Gamma \) is discrete and \( \ker(\pi_n \) form a basis of neighbourhoods of \( \mathrm{Id} \) in \( N_p \) (in fact \( \ker(\pi_n) \) is the subgroup of the stabiliser \( K \) of \( e \) in \( \mathrm{Isom}(T_p) \) fixing \( B_n \) pointwise) it follows that \( \Gamma_n \cong \Gamma_{O_n} \) for \( n \) large enough. Moreover, as it is cocompact a fundamental domain for \( \Gamma \) in \( O \) will be contained in \( O_n \) for \( n \) large enough. We fix the Haar measure on \( O \) so that \( K \) has mass 1, putting:
\[
c = \mathrm{vol}(O/\Gamma),\, c_n = \mathrm{vol}(O_n/\Gamma_n),\, a_n = |\mathrm{Isom}(B_n)|
\]
these two facts then imply that:
\[
c = c_n = \frac{\mathfrak S(k_n)}{a_n}
\]
for large enough \( n \) (as \( a_n \) is the index of \( \ker(\pi_n) \) in \( K \)).

It follows that for these \( n \) we have
\[
[\mathfrak S(k_n) : \Gamma_n] = c\cdot a_n.
\]
We note that the right-hand side has bounded prime factors as \( a_n \) is a power of \( p! \). A result from finite group theory then implies the following.

The subgroup \( \Gamma_n \subset \mathfrak S(k_n) \) contains \( \mathrm{Alt}(X_n) \) for some \( X_n \subset \partial B_n \) with \( |X_n| \ge \frac{k_n}2 + 2 \).

The subset \( X_n \) must contain at least two pairs \( (x_i, y_i) \) of leaves of \( B_n \) such that \( x_i \) and \( y_i \) have the same parent in \( B_n \). The permutation
\[
(x_1\, y_1)(x_2\, y_2)
\]
is then induced by a nontrivial automorphism of \( B_n \) which fixes pointwise \( B_{n-1} \). But for \( n \) large enough this is a contradiction to the fact that such automorphisms do not intersect \( O_n \).

The cocompact case uses similar argument, but the finite group theory involved is more complicated as in this case the equality \( [\mathfrak S(k_n) : \Gamma_n] = c\cdot a_n \) does not hold, only the asymptotic statement \( [\mathfrak S(k_n) : \Gamma_n] \sim c\cdot a_n \).

This is a transcript of my notes from Uri Bader’s lectures. They may not always accurately reflect the content of the lectures, especially in places I did not put down everything that was said, and some comments or details might have been added by me. In particular the last section does not contain the proofs of the Ornstein–Weiss theorem, Dye’s theorem and Rokhlin lemma that Uri explained in his lecture.

1. First lecture
2. Second lecture
3. Third lecture
4. Fourth lecture
5. Fifth lecture

Lecture I

From quasi-isometry to measured equivalence

In all of the following lectures, when no further precision is given all groups will be assumed to belong to the class of locally compact and second countable groups (abbreviated as lcsc groups).

A topological coupling between two groups \( \Gamma, \Lambda \) is an action of \( \Gamma \times \Lambda \) on a locally compact space \( X \) such that both actions \( \Gamma \curvearrowright X \) and \( \Lambda \curvearrowright X \) are properly discontinuous and cocompact.

Theorem (Gromov): The two groups \( \Gamma \) and \( \Lambda \) are quasi-isometric to each other if and only if they admit a topological coupling.

Examples

• If \( \Gamma, \Lambda \) are both cocompact discrete subgroups of a locally compact group \( G \) then \( \Gamma \curvearrowright G \curvearrowleft \Lambda \) (where the actions are by right- and left-translations) is a topological coupling.
• Let \( G = {\mathbb{R}} \) and \( \Gamma = {\mathbb{Z}} \), then the quotient \( \Gamma \backslash G \) is the circle \( \mathbb S^1 \) and a rotation of angle \( \alpha \) gives a coupling of \( \Lambda = \alpha{\mathbb{Z}} \) with \( \Gamma \).
• Let \( U \) be a proper metric space and \( \Gamma, \Lambda \subset \mathrm{Isom}(U) \) act properly discontinuously and cocompactly then they are quasi-isometric by the Milnor–Schwarz lemma. A topological coupling is given by \( G = \mathrm{Isom}(U) \).

Proof of the theorem

Recall that given \( K, C \) the set of \( (K, C) \)-quasi-isometries between two discrete groups \( \Gamma, \Lambda \) is locally compact in the pointwise convergence topology. Moreover the action of \( \Gamma \times \Lambda \) on \( \Lambda^\Gamma \) given by
\[
(\gamma, \lambda) \cdot f = \lambda f(\cdot \gamma)
\]
preserves the subset of \( (K, C) \)-quasi-isometries and the action on both sides is properly discontinuous and cocompact on both sides. If \( \Gamma \) is quasi-isometric to \( \Lambda \) then this set is nonempty for large \( K, C \) and it is a coupling.

Now suppose \( X \) is a topological coupling between \( \Gamma \) and \( \lambda \). Let \( X_0 \) be a relatively compact fundamental domain for the \( \Lambda \)-action. We get a bijection
\[
X_0 \times \Lambda \to X, \, (x, \lambda) \mapsto x \cdot \lambda
\]
and thus for any \( \gamma \in \Gamma \) and \( x \in X \) there exists a unique pair \( c(\gamma, x) \in \Lambda, a(\gamma, x) \in X_0 \) such that
\[
\gamma \cdot x = a(\gamma, x) \cdot c(\gamma, x).
\]
The map
\[
(\gamma, x) = a(\gamma, x)
\]
encodes an action of \( \Gamma \) on \( X / \Lambda \) via the bijection \( X_0 \cong X / \Lambda \). The map \( a \) thus satisfies the relation
\[
a(\gamma_1\gamma_2, x) = a(\gamma_1, \gamma_2 x).
\]
It follows that we have:
\[
c(\gamma_1\gamma_2, x) a(\gamma_1\gamma_2, x) = c(\gamma_2, x) c(\gamma_1, a(\gamma_2, x)) \cdot a(\gamma_1, a(\gamma_2, x)) = c(\gamma_2, x) x(\gamma_1, a(\gamma_2, x)) a(\gamma_1\gamma_2, x)
\]
and it follows that the map \( c : \Gamma \times X / \Lambda \to \Lambda \) satisfies the cocycle relation
\[
c(\gamma_1\gamma_2, \bar x) = c(\gamma_2, \bar x) c(\gamma_1, \gamma_2 \bar x).
\]

To finish the proof of the theorem one can then show that for any two maps \( c( \cdot, x) \) are at bounded distance from each other, and then use the above to see that any of them gives a quasi-isometry \( \Gamma \to \Lambda \).

Advantages of the definition via couplings

1. The definition this gives is purely topological and not metric. It can easily be extended to define quasi-isometry between lovally compact groups. (Note that it is a ctually not entirely trivial to check that this yields the natural definition for compactly generated groups).
2. The notion of coupling makes for group objects in many categories. For example in the \( \mathbf{Set} \) category, one can define couplings as actions of \( \Gamma \times \Lambda \) which are transitive, and such that both actions of \( \Gamma \) and \( \Lambda \) are cofinite and free (respectively have finite stabilisers). This recovers the notion of commensurability (respectively virtual isomorphism) between discrete groups.

Couplings in the measurable category and measure equivalence

In the measurable category one considers the actions of a locally compact group \( G \) on a measure space \( (X, \mu) \) which preserves the measure \( \mu \) and such that there exists a finite measure space \( X_0 \) and a \( G \) -equivariant measure isomorphism
\[
G \times X_0 \to X
\]
(where \( G \) is endowed with its Haar measure). Such an action is called free-proper-cofinite.

Note that when \( G \) is a discrete group one can view \( X_0 \) as a fundamental domain for \( G \) in \( X \) but not in general (as the measure of \( X_0 \) might have to be zero ; in general it is not possible to find a measurable transversal for the action).

The notion of equivalence obtained in this category is via measured couplings which are defined as follows: a measurable coupling between two locally compact groups \( G, H \) is a measure-preserving action of \( G \times H \) on a measured space \( (X, \mu) \) such that both actions of \( G \) and \( H \) are free-proper-cofinite. Whenever there exists such a coupling the groups \( G \) and \( H \) are said to be measure equivalent.

The fundamental example example of a measurable coupling is when \( \Gamma, \Lambda \) are both lattices in a locally compact group \( G \), and the coupling is given by the right- and left-action of \( \Gamma \) and \( \Lambda \) :
\[
(\gamma, \lambda) \cdot x = \gamma x \lambda.
\]

A particular case of measure equivalence is when the action of \( G \times H \) is transitive. Then one can write \( X = G \times H / \Gamma \) for a discrete group \( \Gamma \subset G \times H \), and both projections of \( \Gamma \) into \( G \) and \( H \) are lattices.

Lecture II

The categorical viewpoint

Measure equivalence between locally compact unimodular groups is an equivalence relation:

• For any such group \( G \) we have \( G \sim G \) via the trivial coupling \( G \curvearrowright G \curvearrowleft G \);
• The relation is trivially reflexive, if \( G \sim H \) via a coupling \( G \curvearrowright X \curvearrowleft H \) then also \( H \sim G \) via the opposite coupling.
• The relation is transitive: if \( G \curvearrowright X \curvearrowleft H \) and \( H \curvearrowright Y \curvearrowleft I \) are two measured couplings then the fibered product:
  \[
  X \times_H Y := (X \times Y)/ H
  \]
  (where \( H \) acts by \( h \cdot (x, y) = (x \cdot h, h^{-1} \cdot y) \) is a measurable coupling between \( G \) and \( I \).

Warnings

These operations do not form a monoid:

• If \( G \curvearrowright X \curvearrowleft H \) is a coupling then the coupling \( X \times_G X^{\mathrm{op}} \) is not the trivial \( H \)-coupling.
• The « composition » of couplings via the fibered product is not associative.

Equivalence of couplings

Two couplings \( X, X’ \) between \( G \) and \( H \) are said to be equivalent if there exists a \( G \times H \)-equivariant measure isomorphism between \( X \) and \( X’ \).

More generally, a morphism between two couplings \( X, Y \) between \( G \) and \( H \) is a measure-preserving, \( G \times H \)-equivariant map \( X \to Y \).

Definition: Two couplings \( X, Y \) are weakly equivalent if there exists a third coupling \( Z \) and two morphisms \( Z \to X \) and \( Z \to Y \).

Remark: This can be formalised as a 2-category on the set of free-proper-cofinite actions: a morphism is a coupling, and there is a notion of a « morphism between morphisms ». (Another example of this is given by the category of cobordisms in topology.)

Lemma: Weak equivalence is an equivalence relation on the sets of couplings.

This follows from the construction of fibered products in the category we work in: this gives an equivalence between the conditions
\[
\exists Y_1 \to X \leftarrow Y_2
\]
and
\[
\exists Y_1 \leftarrow Z \to Y_2
\]
for two couplings \( Y_1, Y_2 \).

Construction of fibered products

Let \( \pi: X \to Y \) a measurable map and \( \mu \) a measure on \( X \). Let \( \mu \) a measure on \( X \) and \( \nu = \pi_*\mu \). Disintegration of measures gives a decomposition
\[
\mu = \int_Y \mu_y d\nu(y)
\]
where \( \mu_y \) is a measure on the fiber \( X_y \) above \( y \in Y \). A diagram \( \exists Y_1 \to X \leftarrow Y_2 \) where \( X \) has a measure \( \nu \) thus yields two decompositions
\[
\mu_1 = \int_Y \mu_{1, y} d\nu(y), \, \mu_2 = \int_Y \mu_{2, y} d\nu(y)
\]
and this gives a measure \( \int_D \mu_{1, x} \otimes \mu_{2, x} d\nu(x) \) on the space
\[
Z = \int_X Y_{1, x} \times Y_{2, x} d \nu(x).
\]
One can then check that \( Z \) satisfies the property of a fibered product.

Remark: There is a contravariant functor from the measurable category to the category of von Neumann algebras given by \( X \mapsto L^\infty(X) \). It is in fact an equivalence of categories, and the diagram corresponding to a fibered product
\[
\begin{array}{ccc}
Z & \to & Y_1 \\
\downarrow & & \downarrow \\
Y_2 & \to & X
\end{array}
\]
is the following:
\[
\begin{array}{ccc}
L^\infty(Z) & \supset & L^\infty(Y_1) \\
\cup & & \cup \\
L^\infty(Y_2) & \supset & L^\infty(X)
\end{array}
\]

Observation: A self-coupling \( X \) of a locally compact group \( G \) is weakly equivalent to the trivial coupling \( G \) if and only if there exists a morphism \( X \to G \). Such an \( X \) is called trivial, and a morphism \( X \to G \) is called a Trivialisation.

Example

Let \( \Gamma \) be a discrete group and \( G \) a locally compact group. Let \( \rho_1, \rho_2 \) be two embeddings of \( \Gamma \) into \( G \), whose images \( \rho_i(\Gamma) \) are both lattices in \( G \). Then the fibered product \( X = G \times_\Gamma G \) is trivial if and only if \( \rho_2 \) is conjugated to \( \rho_1 \), that is there exists \( g \in G \) such that
\[
\forall \gamma \in \Gamma : \rho_2(\gamma) = g\rho_1(\gamma)g^{-1}.
\]
Indeed, if is trivial and \( G \times_\Gamma G \to X \) is a trivialisation, denote by \( (g_1, g_2) \mapsto (\overline{g_1, g_2}) \) the composition
\[
G \times G \to G \times_\Gamma G \to X.
\]
Let \( g = (\overline{e, e}) \). Then the definition of \( G \times_\Gamma G \) gives that for any \( \gamma \in \Gamma \) we have:
\[
\rho_0(\gamma)g = (\overline{\rho_0(\gamma)e, e}) = (\overline{e, e\rho_1(\gamma)}) = g\rho_1(\gamma).
\]

Let \( S \) be a closed surface. As an application of the example above it is possible to show that the space
\[
\{ (X, Y) \in \mathrm{Teich}(S) \times \mathrm{Teich}(S) : X \not\cong Y \}
\]
embeds into the space of self-couplings of \( \mathrm{PSL}_2({\mathbb{R}}) \) modulo weak equivalence.

In higher dimensions Mostow rigidity shows that there is no deformation space of lattices. Thus a way to interpret the space of weak equivalence classes of self-coupling of a group is as a measure of failure of rigidity.

ME-rigid groups

Say that a group \( G \) is ME-rigid if any self-coupling of \( G \) admits a unique trivialisation.

Theorem II (Furman): If \( G \) is a measure-rigid group and \( H \) is measure-equivalent to \( G \) then there exists a morphism \( G \to H \) with compact kernel and cofinite, closed image.

The converse of this theorem is trivially true. The next result gives an important family of examples of ME-rigid groups.

Theorem I (Furman): The groups \( \mathrm{PGL}_n({\mathbb{R}}) = \mathrm{Aut}(\mathrm{SL}_n({\mathbb{R}})) \) are rigid for \( n \ge 3 \).

An immediate corollary of Theorems I and II is the following rigidity statement for lattices.

Corollary: Let \( n \ge 3 \) and let \( \Gamma \) be a lattice in \( \mathrm{PGL}_n({\mathbb{R}}) \). If \( \Lambda \) is measure equivalent to \( \Gamma \) then there exists a finite index subgroup in \( \Lambda \) which embeds as a lattice in \( \mathrm{PGL}_n({\mathbb{R}}) \).

Lecture III

Remark: If \( G \) is ME-rigid then \( \mathrm{Aut}(G) = \mathrm{Inn}(G) \) (all automorphisms are inner) and \( Z(G) = \{ e \} \). These conditions amount to the map \( G \to \mathrm{Aut}(G) \) being an isomorphism. The proof of both is very simple:

1. Suppose that \( Z(G) \ni z \not= e \). Then the map \( g \mapsto zg \) is \( G \times G \)-equivariant, that is it is a endomorphism of the trivial coupling. Thus trivialisations will not be unique.
2. Let \( \varphi \in \mathrm{Aut}(G) \). Then we can get a « twisted » self coupling \( G \curvearrowright G \curvearrowleft G \) defined by
   \[
   g \cdot x \cdot h = \varphi(g) x h^{-1}.
   \]
   Then there exists a trivialisation for this coupling if and only if \( \varphi \) is inner (the proof is similar to this one above).

On the unicity of trivialisations in the definition of ME-rigidity

Lemma: Let \( G \) be a lcsc group. The following conditions are equivalent:

• Any trivial self-coupling of \( G \) admits a unique trivialisation.
• The only conjugacy-invariant probability measure on \( G \) is the Dirac mass at the identity:
  \[
  \mathrm{Prob}(G)^G = \{ \delta_e \}.
  \]

Proof: Let \( \mu \in \mathrm{Prob}(G)^G \) and \( X_0 = (G, \mu) \). Then \( X = G \times X_0 \) (where the right factor \( G \) has the Haar measure) is a self-coupling of \( G \) in two ways:

1. \( X \curvearrowleft G \) by the diagonal action
   \[
   (y, x) \cdot g = (yg, g^{-1}xg)
   \]
   and \( G \curvearrowright X \) by left-multiplication:
   \[
   h\cdot(y, x) = (hy, x).
   \]
2. \( X \curvearrowleft G \) by right-multiplication
   \[
   (y, x) \cdot g = (yg, x)
   \]
   and \( G \curvearrowright X \) by left-multiplication:
   \[
   h \cdot (y, x) = (y, hxh^{-1}).
   \]

Then if \( \mu \neq \delta_{\{e\}} \) the maps \( X \to G \) given respectively by \( (x, y) \mapsto xy \) and by \( (x, y) \mapsto x \) are two distinct trivialisations.

Conversely, suppose that there exists a trivial self coupling \( X \) with two distinct trivialisations
\[
G \overset{\varphi_1}{\leftarrow} X \overset{\varphi_2}{\to}.
\]
Define a map \( X \to G \) by:
\[
\psi(x) = \phi_1(x) \phi_2(x)^{-1}.
\]
Then \( \psi \) is right-\( G \)-invariant and we get a \( G \)-equivariant map \( X/G \to G \). The pushforward of the finite \( G \)-invariant measure on \( X / G \) to \( G \) is a conjugacy-invariant measure on \( G \) with finite mass, and it is not supported on the identity.

Remark

If \( G \) is a discrete group then the condition that \( \mathrm{Prob}(G)^G = \{ \delta_{\{e\}} \} \) is equivalent to the condition that all nontrivial conjugacy classes of \( G \) be infinite (ICC). The former condition is thus sometimes called « strong ICC ».

Invariant probability measures in algebraic actions

Theorem: Let \( G \) be a semisimple \( {\mathbb{R}} \)-algebraic group such that the Lie group \( G_{\mathbb{R}} \) has no compact factor and let \( G \curvearrowright V \) an algebraic action (action by regular automorphisms of the variety \( V \)). Then the natural map
\[
\mathrm{Prob}(V^G) \to \left( \mathrm{Prob}(V) \right)^G
\]
is surjective.

Applied to the action of \[ G \) by conjugation on itself this result has the following consequence.

Corollary: If \( G \) is an adjoint Lie group then \( \mathrm{Prob}(G)^G = \{ \delta_{\{e\}} \).

Demonstration of Theorem I

We will prove that \( G = \mathrm{PGL}_n({\mathbb{R}}) \) is ME-rigid, admitting a deep result due to R. Zimmer.

Since \( G \) is an adjoint group it follows from the corollary above that trivialisations are unique. We need to prove that any self-coupling of \( G \) is trivial. Let \( G \curvearrowright G \curvearrowleft G \) be such a coupling.

As was the case in the metric setting (see above) we can write \( X = G \times X_0 \) where the right-action is by right-multiplication on the \( G \) factor. The left action is given by a measurable cocycle \( c : G \times X_0 \to G \) so that
\[
g \cdot (h, x_0) = (hc(g, x_0), g\cdot x_0).
\]
We apply the following theorem.

Theorem (Cocycle superrigidity, Zimmer): Let \( G \) be a simple, adjoint Lie group of higher rank. If \( c: G \times X_0 \to \mathrm{Gl}_m({\mathbb{C}}) \) is a measurable cocycle whose image is unbounded and has a semisimple Zariski closure, then \( c \) is cohomologous to a morphism \( G \to \mathrm{GL}_m({\mathbb{C}}) \).

Two cocycles \( c, c’ \) are cohomologous if there exists \( \psi : X_0 \to H \) such that for any \( g \in G \) we have \( c'(g, x) = \psi(x)c(g, x)\psi(g\cdot x) \). Thus the theorem above implies that there exists a morphism \( \varphi: G \to G \) such that
\[
g \cdot (h, x_0) = (\varphi(g)h, g \cdot x_0)
\]
and it follows that the projection \( G \times X_0 \to G \) is a morphism, where \( G \) has the twisted coupling structure given by \( g \cdot x \cdot h = \varphi(g) x h \). Since \( G \) is adjoint the automorphism \( \varphi \) is inner, and it follows that the latter coupling admits a trivialisation.

ME-rigidity in rank 1

The group \( \mathrm{PSL}_2({\mathbb{R}}) \) is clearly nonrigid since there are nontrivial deformations of lattices given by the Teichmüller spaces. The ME-rigidity of other rank 1 Lie groups is an open question.

It is however possible to prove rigidity results for a stricter notion of measure equivalence. We will define this notion in analogy with the QI-case seen in the first lecture. Recall that in this setting a cocycle \( c : \Gamma \times X_0 \to \Lambda \) gives a family of quasi-isometries \( \Gamma \to \Lambda \), the maps \( c(\cdot, x_0 \) for \( x_0 \in X_0 \). In the measurable setting, if a coupling between \( G \) and \( H \) gives rise to a cocycle \( c : G \times X_0 \to H \) we say that the former is \( p \)-integrable or \( L^p \) if for any \( g \in G \) the integral
\[
\int_{X_0} |c(g, x)|_H^p dx
\]
is finite (where \( |\cdot|_H \) is a left-invariant metric on \( H \)). Then there are rigidity results in rank 1 for such couplings due to Bader–Furman–Monod.

Lecture IV

Sketch of proof of Theorem II

Suppose that \( G \) is a ME-rigid group and that \( H \) is measure equivalent to \ (G \). We want to construct a morphism \( \rho : G \to H \) which has a compact kernel and a closed cofinite image.

Let \( G \curvearrowright X \curvearrowleft H \) be a measured coupling between \( G \) and \( H \). Then \( X \times X^{\mathrm{op}} \) is a self-coupling of \( G \), and since \( G \) is ME-rigid there exists a unique trivialising morphism
\[
\bar\Phi : X \times X^{\mathrm{op}} \to G,
\]
which can be lifted to an application \( \Phi : X \times X \to G \) which is \( H \)-invariant and \( G \)-equuivariant. Moreover \( \Phi \) is uniquely determined by these properties. For the rest of the proof we supposed that we have chosen a point \( x_0 \in X \) which is « generic », in a sense to be precised later, and we define a map \( \Psi : X \to G \) by \( \Psi(x) = \Phi(x, x_0) \).

We will make use of the following notion.

Definition: Let \( G_i, i=1, 2, 3 \) be lcsc groups. A tripling between them is a measure presering action of \( G_1 \times G_2 \times G_ 3 \) on a space \( Y \) such that for any \( 1 \le i < j \le 3 \) the restriction of the action to \( G_i \times G_j \) is a coupling between \( G_i \) and \( G_j \).

Taking \( G_i = G \) and \( Y = (X \times X \times X) / H \) (where \( H \) acts by the diagonal action) we get a tripling. For any \( \{i, j, k\} = \{1, 2, 3\} \), by ME-rigidity of \( G \) applied to the coupling between \( G_j \) and \( G_k \) we obtain a map \( F_{j, k} : X^3 \to G \) which is \( H \times G_i \)-invariant and \( G_j \times G_k \)-equivariant (for the action \( G_j \curvearrowright G \curvearrowleft G_k \) at the target), and which is uniquely determined by these properties. Now the application given by
\[
X \times X \times X \overset{\pi_{j, k}}{\to} X \times X \overset{\Phi}{\to} G \]
(where \( \pi_{j, k} \) is the projection onto the product of the \( j\)th with the \( k \)th factors) satisfies the same properties as \( F_{j, k} \) and hence
\[
F_{j, k} = \Phi \circ \pi_{j, k}.
\]
In the same way we obtain
\[
F_{j, k} = \iota \circ F_{k, j} =: F_{j, k}^{-1}
\]
(where \( \iota \) is the self-map \( g \mapsto g^{-1} \) of \( G \)) and also
\[
F_{j, k} = F_{j, i} \cdot F_{i, k}
\]
(where \( \cdot \) denotes multiplication in \( G \)). It follows that, for generic \( x_1, x_2, x_3 \in X \), we have:
\[
\begin{array}{rl}
\Phi(x_1, x_2) &= F_{1,2}(x_1, x_2, x_3) \\
&= F_{1,3}(x_1, x_2, x_3) F_{3,2}(x_1, x_2, x_3) \\
&= F_{1,3}(x_1, x_2, x_3) F_{2,3}(x_1, x_2, x_3)^{-1} \\
&= \Phi(x_1, x_3) \Phi(x_2, x_3)^{-1}.
\end{array}
\]
We suppose that \( x_0 \) has been chosen so that the equation above holds for \( x_3 = x_0 \) and we get that:
\[
\Phi(x_1, x_2) = \Psi(x_1) \Psi(x_2)^{-1}.
\]
As \( \Phi \) is \( H \)-invariant it follows that for any \( h \in H \) we have:
\[
\Psi(x_1 \cdot h) \Psi(x_2 \cdot h)^{-1} = \Psi(x_1) \Psi(x_2)^{-1}
\]
which we rewrite as
\[
\Psi(x_1)^{-1} \Psi(x_1 \cdot h) = \Psi(x_2)^{-1} \Psi(x_2 \cdot h).
\]
In other words the element \( \Psi(x)^{-1}\Psi(x \cdot h) \in G \) does not depend on \( x \in X \) in a generic subset. Thus we can define
\[
\rho(h) = \Psi(x)^{-1}\Psi(x \cdot h)
\]
and it is easy to see that this defines a morphism \( H \to G \). It is measurable and hence continuous. It remains to check the compactness of the kernel and properties of the image, which we won’t do here.

A simplicial category

For \( n \ge 1 \) a « \( n \)-upling » between lcsc groups \( G_1, \ldots, G_n \) is defined inductively as follows :

• For \( n= 1 \) it is a probability measure preserving (pmp) action;
• For \( n > 1 \) it is a measure-preserving action of \( G_1 \times \cdots \times G_n \) on a space \( Y \) such that for any \( i = 1, \ldots, n \) the action
  \[
  \prod_{j \not= i} G_j \curvearrowright X / G_i
  \]
  is an \( (n-1) \)-upling.

For \( n = 2 \) this recovers the notion of coupling and for \( n = 3 \) that of a tripling in the proof above. This gives the set of pmp actions of lcsc groups the structure of a simplicial category, the study of which could lead to new developments.

Invariants of measure equivalence

Induction from a measured equivalence

Let \( G \curvearrowright X \curvearrowleft H \) be a measured coupling between \( G \) and \( H \) and \( \rho : H \to \mathcal U(V) \) a unitary representation of \( H \) on a Hilbert space \( V \). Then the induced representation \( I(\rho) \) of \( G \) from \( H \) via the coupling is defined on the space:
\[
I(V) = \{ f : X \to V, f \text{ is \( H \)-equivariant and } \int_{X / H} \|f(x)\|_V^2 dx < +\infty \}
\]
which is a Hilbert space with the norm given by \( \|f\|_{I(V)}^2 = \int_{X / H} \|f(x)\|_V^2 dx \), on which \( G \) acts unitarily by
\[
I(\rho)(g)f(x) = f(g \cdot x).
\]

Recall that a representation of \( G \) on a Hilbert space \( W \) is said to have almost invariant vectors if for any compact subset \( S \) of \( G \) and any \( \varepsilon > 0 \) there exists a \( w \in W \) with \( \|w\|_W = 1 \) and
\[
\forall s \in S : \| sw – w \|_W \le \varepsilon.
\]
The following lemma is left as an exercise.

Lemma: The induced representation \( I(\rho) \) has almost invariant vectors if and only if \( \rho \) itself does.

Amenability

Recall that a characterisation of amenability for lcsc groups is as follows: \( G \) is amenable if and only if its left- or right-regular representation on \( L^2(G) \) admits almost-invariant vectors.

Let \( G, H \) be ME-equivalent and \( V = L^2(H) \). Then as a \( G \)-representation we have \( I(V) \cong L^2(X) \). Now the measurable decomposition \( X = G \times X_0 \) gives an isomorphism of \( G \)-representations
\[
L^2(X) \cong L^2(G) \otimes W
\]
where \( W \) is a separable Hilbert space on which \( G \) acts trivially.

Now if \( H \) is amenable, \( L^2(H) \) admits almost invariant and hence so does \( L^2(G) \times W \). But this is easily seen to mean that \( L^2(G) \) itself admits almost invariant vectors, and so \( G \) is amenable as well.

A compilation of invariants

Beyond amenability, the lemma about almost invariant vectors allows to prove that the following properties are ME-invariant:

1. Kazhdan’s property (T);
2. a-T-menability or Haagerup’s property

In addition a theorem of Gaboriau gives a « proportionality principle » for \( L^2 \)-Betti numbers of ME-equivalent groups. A corollary which is easier to state is the following (see Marc Bourdon’s lectures for the definition of the reduced \( \ell_2 \)-cohomology groups \( \overline H^k(\cdot, \ell_2\cdot) \)).

Theorem: If \( \Gamma, \Lambda \) are two discrete groups which are measure equivalent and \( \overline H^k(\Gamma, \ell_2\Gamma) = 0 \) for some \( k \ge 0 \) then also \( \overline H^k(\Lambda, \ell_2\Lambda) = 0 \).

Measure equivalence and quasi-isometry

At first sight it appears that measure equivalence is a weaker form of equivalence than quasi-isometry. For example two lattices in a lcsc group are always ME-equivalent but if one is uniform and the other not they cannot be QI to each other. (And, as seen in Kapovich’s lectures, Schwartz’s theorem shows that the QI-classification of nonuniform lattices in rank 1 Lie groups is quite fine.)

For general groups there is no hierarchical relation between the two notions. In fact there exists examples of two groups quasi-isometric to each other such that one has property (T) and not the other. The construction is subtle and goes as follows. Let \( \Gamma \) be a uniform lattice in the Lie group \( \mathrm{Sp}(n, 1) \). Then \( \Gamma \) has property (T) since \( \mathrm{Sp}(n, 1) \) does. The cohomology space \( H^2(\Gamma, {\mathbb{Q}}) \) can be nonzero and in this case there exists a nontrivial central \( {\mathbb{Z}} \)-extension \( \Gamma_c \) of \( \Gamma \) corresponding to an infinite-order class \( c \in H^2(\Gamma, {\mathbb{Z}}) \). For a well-chosen \( c \) the groups \( \Gamma_c \) and \( \Gamma \times {\mathbb{Z}} \) are quasi-isometric to each other. On the other hand the first has property (T) but not the second. In particular these two groups cannot be measure equivalent.

Lecture V

We have seen the amenability is ME-invariant. It is easy to prove that a lcsc group is measure equivalent to the trivial group if and only if it is compact. The following result shows that there is only one ME-class of non-compact lcsc amenable groups.

Theorem (Dye, Ornstein–Weiss, Connes–Feldman–Weiss, Furman): A lcsc group \( G \) is ME-equivalent to \( {\mathbb{Z}} \) if and only if it is amenable and non-compact.

Since \( {\mathbb{Z}} \) is amenable a group ME to it will be so as well. We will partly explain how to prove the « only if » statement, which requires the following notion.

Definition: Two groups \( G, H \) are orbit equivalent is there exists a finite measure space \( \Omega \) and pmp actions of \( G \) and \( H \) on \( \Omega \) which have the same measured equivalence relation, that is \( H \cdot x = G \cdot x \) for almost all \( x \in \Omega \).

If we have such an \( \Omega \) then we get « rearrangement cocycles » \( c : G \times \Omega \to H \) and \( c’ : H \times \Omega \to G \). In particular \( G \) and \( H \) are ME-equivalent. Thus the following theorem, due to Orntein–Weiss for discrete groups and Connes–Feldman–Weiss in general, implies the theorem above.

Theorem: Let \( G \) be an amenable non-compact lcsc group and \( G \curvearrowright \Omega \) an essentially free pmp action. Then if \( G \) is countable there exists a pmp action of \( {\mathbb{Z}} \) on \( \Omega \) with the same orbits as that of \( G \). If \( G \) is uncountable then the same statement holds with \( {\mathbb{Z}} \) replaced by \( {\mathbb{R}} \).

For a proof in the case where \( G \) is countable see this previous post.