\( \def \rtimes{|\mspace{-10mu} \times \mspace{3mu}} \) \( \def \curvearrowright{\mspace{3mu} \cap \mspace{-10mu} \downarrow} \)

• 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.