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 \( [\mathfrak S(k_n) : \Gamma_n] \sim c\cdot a_n \).