About one week ago, Serge Cantat gave a beautiful talk in Bourbaki seminar about the recent works of Brown–Fisher–Hurtado on Zimmer’s program. The video of this talk and the corresponding lecture notes are available here and here.

In this post, I will transcript my notes of this talk: as usual, all errors/mistakes are my sole responsibility.

**1. Introduction**

General philosophy behind Zimmer’s program: given a compact manifold (say, the -dimensional sphere), we would like to describe the geometrical and algebraic properties of groups of finite type acting faithfully on ; conversely, given our favorite group of finite type, we want to know the class of compact manifolds on which acts faithfully; in this context, Zimmer’s program proposes some answers to these problems when is a lattice in a Lie group.

More precisely, let be a connected Lie group with finite center whose Lie algebra is semi-simple, and let be a connected maximal split torus of . The dimension of , or equivalently, the dimension of the Lie algebra of , is the so-called *real rank* of , and it is denoted by . Let be a lattice of , i.e., a discrete subgroup such that the quotient has finite Haar measure.

For the sake of concreteness, today we will deal exclusively with the prototypical case of and is subgroup of diagonal matrices in with positive entries.

In this setting, Zimmer’s program offers restrictions on the dimension of compact manifolds admitting non-trivial actions of by smooth diffeomorphisms. In this direction, Aaron Brown, David Fisher and Sebastian Hurtado showed here that

Theorem 1Let be a connected compact manifold. Suppose that the lattice of is uniform (i.e., is compact).If there exists a homomorphism with infinite image, then

As we are going to see below, the proof of this theorem is a beautiful blend of ideas from geometric group theory and dynamical systems.

Before describing the arguments of Brown–Fisher–Hurtado, let us make a few comments of the statement of their theorem.

Remark 1The assumption of compactness of is important: indeed, any countable group acts faithfully by biholomorphisms of a connected non-compact Riemann surface (see the footnote 1 of Cantat’s text for a short proof of this fact).

Remark 2The hypothesis of uniformity of is technical: there is some hope to treat non-uniform lattices and, in fact, Brown–Fisher–Hurtado managed to recently extend their result to the case of .

Remark 3The conclusion is optimal: (and, a fortiori, any lattice of ) acts on the real projective space by projective linear transformations. On the other hand, if one changes , then the inequality can be improved: for example, Brown–Fisher–Hurtado proves that when is the symplectic group of real rank .

Remark 4Concerning the regularity of the elements of , even though one expects similar statements for actions by homeomorphisms, Brown–Fisher–Hurtado deals only with -diffeomorphisms because they need to employ the so-called Pesin theory of non-uniform hyperbolicity.Nevertheless, we shall assume that in the sequel for a technical reason explained later.

Remark 5This theorem is obvious when : indeed, a compact manifold whose group of diffeomorphisms is infinite has dimension .Hence, we can (and do) assume without loss of generality that in what follows.

Remark 6The statement of Brown–Fisher–Hurtado theorem is comparable to Margulis super-rigidity theorem providing a control on the dimension of linear representations of .

**2. Preliminaries**

Fix a Riemannian metric on . Denote by a finite set of generators of . For the sake of convenience, we suppose that contains the identity element and is symmetric (i.e., if and only if ).

The length of a word on the letters of the alphabet is denoted by : in other words, is the distance between and in the Cayley graph of .

The ball of radius is

We say that an action is *feeble* whenever if , there exists such that

for all . (Here, stands for the derivative, and the norm is measured with respect to the Riemannian metric fixed above.) Also, we say that an action is *vigorous* if it is not feeble.

The proof of Theorem 1 is naturally divided into two regimes depending on whether is feeble or vigorous.

**3. Feeble actions**

Our goal in this section is to show that if is *feeble*, then there exists a -invariant Riemannian metric on .

Before proving this claim, let us see why it allows to establish Theorem 1 for feeble actions. The claim implies that is a subgroup of isometries of . Since is compact, Myers–Steenrod theorem says that its group of isometries is a compact Lie group. This permits to apply Margulis super-rigidity theorem and the classical theory of compact Lie groups to get the desired inequality .

Let us now prove the claim.

The first ingredient is a lemma of Fisher–Margulis ensuring that a feeble action is “feeble to all orders”, i.e., for all and , there exists such that

for all .

The second ingredient is provided by the so-called *reinforced property (T)* of Lafforgue. In a nutshell, this property says the following. Given a Hilbert space, denote by the group of continuous linear operators. Let be a parameter. We say that a representation is –*moderate* if there exists such that

for all . Given such a representation , we denote by the set of -invariant vectors. The next statement described the reinforced property (T).

Theorem 2 (Lafforgue, de Laat–de la Salle)Let be a uniform lattice of of . Then, there exist

- (1)
- (2) probability measures on supported on

such that for all -moderate , one has a projection with

Remark 7de la Salle is currently working on extending this result to non-uniform lattices.

We want to explore this theorem to produce the desired invariant Riemannian metric in the claim.

Since any Riemannian metric is a section of , let us consider the action of on induced by .

Denote by the Hilbert space of sections of whose first derivatives are (i.e., is a Sobolev space of type ).

Remark 8Sobolev embedding theorem implies that an element of is when .

Observe that the action of on gives a representation

Take a small parameter, where and are the quantities provided the reinforced property (T). By Theorem 2, we have a projection such that

In other words, is a -invariant, -section of which is the limit of the Riemannian metrics

In particular, is non-negative definite. At this point, our task is reduced to prove that is a Riemannian metric, i.e., for all . For this sake, note that if , then the inequality above would give

for all . On the other hand, the action of is feeble of all orders, so that

Since , we get a contradiction unless , i.e., .

This completes the proof of Theorem 1 for feeble actions.

Remark 9We used that here: indeed, we took sufficiently large to apply Sobolev embedding theorem in order to obtain a -smooth object and we exploited the fact that is feeble of order to conclude that is a Riemannian metric.In the case of actions , one replaces the Hilbert spaces by Banach spaces , and one employs the version of the reinforced property (T) for Banach spaces.

**4. Vigorous actions**

In this section, we assume that is vigorous.

Roughly speaking, we are going to treat the case of vigorous actions by exploring the tension between the vigour of the action (creating non-zero Lyapunov exponents) and Zimmer’s super-rigidity theorem for cocycles (saying that the Lyapunov exponents of the action with respect to any invariant probability measure vanish when ).

Logically, the problem with this strategy is the fact that is not amenable, so that the existence of invariant probability measures (required by Zimmer’s super-rigidity theorem) is far from being automatic. In particular, this partly explains why the first versions of Zimmer’s program dealt exclusively with actions of by *volume-preserving* diffeomorphisms of . Also, even if we disposed of invariant probability measures, their supports could be very “thin”, so that their generic points would not “feel” the vigour of the action (and hence no contradiction could be derived).

Anyhow, we will discuss how to overcome the difficulties in the previous paragraph: we shall use the vigour of the action in order to construct an invariant probability measure with some positive Lyapunov exponent, so that the desired conclusion will follow from Zimmer’s super-rigidity.

**4.1. Suspensions**

We start by replacing the action of by a `cousin’ action of . More concretely, consider the product space . Note that acts on via

and acts on via

In particular, acts on the space (because the actions of and commute).

Observe that the action of on is a suspension of the action of on with respect to the natural projection .

We denote by is the vertical tangent bundle (i.e., the tangent space to the fibers of ). Let be the restriction of the derivative of to . Given and a -invariant probability measure on , we define

the maximal vertical Lyapunov exponent of with respect to .

Remark 10For each fixed and , the quantity is a continuous function of . Therefore, for each fixed , the Lyapunov exponent is a upper semi-continuous function of .

Our goal is to exhibit a probability measure on which is -invariant and possessing a positive Lyapunov exponent, i.e.,

for some .

**4.2. -invariant measures**

The first step towards our goal consists in constructing a probability measure on which is -invariant, -ergodic and possessing a positive Lyapunov exponent in the sense that

for some .

For this sake, we recall that a vigorous action has the property that for some and for a sequence , , one has

By Cartan’s decomposition , where is a maximal compact subgroup. Thus, we can write . By compactness, is uniformly bounded for all , so that

where , . (Here we used that and the size of are comparable [because is compact].)

By extracting a subsequence of , we can assume that goes to infinity in a fixed direction with a fixed speed. In particular, this allows us to replace by the iterates of a single element , i.e., we found an element of with

If we take such that , then it is not hard to see that the sequence of probability measures

accumulate into some -invariant probability measure with

Since is amenable and commutes with , we can replace by an -invariant probability measure (by taking averages along F{\o}lner sequences) whose Lyapunov exponent is positive (thanks to the upper semi-continuity property mentioned in Remark 10). Finally, by taking an appropriate ergodic component, we can also assume that is -ergodic.

**4.3. Ratner theory and higher rank groups**

We affirm that the probability measure constructed above can be chosen so that its projection to is

Indeed, the assumption that implies that contains unipotent subgroups commuting with . Since an unipotent subgroup is amenable, we can repeat the argument of the previous subsection (with replaced by such unipotents) to get an *additional* invariance, i.e., we can assume that is invariant under *and* some unipotent subgroup. At this point, Ratner’s theory permits to control the projection and, in particular, to assert that is the Haar measure.

**4.4. Entropy argument**

Let us now show that described above is -invariant on *if* . Note that this completes the proof of Theorem 1 for vigorous actions because the positivity of the Lyapunov exponent contradicts Zimmer’s super-rigidity theorem *unless* .

The vertical Lyapunov exponents of the elements of with respect to define *linear (Lyapunov) forms*

The total number of linear forms (counting multiplicities) is . Since we are assuming that , there exists with

for all .

Let . Recall that the Lyapunov forms associated to the action of on are the trivial form and the roots of . Thus, Pesin entropy formula says that the entropy of the action of on *coincides* with the sum of positive Lyapunov exponent:

(where is the root space of ).

On the other hand, Margulis–Ruelle inequality says that the entropy of action of on is *bounded* by the sum of positive Lyapunov exponents. Since the vertical Lyapunov exponents of vanish, we conclude that

Because projects onto , we also have

In summary, we obtain that Margulis–Ruelle inequality is actually an *equality*. By the invariance theorem of Ledrappier–Young, we derive that is invariant by for all roots with .

By reversing the time (i.e., replacing by ) in the previous argument, we also obtain that is invariant by for all roots with .

Since is the smallest group containing all with (as is a simple Lie group), we conclude the desired -invariance of .

This ends the proof of Theorem 1.

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