Posted by: matheuscmss | October 29, 2017

## Serge Cantat’s Bourbaki seminar talk 2017

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 ${M}$ (say, the ${3}$-dimensional sphere), we would like to describe the geometrical and algebraic properties of groups ${\Gamma}$ of finite type acting faithfully on ${M}$; conversely, given our favorite group ${G}$ of finite type, we want to know the class of compact manifolds ${M}$ on which ${G}$ acts faithfully; in this context, Zimmer’s program proposes some answers to these problems when ${\Gamma}$ is a lattice in a Lie group.

More precisely, let ${G}$ be a connected Lie group with finite center whose Lie algebra ${\mathfrak{g}}$ is semi-simple, and let ${A}$ be a connected maximal split torus of ${G}$. The dimension of ${A}$, or equivalently, the dimension of the Lie algebra ${\mathfrak{a}}$ of ${A}$, is the so-called real rank of ${G}$, and it is denoted by ${rg(G)}$. Let ${\Gamma}$ be a lattice of ${G}$, i.e., a discrete subgroup such that the quotient ${G/\Gamma}$ has finite Haar measure.

For the sake of concreteness, today we will deal exclusively with the prototypical case of ${G=SL_{k+1}(\mathbb{R})}$ and ${A\simeq (\mathbb{R}_+)^k}$ is subgroup of diagonal matrices in ${G}$ with positive entries.

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

Theorem 1 Let ${M}$ be a connected compact manifold. Suppose that the lattice ${\Gamma}$ of ${G=SL_{k+1}(\mathbb{R})}$ is uniform (i.e., ${G/\Gamma}$ is compact).If there exists a homomorphism ${\alpha:\Gamma\rightarrow\textrm{Diff}(M)}$ with infinite image, then

$\displaystyle \textrm{dim}(M)\geq \textrm{rg}(G)$

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 1 The assumption of compactness of ${M}$ 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 2 The hypothesis of uniformity of ${\Gamma}$ 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 ${\Gamma=SL_{k+1}(\mathbb{Z})}$.

Remark 3 The conclusion is optimal: ${G=SL_{k+1}(\mathbb{R})}$ (and, a fortiori, any lattice ${\Gamma}$ of ${G}$) acts on the real projective space ${M=\mathbb{P}^k(\mathbb{R})}$ by projective linear transformations. On the other hand, if one changes ${G}$, then the inequality ${\textrm{dim}(M)\geq \textrm{rg}(G)}$ can be improved: for example, Brown–Fisher–Hurtado proves that ${\textrm{dim}(M)\geq 2n-1}$ when ${G}$ is the symplectic group ${Sp(2n,\mathbb{R})}$ of real rank ${n}$.

Remark 4 Concerning the regularity of the elements of ${\textrm{Diff}(M)}$, even though one expects similar statements for actions by homeomorphisms, Brown–Fisher–Hurtado deals only with ${C^2}$-diffeomorphisms because they need to employ the so-called Pesin theory of non-uniform hyperbolicity.Nevertheless, we shall assume that ${\textrm{Diff}(M)=\textrm{Diff}^{\infty}(M)}$ in the sequel for a technical reason explained later.

Remark 5 This theorem is obvious when ${\textrm{rg}(G)=1}$: indeed, a compact manifold ${M}$ whose group of diffeomorphisms is infinite has dimension ${\textrm{dim}(M)\geq 1}$.Hence, we can (and do) assume without loss of generality that ${\textrm{rg}(G)\geq 2}$ in what follows.

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

2. Preliminaries

Fix ${h}$ a Riemannian metric on ${M}$. Denote by ${S}$ a finite set of generators of ${\Gamma}$. For the sake of convenience, we suppose that ${S}$ contains the identity element ${e_{\Gamma}}$ and ${S}$ is symmetric (i.e., ${g\in S}$ if and only if ${g^{-1}\in S}$).

The length of a word ${\gamma}$ on the letters of the alphabet ${S}$ is denoted by ${\ell_S(\gamma)}$: in other words, ${\ell_S(\gamma)}$ is the distance between ${\gamma}$ and ${e_{\Gamma}}$ in the Cayley graph of ${(\Gamma, S)}$.

The ball of radius ${n}$ is

$\displaystyle \textrm{Ball}_S(n):=\{\gamma| \ell_S(\gamma)\leq n\} = \{s_1\dots s_n| s_i\in S\,\forall\, 1\leq i\leq n\}:=\underbrace{S\cdots S}_{n \textrm{ times}}$

We say that an action ${\alpha:\Gamma\rightarrow \textrm{Diff}(M)}$ is feeble whenever if ${\forall \varepsilon>0}$, there exists ${C_{\varepsilon}>0}$ such that

$\displaystyle \|D\alpha(\gamma)\|_M\leq C_{\varepsilon} \exp(\varepsilon \ell_S(\gamma))$

for all ${\gamma\in\Gamma}$. (Here, ${D}$ stands for the derivative, and the norm is measured with respect to the Riemannian metric ${h}$ fixed above.) Also, we say that an action ${\alpha:\Gamma\rightarrow \textrm{Diff}(M)}$ is vigorous if it is not feeble.

The proof of Theorem 1 is naturally divided into two regimes depending on whether ${\alpha:\Gamma\rightarrow \textrm{Diff}(M)}$ is feeble or vigorous.

3. Feeble actions

Our goal in this section is to show that if ${\alpha:\Gamma\rightarrow \textrm{Diff}(M)}$ is feeble, then there exists a ${\alpha(\Gamma)}$-invariant Riemannian metric on ${M}$.

Before proving this claim, let us see why it allows to establish Theorem 1 for feeble actions. The claim implies that ${\alpha(\Gamma)}$ is a subgroup of isometries of ${M}$. Since ${M}$ 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 ${\textrm{dim}(M)\geq \textrm{rg}(G)}$.

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 ${m\in\mathbb{N}}$ and ${\varepsilon>0}$, there exists ${C_{\varepsilon,m}>0}$ such that

$\displaystyle \|D^{(m)}\alpha(\gamma)\|_M \leq C_{\varepsilon, m} \exp(\varepsilon \ell_S(\gamma))$

for all ${\gamma\in\Gamma}$.

The second ingredient is provided by the so-called reinforced property (T) of Lafforgue. In a nutshell, this property says the following. Given ${X}$ a Hilbert space, denote by ${G(X)}$ the group of continuous linear operators. Let ${\tau>0}$ be a parameter. We say that a representation ${\rho:\Gamma\rightarrow G(X)}$ is ${\tau}$moderate if there exists ${c_{\tau}>0}$ such that

$\displaystyle \|\rho(\gamma)\|_X\leq C_{\tau} \exp(\tau \ell_S(\gamma))$

for all ${\gamma\in\Gamma}$. Given such a representation ${\rho}$, we denote by ${X^{\Gamma}}$ the set of ${\rho(\Gamma)}$-invariant vectors. The next statement described the reinforced property (T).

Theorem 2 (Lafforgue, de Laat–de la Salle) Let ${\Gamma}$ be a uniform lattice of ${G}$ of ${\textrm{rg}(G)\geq 2}$. Then, there exist

• (1) ${\tau, \chi>0}$
• (2) probability measures ${(m_n)_{n\in\mathbb{N}}}$ on ${\Gamma}$ supported on ${\textrm{supp}(m_n)\subset \textrm{Ball}_S(n)}$

such that for all ${\tau}$-moderate ${\rho:\Gamma\rightarrow G(X)}$, one has a projection ${P:X\rightarrow X^{\Gamma}}$ with

$\displaystyle \|P-\sum\limits_{\gamma\in\textrm{Ball}_S(n)} m_n(\gamma)\rho(\gamma)\|\leq \exp(-\chi n)$

Remark 7 de 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 ${Sym^2(T^*M)}$, let us consider the action of ${\Gamma}$ on ${Sym^2(T^*M)}$ induced by ${\alpha}$.

Denote by ${X}$ the Hilbert space of sections of ${Sym^2(T^*M)}$ whose ${m}$ first derivatives are ${L^2}$ (i.e., ${X}$ is a Sobolev space of type ${W^{m,2}}$).

Remark 8 Sobolev embedding theorem implies that an element of ${X}$ is ${C^2}$ when ${3+\frac{\textrm{dim}(M)}{2}.

Observe that the action of ${\Gamma}$ on ${Sym^2(T^*M)}$ gives a representation

$\displaystyle \rho:\Gamma\rightarrow G(X)$

Take ${\varepsilon\ll\tau,\chi}$ a small parameter, where ${\tau}$ and ${\chi}$ are the quantities provided the reinforced property (T). By Theorem 2, we have a projection ${P:X\rightarrow X^{\Gamma}}$ such that

$\displaystyle \|P(h) - \sum \limits_{\gamma\in\textrm{Ball}_S(n)} m_n(\gamma)\rho(\gamma)(h)\|\leq \exp(-\chi n)$

In other words, ${P(h)}$ is a ${\alpha(\Gamma)}$-invariant, ${C^2}$-section of ${Sym^2(T^*M)}$ which is the limit of the Riemannian metrics

$\displaystyle \sum \limits_{\gamma\in\textrm{Ball}_S(n)} m_n(\gamma)\rho(\gamma)(h) := \sum \limits_{\gamma\in\textrm{Ball}_S(n)} m_n(\gamma)\alpha(\gamma)_*h$

In particular, ${P(h)}$ is non-negative definite. At this point, our task is reduced to prove that ${P(h)}$ is a Riemannian metric, i.e., ${P(h)(u)\neq 0}$ for all ${u\neq 0}$. For this sake, note that if ${P(h)(u)=0}$, then the inequality above would give

$\displaystyle \|\sum \limits_{\gamma\in\textrm{Ball}_S(n)} m_n(\gamma)h(D\alpha(\gamma)u)\|\leq \exp(-\chi n) \|u\|$

for all ${n\in\mathbb{N}}$. On the other hand, the action of ${\alpha}$ is feeble of all orders, so that

$\displaystyle \|\sum \limits_{\gamma\in\textrm{Ball}_S(n)} m_n(\gamma)h(D\alpha(\gamma)u)\|\geq C_{\varepsilon,m}^{-1} \exp(-\varepsilon n) \|u\|$

Since ${\varepsilon\ll\chi}$, we get a contradiction unless ${\|u\|=0}$, i.e., ${u=0}$.

This completes the proof of Theorem 1 for feeble actions.

Remark 9 We used that ${\alpha:\Gamma\rightarrow \textrm{Diff}^{\infty}(M)}$ here: indeed, we took ${m}$ sufficiently large to apply Sobolev embedding theorem in order to obtain a ${C^2}$-smooth object ${P(h)}$ and we exploited the fact that ${\alpha}$ is feeble of order ${m}$ to conclude that ${P(h)}$ is a Riemannian metric.In the case of actions ${\alpha:\Gamma\rightarrow\textrm{Diff}^2(M)}$, one replaces the Hilbert spaces ${X=W^m,2}$ by Banach spaces ${W^{m,p}}$, and one employs the version of the reinforced property (T) for Banach spaces.

4. Vigorous actions

In this section, we assume that ${\alpha:\Gamma\rightarrow\textrm{Diff}(M)}$ 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 ${\alpha:\Gamma\rightarrow\textrm{Diff}(M)}$ with respect to any invariant probability measure vanish when ${\textrm{dim}(M)<\textrm{rg}(G)}$).

Logically, the problem with this strategy is the fact that ${\Gamma}$ 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 ${\Gamma}$ by volume-preserving diffeomorphisms of ${M}$. 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 ${\textrm{dim}(M)\geq \textrm{rg}(G)}$ will follow from Zimmer’s super-rigidity.

4.1. Suspensions

We start by replacing the action of ${\Gamma}$ by a `cousin’ action of ${G}$. More concretely, consider the product space ${G\times M}$. Note that ${\gamma\in\Gamma}$ acts on ${G\times M}$ via

$\displaystyle (g,y)\mapsto (g\gamma^{-1}, \alpha(\gamma)y)$

and ${g'\in G}$ acts on ${G\times M}$ via

$\displaystyle (g,y)\mapsto (g'g,y)$

In particular, ${G}$ acts on the space ${M_{\alpha} = (G\times M)/\Gamma}$ (because the actions of ${G}$ and ${\Gamma}$ commute).

Observe that the action of ${G}$ on ${M_{\alpha}}$ is a suspension of the action of ${G}$ on ${G/\Gamma}$ with respect to the natural projection ${\pi:M_{\alpha}\rightarrow G/\Gamma}$.

We denote by ${T^{\pi} M_{\alpha}}$ is the vertical tangent bundle (i.e., the tangent space to the fibers of ${\pi}$). Let ${D^{\pi} g}$ be the restriction of the derivative of ${g}$ to ${T^{\pi}M_{\alpha}}$. Given ${g\in G}$ and ${\mu}$ a ${g}$-invariant probability measure on ${M_{\alpha}}$, we define

$\displaystyle \lambda_+^{\pi}(g;\mu):=\inf\limits_{n\geq 1}\int\frac{1}{n}\log\|D^{\pi}g^n(x)\|d\mu(x)$

the maximal vertical Lyapunov exponent of ${g}$ with respect to ${\mu}$.

Remark 10 For each fixed ${g\in G}$ and ${n\in\mathbb{N}}$, the quantity ${\int\frac{1}{n}\log\|D^{\pi}g^n(x)\|d\mu(x)}$ is a continuous function of ${\mu}$. Therefore, for each fixed ${g\in G}$, the Lyapunov exponent ${\lambda_+^{\pi}(g;\mu)}$ is a upper semi-continuous function of ${\mu}$.

Our goal is to exhibit a probability measure ${\mu}$ on ${M_{\alpha}}$ which is ${G}$-invariant and possessing a positive Lyapunov exponent, i.e.,

$\displaystyle \lambda_+^{\pi}(g,\mu)>0$

for some ${g\in G}$.

4.2. ${A}$-invariant measures

The first step towards our goal consists in constructing a probability measure ${\mu}$ on ${M_{\alpha}}$ which is ${A}$-invariant, ${A}$-ergodic and possessing a positive Lyapunov exponent in the sense that

$\displaystyle \lambda_+^{\pi}(b,\mu)>0$

for some ${b\in A}$.

For this sake, we recall that a vigorous action ${\alpha:\Gamma\rightarrow\textrm{Diff}(M)}$ has the property that for some ${\varepsilon_0>0}$ and for a sequence ${\gamma_n\in\Gamma}$, ${\gamma_n\rightarrow\infty}$, one has

$\displaystyle \|D^{\pi}\gamma_n\|_{M_{\alpha}}\geq \exp(\varepsilon_0 \ell_S(\gamma_n))$

By Cartan’s decomposition ${G=KAK}$, where ${K}$ is a maximal compact subgroup. Thus, we can write ${\gamma_n = k_n a_n k_n'}$. By compactness, ${\|Dk\|_{M_{\alpha}}}$ is uniformly bounded for all ${k\in K}$, so that

$\displaystyle \|D^{\pi} a_n\|_{M_{\alpha}}\geq \exp(\frac{\varepsilon_0}{2} |u_n|)$

where ${a_n=\exp(u_n)}$, ${u_n\in\mathfrak{a}}$. (Here we used that ${\ell_S(\gamma_n)}$ and the size of ${|u_n|}$ are comparable [because ${G/\Gamma}$ is compact].)

By extracting a subsequence of ${u_n\in\mathfrak{a}\simeq\mathbb{R}^k}$, we can assume that ${u_n}$ goes to infinity in a fixed direction with a fixed speed. In particular, this allows us to replace ${a_n}$ by the iterates ${b^n}$ of a single element ${b\in A}$, i.e., we found an element ${b}$ of ${A}$ with

$\displaystyle \|D^{\pi}b^n\|_{M_{\alpha}}\geq \exp(\frac{\varepsilon_0}{2} n)$

If we take ${x_n\in M_\alpha}$ such that ${\|D^{\pi} b^n(x_n)\| = \|D^{\pi} b^n\|}$, then it is not hard to see that the sequence of probability measures

$\displaystyle \frac{1}{n}\sum\limits_{j=0}^{n-1}\delta_{b^j(x_n)}$

accumulate into some ${b}$-invariant probability measure ${\nu}$ with

$\displaystyle \lambda_+^{\pi}(b;\mu)>0$

Since ${A}$ is amenable and ${A}$ commutes with ${b}$, we can replace ${\nu}$ by an ${A}$-invariant probability measure ${\mu}$ (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 ${\mu}$ is ${A}$-ergodic.

4.3. Ratner theory and higher rank groups

We affirm that the probability measure ${\mu}$ constructed above can be chosen so that its projection to ${G/\Gamma}$ is

$\displaystyle \pi_*(\mu) = \textrm{ Haar }$

Indeed, the assumption that ${\textrm{rg}(G)\geq 2}$ implies that ${G}$ contains unipotent subgroups commuting with ${b}$. Since an unipotent subgroup is amenable, we can repeat the argument of the previous subsection (with ${A}$ replaced by such unipotents) to get an additional invariance, i.e., we can assume that ${\mu}$ is invariant under ${A}$ and some unipotent subgroup. At this point, Ratner’s theory permits to control the projection ${\pi_*(\mu)}$ and, in particular, to assert that ${\pi_*(\mu)}$ is the Haar measure.

4.4. Entropy argument

Let us now show that ${\mu}$ described above is ${G}$-invariant on ${M_{\alpha}}$ if ${\textrm{dim}(M) < \textrm{rg}(G)}$. Note that this completes the proof of Theorem 1 for vigorous actions because the positivity of the Lyapunov exponent ${\lambda_+^{\pi}(b;\mu)}$ contradicts Zimmer’s super-rigidity theorem unless ${\textrm{dim}(M) \geq \textrm{rg}(G)}$.

The vertical Lyapunov exponents of the elements of ${A}$ with respect to ${\mu}$ define linear (Lyapunov) forms

$\displaystyle \lambda_i^{\pi}:\mathfrak{a}\rightarrow\mathbb{R}$

The total number of linear forms (counting multiplicities) is ${\textrm{dim}(M)}$. Since we are assuming that ${\textrm{dim}(M)<\textrm{rg}(G)}$, there exists ${u\in\mathfrak{a}\setminus\{0\}}$ with

$\displaystyle \lambda_i^{\pi}(u)=0$

for all ${1\leq i\leq \textrm{dim}(M)}$.

Let ${c:=\exp(u)\in A}$. Recall that the Lyapunov forms associated to the action of ${A}$ on ${G/\Gamma}$ are the trivial form ${0}$ and the roots of ${A}$. Thus, Pesin entropy formula says that the entropy of the action of ${c}$ on ${G/\Gamma}$ coincides with the sum of positive Lyapunov exponent:

$\displaystyle h(c;\textrm{ Haar on }G/\Gamma) = \sum\limits_{\substack{L \textrm{ root}, \\ L(u)>0}} \textrm{dim}(\mathfrak{g}^L) L(u)$

(where ${\mathfrak{g}^L}$ is the root space of ${L}$).

On the other hand, Margulis–Ruelle inequality says that the entropy of action of ${c}$ on ${M_{\alpha}}$ is bounded by the sum of positive Lyapunov exponents. Since the vertical Lyapunov exponents of ${c}$ vanish, we conclude that

$\displaystyle h(c; \mu \textrm{ on } M_{\alpha}) \leq \sum\limits_{\substack{L \textrm{ root}, \\ L(u)>0}} \textrm{dim}(\mathfrak{g}^L) L(u) = h(c;\textrm{ Haar on }G/\Gamma)$

Because ${\mu}$ projects onto ${\pi_*(\mu) = \textrm{Haar}}$, we also have

$\displaystyle h(c;\textrm{ Haar on }G/\Gamma)\leq h(c; \mu \textrm{ on } M_{\alpha})$

In summary, we obtain that Margulis–Ruelle inequality is actually an equality. By the invariance theorem of Ledrappier–Young, we derive that ${\mu}$ is invariant by ${G^L:=\exp(\mathfrak{g}^L)}$ for all roots ${L}$ with ${L(u)>0}$.

By reversing the time (i.e., replacing ${c}$ by ${c^{-1}}$) in the previous argument, we also obtain that ${\mu}$ is invariant by ${G^L:=\exp(\mathfrak{g}^L)}$ for all roots ${L}$ with ${L(u)<0}$.

Since ${G}$ is the smallest group containing all ${G^L}$ with ${L(u)\neq 0}$ (as ${G}$ is a simple Lie group), we conclude the desired ${G}$-invariance of ${\mu}$.

This ends the proof of Theorem 1.