Posted by: matheuscmss | January 23, 2013

## First Bourbaki seminar of 2013

Last Saturday (January 19, 2013), the first Bourbaki seminar of this year took place at amphithéâtre Hermite of Institut Henri Poincaré (as usual), and the following topics were discussed:

The speakers did a great job in explaining these topics in a language that was accessible to the audience, and, for this reason, I decided to make a post about one of these talk.

Of course, I used my personal taste as a guide for my choice, and, for this reason, I spent some time hesitating between the talks of F. Charles and Y. de Cornulier.

In fact, the talk of F. Charles pointed in a direction that I learned recently to admire, namely, variations of Hodge structures, (intermediate) Jacobians, Hodge classes and Hodge conjecture, but, in the end, I’ve chosen to post about Y. de Cornulier’s talk by a simple reason: in the final part of his talk, F. Charles needed some material from Algebraic Geometry that I’m not comfortable with (such as perverse sheaves, intersection complexes, etc.), while Y. de Cornulier’s talk was entirely accessible for someone with a mild Dynamical Systems background (like me).

So, below I will transcript my notes of Y. de Cornulier’s talk about topologically full groups (after the works of Matui and Juschenko-Monod). As usual, the eventual mistakes in what follows are my entire responsibility.

1. Amenability

Definition 1 (J. von Neumann (1929)) A group ${G}$ is amenable if it admits a mean ${m}$ (i.e., a finitely additive function ${m}$ from the class of subsets ${\mathbb{P}(G)}$ taking values in the interval ${[0,1]}$ such that ${m(G)=1}$) that is invariant under left translations by elements of ${g\in G}$ (i.e., ${m(g.A)=m(A)}$ for all ${A\subset G}$ and ${g\in G}$).

After the works of M. Day, E. Fölner and H. Kesten (among others), we know how to characterize amenability in several equivalent ways. For example:

• Reasonable actions on convex compact subsets have fixed points: ${G}$ is amenable if and only if any continuous affine action of ${G}$ on a convex compact subset of a locally compact space has a fixed point.
• Existence of Fölner “sequences”: ${G}$ is amenable if and only if for all finite subset ${S\subset G}$ and all ${\varepsilon>0}$, there exists a finite non-empty subset ${\emptyset\neq F\subset G}$ that is ${\varepsilon}$-almost invariant under left-translation by elements of ${S}$, i.e.,

$\displaystyle \frac{|S\cdot F-F|}{|F|}<\varepsilon$

• Random walks have “small” “large” probability to come back to the origin: given ${\mu}$ a probability measure on ${G}$ supported on a finite symmetric subset of ${G}$, the probability that a random walk with law ${\mu}$ comes backs to the identity element ${1\in G}$ after ${n}$ steps decays sub-exponentially fast as ${n\rightarrow\infty}$.

The most basic examples of amenable groups are finite groups: indeed, if ${G}$ is a finite group, then ${\frac{1}{|G|}\sum\limits_{g\in G}\delta_g}$ is a invariant mean. Also, ${\mathbb{Z}}$ is amenable, and, more generally, Abelian groups are amenable.

From these basic examples, one can construct more examples of amenable groups by noticing that amenability is a property that is stable under extensions and direct limits, that is:

• Stability of amenability under extensions: if ${N}$ and ${G/N}$ are amenable groups, then ${G}$ is an amenable group.
• Stability of amenability under direct limits: if ${G=\bigcup\limits_{i\in I} G_i}$ and, for all ${i}$, ${G_i}$ is amenable, then ${G}$ is amenable.

For more details on these properties (in the case of countable groups), the reader might want to consult Terence Tao’s notes on this subject.

In any case, the smallest class of groups that is stable under extensions and direct limits containing all finite and Abelian groups is called the class of elementary amenable groups. For example, it is possible to check that virtually solvable groups are elementary amenable.

In 1957, M. Day asked whether there are non-elementary amenable groups ${G}$.

In 1984, R. Grigorchuk constructed a group with sub-exponential non-polynomial growth. Since the subexponential growth property implies amenability and all elementary amenable groups have polynomial growth, it follows that Grigorchuk’s group solves positively M. Day’s question.

In 2005, L. Bartholdi and B. Virág showed the amenability of other non-elementary examples of groups (constructed by R. Grigorchuk and A. Zuk) whose growth is not subexponential based on considerations about the probability of return of random walks.

A common feature of these amenable non-elementary groups is that they have “plenty” of their quotients are finite groups because their “self-similar” nature (coming from the fact that they act on trees or automatas).

In particular, until recently, it was not known whether there were non-trivial amenable groups of finite type without non-trivial finite quotients (and, thus, a fortiori, simple, and infinite).

In order to appreaciate the difficulty of the question, Y. Cornulier recalled that, if ${G}$ contains a copy of the free group in two generators, then ${G}$ is not amenable (as it was shown by von Neumann: see, e.g., Terence Tao’s notes on amenability for more details).

The main goal of Y. Cornulier’s Bourbaki seminar talk was the presentation of the results of H. Matui, and K. Juschenko and N. Monod leading to the construction of infinite simple amenable groups of finite type. As we’ll see by the end of this post, the construction of the desired groups will follow from dynamical considerations.

Anyway, as Y. Cornulier explained to us, the first step towards the construction of such groups is the following criterion of Juschenko-Monod:

Theorem 2 (Juschenko-Monod criterion) Let ${BD(\mathbb{Z})}$ be bounded displacement group, i.e., the group of permutations ${\sigma:\mathbb{Z}\rightarrow \mathbb{Z}}$ such that ${\|\sigma-\textrm{id}\|_{\infty}<\infty}$. Let ${G\rightarrow BD(\mathbb{Z})}$ be an action of ${G}$ on ${\mathbb{Z}}$ via permutations of bounded displacement such that ${\textrm{Stab}_{G}(\mathbb{N})}$ is amenable. Then, ${G}$ is amenable.

This criterion seems strange at first sight: we assume amenability of the group ${\textrm{Stab}_{G}(\mathbb{N})}$ in order to conclude the amenability of ${G}$. However, we will see that this criterion is flexible enough for our purposes.

For now, let us give a sketch of proof of this criterion. We will need the following lemma:

Lemma 3 The group ${BD(\mathbb{Z})\times\{0,1\}^{(\mathbb{Z})}}$ preserves a mean on ${\{0,1\}^{(\mathbb{Z})}}$. Here, ${\{0,1\}^{(\mathbb{Z})}}$ is the set of bi-infinite sequences of ${0}$‘s and ${1}$‘s consisting of finitely many ${1}$‘s, or, equivalently, ${\{0,1\}^{(\mathbb{Z})}}$ is the set of all finite subsets of ${\mathbb{Z}}$.

Proof: We will construct a mean preserved by ${BD(\mathbb{Z})\times\{0,1\}^{(\mathbb{Z})}}$ via a compactness argument. More precisely, let us consider the functions ${f_n:\{0,1\}^{\mathbb{Z}}\rightarrow(0,1]}$ given by

$\displaystyle f_n((x_j)_{j\in\mathbb{Z}}) = \exp\left(-n\sum\limits_{j}x_j e^{-|j|/n}\right)$

Note that, for each ${n}$, the function ${f_n}$ is continuous on ${\{0,1\}^{\mathbb{Z}}}$ and, a fortiori, ${f_n}$ belongs to ${L^2(\{0,1\}^{\mathbb{Z}})}$. Here, we equip ${\{0,1\}^{\mathbb{Z}}}$ with the Bernoulli measure (i.e., we assign weight 1/2 to the symbols ${0}$ and ${1}$, and we consider the product measure).

Let us now replace ${f_n}$ by ${u_n:=f_n/\|f_n\|_2}$ in order to get a sequence of ${L^2}$-functions with unit ${L^2}$-norm. The reader is invited to check that the sequence ${u_n}$ satisfies the following properties:

• 1) ${\|u_n - g u_n\|_2\rightarrow 0}$ as ${n\rightarrow\infty}$ for all ${g\in BD(\mathbb{Z})}$.
• 2) ${\|u_n\cdot \chi_{H_0}\|_2\rightarrow 1}$ as ${n\rightarrow\infty}$, where ${H_0}$ is the hyperplane ${H_0=\{(x_j):x_0=0\}}$.

Using these properties, we take ${\widehat{u}_n\in\ell^2(\{0,1\}^{(\mathbb{Z})})}$ the Fourier transform of ${u_n}$. By Plancherel formula, ${\|\widehat{u}_n\|_2=1}$. In particular, ${\widehat{u}_n^2\in\ell^1(\{0,1\}^{(\mathbb{Z})})\subset \ell^{\infty}(\{0,1\}^{(\mathbb{Z})})^*}$.

By compactness of the unit ball of ${\ell^{\infty}(\{0,1\}^{(\mathbb{Z})})^*}$ in the weak-* topology (Banach-Alaoglu theorem), we can select a subsequence of ${\widehat{u}_n^2}$ accumulating some ${m\in\ell^{\infty}(\{0,1\}^{(\mathbb{Z})})^*}$.

Note that ${m}$ induces a mean on ${\{0,1\}^{(\mathbb{Z})}}$. Furthermore, 1) implies that ${m}$ is ${BD(\mathbb{Z})}$-invariant, while 2) implies that ${m}$ is invariant under translations. It follows that ${m}$ is a ${BD(\mathbb{Z})\times\{0,1\}^{(\mathbb{Z})}}$-invariant mean, and this ends the proof of the lemma. $\Box$

Using this lemma, we can complete the sketch of proof of Juschenko-Monod criterion (Theorem 2): we consider the following twisted action of ${G}$ on ${\{0,1\}^{(\mathbb{Z})}}$

$\displaystyle g\widehat{\cdot} M = g\cdot M + \chi_{\mathbb{N}}-\chi_{g\mathbb{N}}$

In other words, the action ${\widehat{\cdot}}$ is obtained from the usual action by adding a coboundary ${\chi_{\mathbb{N}}-\chi_{g\mathbb{N}}}$.

Note that the stabilizer of ${0}$ for ${\widehat{\cdot}}$ is the stabilizer ${\textrm{Stab}_{G}(\mathbb{N})}$ of ${\mathbb{N}}$ for the usual action of ${G}$, and, by assumption, this subgroup is amenable. More generally, by a similar reasoning, one can show that, from our assumption on ${G}$, the stabilizers of all points of ${\{0,1\}^{(\mathbb{Z})}}$ for ${\widehat{\cdot}}$ are amenable. Furthermore, by the previous lemma, we know that the twisted action ${\widehat{\cdot}}$ of ${G}$ preserves a mean on ${\{0,1\}^{(\mathbb{Z})}}$. At this point, the sketch of proof is complete because it is a known fact that a group ${G}$ whose action on a space ${X}$ preserves a mean and has amenable stabilizers for all points of ${X}$ is amenable.

2. Topologically full groups

Let ${\Gamma}$ be a group acting on a topological space ${X}$ by homeomorphisms.

Definition 4 The topological full group of ${\Gamma}$ is

$\displaystyle [[\Gamma]]:=\{h\in Homeo(X): h \textrm{ locally belongs to } \Gamma\}$

Here, by “locally belongs to ${\Gamma}$” we mean that ${\exists\,\kappa:X\rightarrow\Gamma}$ continuous (i.e., locally constant) such that ${h(x)=\kappa(x).x}$.

This definition was introduced by W. Krieger (in 1980), and T. Giordano, I. Putman and C. Skau (in 1995 and 1999).

For later use, we will need a couple of definitions coming from Dynamical Systems/Ergodic Theory.

Definition 5 The action of ${\Gamma}$ on ${X}$ is minimal if all orbits are dense.

Definition 6 Let ${A}$ be a finite alphabet. The natural action of ${\Gamma}$ on ${A^{\Gamma}}$ (by left-translation of indices) is a shift, and any closed shift-invariant subset of ${A^{\Gamma}}$ is a subshift.

It is not hard to characterize subshifts:

Proposition 7 An action of ${\Gamma}$ on ${X}$ is (equivariantly) isomorphic to a subshift if and only if ${X}$ is compact and there is a finite number of clopen subsets of ${X}$ whose ${\Gamma}$-translates separate points. Equivalenty, the action is isomorphic to a subshift if and only if ${X}$ is totally disconnected and the Boolean algebra ${Clopen(X)}$ (of clopen subsets) has finite type as an algebra equipped with a ${\Gamma}$-action.

The main result of today’s discussion is:

Theorem 8 (Matui and Juschenko-Monod) Let ${X}$ be a Cantor space and let ${\phi\in Homeo(X)}$ be a minimal homeomorphism (i.e., the ${\mathbb{Z}}$-action induced by ${\phi}$ is minimal). Then, the derived group

$\displaystyle [[\phi]]'$

is an infinite simple amenable group. If, moreover, ${(\phi,X)}$ is isomorphic to a subshift, then ${[[\phi]]'}$ has finite type.

The fact that ${[[\phi]]'}$ is simple was shown by Matui and the amenability of ${[[\phi]]'}$ (conjectured by Grigorchuk-Medynets) was shown by Juschenko-Monod. Finally, Matui showed that ${[[\phi]]'}$ has finite type whenever ${(\phi,X)}$ is isomorphic to a subshift.

Before saying a few words on the proof of this theorem, let us give some examples of dynamical systems ${(\phi,X)}$ naturally related to this theorem. Let ${C=\mathbb{R}/\mathbb{Z}}$ be the circle and consider the rotation ${r_{\alpha}:C\rightarrow C}$, ${x\mapsto x+\alpha}$ for ${\alpha\notin\mathbb{Q}}$. The rotation ${r_{\alpha}}$ is minimal, but this example is not “valid” because ${C}$ is not a Cantor space. However, Denjoy’s construction permits to overcome this technical difficulty: concretely, we will produce a Cantor space where ${r_{\alpha}}$ by blowing up the circle along the orbit of a point (say ${0}$); more precisely, let ${C_{\alpha}=(C-\mathbb{Z}\alpha)\cup (\mathbb{Z}\alpha\times\{\pm\})}$; the cyclic order of ${C}$ induces a natural cyclic order on ${C_{\alpha}}$, and, from this, it can be checked that ${C_{\alpha}}$ is a Cantor space; moreover, the action of ${r_{\alpha}}$ on ${C}$ induces a minimal homeomorphism ${\widetilde{r}_{\alpha}}$ on ${C_{\alpha}}$ that is isomorphic to a subshift (because its translates separate points).

Now, let us make a few comments on the proof of theorem.

Firstly, it is not hard to see that ${[[\phi]]'}$ is infinite. Indeed, given ${n\in\mathbb{N}}$, we select ${x\in X}$ and we fix a clopen neighborhood ${U}$ of ${x}$ such that ${\phi^k(U)\cap\phi^j(U)=\emptyset}$ for all ${0\leq k\neq j\leq n}$: note that the existence of ${U}$ follows from the fact that the orbit of ${x}$ is not periodic (as ${\phi}$ is minimal and ${X}$ is a Cantor space [hence infinite]). Using ${U, \dots, \phi^n(U)}$, it is not hard to see that we have an injection from the symmetric group ${Sym(n)}$ into ${[[\phi]]}$. Therefore, the derived group ${[[\phi]]'}$ contains a copy of the alternate group ${Alt(n)}$. Since ${n\in\mathbb{N}}$ is arbitrary, we conclude that ${[[\phi]]'}$ is infinite.

Secondly, it is important to assume that ${\phi}$ is minimal to deduce that ${[[\phi]]'}$ is simple. In fact, if ${\phi}$ is not minimal, we can find a ${\phi}$-invariant subset ${\emptyset \neq Y\neq X}$. In general, it is rare that the natural homomorphism ${[[\Gamma,X]]\rightarrow [[\Gamma,Y]]}$ from the group ${[[\Gamma,X]]}$ of homeomorphism of ${X}$ locally in ${\Gamma}$ to the group ${[[\Gamma,Y]]}$ of homeomorphism of ${Y}$ locally in ${\Gamma}$ is trivial or injective. On the other hand, all homomorphisms of a simple group ${G}$ are trivial or injective. So, this suggests that the minimality of ${\phi}$ has something to do with the simplicity of ${[[\phi]]'}$.

Thirdly, it is important that ${\phi}$ is a subshift to get that ${[[\phi]]'}$ has finite type: otherwise, if ${\phi}$ is not a subshift, then we could write

$\displaystyle Clopen(X)=\bigcup_n A_n$

where ${A_n}$ are certain Boolean algebras; then, in general, we have that ${[[\phi]]_{A_n}\subset[[\phi]]}$ (where ${[[\phi]]_{A_n}}$ consists of homeomorphisms locally coinciding with powers of ${\phi}$ on clopen sets in ${A_n}$), but it is rare that ${[[\phi]]_{A_n}=[[\phi]]}$ and this would prevent finiteness of the type.

Finally, the amenability of ${[[\phi]]'}$ depends on a lemma of Putnam. More concretely, given ${x\in X}$, ${[[\phi]]}$ acts on the orbit ${\phi^{\mathbb{Z}}(x)\simeq\mathbb{Z}}$: indeed, if ${h(x)=k(x).x}$, then define ${h.n=k(\phi^n(x))+n}$. Note that the function ${k}$ is bounded (it is locally constant on the Cantor space ${X}$), and, thus ${h}$ acts via a bounded displacement permutation of ${\mathbb{Z}}$.

In particular, the amenability of ${[[\phi]]'}$ follows from Juschenko-Monod criterion (Theorem 2) if we can show the amenability of a certain stabilizer, and, for this task, the following lemma of Putnam is useful:

Lemma 9 (Putnam (1989)) The stabilizer of ${\phi^{\mathbb{N}}(x)}$ in ${[[\phi]]}$ is locally finite.

Closing our discussion, let us mention that the idea behind Putnam’s lemma is that any finite symmetric subset ${S}$ of the stabilizer of ${\phi^{\mathbb{N}}(x)}$ acts with a uniformly bounded displacement, say ${\leq k}$, and, from this, one can show that the “partial action” of ${S}$ on ${\{-k,\dots,k\}}$ can be found in a cobounded subset ${J}$ of ${\mathbb{Z}}$ (i.e., a subset ${J}$ with ${\sup\limits_{n\in \mathbb{Z}} d(n,J)<\infty}$). Using this fact, one can deduce that ${S}$ acts with bounded orbits, and, since ${S}$ is arbitrary, this implies that the stabilizer of ${\phi^{\mathbb{N}}(x)}$ in ${[[\phi]]}$ is locally finite.

## Responses

1. Update (March 10, 2013): As Y. Cornulier pointed out to me, the previous version of this post had the following errors: after Definition 1, the probability of return to origin in an amenable group is “large” rather than “small”; in the proof of Lemma 3, there was a confusion between $\{0,1\}^{\mathbb{Z}}$ and $\{0,1\}^{(\mathbb{Z})}$ at some points; and the name of “Giordano” was misspelled as “Giordaneo”.