- F. Charles talked about recent progresses on normal functions,
- Y. de Cornulier talked about topologically full groups,
- F. Santambrogio talked about gradient flows on metric spaces, and
- L. Zambotti talked about the KPZ (Kardar-Parisi-Zhang) equation.
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.
Definition 1 (J. von Neumann (1929)) A group is amenable if it admits a mean (i.e., a finitely additive function from the class of subsets taking values in the interval such that ) that is invariant under left translations by elements of (i.e., for all and ).
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: is amenable if and only if any continuous affine action of on a convex compact subset of a locally compact space has a fixed point.
- Existence of Fölner “sequences”: is amenable if and only if for all finite subset and all , there exists a finite non-empty subset that is -almost invariant under left-translation by elements of , i.e.,
- Random walks have
“small”“large” probability to come back to the origin: given a probability measure on supported on a finite symmetric subset of , the probability that a random walk with law comes backs to the identity element after steps decays sub-exponentially fast as .
The most basic examples of amenable groups are finite groups: indeed, if is a finite group, then is a invariant mean. Also, 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 and are amenable groups, then is an amenable group.
- Stability of amenability under direct limits: if and, for all , is amenable, then 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 .
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 order to appreaciate the difficulty of the question, Y. Cornulier recalled that, if contains a copy of the free group in two generators, then 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 be bounded displacement group, i.e., the group of permutations such that . Let be an action of on via permutations of bounded displacement such that is amenable. Then, is amenable.
This criterion seems strange at first sight: we assume amenability of the group in order to conclude the amenability of . 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 preserves a mean on . Here, is the set of bi-infinite sequences of ‘s and ‘s consisting of finitely many ‘s, or, equivalently, is the set of all finite subsets of .
Proof: We will construct a mean preserved by via a compactness argument. More precisely, let us consider the functions given by
Note that, for each , the function is continuous on and, a fortiori, belongs to . Here, we equip with the Bernoulli measure (i.e., we assign weight 1/2 to the symbols and , and we consider the product measure).
Let us now replace by in order to get a sequence of -functions with unit -norm. The reader is invited to check that the sequence satisfies the following properties:
- 1) as for all .
- 2) as , where is the hyperplane .
Using these properties, we take the Fourier transform of . By Plancherel formula, . In particular, .
By compactness of the unit ball of in the weak-* topology (Banach-Alaoglu theorem), we can select a subsequence of accumulating some .
Note that induces a mean on . Furthermore, 1) implies that is -invariant, while 2) implies that is invariant under translations. It follows that is a -invariant mean, and this ends the proof of the lemma.
Using this lemma, we can complete the sketch of proof of Juschenko-Monod criterion (Theorem 2): we consider the following twisted action of on
In other words, the action is obtained from the usual action by adding a coboundary .
Note that the stabilizer of for is the stabilizer of for the usual action of , and, by assumption, this subgroup is amenable. More generally, by a similar reasoning, one can show that, from our assumption on , the stabilizers of all points of for are amenable. Furthermore, by the previous lemma, we know that the twisted action of preserves a mean on . At this point, the sketch of proof is complete because it is a known fact that a group whose action on a space preserves a mean and has amenable stabilizers for all points of is amenable.
2. Topologically full groups
Let be a group acting on a topological space by homeomorphisms.
Definition 4 The topological full group of is
Here, by “locally belongs to ” we mean that continuous (i.e., locally constant) such that .
For later use, we will need a couple of definitions coming from Dynamical Systems/Ergodic Theory.
Definition 5 The action of on is minimal if all orbits are dense.
Definition 6 Let be a finite alphabet. The natural action of on (by left-translation of indices) is a shift, and any closed shift-invariant subset of is a subshift.
It is not hard to characterize subshifts:
Proposition 7 An action of on is (equivariantly) isomorphic to a subshift if and only if is compact and there is a finite number of clopen subsets of whose -translates separate points. Equivalenty, the action is isomorphic to a subshift if and only if is totally disconnected and the Boolean algebra (of clopen subsets) has finite type as an algebra equipped with a -action.
The main result of today’s discussion is:
is an infinite simple amenable group. If, moreover, is isomorphic to a subshift, then has finite type.
The fact that is simple was shown by Matui and the amenability of (conjectured by Grigorchuk-Medynets) was shown by Juschenko-Monod. Finally, Matui showed that has finite type whenever is isomorphic to a subshift.
Before saying a few words on the proof of this theorem, let us give some examples of dynamical systems naturally related to this theorem. Let be the circle and consider the rotation , for . The rotation is minimal, but this example is not “valid” because is not a Cantor space. However, Denjoy’s construction permits to overcome this technical difficulty: concretely, we will produce a Cantor space where by blowing up the circle along the orbit of a point (say ); more precisely, let ; the cyclic order of induces a natural cyclic order on , and, from this, it can be checked that is a Cantor space; moreover, the action of on induces a minimal homeomorphism on 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 is infinite. Indeed, given , we select and we fix a clopen neighborhood of such that for all : note that the existence of follows from the fact that the orbit of is not periodic (as is minimal and is a Cantor space [hence infinite]). Using , it is not hard to see that we have an injection from the symmetric group into . Therefore, the derived group contains a copy of the alternate group . Since is arbitrary, we conclude that is infinite.
Secondly, it is important to assume that is minimal to deduce that is simple. In fact, if is not minimal, we can find a -invariant subset . In general, it is rare that the natural homomorphism from the group of homeomorphism of locally in to the group of homeomorphism of locally in is trivial or injective. On the other hand, all homomorphisms of a simple group are trivial or injective. So, this suggests that the minimality of has something to do with the simplicity of .
Thirdly, it is important that is a subshift to get that has finite type: otherwise, if is not a subshift, then we could write
where are certain Boolean algebras; then, in general, we have that (where consists of homeomorphisms locally coinciding with powers of on clopen sets in ), but it is rare that and this would prevent finiteness of the type.
Finally, the amenability of depends on a lemma of Putnam. More concretely, given , acts on the orbit : indeed, if , then define . Note that the function is bounded (it is locally constant on the Cantor space ), and, thus acts via a bounded displacement permutation of .
In particular, the amenability of 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 in is locally finite.
Closing our discussion, let us mention that the idea behind Putnam’s lemma is that any finite symmetric subset of the stabilizer of acts with a uniformly bounded displacement, say , and, from this, one can show that the “partial action” of on can be found in a cobounded subset of (i.e., a subset with ). Using this fact, one can deduce that acts with bounded orbits, and, since is arbitrary, this implies that the stabilizer of in is locally finite.