The workshop was very interesting in several aspects. First, the topics of the talks concerned different research specialities (as you can see from the schedule here), so that it was an excellent opportunity to learn about advances in other related areas. Secondly, the schedule gave sufficient free time so that we could talk to each other. Also, I was happy to meet new people that I knew previously only through their work (e.g., Alex Kontorovich and John Pardon).
In particular, we had two free afternoons on Wednesday and Friday, and I certainly enjoyed both of them: on Wednesday Alex Eskin drove me to the beach and we spent a significant part of the afternoon talking to each other there, and on Friday I went to Getty Center with Sasha Bufetov, Ursula Hamenstadt, Pat Hooper, John Pardon, Federico Rodriguez-Hertz, John Smillie, and Anton Zorich, where, besides classical painters like Monet, Renoir, etc., I saw
- a painting of Fointainebleau forest by Theodore Rousseau (especially appealing to me because I’m living near Fointanebleau since last January…),
- this beautiful painting of Fernand Khnopff, and
- the Dancer sculputure of Paolo Troubetzkoy (little curiosity: Anton Zorich is almost sure that he is a great-grand father of the mathematician Serge Troubetzkoy).
As usual, the talks were very nice (and they will be available at IPAM website here in a near future), and hence I decided to transcript in this post my notes of one of the talks, namely, John Pardon’s talk on his solution of Hilbert-Smith conjecture for 3-manifolds. Of course, the eventual mistakes in what follows are my entire responsibility.
1. Statement of Hilbert-Smith conjecture
In this section, we will quickly review some of the history behind the Hilbert-Smith conjecture. For a more serious reading, we recommend consulting Terence Tao’s notes on Hilbert’s 5th problem (as well as his post here on Hilbert-Smith conjecture).
The 5th problem in the famous list of Hilbert’s problems (stated in 1900) is the following conjecture.
An important step towards the solution of Hilbert’s 5th problem is the following theorem of Gleason and Yamabe:
Theorem 1 (Gleason-Yamabe) Let be locally compact group. Let be an open set containing the identity element . Then, there exists compact and an open subgroup such that is a Lie group.
Remark 1 Another way of stating this theorem is: we have an exact sequence
where is the inverse limit of the family of Lie groups obtained by shrinking towards and “discrete” stands for a discrete
groupspace (cf. Terence Tao’s comment below).
An important corollary of Gleason-Yamabe theorem is:
Corollary 2 is NSS (no small subgroups) if and only if is a Lie group.
In other words, this corollary provides a criterion to recognize Lie groups (and thus it explains the interest of Gleason-Yamabe theorem to Hilbert’s 5th problem). Namely, if has no small subgroups (i.e., there exists a neighborhood of the identity element containing no non-trivial subgroup of ), then is a Lie group.
As it turns out, Hilbert-Smith conjecture is a generalization of Hilbert’s 5th problem where one asks whether Lie groups are the sole (locally compact) groups to act faithfully on manifolds:
Conjecture (Hilbert-Smith). If a (locally compact) group acts faithfully on a manifold (i.e., we have an injective continuous homomorphism ), then is a Lie group.
It is known (see, e.g., this post of Terence Tao for further explanations) that the Hilbert-Smith is equivalent to the following “-adic version”:
Conjecture (Hilbert-Smith for -adic actions). There is no injective continuous homomorphism where is the group of p-adic integers.
It is not hard to convince oneself that the p-adic case is important for Hilbert-Smith conjecture: let generate and, by abusing notation, denote by the corresponding homeomorphism; then, the sequence converges to the identity map . Of course, this occurs because (for ) are small subgroups of (a non-Lie group!).
In summary, the philosophy behind the Hilbert-Smith conjecture is that a compact group acting non-trivially on can not act very close to .
As it is nicely explained in this post of Terence Tao, the reduction of Hilbert-Smith conjecture to the p-adic case uses the following result of M. Newman in 1931:
Theorem 3 (Newman) Let be an open set containing the unit (Euclidean) closed ball . Suppose that has a -action whose orbits have diameter bounded by , i.e., we have an homeomorphism of period (that is, ) such that all -orbits have diameter . Then, the action is trivial.
Proof: A rough sketch of proof goes like this.
Assume that the action is not trivial and consider the map , .
From the fact that -orbits have diameter , one can check that ( is homotopic to the identity map) and thus its degree is .
On the other hand, factors through a map via the natural projection map . Since the projection has degree , it follows that the degree of is , a multiple of .
This contradiction “proves” the theorem.
Using this type of argument, one can also show that:
Theorem 4 Let be a manifold with a metric. Then, there exists such that, if is a compact Lie group acting on with orbits of diameter , then the action is trivial.
After this short discussion of the reduction of Hilbert-Smith conjecture to the p-adic case, let us close this section by pointing out that the general case of Hilbert-Smith conjecture is open. Nevertheless, it was known to be true for low-dimensional manifolds, namely, Montgomery-Zippin showed in 1955 that the conjecture is true for and dimensional manifolds.
In next section, we will discuss the case of -manifolds (after Pardon).
2. Hilbert-Smith conjecture in dimension 3
For the sake of this exposition, let be a connected, orientable, irreducible -manifold with and exactly two ends, e.g., where is a genus surface.
Using the orientation, it makes sense to call one end of the “ end” and the other end of the “ end”.
The basic idea of Pardon to show the Hilbert-Smith conjecture in dimension is to reduce it to a -dimensional problem (that one can handle using our knowledge of the mapping class group of surfaces). In this direction, let be the set of surfaces such that is incompressible (i.e., injects into ) and separates the – end from the + end (i.e, generates ) modulo isotopies (or, equivalently, homotopies).
Definition 5 Given two surfaces , we say that if and only if there are surfaces isotopic to (resp.) such that is contained in the “- end” , that is, is to the “left” of . For example, when , the surface is to the left of .
The first important fact about is:
Lemma 6 is a partially ordered set.
The second (crucial) fact about is the following lemma suggested by Ian Agol to John Pardon:
has a least element.
Proof: A rough sketch of proof goes as follows. By looking at the figure below
one sees that is a natural choice (where , resp. , is the “+ end”, resp. “- end” of , resp. ).
However, this might not be a good choice because the intersection between and might be “artificially complicated” like in the figure below:
Here, J. Pardon overcomes this difficulty by using the following result of M. Freedman, J. Hass and P. Scott saying that if the representatives and of and minimize area, then the intersection between and is “minimal”:
Theorem 8 (Freedman-Hass-Scott) Let be incompressible. Assume that and are area-minimizing representatives of their homology classes. If and can be isotoped to be disjoint, then and are already disjoint unless they coincide.
Using this result, Pardon shows that is a least element of (where and are area-minimizing representatives of and ).
Remark 2 It is implicit in Pardon’s arguments above that the topological and PL (piecewise linear) categories coincide for -dimensional manifolds (that is, a topological -manifold can be triangulated), a profound theorem of E. Moise. Of course, this result doesn’t extend to higher dimensions and this partly explains why Pardon’s arguments really are “-dimensional”.
At this point, we are ready to give a sketch of proof of Pardon’s theorem:
Theorem 9 (Pardon) There is no injective continuous homomorphism
Proof: Suppose by contradiction that there exists a -action on . Up to replacing by for some large , we can assume that this action is very close to the identity.
Let be a handlebody of genus 2 and denote by its orbit under the -action. Consider now a small arc connecting two boundary points of , denote by its orbit under the -action, and define :
Since acts very close to the identity:
- (1) looks like a handlebody of genus 2 in a coarse scale.
On the other hand, Pardon shows that:
- (2) is non-trivial (here, stands for Cech cohomology).
Now, let be the -neighborhood of (for some -invariant metric) with very small, and define .
By definition, acts on the (invariant) -manifold , and, a fortiori, on the set of incompressible separating surfaces on modulo isotopies.
Since is a lattice (cf. Lemma 7) and a least element of is fixed up to isotopy by the action, we get an action
of on the mapping-class group of .
At this point, we get a contradiction as follows. By item (1), if we look at the projections to of the curves shown in the figure below
Using these informations, one can deduce the existence of a cyclic subgroup of (essentially the image of under the map ) such that the module annihilated by has a submodule where the intersection form is given by equation (1).
But, Pardon proves that this is a contradiction as follows. Using Nielsen’s classification of cyclic subgroups of the mapping class group (saying that any is realized by a -action on by isometries in some metric), he shows that the intersection form on the module is:
where is the genus of . Thus, there is no submodule of where the intersection form is given by equation (1) and this completes the sketch of proof of Hilbert-Smith conjecture for -manifolds.