Introduction to approximate groups, V
Today, I explain mathematics from the 1950’s: the solution of Hilbert’s 5th problem. Indeed, ideas from this old theory enters our structure theory of approximate groups.
1. Hilbert’s 5th problem
In 1900, Hilbert wondered wether one can remove differentiability from the theory of Lie groups. Before him, Lie had shown that Lie groups are the same as analytic Lie groups. At that time, groups were always thought of as acting on some manifold, so Hilbert asked wether a continuous action of a manifold must be differentiable.
A Bourbaki report by Serre in 1951 (a year before the problem was solved) identifies two problems:
- A locally Euclidean topological group is a Lie group.
- A locally compact subgroup of homeomorphisms of a manifold is a Lie group.
(2) implies (1). (1) is now a theorem by Montgomery-Zippin, based on work by Gleason.
Problem (2) is still open (sometimes called Hilbert-Smith conjecture). It is known in dimension 2 (Montgomery-Zippin) and 3 (Pardon). It is known for Lipschitz homeomorphisms.
Problem (2) is equivalent to wether can act faithfully by homeomorphisms on a manifold.
In 3 Annals papers, Gleason proves that an NSS group is a Lie group. NSS = no small subgroups.
Montgomery-Zippin showed that locally Euclidean groups are NSS. Also, they solved problem (2) for transitive actions: if a locally compact group acts faithfully and transitively on a locally connected finite dimensional topological space, then it is NSS.
Yamabe proved a structure theorem for locally compact groups.
Theorem 1 (Yamabe) Every locally compact group has an open subgroup which is approximated by Lie groups: every neighborhood of the identity in there exists a normal subgroup of , contained in , such that is Lie with finitely many components.
In 1955, Montgomery-Zippin wrote a book account of these results. Nowadays, Yamabe’s theorem rather directly implies all others.
1.4. Previous results
Haar (1930) constructed the Haar measure. Cartan-von Neumann showed that closed subgroups of Lie groups are Lie.
Peter-Weyl proved the compact case of Yamabe’s theorem. They show that
where is finite dimensional. It follows that embeds in a (countable product of) general linear groups. It is based on the spectral theorem. Indeed, construct a continuous function with small support. Make it inversion and conjugation invariant. Show that the operator of convolution with is compact (Arzela-Ascoli). Hence it has a sequence of finite dimensional eigenspaces. The spectral theorem says that
Projection to finitely many summands gives a homorphism to with small kernel.
The abelian case follows from Pontrijagin’s duality theory.
Gelfand-Raikov tried to use representation theory in the noncompact case as well, but their efforts did not have a posterity.
2. Sketch of Yamabe Montgomery-Zippin
Note that there exist connected compact groups which are not locally connected: the solenoid .
Let act faithfully transitively on a locally connected and finite dimensional topological space . Up to taking an open subgroup, by Yamabe, one can assume that where and is Lie. is a manifold, and is a projective limit of . Since is finite dimensional, dimension stabilizes, is a finite covering space of , hence is profinite. By local connectedness of , is finite, is Lie.
2.1. Isometry groups
Note that Gromov uses a slightly different version, dealing with isometry groups. It follows easily from Yamabe’s theorem.
3. Proof of Yamabe’s Theorem
Yamabe’s theorem is a local statement. Hence versions for local groups appeared. Jacoby’s version from the 1950’s was flawed. Isa Goldbring finally produced a correct proof in 2010. This is the result of decades of works which enlightened the fact that the proof goes smoothly in the language of nonstandard analysis (Hirschfeld in the 1970’s) and model theory (van den Dries and Goldbring’s seminar notes, appeared in L’Enseignement Mathématique). Tao’s 2013 book gives a modern treatment.
3.1. Gleason-Yamabe lemmas
Let be a locally compact group. Let be a neighborhood of 1. For , let
Analogously for subsets ,
The escape norm is the inverse
Lemma 2 Assume that is NSS. Let be sufficiently small compact neighborhoods of 1. Then
- For all compact sets , .
- For all ,
This is easy for Lie groups, using differentiability properties of the exponential maps. If (1) fails, then extracting subsequences, one would obtain a nontrivial subgroup contained in . In his 1951 survey, Serre alludes to the fact that if (2) would hold, one would be finished. More precisely, he means the following lemma, used in the proof of Lemma 1.
Lemma 3 Let be a locally compact group. Let be a neighborhood of 1. Then there exists such that for all , there exists a neighborhood of such that for all compact subsets ,
- either ,
- or such that .
This lemma, combined with the Peter-Weyl Theorem, implies approximability of locally compact groups by NSS groups. Indeed, it has the following
Corollary 4 (Subgroup trapping) For all there exists such that the subgroup generated by all subgroups of contained in is contained in .
3.2. Proof of Gleason’s theorem
Gleason’s theorem NSS Lie uses Lemma 1 as follows. If is NSS, then every element has a unique square root. Going to limits shows that every element belongs to some 1-parameter subgroups. The set of 1-parameter subgroups is a candidate for a Lie algebra. Addition is defined by
Lemma 1 tells that the -th power stays bounded. The bracket estimate implies that the limit exists. A locally compact vectorspace is finite dimensional. The adjoint action is well defined, it maps to a Lie group, with a kernel which is abelian, hence Lie.
3.3. Proof of Lemma 1
As is Peter-Weyl’s Theorem, pick a bump function . Let
Also . This implies that . The key point is to get the reverse inequality .
If were , Taylor’s expansion
would prove it. In absence of regularity, Gleason uses convolution. For instance, if , a second derivative
is expressed in terms of first derivatives.
These techniques still work for locally compact local groups, and allow to conclude that any locally compact local group has the same germ as a group. This fails in infinite dimensions.