## Notes of Emmanuel Breuillard’s fifth Cambridge lecture 19-04-2017

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 ${C^2}$ 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.

1.1. 1951

A Bourbaki report by Serre in 1951 (a year before the problem was solved) identifies two problems:

1. A locally Euclidean topological group is a Lie group.
2. 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 ${{\mathbb Z}_p}$ can act faithfully by homeomorphisms on a manifold.

1.2. 1952

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.

1.3. 1953

Yamabe proved a structure theorem for locally compact groups.

Theorem 1 (Yamabe) Every locally compact group ${G}$ has an open subgroup ${G'}$ which is approximated by Lie groups: every neighborhood ${U}$ of the identity in ${G'}$ there exists a normal subgroup ${K}$ of ${G'}$, contained in ${U}$, such that ${G'/K}$ 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

$\displaystyle \begin{array}{rcl} L^2(G)=\bigoplus_{\pi\in\hat G}\mathcal{H}_\pi \end{array}$

where ${\mathcal{H}_\pi}$ is finite dimensional. It follows that ${G}$ embeds in a (countable product of) general linear groups. It is based on the spectral theorem. Indeed, construct a continuous function ${\phi}$ with small support. Make it inversion and conjugation invariant. Show that the operator of convolution with ${\phi}$ is compact (Arzela-Ascoli). Hence it has a sequence of finite dimensional eigenspaces. The spectral theorem says that

$\displaystyle \begin{array}{rcl} L^2(G)=\bigoplus_{\lambda}\mathcal{H}_\lambda \end{array}$

Projection to finitely many summands gives a homorphism to ${Gl(N)}$ 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 ${\Rightarrow}$ Montgomery-Zippin

Note that there exist connected compact groups which are not locally connected: the solenoid ${{\mathbb R}\times{\mathbb Q}_2/{\mathbb Z}[\frac{1}{2}]}$.

Let ${G}$ act faithfully transitively on a locally connected and finite dimensional topological space ${X}$. Up to taking an open subgroup, by Yamabe, one can assume that ${G=\lim G_n}$ where ${X_n=G_n/H_n}$ and ${G_n}$ is Lie. ${X_n}$ is a manifold, and ${X}$ is a projective limit of ${X_n}$. Since ${X}$ is finite dimensional, dimension stabilizes, ${X_n}$ is a finite covering space of ${X_{n_0}}$, hence ${H_n}$ is profinite. By local connectedness of ${X}$, ${H_n}$ is finite, ${G}$ 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 ${G}$ be a locally compact group. Let ${U}$ be a neighborhood of 1. For ${g\in U}$, let

$\displaystyle n_U(g)=\sup\{n\in{\mathbb N}\,;\,1,g,\ldots,g^n\in U\}.$

Analogously for subsets ${Q\subset U}$,

$\displaystyle n_U(Q)=\sup\{n\in{\mathbb N}\,;\,1,Q,\ldots,Q^n\subset U\}.$

The escape norm is the inverse

$\displaystyle \begin{array}{rcl} \|g\|_U=\frac{1}{n_U(g)}. \end{array}$

Lemma 2 Assume that ${G}$ is NSS. Let ${U,V}$ be sufficiently small compact neighborhoods of 1. Then

1. For all compact sets ${Q\subset G}$, ${\frac{1}{C}n_U(Q)\leq n_V(Q)\leq C\,n_U(Q)}$.
2. For all ${g,h\in U}$,

$\displaystyle \begin{array}{rcl} \|hgh^{-1}\|_U&\leq& C\|g\|_U.\\ \|gh\|_U&\leq& C(\|g\|_U+\|h\|_U).\\ \|[g,h]\|_U&\leq& C\|g\|_U\|h\|_U. \end{array}$

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 ${U}$. 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 ${G}$ be a locally compact group. Let ${V}$ be a neighborhood of 1. Then there exists ${C}$ such that for all ${K\geq 1}$, there exists a neighborhood ${U\subset V}$ of ${1}$ such that for all compact subsets ${Q\subset U}$,

• either ${n_{V^4}(Q)\geq K\,n_V(Q)}$,
• or ${\exists g\in Q}$ such that ${n_U(g)\leq KC\,n_V(Q)}$.

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 ${V}$ there exists ${U\subset V}$ such that the subgroup generated by all subgroups of ${G}$ contained in ${U}$ is contained in ${V}$.

3.2. Proof of Gleason’s theorem

Gleason’s theorem NSS ${\Rightarrow}$ Lie uses Lemma 1 as follows. If ${G}$ 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

$\displaystyle \begin{array}{rcl} X(t)+Y(t)=\lim_{n\rightarrow\infty}(X(\frac{t}{n})Y(\frac{t}{n}))^{n}. \end{array}$

Lemma 1 tells that the ${n}$-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 ${G}$ 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 ${\phi}$. Let

$\displaystyle \begin{array}{rcl} \partial_g \phi(x)=\phi(g^{-1}x)-\phi(x). \end{array}$

Then trivially

$\displaystyle \begin{array}{rcl} \|\partial_{gh} \phi\|_\infty\leq \|\partial_g \phi\|_\infty+\|\partial_h \phi\|_\infty. \end{array}$

Also ${\|\partial_g \phi\|_\infty<1\Rightarrow g\in U}$. This implies that ${\|g\|_U\leq 2\|\partial_g \phi\|_\infty}$. The key point is to get the reverse inequality ${\|\partial_g \phi\|_\infty\leq C\|g\|_U}$.

If ${\phi}$ were ${C^2}$, Taylor’s expansion

$\displaystyle \begin{array}{rcl} \partial_{g^n} \phi=n\partial_{g} \phi+\sum_{i=0}^{n-1}\partial_{g^i}\partial_{g} \phi \end{array}$

would prove it. In absence of regularity, Gleason uses convolution. For instance, if ${\phi=\psi_1\star\psi_2}$, a second derivative

$\displaystyle \begin{array}{rcl} \partial_g\partial_h\phi=\int_{G}\partial_g\psi_1(y)\partial_{yhy^{-1}}\psi_2(y^{-1}x)\,dy \end{array}$

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.