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In this set of notes we will be able to finally prove the Gleason-Yamabe theorem from Notes 0, which we restate here:

Theorem 1 (Gleason-Yamabe theorem) Let ${G}$ be a locally compact group. Then, for any open neighbourhood ${U}$ of the identity, there exists an open subgroup ${G'}$ of ${G}$ and a compact normal subgroup ${K}$ of ${G'}$ in ${U}$ such that ${G'/K}$ is isomorphic to a Lie group.

In the next set of notes, we will combine the Gleason-Yamabe theorem with some topological analysis (and in particular, using the invariance of domain theorem) to establish some further control on locally compact groups, and in particular obtaining a solution to Hilbert’s fifth problem.

To prove the Gleason-Yamabe theorem, we will use three major tools developed in previous notes. The first (from Notes 2) is a criterion for Lie structure in terms of a special type of metric, which we will call a Gleason metric:

Definition 2 Let ${G}$ be a topological group. A Gleason metric on ${G}$ is a left-invariant metric ${d: G \times G \rightarrow {\bf R}^+}$ which generates the topology on ${G}$ and obeys the following properties for some constant ${C>0}$, writing ${\|g\|}$ for ${d(g,\hbox{id})}$:

• (Escape property) If ${g \in G}$ and ${n \geq 1}$ is such that ${n \|g\| \leq \frac{1}{C}}$, then ${\|g^n\| \geq \frac{1}{C} n \|g\|}$.
• (Commutator estimate) If ${g, h \in G}$ are such that ${\|g\|, \|h\| \leq \frac{1}{C}}$, then

$\displaystyle \|[g,h]\| \leq C \|g\| \|h\|, \ \ \ \ \ (1)$

where ${[g,h] := g^{-1}h^{-1}gh}$ is the commutator of ${g}$ and ${h}$.

Theorem 3 (Building Lie structure from Gleason metrics) Let ${G}$ be a locally compact group that has a Gleason metric. Then ${G}$ is isomorphic to a Lie group.

The second tool is the existence of a left-invariant Haar measure on any locally compact group; see Theorem 3 from Notes 3. Finally, we will also need the compact case of the Gleason-Yamabe theorem (Theorem 8 from Notes 3), which was proven via the Peter-Weyl theorem:

Theorem 4 (Gleason-Yamabe theorem for compact groups) Let ${G}$ be a compact Hausdorff group, and let ${U}$ be a neighbourhood of the identity. Then there exists a compact normal subgroup ${H}$ of ${G}$ contained in ${U}$ such that ${G/H}$ is isomorphic to a linear group (i.e. a closed subgroup of a general linear group ${GL_n({\bf C})}$).

To finish the proof of the Gleason-Yamabe theorem, we have to somehow use the available structures on locally compact groups (such as Haar measure) to build good metrics on those groups (or on suitable subgroups or quotient groups). The basic construction is as follows:

Definition 5 (Building metrics out of test functions) Let ${G}$ be a topological group, and let ${\psi: G \rightarrow {\bf R}^+}$ be a bounded non-negative function. Then we define the pseudometric ${d_\psi: G \times G \rightarrow {\bf R}^+}$ by the formula

$\displaystyle d_\psi(g,h) := \sup_{x \in G} |\tau(g) \psi(x) - \tau(h) \psi(x)|$

$\displaystyle = \sup_{x \in G} |\psi(g^{-1} x ) - \psi(h^{-1} x)|$

and the semi-norm ${\| \|_\psi: G \rightarrow {\bf R}^+}$ by the formula

$\displaystyle \|g\|_\psi := d_\psi(g, \hbox{id}).$

Note that one can also write

$\displaystyle \|g\|_\psi = \sup_{x \in G} |\partial_g \psi(x)|$

where ${\partial_g \psi(x) := \psi(x) - \psi(g^{-1} x)}$ is the “derivative” of ${\psi}$ in the direction ${g}$.

Exercise 1 Let the notation and assumptions be as in the above definition. For any ${g,h,k \in G}$, establish the metric-like properties

1. (Identity) ${d_\psi(g,h) \geq 0}$, with equality when ${g=h}$.
2. (Symmetry) ${d_\psi(g,h) = d_\psi(h,g)}$.
3. (Triangle inequality) ${d_\psi(g,k) \leq d_\psi(g,h) + d_\psi(h,k)}$.
4. (Continuity) If ${\psi \in C_c(G)}$, then the map ${d_\psi: G \times G \rightarrow {\bf R}^+}$ is continuous.
5. (Boundedness) One has ${d_\psi(g,h) \leq \sup_{x \in G} |\psi(x)|}$. If ${\psi \in C_c(G)}$ is supported in a set ${K}$, then equality occurs unless ${g^{-1} h \in K K^{-1}}$.
6. (Left-invariance) ${d_\psi(g,h) = d_\psi(kg,kh)}$. In particular, ${d_\psi(g,h) = \| h^{-1} g \|_\psi = \| g^{-1} h \|_\psi}$.

In particular, we have the norm-like properties

1. (Identity) ${\|g\|_\psi \geq 0}$, with equality when ${g=\hbox{id}}$.
2. (Symmetry) ${\|g\|_\psi = \|g^{-1}\|_\psi}$.
3. (Triangle inequality) ${\|gh\|_\psi \leq \|g\|_\psi + \|h\|_\psi}$.
4. (Continuity) If ${\psi \in C_c(G)}$, then the map ${\|\|_\psi: G \rightarrow {\bf R}^+}$ is continuous.
5. (Boundedness) One has ${\|g\|_\psi \leq \sup_{x \in G} |\psi(x)|}$. If ${\psi \in C_c(G)}$ is supported in a set ${K}$, then equality occurs unless ${g \in K K^{-1}}$.

We remark that the first three properties of ${d_\psi}$ in the above exercise ensure that ${d_\psi}$ is indeed a pseudometric.

To get good metrics (such as Gleason metrics) on groups ${G}$, it thus suffices to obtain test functions ${\psi}$ that obey suitably good “regularity” properties. We will achieve this primarily by means of two tricks. The first trick is to obtain high-regularity test functions by convolving together two low-regularity test functions, taking advantage of the existence of a left-invariant Haar measure ${\mu}$ on ${G}$. The second trick is to obtain low-regularity test functions by means of a metric-like object on ${G}$. This latter trick may seem circular, as our whole objective is to get a metric on ${G}$ in the first place, but the key point is that the metric one starts with does not need to have as many “good properties” as the metric one ends up with, thanks to the regularity-improving properties of convolution. As such, one can use a “bootstrap argument” (or induction argument) to create a good metric out of almost nothing. It is this bootstrap miracle which is at the heart of the proof of the Gleason-Yamabe theorem (and hence to the solution of Hilbert’s fifth problem).

The arguments here are based on the nonstandard analysis arguments used to establish Hilbert’s fifth problem by Hirschfeld and by Goldbring (and also some unpublished lecture notes of Goldbring and van den Dries). However, we will not explicitly use any nonstandard analysis in this post.

This is another installment of my my series of posts on Hilbert’s fifth problem. One formulation of this problem is answered by the following theorem of Gleason and Montgomery-Zippin:

Theorem 1 (Hilbert’s fifth problem) Let ${G}$ be a topological group which is locally Euclidean. Then ${G}$ is isomorphic to a Lie group.

Theorem 1 is deep and difficult result, but the discussion in the previous posts has reduced the proof of this Theorem to that of establishing two simpler results, involving the concepts of a no small subgroups (NSS) subgroup, and that of a Gleason metric. We briefly recall the relevant definitions:

Definition 2 (NSS) A topological group ${G}$ is said to have no small subgroups, or is NSS for short, if there is an open neighbourhood ${U}$ of the identity in ${G}$ that contains no subgroups of ${G}$ other than the trivial subgroup ${\{ \hbox{id}\}}$.

Definition 3 (Gleason metric) Let ${G}$ be a topological group. A Gleason metric on ${G}$ is a left-invariant metric ${d: G \times G \rightarrow {\bf R}^+}$ which generates the topology on ${G}$ and obeys the following properties for some constant ${C>0}$, writing ${\|g\|}$ for ${d(g,\hbox{id})}$:

The remaining steps in the resolution of Hilbert’s fifth problem are then as follows:

Theorem 4 (Reduction to the NSS case) Let ${G}$ be a locally compact group, and let ${U}$ be an open neighbourhood of the identity in ${G}$. Then there exists an open subgroup ${G'}$ of ${G}$, and a compact subgroup ${N}$ of ${G'}$ contained in ${U}$, such that ${G'/N}$ is NSS and locally compact.

Theorem 5 (Gleason’s lemma) Let ${G}$ be a locally compact NSS group. Then ${G}$ has a Gleason metric.

The purpose of this post is to establish these two results, using arguments that are originally due to Gleason. We will split this task into several subtasks, each of which improves the structure on the group ${G}$ by some amount:

Proposition 6 (From locally compact to metrisable) Let ${G}$ be a locally compact group, and let ${U}$ be an open neighbourhood of the identity in ${G}$. Then there exists an open subgroup ${G'}$ of ${G}$, and a compact subgroup ${N}$ of ${G'}$ contained in ${U}$, such that ${G'/N}$ is locally compact and metrisable.

For any open neighbourhood ${U}$ of the identity in ${G}$, let ${Q(U)}$ be the union of all the subgroups of ${G}$ that are contained in ${U}$. (Thus, for instance, ${G}$ is NSS if and only if ${Q(U)}$ is trivial for all sufficiently small ${U}$.)

Proposition 7 (From metrisable to subgroup trapping) Let ${G}$ be a locally compact metrisable group. Then ${G}$ has the subgroup trapping property: for every open neighbourhood ${U}$ of the identity, there exists another open neighbourhood ${V}$ of the identity such that ${Q(V)}$ generates a subgroup ${\langle Q(V) \rangle}$ contained in ${U}$.

Proposition 8 (From subgroup trapping to NSS) Let ${G}$ be a locally compact group with the subgroup trapping property, and let ${U}$ be an open neighbourhood of the identity in ${G}$. Then there exists an open subgroup ${G'}$ of ${G}$, and a compact subgroup ${N}$ of ${G'}$ contained in ${U}$, such that ${G'/N}$ is locally compact and NSS.

Proposition 9 (From NSS to the escape property) Let ${G}$ be a locally compact NSS group. Then there exists a left-invariant metric ${d}$ on ${G}$ generating the topology on ${G}$ which obeys the escape property (1) for some constant ${C}$.

Proposition 10 (From escape to the commutator estimate) Let ${G}$ be a locally compact group with a left-invariant metric ${d}$ that obeys the escape property (1). Then ${d}$ also obeys the commutator property (2).

It is clear that Propositions 6, 7, and 8 combine to give Theorem 4, and Propositions 9, 10 combine to give Theorem 5.

Propositions 610 are all proven separately, but their proofs share some common strategies and ideas. The first main idea is to construct metrics on a locally compact group ${G}$ by starting with a suitable “bump function” ${\phi \in C_c(G)}$ (i.e. a continuous, compactly supported function from ${G}$ to ${{\bf R}}$) and pulling back the metric structure on ${C_c(G)}$ by using the translation action ${\tau_g \phi(x) := \phi(g^{-1} x)}$, thus creating a (semi-)metric

$\displaystyle d_\phi( g, h ) := \| \tau_g \phi - \tau_h \phi \|_{C_c(G)} := \sup_{x \in G} |\phi(g^{-1} x) - \phi(h^{-1} x)|. \ \ \ \ \ (3)$

One easily verifies that this is indeed a (semi-)metric (in that it is non-negative, symmetric, and obeys the triangle inequality); it is also left-invariant, and so we have ${d_\phi(g,h) = \|g^{-1} h \|_\phi = \| h^{-1} g \|_\phi}$, where

$\displaystyle \| g \|_\phi = d_\phi(g,\hbox{id}) = \| \partial_g \phi \|_{C_c(G)}$

where ${\partial_g}$ is the difference operator ${\partial_g = 1 - \tau_g}$,

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

This construction was already seen in the proof of the Birkhoff-Kakutani theorem, which is the main tool used to establish Proposition 6. For the other propositions, the idea is to choose a bump function ${\phi}$ that is “smooth” enough that it creates a metric with good properties such as the commutator estimate (2). Roughly speaking, to get a bound of the form (2), one needs ${\phi}$ to have “${C^{1,1}}$ regularity” with respect to the “right” smooth structure on ${G}$ By ${C^{1,1}}$ regularity, we mean here something like a bound of the form

$\displaystyle \| \partial_g \partial_h \phi \|_{C_c(G)} \ll \|g\|_\phi \|h\|_\phi \ \ \ \ \ (4)$

for all ${g,h \in G}$. Here we use the usual asymptotic notation, writing ${X \ll Y}$ or ${X=O(Y)}$ if ${X \leq CY}$ for some constant ${C}$ (which can vary from line to line).

The following lemma illustrates how ${C^{1,1}}$ regularity can be used to build Gleason metrics.

Lemma 11 Suppose that ${\phi \in C_c(G)}$ obeys (4). Then the (semi-)metric ${d_\phi}$ (and associated (semi-)norm ${\|\|_\phi}$) obey the escape property (1) and the commutator property (2).

Proof: We begin with the commutator property (2). Observe the identity

$\displaystyle \tau_{[g,h]} = \tau_{hg}^{-1} \tau_{gh}$

whence

$\displaystyle \partial_{[g,h]} = \tau_{hg}^{-1} ( \tau_{hg} - \tau_{gh} )$

$\displaystyle = \tau_{hg}^{-1} ( \partial_h \partial_g - \partial_g \partial_h ).$

From the triangle inequality (and translation-invariance of the ${C_c(G)}$ norm) we thus see that (2) follows from (4). Similarly, to obtain the escape property (1), observe the telescoping identity

$\displaystyle \partial_{g^n} = n \partial_g + \sum_{i=0}^{n-1} \partial_g \partial_{g^i}$

for any ${g \in G}$ and natural number ${n}$, and thus by the triangle inequality

$\displaystyle \| g^n \|_\phi = n \| g \|_\phi + O( \sum_{i=0}^{n-1} \| \partial_g \partial_{g^i} \phi \|_{C_c(G)} ). \ \ \ \ \ (5)$

But from (4) (and the triangle inequality) we have

$\displaystyle \| \partial_g \partial_{g^i} \phi \|_{C_c(G)} \ll \|g\|_\phi \|g^i \|_\phi \ll i \|g\|_\phi^2$

and thus we have the “Taylor expansion”

$\displaystyle \|g^n\|_\phi = n \|g\|_\phi + O( n^2 \|g\|_\phi^2 )$

which gives (1). $\Box$

It remains to obtain ${\phi}$ that have the desired ${C^{1,1}}$ regularity property. In order to get such regular bump functions, we will use the trick of convolving together two lower regularity bump functions (such as two functions with “${C^{0,1}}$ regularity” in some sense to be determined later). In order to perform this convolution, we will use the fundamental tool of (left-invariant) Haar measure ${\mu}$ on the locally compact group ${G}$. Here we exploit the basic fact that the convolution

$\displaystyle f_1 * f_2(x) := \int_G f_1(y) f_2(y^{-1} x)\ d\mu(y) \ \ \ \ \ (6)$

of two functions ${f_1,f_2 \in C_c(G)}$ tends to be smoother than either of the two factors ${f_1,f_2}$. This is easiest to see in the abelian case, since in this case we can distribute derivatives according to the law

$\displaystyle \partial_g (f_1 * f_2) = (\partial_g f_1) * f_2 = f_1 * (\partial_g f_2),$

which suggests that the order of “differentiability” of ${f_1*f_2}$ should be the sum of the orders of ${f_1}$ and ${f_2}$ separately.

These ideas are already sufficient to establish Proposition 10 directly, and also Proposition 9 when comined with an additional bootstrap argument. The proofs of Proposition 7 and Proposition 8 use similar techniques, but is more difficult due to the potential presence of small subgroups, which require an application of the Peter-Weyl theorem to properly control. Both of these theorems will be proven below the fold, thus (when combined with the preceding posts) completing the proof of Theorem 1.

The presentation here is based on some unpublished notes of van den Dries and Goldbring on Hilbert’s fifth problem. I am indebted to Emmanuel Breuillard, Ben Green, and Tom Sanders for many discussions related to these arguments.