# Monotone functions are differentiable a.e. and Hilbert's Fifth Problem: what's the connection?

In Andrew Gleason's interview for More Mathematical People, there is the following exchange concerning Gleason's work on Hilbert's fifth problem on whether every locally Euclidean topological group is a Lie group (page 92).

MP: Is there some "human" story you can tell us about the breakthrough when it came?

Gleason: Yes, there's a really remarkable story about that. Sometime -- I can't tell you the exact date but let's say around 1949 -- I was doing other things too, and one of the things that I found very interesting and very curious and which I really felt I should try to understand better was a very famous theorem to the effect that a monotonic function is almost everywhere differentiable. It's a rather remarkable and very difficult theorem -- it's not easy to prove. A very very hard theorem of analysis and a really surprising theorem. Well, at the time I was sort of speculating about this theorem, but it wasn't for at least two years that I suddenly realized that that would solve the problem I was dealing with! Knowing that, in connection with some other stuff I had been working on, really put the whole thing together. It was a realization that although this theorem had been on my mind for maybe two years, I had never recognized that it was crucial to the arguments that I was trying to work through in the Hilbert problem. I hadn't realized it. Then suddenly it just came to me.

MP: It just came to you?

Gleason: That's right. It just came to me that I could use this technique, this theorem, in connection with these curves in Hilbert space that I was dealing with -- and get the answer! ...

I've never studied Hilbert's Fifth Problem or its solutions, but I've always been curious what Gleason meant by this connection. Can anyone shed some light on this?

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Just a curious question in passing, whether the flip-side: Everywhere differentiable, but nowhere monotone functions, are also of interest? See e.g., jstor.org/stable/2318996 –  Suvrit Nov 2 '10 at 9:46
I don't see how it is relevant here, but such wiggly things are indeed curious and interesting. –  Todd Trimble Nov 2 '10 at 11:22
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## 1 Answer

Well, I cannot say for certain, but I did know Gleason well (he was my thesis advisor, and we wrote a paper together after that) and I have written an essay about Gleason's work on the Fifth Problem (in the Gleason Memorial article in the AMS Notices --- http://www.ams.org/notices/200910/rtx091001236p.pdf ) and based on that I think I can make a reasonable guess about what he had in mind. Recall that what Gleason actually proved was that a locally compact group without arbitrarily small subgroups is a Lie group (then Montomery and Zippin proved that a locally Euclidean group did not have small subgroups). A key idea in Gleason's proof was the construction of a unique one-parameter subgroup through any point sufficiently close to the identity (i.e., essentially, constructing the exponential map) and this in turn depended on showing the existence of a unique square root for elements near the identity (see his paper "Square roots in locally Euclidean groups"). I believe that it is the step going from square roots to one-parameter subgroups that used ideas from the monotonic implies differentiable a.e. theorem. The "these curves" that he mentions in that More Mathematical People article, can only be the one-parameter subgroups. For more details, see my Notices article above, particularly the section called "Following in Gleason’s Footsteps".

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@Dick: I was aware of your having written that paper through another MO exchange: mathoverflow.net/questions/31387/…, and so I was hoping you'd weigh in! Your answer here makes a lot of sense, and now let me have a look at your article. Thanks very much! –  Todd Trimble Nov 1 '10 at 19:21
I have to admit that I'm still murky on where a monotone function enters here, but perhaps you're suggesting that it's more ideas from the proof of this theorem of analysis than the statement itself that is relevant to Gleason's thinking? Maybe at this point I need to dig into the literature more... thanks again for your response. –  Todd Trimble Nov 2 '10 at 11:42
@Todd: "...it's more ideas from the proof of this theorem of analysis than the statement itself that is relevant". Yes. Recall from my article that, in attempting to define a $1$-parameter group $f$ with $f(1)$ a given group element near the identity, from the existence of unique square roots near $e$ it is easy to see how to define $f$ at all dyadic rationals $m/2^n$. More difficult is to see how to prove that $f$ so defined is continuous---and so extends to a $1$-parameter group. This is where I think Gleason is saying that he used ideas from the proof of the monotone function theorem. –  Dick Palais Nov 2 '10 at 16:32
I was reading again through Gleason's Annals paper "Groups without small subgroups" jstor.org/stable/1969795 and found that he used the almost everywhere differentiability of Lipschitz functions on page 205, in order to create a one-parameter subgroup out of a Lipschitz action on L^2. (This particular trick is however not used in more recent treatments of Hilbert's fifth.) –  Terry Tao Feb 12 at 21:29
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