Hilbert’s fifth problem and related topics.

Terence TaoGraduate Studies in Mathematics, Vol. 153

American Mathematical Society, 2014

338 pp., hardcover

ISBN-10: 1-4704-1564-X

ISBN-13: 978-1-4704-1564-8

Last updated: May 13, 2023

This continues my series of books derived from my blog, and is based on the lecture notes for my graduate course of the same name. The preceding books in this series were “Structure and Randomness“, “Poincaré’s legacies“, “An epsilon of room“, “An introduction to measure theory“, “Topics in random matrix theory“, “Higher order Fourier analysis“, and “Compactness and contradiction“.

An online version of the MS can be found here.

This book received the 2015 PROSE award in the mathematics category.

Pre-errata (to be corrected in the published version):

- Page 5: In the proof of Theorem 1.1.2, should be taken to be , rather than .
- Page 10: In Exercise 1.1.7, should be , and should be .
- Page 20: In Problem 1.1.14, the hypothesis that G has polynomial growth is missing and should be inserted.
- Page 79: In the last paragraph, a right parenthesis is missing after “Exercise 1.4.3”.

Errata:

- Page 16: In Example 1.2.5, the series should be .
- Page ???: In Exercise 1.2.11(viii), should be (two occurrences).
- Page 18: In Definition 1.2.8, all occurrences of G should be replaced with M.
- Page 54: In Exercise 3.0.8, should be assumed to be Hausdorff.
- Page 105: After (5.9), “ and so ” should be “for all , and so ” and similarly for .
- Page 109: After (5.13), a “the” is missing in “takes values in obeys the Lipschitz bound”.
- Pages 115-116: After (5.20), should lie in rather than , so references to the latter should be replaced with the former. In the first display of page 116, an is missing in the integrand, and the second should just be .
- Page 117: In the proof of Proposition 5.5.1: “ is small enough” should be “ are small enough”.
- Page ???: For Exercise 6.1.2(ii), add the additional hint: “It is somewhat tricky to establish that is NSS (and hence Lie). To do this, lift an NSS open neighbourhood of the identity in to an open set in containing with the property that any subgroup of contained in is in fact contained in . Use an intersection of finitely many conjugates of to establish the NSS property for . For part (iii), add the additional hint: “Argue as in Exercise 4.2.9, but working with the NSS property instead of Cartan’s theorem, and the open mapping theorem for topological groups instead of the fact that a continuous injection from compact spaces to Hausdorff spaces is a homeomorphism onto the image”.
- Page ???: Replace Exercise 6.1.3 and the paragraph ensuing with “Exercise: Suppose we iterate the above maps and pass to the direct limit as sets (defined similarly to inverse limits, but with all arrows reversed), identified with in the obvious fashion. Show that for all , the canonical maps to the direct limit are continuous when is given the compact-open topology. Use this together with the exponential map and the evaluation map from to to show that there exists a continuous injective map from an open neighbourhood of the identity in to that is a right inverse of theq uotient map from to on this neighbourhood.”
- Page 181: In remark 8.2.3, it should be added that in the more general non-symmetric case discussed here, needs to be replaced by . Also, in the analogue of Exercise 8.2.2 in this more general case, needs to be replaced by an arbitrary translate of .
- Page 302: In most of this section should be and vice versa (also the subscript of by is missing in several places).

Thanks to Michael Cowling, Frederik vom Ende, Mikhail Katz, Zhigang Li, Hagen Papenburg, Lam Pham, Arturo Rodríguez Fanlo, Fan Zheng, and an anonymous contributor for corrections.

## 12 comments

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27 March, 2012 at 11:00 am

Hilbert’s fifth problem and related topics « What’s new[…] finished the first draft of the the first of my books based on my 2011 blog posts, entitled “Hilbert’s fifth problem and related topics“, based on the lecture notes for my graduate course of the same name. The PDF of this draft […]

29 March, 2012 at 11:54 pm

Mads SørensenTypographical suggestions:

P. 5, l. 5 in the proof:

The ‘s in shouldn’t be in italic mode.

P. 6, l. 2 in exercise 1.1.4:

The ‘s in shouldn’t be in italic mode.

P. 8, l. 1 after the two times two matrix:

Is the ‘s in supposed to be in italic mode?

P. 17, footnote 4:

Use \dots instead of “…” and remember a comma after the definition of the group G_{3}.

In general:

When typing a map, use \colon instead of a normal colon. (The spacing is incorrect.) Also, use \coloneqq from the mathtools package instead of “:=”.

[Thanks, this will be incorporated into the next revision of the ms. -T.]1 February, 2013 at 10:19 am

Small doubling in groups « What’s new[…] abelian case is discussed in this book of mine with Vu, and the nonabelian case discussed in this more recent book of mine), but instead focuses on the statements of the various known results, as well as some remaining […]

7 December, 2013 at 4:06 pm

Ultraproducts as a Bridge Between Discrete and Continuous Analysis | What's new[…] of Hilbert’s fifth problem (which classifies the latter) to study approximate groups; see this text of mine for more […]

8 July, 2014 at 11:34 am

Hilbert’s fifth problem and approximate groups | What's new[…] The slides cover essentially the same range of topics in this series of lecture notes, or in this text of mine, though of course in considerably less detail, given that the slides are meant to be presented in […]

18 February, 2016 at 1:38 pm

LamDear Terry,

I have a few questions on Section 8.2 (Sanders-Croot-Sisask-theory), related to remark 8.2.3.

– p.180: exercise 8.2.1. S^m is contained in A^2A^{-2}=A^4 if A is assumed either symmetric, or an approximate subgroup, but in remark 8.2.3, you mention that small doubling+finite+non-empty suffices. But in that case, we can’t get S^m contained in A^4, can we?

– p.180: exercise 8.2.2. If f is supported on A^2, Sf, Tf would be supported on A^3, in which case the small doubling assumption is not quite enough to guarantee the variance bound (we would need small tripling in full generality, of course if A is assumed to be an approximate group, that should be ok). It seemed that it is important that f be supported on gA (which is the case if f=1_{gA}), since then the support of Tf would be of size |A^2|<K|A|.

– p.181: you say that |A^2|/M 1_{y_iA} has an l^2-norm of O_K(|A|^3/2), but I found a bound depending on both K and M.

Is there a natural way that one can, without any symmetry assumption on A, have A^2A^{-2} contained in some higher iterated power A^k?

Thanks!

-L

18 February, 2016 at 3:55 pm

Terence TaoThanks for the corrections. In the non-symmetric case one indeed has to replace by . This need not be contained in for moderately large , even in the abelian case, consider e.g. the case when for some finite group and some , all of whose multiples lie outside of .

25 August, 2017 at 5:01 am

AnonymousHow much theory of Lie groups and Lie algebras do you assume for a student attending this course? In UCLA is the Lie theory a prerequisite for this course?

[Prior knowledge of Lie groups and Lie algebras would be helpful, but all the material required is covered in the text also. -T]13 October, 2019 at 11:44 pm

TomI have a question concerning the copyright of your Hilbert’s 5th problem book. Is it allowed to print copies of the version on this website for teaching purposes, or does this require permission e.g. from the AMS?

14 October, 2019 at 8:50 am

Terence TaoHmm, technically my contract gives the AMS control over “all methods of publication” of the book, though it is unclear whether to me if printing out multiple copies for teaching purposes counts as “publication”. I do have an explicit right to maintain a PDF copy on my web page and there should be no difficulty in directing students to that link where they can print a copy for their personal use.

19 October, 2019 at 2:29 pm

AnonymousHow can a print of the PDF file – for whatever reason – not be a copyrightable version? :-) (I myself am against copyright and I’m freeware enthusiast …)

12 March, 2023 at 5:04 am

LZG (@ZhigangLi_NK)Dear Mr.Tao :It’s my honor to find a typo.

Page 34,Exercise1.2.11(viii) I think (xy) should be (x+y)

[Erratum added, thanks – T.]