Hilbert’s fifth problem and related topics.
Terence Tao

Graduate 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, $\varepsilon$ should be taken to be $1/10n$, rather than $1/10d$.
• Page 10: In Exercise 1.1.7, $(x+1,0)$ should be $(x+1,1)$, and $(x,1)$ should be $(x,0)$.
• 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 $x - x^2 + x^3 -\dots$ should be $-x + x^2 - x^3 + \dots$.
• Page ???: In Exercise 1.2.11(viii), $xy$ should be $x*y$ (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, $G$ should be assumed to be Hausdorff.
• Page 105: After (5.9), “ $\|g^n\| \leq \varepsilon$ and so $g^n \in B(0,\varepsilon)$” should be “for all $m \leq n$, $\|g^n\| \leq \varepsilon$ and so $g^n \in B(0,\varepsilon)$” and similarly for $(g^y)^n \in B(0,5\varepsilon)$.
• Page 109: After (5.13), a “the” is missing in “takes values in $[0,1]$ obeys the Lipschitz bound”.
• Pages 115-116: After (5.20), $q$ should lie in $Q$ rather than $Q[V]$, so references to the latter should be replaced with the former.  In the first display of page 116, an $\eta$ is missing in the integrand, and the second $\partial_{q^i}$ should just be $\partial_q$.
• Page 117: In the proof of Proposition 5.5.1: “ $\varepsilon$ is small enough” should be “ $U_1, B, \varepsilon$ are small enough”.
• Page ???: For Exercise 6.1.2(ii), add the additional hint: “It is somewhat tricky to establish that $G/K$ is NSS (and hence Lie).  To do this, lift an NSS open neighbourhood of the identity in $G'/K'$ to an open set $W'$ in $G'$ containing $K'$ with the property that any subgroup of $G'$ contained in $W'$ is in fact contained in $K'$.  Use an intersection of finitely many conjugates of $W'$ to establish the NSS property for $G/K$.  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 $L(G)$ in the obvious fashion.  Show that for all $n$, the canonical maps to the direct limit $L(L_n) \to L(G)$ are continuous when $L(G)$ is given the compact-open topology.  Use this together with the exponential map $\exp: L(L_n) \to L_n$ and the evaluation map from $L(G)$ to $G$ to show that there exists a continuous injective map $s_n$ from an open neighbourhood of the identity in $L_n$ to $G$ that is a right inverse of theq uotient map from $G$ to $L_n$ on this neighbourhood.”
• Page 181: In remark 8.2.3, it should be added that in the more general non-symmetric case discussed here, $A^4$ needs to be replaced by $A^2 \cdot (A^{-1})^2$.   Also, in the analogue of Exercise 8.2.2 in this more general case, $A^2$ needs to be replaced by an arbitrary translate $g \cdot A$ of $A$.
• Page 302: In most of this section $\rho_\xi(g)$ should be $\rho_\xi(g)^*$ and vice versa (also the subscript of $\rho$ by $\xi$ 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.