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Hilbert’s fifth problem and related topics

Terence Tao · original page

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. It received the 2015 PROSE award in the mathematics category.

• Online draft
• Publisher (AMS)
• Lecture notes

Errata

13 corrections · 3 awaiting a page number

1. p. 16In Example 1.2.5, the series $x - x^2 + x^3 -\dots$ should be $-x + x^2 - x^3 + \dots$.
2. p. ?In Exercise 1.2.11(viii), $xy$ should be $x*y$ (two occurrences).
3. p. 18In Definition 1.2.8, all occurrences of G should be replaced with M.
4. p. 54In Exercise 3.0.8, $G$ should be assumed to be Hausdorff.
5. p. 105After (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 $(g^y)^n \in B(0,5\varepsilon)$.
6. p. 109After (5.13), a “the” is missing in “takes values in $[0,1]$ obeys the Lipschitz bound”.
7. p. 115-116After (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 $\partial_q$.
8. p. 117In the proof of Proposition 5.5.1: “$\varepsilon$ is small enough” should be “$U_1, B, \varepsilon$ are small enough”.
9. p. ?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”.
10. p. ?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.”
11. p. 181In 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 $A$.
12. p. 270In the proof of Lemma 2.7.6, “dense subset of the preimage” should be “open subset of the preimage”.
13. p. 302In 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).

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

4 corrections

1. p. 5In the proof of Theorem 1.1.2, $\varepsilon$ should be taken to be $1/10n$, rather than $1/10d$.
2. p. 10In Exercise 1.1.7, $(x+1,0)$ should be $(x+1,1)$, and $(x,1)$ should be $(x,0)$.
3. p. 20In Problem 1.1.14, the hypothesis that G has polynomial growth is missing and should be inserted.
4. p. 79In the last paragraph, a right parenthesis is missing after “Exercise 1.4.3”.

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Last updated: July 9, 2026.