Expansion in finite simple groups of Lie type.
Terence Tao

Graduate Studies in Mathematics, 164. American Mathematical Society, Providence, RI, 2015.

Last updated: Apr 10, 2021

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“, “Compactness and contradiction“, “Hilbert’s fifth problem and related topics“, and “Spending symmetry“.

A draft copy of the book can be found here.

Errata:

• Page ???: at the end of the implication of (ii) from (i) in Proposition 1.2.1, “ $\varepsilon k |F|/2$ pairs $v,w$” should be “ $\varepsilon k |F|/2$ ordered pairs $(v,w)$“, the right-hand side of $\varepsilon |F|/2$ in the next display should be $\varepsilon k |F|/4$,  and the lower bound for $h(G_n)$ should be $\varepsilon k/4$ rather than $\varepsilon/2$.
• Page ???: In the proof of Proposition 2.2.8, the Haar measure on $GL_d$ needs to be Lebesgue measure divided by $|v_1 \wedge \dots \wedge v_d|$.  The Haar measure of the $w_1,\dots,w_{d-1}$ should then be $O(1)$ rather than $O(1/|v_d|)$, and similarly for the integral in the next display; but to compensate for this, the integral should be against $\frac{dv_d}{|v_d|}$ rather than $dv_d$.
• Page ???: In Exercise 2.0.4, replace the hint with “you may assume without proof Petersen’s 2-factor theorem, which asserts that every $k$-regular graph with $k$ even can be decomposed into $k/2$ edge-disjoint $2$-regular graphs. Now use the previous example.”.
• Page 51: Before Exercise 2.3.2, “total mass” should be “total variation”,
• Page 74, Remark 3.3.7, second line: “(pseudo-)randomly sowing” should be “(pseudo-)randomly sewing”
• Page 89, Lemma 4.1.3, fourth line: “ $a\in A$ and $b\in B$” should be “ $a\in A'$ and $b\in B'$
• Page 93, Lemma 4.1.7, last line: “ $g\in H$” should be “ $g\in G$”.
• Page 95, Proof of Lemma 4.1.7, last line: “ $|gH\cap A|\ll K^{-O(1)}|A|$” should be “ $|gH\cap A|\gg K^{-O(1)}|A|$
• Page 98, Proof of Lemma 4.2.2, eighth line: $\mu^{*n} \|)_\infty$ should be $\|\mu^{*n} \|)_\infty$.
• Page 102, Exercise 5.0.1, seventh line: add right parenthesis after “should not actually depend on $d$“.  Also, “Girth bound” should be “Diameter bound”.
• Page 103-104, Proof of Theorem 5.0.2, last line of 103 and first two lines of 104: “If $h\in H'$, then $A$ and $Ah$ both lie in $H'$” should read “If $h\in H'$, then $A$ and $Ah$ both lie in $gH'$“, and both occurrences of $AA^{-1}$ should be $A^{-1} A$.
• Page 109, paragraph 3, first line and third-to-last line: “Lemma 5.1.3” should be “Lemma 5.1.7”
• Page 121, proof of Lemma 5.3.2:” $A^2$ can be covered by $K$ copies of $A$” should be “ $A^2$ can be covered by $K$ translates $aA$ of $A$, and we can restrict to those $a$ in $A^3$ since otherwise $aA$ will not intersect $A$“.  Then, later in the proof, $a \in A$ should be $a \in A^3$, $A^7$ should be $A^{10}$, $K^6 |A|$ should be $K^9 |A|$, and $K^{-2C-11}$ should be $K^{-2C-14}$ (three occurrences). In Exercise 5.3.2, add the hypothesis that $V$ is not the one-dimensional space spanned by the identity element.
• Page 122, Proof of Proposition 5.3.3, third line: “ $|A^2\cap B|\leq K^{-C}|B|$” should be “ $|A^2\cap B|\leq K^{-C}|A|$
• Page 140, Exercise 6.1.3, part (ii): Add the hypothesis that $a$ is not the identity element of $\Gamma$
• Page 140, last paragraph, second line: The definition of $\mu$ is missing a right parenthesis.
• Page 142, second paragraph, fifth line: “sufficiently small $\mu$” should be “sufficiently small $c$“.
• Page 142: In (1.66), $D_+$ should be ${\mathcal D}_+$.
• Page 152,  third line: “in the $\mathcal{D}_{+}$” and “in the $\mathcal{D}_{-}$ case”, should be interchanged.
• Page 155, Exercise 7.1.4, part (i), displayed equation: “ $\pi_2(x,z)$” should be “ $\pi_{*}(x,z)$
• Page 155, Exercise 7.1.4, part (i): $\sum_{p\leq x,p\equiv 1\pmod{4}} \frac{1}{p}$ is $\frac{1}{2}\log\log{x}+O(1)$, not $\frac{1}{2}\log{x}+O(1)$
• Page 155, Exercise 7.1.4, part (ii): “infinitely natural numbers” should be “infinitely many natural numbers”
• Page 157, Theorem 7.2.1, second-to-last line: $K_{\phi(i+1)}/K_{\phi(i)}$ should be $K_{\sigma(i+1)}/K_{\sigma(i)}$
• Pages 160-161, after the proof of Corollary 7.2.3: All appearances of $\Gamma$ and $\Gamma'$ should instead be $\Lambda$ and $\Lambda'$.
• Page 162, last paragraph, first sentence: “Theorem 7.0.1” should be “Theorem 7.3.1”
• Page 164, fourth line: “hyopthesis” should be “hypothesis”
• Page 208: In Exercise 10.1.2, “defined as the closure” should be “defined as the completion”.
• Page 213: In Exercise 10.2.1(i), “Liouville’s theorem” should be “the maximum principle”, and the phrase “viewed as a function of $x$ and $t$ should be added.
• Page 252: In Corollary 11.7.2, the conclusions about certain combinations of $\alpha,\beta$ not being roots is incorrect and should be deleted.
• Page 254: In Lemma 11.7.8, a fourth case needs to be excluded involving three chains of simple edges of length 1,3,3, which can be treated by essentially the same argument as in (c) (or (b)).

Thanks to Jean-Phillipe Burelle, Justin Ciecerbach, Claudeh5, Pietro Gheri, Constantin Kogler, Sarah Peluse, Doron Puder, Daniel Raban, Matthew Tointon, and an anonymous commenter for corrections.

A (second) draft version of the MS can be found here.  Note that the section numbering in the published version differs from that in the draft.F