Expansion in finite simple groups of Lie type.

Terence TaoGraduate 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, “ pairs ” should be “ ordered pairs “, the right-hand side of in the next display should be , and the lower bound for should be rather than .
- Page ???: In the proof of Proposition 2.2.8, the Haar measure on needs to be Lebesgue measure divided by . The Haar measure of the should then be rather than , and similarly for the integral in the next display; but to compensate for this, the integral should be against rather than .
- Page ???: In Exercise 2.0.4, replace the hint with “you may assume without proof Petersen’s 2-factor theorem, which asserts that every -regular graph with even can be decomposed into 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: “ and ” should be “ and ”
- Page 93, Lemma 4.1.7, last line: “” should be “”.
- Page 95, Proof of Lemma 4.1.7, last line: “” should be “”
- Page 98, Proof of Lemma 4.2.2, eighth line: should be .
- Page 102, Exercise 5.0.1, seventh line: add right parenthesis after “should not actually depend on “. 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 , then and both lie in ” should read “If , then and both lie in “, and both occurrences of should be .
- 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:” can be covered by copies of ” should be “ can be covered by translates of , and we can restrict to those in since otherwise will not intersect “. Then, later in the proof, should be , should be , should be , and should be (three occurrences). In Exercise 5.3.2, add the hypothesis that is not the one-dimensional space spanned by the identity element.
- Page 122, Proof of Proposition 5.3.3, third line: “” should be ““
- Page 140, Exercise 6.1.3, part (ii): Add the hypothesis that is not the identity element of
- Page 140, last paragraph, second line: The definition of is missing a right parenthesis.
- Page 142, second paragraph, fifth line: “sufficiently small ” should be “sufficiently small “.
- Page 142: In (1.66), should be .
- Page 152, third line: “in the ” and “in the case”, should be interchanged.
- Page 155, Exercise 7.1.4, part (i), displayed equation: “” should be ““
- Page 155, Exercise 7.1.4, part (i): is , not
- 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: should be
- Pages 160-161, after the proof of Corollary 7.2.3: All appearances of and should instead be and .
- 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 and should be added.
- Page 252: In Corollary 11.7.2, the conclusions about certain combinations of 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

## 8 comments

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31 October, 2013 at 10:41 am

Expansion in finite simple groups of Lie type | What's new[…] As always, corrections or comments are greatly appreciated (and errata will be collected at this page). […]

6 November, 2013 at 6:32 am

pg261page 219 theorem 2.4.1 (iii) g x g has become g \to x g.

page 215 bottom, weird grammar “reduced to that of understanding that of its factors” (should be “that of understanding its factors, I guess).

[Corrected, thanks – T.]10 November, 2013 at 5:29 pm

Mikhail OstrovskiiSome minor corrections/suggestions:

Page 23, formula (1.10). Square the norm

Page 30, Exercise 1.2.11, hint. Replace the right-hand side by $2-2Re\langle v,w\rangle_H$.

2 lines below that `form’ should be `forms’

page 54,third line: ‘this set of notes’ should be `this section’

Lemma 1.3.3 and Exercise 1.3.7, $F$ should be $\mathbf{F}$

[Thanks, this will be corrected in the next revision of the ms – T.]31 March, 2014 at 3:11 pm

Wolfgang MoensOn page 37, just above proposition 1.2.22, the word “index” seems to be missing:

“… pass from a group to a finite subgroup as far as property (T) is concerned” > “… pass from a group to a finite-index subgroup as far as property (T) is concerned”.

[This will be corrected in the next revision of the ms, thanks – T.]29 April, 2016 at 1:33 pm

Sarah PeluseHere are some (potential) typos that I found. Maybe some of them are not typos and I am just confused. Sorry if this is not the right post to comment on. The page, theorem, paragraph, and line numbers are for the print GSM book.

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: Should “$g\in H$” be “$g\in G$”? It’s not clear to me that the $g$ found in the proof should be in $H$.

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: $\mu^{*n}$ is missing the left norm bar around it

Page 102, Exercise 5.0.1, seventh line: A close parenthesis is missing where the sentence ends in the middle of the line

Page 103-104, Proof of Theorem 5.0.2, last line of 103 and first two lines of 104: At this point in the proof we’ve already shown that $A$ is contained in a left coset $gH’$ of a subgroup $H’\leq G$ with $|H’|<2|A|$. Here $H'=g^{-1}Hg$ for some $g\in G$ and $H=AA^{-1}$, which we established earlier is a subgroup of $G$. First, I think that "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 $gH'$" in the sentence that starts on page 103 and ends on page 104. Continuing the argument, since $|A|=|Ah|$ and $|gH'|<2|A|$, $A\cap Ah$ must be nonempty. That is, we can write $a_1=a_2h$ for some $a_1,a_2\in A$. This implies that $a_2^{-1}a_1=h$, i.e. that $h\in A^{-1}A$. The first line on page 104 says that "$h\in AA^{-1}$", and unless I'm missing something this should read "$h\in A^{-1}A$". For the same reason, I think that in the next line "$AA^{-1}=H'$" should be "$A^{-1}A=H'$", especially since we want to conclude that $|H'|<\frac{3}{2}|A|$, so we presumably want to be able to use the bound for $|A^{-1}A|$ instead of the double counting bound for $|AA^{-1}|$.

Page 109, paragraph 3, first line and third-to-last line: Should "Lemma 5.1.3" instead be "Lemma 5.1.7" in these two spots? I can see how to prove these things by arguments similar to Lemma 5.1.7, but can't see how the proof of Lemma 5.1.3 might help.

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): Here $a$ should not be the identity element of $\Gamma$

Page 140, last paragraph, second line: The definition of $\mu$ is missing a close parenthesis

Page 142, second paragraph, fifth line: "sufficiently small $\mu$" should be "sufficiently small $c$"

Page 152, beginning of the third line: "in the $\mathcal{D}_{+}$" should be "in the $\mathcal{D}_{-}$ case"

Page 152, end of the third line: "in the $\mathcal{D}_{-}$ case" should be "in the $\mathcal{D}_{+}$ case"

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): $\sum_{p\leq x,p\equiv 1\pmod{4}}$ 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" or something

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: Here $\Lambda$ is some subgroup of $SL_2$ which is not virtually solvable and $\Lambda'\leq\Lambda$ is the free subgroup on two generators furnished by the Tits alternative, but in a lot of places $\Gamma$ and $\Gamma'$ are written instead of $\Lambda$ and $\Lambda'$. Sometimes both the $\Gamma$'s and $\Lambda$'s appear in the same sentence, like in the first sentence of the last paragraph of section 7.2. Maybe there is something I'm missing and $\Lambda$ and $\Gamma$ are supposed to be different things?

Page 162, last paragraph, first sentence: "Theorem 7.0.1" should maybe be "Theorem 7.3.1", since we just proved Theorem 7.0.1 assuming Theorem 7.3.1

Page 164, fourth line: "hyopthesis" should be "hypothesis"

[Corrections added, thanks – T.]29 April, 2016 at 4:01 pm

Sarah PeluseSorry, I forgot to write what is being summed (the reciprocal of the prime $p$) in my comment above about the sum in Exercise 7.1.4:

$\sum_{p\leq x,p\equiv 1\pmod{4}}\frac{1}{p}$ is $\frac{1}{2}\log\log{x}+O(1)$, not $\frac{1}{2}\log{x}+O(1)$

4 January, 2017 at 1:03 am

DoronHi,

Page 121, proof of Lemma 5.3.2:

You write “As A^2 can be covered by K copies of A, we can find a\in A such that …” (just before equation (5.6)). I can see why you can find such g\in G, but not why you can find a\in A. (The translates of A are by arbitrary elements of the group, aren’t they?)

Also, in Exercise 5.3.2, shouldn’t the vector space spanned by the identity be excluded?

[Corrections added, thanks – T.]7 January, 2017 at 3:02 pm

AlonOn page 124, in the proof of Lemma 5.3.4, you write “One has $|gA \cap aTa^{-1}| \gg K^{-O(1)} |A|^{1/3}$ for some $g \in G$, which implies $|A^2 \cap aTa^{-1}| \gg K^{-O(1)} |A|^{1/3}$.

How is this implied?