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(This is in part a request for references and in part a somewhat pedagogical question.)

I gave a course on expanders seven years ago, and I am giving a course on expanders again now. We will soon do Margulis's constructions of expander graphs. I'm minded to first give a self-contained proof in one class (expansion for Schreier graphs of $\mathrm{SL}_2(\mathbb{Z}/N\mathbb{Z})\ltimes (\mathbb{Z}/N \mathbb{Z})^2$, following Gabber-Galil and Jimbo-Maruocka's new proofs of Margulis's results, but in a slightly more abstract way - no reason not to use the Ping-Pong lemma and so forth), then do expansion for Cayley graphs of $\mathrm{SL}_3(\mathbb{Z}/N \mathbb{Z})$ (as I did seven years ago), and then do property $T$.

(a) Whom should I cite for the proof of expansion for Cayley graphs of $\mathrm{SL}_3(\mathbb{Z}/N\mathbb{Z})$ based on Gabber-Galil/Jimbo-Maruocka (as opposed to the usual proof via property $(T)$ for $\mathrm{SL}_3(\mathbb{R})$? (Or did I come up with it myself (not very hard, given what I was given) only to then forget about it?)

(b) Proving full Property $T$ (first for $\mathrm{SL}_2(\mathbb{R})\ltimes \mathbb{R}^2$, then for $\mathrm{SL}_3(\mathbb{R})$) in this way doesn't seem hard, but surely that is also standard?

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    $\begingroup$ $\mathrm{SL}_2(\mathbb{Z}) \ltimes \mathbb{Z}^2$ does not have Property T (but it indeed has Property $\tau$ with respect to congruence subgroups — for $\mathrm{SL}_2(\mathbb{Z})$ this is due to Brooks, I think) $\endgroup$
    – YCor
    Nov 28, 2022 at 16:53
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    $\begingroup$ Ah - surely it has relative property $T$? $\endgroup$
    – H A Helfgott
    Nov 28, 2022 at 16:58
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    $\begingroup$ Yes. But this is not enough to prove the expansion for $\mathrm{SL}_2(\mathbf{Z}/n\mathbf{Z})\ltimes (\mathbf{Z}/n\mathbf{Z})^2$. For this, at the left-hand side you really need Property $\tau$ wrt congruence subgroups. $\endgroup$
    – YCor
    Nov 28, 2022 at 17:01
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    $\begingroup$ Yes, going from relative property T to property T this way is standard. $\endgroup$
    – Moishe Kohan
    Nov 28, 2022 at 17:02
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    $\begingroup$ @MoisheKohan Right - this is clearly a proof of expansion for $\mathrm{SL}_3(\mathbb{Z}/N\mathbb{Z})$, starting from expansion for the Schreier graph of $\mathrm{SL}_2(\mathbb{Z}/N\mathbb{Z}) \ltimes (\mathbb{Z}/N\mathbb{Z})^2$, modelled after the usual proof of property $T$ starting from relative property $T$. But is this proof (a minor variation on something standard) due to my past self, or to someone else I should credit? I don't know! $\endgroup$
    – H A Helfgott
    Nov 28, 2022 at 18:46

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