So the lecture notes I posted a year ago giving a “simpler proof” of expansion for the Margulis-Gabber-Galil graphs turns out to be very similar to arguments of M. Berger (1991) and Y. Shalom (1999) for proving that has property (T). See Section 4.2 of the book of Bekka, de la Harpe, and Valette.

Anyone interested in expander graphs should take a look at Amir Yehudayoﬀ’s recent article in the SIGACT newsletter: Proving expansion in three steps. Amir uses a ping-pong lemma argument (which is essentially the same as the “combinatorial lemma” in the preceding post) to exemplify the “opening” of the three part strategy of Bourgain and Gamburd to proving expansion in Cayley graphs.

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