We prove that all finite simple groups of Lie type, with the exception of the Suzuki groups, can be made into a family of expanders in a uniform way. This confirms a conjecture of Babai, Kantor and Lubotzky from 1989, which has already been proved by Kassabov for sufficiently large rank. The bounded rank case is deduced here from a uniform result for SL2 which is obtained by combining results of Selberg and Drinfeld via an explicit construction of Ramanujan graphs by Lubotzky, Samuels and Vishne.
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Alexander Lubotzky, Finite simple groups of Lie type as expanders. J. Eur. Math. Soc. 13 (2011), no. 5, pp. 1331–1341DOI 10.4171/JEMS/282