Ramanujan complexes and finite groups of Lie tupe as expanders

Alexander Lubotzky
The Hebrew University of Jerusalem

We will use Ramanujan graphs to show that all SL(2,Fq) can be made simoltnously expanders (for all prime powers q). Then Ramanujan complexes are used to make all SL(n,Fq) expanders (all n and all q). Togther with the work of Kassabov and Nikolov it leads to a proof that all non abelian finite simple groups (with the possible exception of the Ree groups) are uniformly expanders w.r.t a bounded number of generators.

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