Abstract
By now, we have a product theorem in every finite simple group G of Lie type, with the strength of the bound depending only in the rank of G . Such theorems have numerous consequences: bounds on the diameters of Cayley graphs, spectral gaps, and so forth. For the alternating group Alt n , we have a quasipolylogarithmic diameter bound (Helfgott-Seress 2014), but it does not rest on a product theorem. We shall revisit the proof of the bound for Alt n , bringing it closer to the proof for linear algebraic groups, and making some common themes clearer. As a result, we will show how to prove a product theorem for Alt n – not of full strength, as that would be impossible, but strong enough to imply the diameter bound.
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