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arXiv · 1804.03049

Growth in linear algebraic groups and permutation groups: towards a unified perspective

Harald A. Helfgott
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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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Harald A. Helfgott. 2018-04-09. Growth in linear algebraic groups and permutation groups: towards a unified perspective. https://arxiv.org/abs/1804.03049

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