Growth in linear algebraic groups and permutation groups: towards a unified perspective
Abstract
By now, we have a product theorem in every finite simple group of Lie type, with the strength of the bound depending only in the rank of . 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.
Cite
@article{arxiv.1804.03049,
title = {Growth in linear algebraic groups and permutation groups: towards a unified perspective},
author = {Harald A. Helfgott},
journal= {arXiv preprint arXiv:1804.03049},
year = {2018}
}
Comments
To appear in the proceedings of the St Andrews conference (2017). A paragraph (reiterating credit) was accidentally omitted from v2