Growth estimates and diameter bounds for classical Chevalley groups

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• Harald Helfgott (CNRS (Centre national de la recherche scientifique), Georg-August-Universität Göttingen)
• Thursday 02 June 2022, 16:00-17:00
• Seminar Room 1, Newton Institute.

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GRA2 - Groups, representations and applications: new perspectives

Babai’s conjecture states that, for any finite simple non-abelian group G, the diameter of G is bounded by (log |G|)^{C for some absolute constant C. We prove that, for any classical Chevalley group G of rank r defined over a field F_q with q not too small with respect to r,   diam(G(F_q)) <= (log |G(F_q)|)}{1947 r^{4 log 2r}.   This bound improves on results by Breuillard-Green-Tao and Pyber-Szabó, and, for q large enough, also on Halasi-Maróti-Pyber-Qiao. Our bound is achieved by way of giving dimensional estimates for certain subvarieties of G, i.e. estimates of the form |A∩V(F_q)| << |A}C|^{dim(V)/dim(G)} valid for all generating sets A. We also provide an explicit dimensional estimate for general subvarieties of G.  (joint with D. Dona and J. Bajpai)

This talk is part of the Isaac Newton Institute Seminar Series series.

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