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SUMMARY:Growth estimates and diameter bounds for classical Chevalley group
 s - Harald Helfgott (CNRS (Centre national de la recherche scientifique)\,
  Georg-August-Universität Göttingen)
DTSTART:20220602T150000Z
DTEND:20220602T160000Z
UID:TALK174974@talks.cam.ac.uk
DESCRIPTION:Babai's conjecture states that\, for any finite simple non-abe
 lian group G\, the diameter of G is bounded by (log |G|)^C for some absolu
 te 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\,\n&nbs
 p\;\ndiam(G(F_q)) <= (log |G(F_q)|)^{1947 r^4 log 2r}.\n&nbsp\;\nThis boun
 d improves on results by Breuillard-Green-Tao and Pyber-Szab&oacute\;\, an
 d\, for q large enough\, also on Halasi-Mar&oacute\;ti-Pyber-Qiao. Our bou
 nd is achieved by way of giving dimensional estimates for certain subvarie
 ties of G\, i.e. estimates of the form |A&cap\;V(F_q)| << |A^C|^{dim(V)/di
 m(G)} valid for all generating sets A. We also provide an explicit dimensi
 onal estimate for general subvarieties of G.\n&nbsp\;(joint with D. Dona a
 nd J. Bajpai)
LOCATION:Seminar Room 1\, Newton Institute
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