Growth estimates and diameter bounds for classical Chevalley groups
Duration: 57 mins 35 secs
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| Description: |
Harald Helfgott (CNRS (Centre national de la recherche scientifique); Georg-August-Universität Göttingen)
02/06/2022 Programme: GRA2 SemId: 36226 |
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| Created: | 2022-06-08 11:49 |
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| Collection: |
Groups, representations and applications: new perspectives
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| Publisher: | Harald Helfgott |
| Copyright: | Isaac Newton Institute |
| Language: | eng (English) |
| Distribution: |
World
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| Categories: |
iTunes - Science iTunes - Mathematics - Advanced Mathematics |
| Explicit content: | No |
| Aspect Ratio: | 16:9 |
| Screencast: | No |
| Bumper: | UCS Default |
| Trailer: | UCS Default |
| Abstract: | 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) |
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