Growth estimates and diameter bounds for untwisted classical groups
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866916520923234304 |
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| author | Bajpai, Jitendra Dona, Daniele Helfgott, Harald Andrés |
| author_facet | Bajpai, Jitendra Dona, Daniele Helfgott, Harald Andrés |
| contents | 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 untwisted classical group $G$ of rank $r$ defined over a field $\mathbb{F}_{q}$ with $q$ not too small with respect to $r$, \begin{equation*} \mathrm{diam}(G(\mathbb{F}_{q}))\leq(\log|G(\mathbb{F}_{q})|)^{408r^{4}}. \end{equation*} This bound improves on results by Breuillard, Green, and Tao [9], Pyber and Szabó [38], and, for $q$ large enough, also by Halasi, Maróti, Pyber, and Qiao [16]. Our approach is in several ways closer to that of preexistent work by Helfgott [20], in that we give dimensional estimates (that is, bounds of the form $|A\cap V(\mathbb{F}_{q})|\ll|A^{C}|^{\dim(V)/\dim(G)}$, where $A$ is any generating set) for varieties $V$ of specific types, and work in the Lie algebra whenever possible. One of our main tools is a new, more efficient form of escape from subvarieties. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2110_02942 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Growth estimates and diameter bounds for untwisted classical groups Bajpai, Jitendra Dona, Daniele Helfgott, Harald Andrés Group Theory Algebraic Geometry Combinatorics 20F69, 20G40, 05C25 (Primary), 14A10, 05C12 (Secondary) 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 untwisted classical group $G$ of rank $r$ defined over a field $\mathbb{F}_{q}$ with $q$ not too small with respect to $r$, \begin{equation*} \mathrm{diam}(G(\mathbb{F}_{q}))\leq(\log|G(\mathbb{F}_{q})|)^{408r^{4}}. \end{equation*} This bound improves on results by Breuillard, Green, and Tao [9], Pyber and Szabó [38], and, for $q$ large enough, also by Halasi, Maróti, Pyber, and Qiao [16]. Our approach is in several ways closer to that of preexistent work by Helfgott [20], in that we give dimensional estimates (that is, bounds of the form $|A\cap V(\mathbb{F}_{q})|\ll|A^{C}|^{\dim(V)/\dim(G)}$, where $A$ is any generating set) for varieties $V$ of specific types, and work in the Lie algebra whenever possible. One of our main tools is a new, more efficient form of escape from subvarieties. |
| title | Growth estimates and diameter bounds for untwisted classical groups |
| topic | Group Theory Algebraic Geometry Combinatorics 20F69, 20G40, 05C25 (Primary), 14A10, 05C12 (Secondary) |
| url | https://arxiv.org/abs/2110.02942 |