Diameter of Cayley graphs of \(\mathrm{SL}(n,p)\) with generating sets containing a transvection. (English) Zbl 1473.20053
Let \(G\) be a group given by a generating set \(X\). Denote by \(\mathrm{diam}(G)\) the diameter of \(G\), i.e. the maximum of diameters of all Cayley graphs \(\mathrm{Cay}(G,X)\) where \(X\) runs through all generating sets of \(G\). A well-known conjecture of Babai states that if \(G\) is a non-abelian finite simple group then \(\mathrm{diam}(G) < (\mathrm{log}|G|)^c\) for some absolute constant \(c\). For the case of \(\mathrm{SL}(2, p)\), the conjecture was proved by H. A. Helfgott [Ann. Math. (2) 167, No. 2, 601–623 (2008; Zbl 1213.20045)]. For finite simple groups of Lie type of bounded rank, the conjecture was verified by L. Pyber and E. Szabó [J. Am. Math. Soc. 29, No. 1, 95–146 (2016; Zbl 1371.20010)], and independently by E. Breuillard et al. [Geom. Funct. Anal. 21, No. 4, 774–819 (2011; Zbl 1229.20045)]. For classical groups of unbounded rank, the conjecture remains open.
In this pape,r it is proved that if a generating set \(X\) of \(G = \mathrm{SL}(n, p)\) contains a transvection then \(\mathrm{diam}(\mathrm{Cay}(G, X)) = \textit{O}((\mathrm{log \, p})^cn^{13})\) for some absolute constant \(c\) (Theorem 1.3). Furthermore, it is shown that if \(K\) is an arbitrary field and a generating set \(X\) of \(\mathrm{SL}(n, K)\) contains \(\{1+ \lambda x \mid \lambda \in K^*\}\), where \(t = 1+x\) is a transvection, then \(\mathrm{diam}(\mathrm{Cay}(G, X)) = \textit{O}(n^{11})\) (Theorem 1.5).
In this pape,r it is proved that if a generating set \(X\) of \(G = \mathrm{SL}(n, p)\) contains a transvection then \(\mathrm{diam}(\mathrm{Cay}(G, X)) = \textit{O}((\mathrm{log \, p})^cn^{13})\) for some absolute constant \(c\) (Theorem 1.3). Furthermore, it is shown that if \(K\) is an arbitrary field and a generating set \(X\) of \(\mathrm{SL}(n, K)\) contains \(\{1+ \lambda x \mid \lambda \in K^*\}\), where \(t = 1+x\) is a transvection, then \(\mathrm{diam}(\mathrm{Cay}(G, X)) = \textit{O}(n^{11})\) (Theorem 1.5).
Reviewer: Alla Detinko (Hull)
MSC:
| 20G40 | Linear algebraic groups over finite fields |
| 05C25 | Graphs and abstract algebra (groups, rings, fields, etc.) |
| 20D06 | Simple groups: alternating groups and groups of Lie type |
| 20F05 | Generators, relations, and presentations of groups |
References:
| [1] | Babai, L., On the diameter of Eulerian orientations of graphs, (Proc. 17th Ann. Symp. on Discrete Algorithms (SODA’06) (2006), ACM-SIAM), 822-831 · Zbl 1192.05084 |
| [2] | Babai, L.; Seress, Á., On the diameter of permutation groups, Eur. J. Comb., 13, 231-243 (1992) · Zbl 0783.20001 |
| [3] | Babai, L.; Beals, R.; Seress, Á., On the diameter of the symmetric group: polynomial bounds, (Proceedings of the Fifteenth Annual ACM-SIAM Symposium on Discrete Algorithms (2004), ACM: ACM New York), 1108-1112 · Zbl 1318.20002 |
| [4] | Bamberg, J.; Gill, N.; Hayes, T. P.; Helfgott, H. A.; Seress, Á.; Spiga, P., Bounds on the diameter of Cayley graphs of the symmetric group, J. Algebraic Comb., 40, 1-22 (2014) · Zbl 1297.05107 |
| [5] | Breuillard, E.; Green, B.; Tao, T., Approximate subgroups of linear groups, Geom. Funct. Anal., 21, 774-819 (2011) · Zbl 1229.20045 |
| [6] | Gorenstein, D., Finite Groups (1980), Chelsea Publishing Co.: Chelsea Publishing Co. New York · Zbl 0185.05701 |
| [7] | Halasi, Z.; Maróti, A.; Pyber, L.; Qiao, Y., An improved diameter bound for finite simple groups of Lie type, Bull. Lond. Math. Soc., 51, 645-657 (2019) · Zbl 1454.20092 |
| [8] | Helfgott, H. A., Growth and generation in \(S L_2(\mathbb{Z} / p \mathbb{Z})\), Ann. Math. (2), 167, 601-623 (2008) · Zbl 1213.20045 |
| [9] | Helfgott, H. A., Growth in linear algebraic groups and permutation groups: towards a unified perspective, (Campbell, C. M.; Parker, C. W.; Quick, M. R.; Robertson, E. F.; Roney-Dougal, C. M., Groups St Andrews 2017 in Birmingham (2019), Cambridge University Press: Cambridge University Press Cambridge), 300-345 · Zbl 1514.20190 |
| [10] | Helfgott, H. A.; Seress, Á., On the diameter of permutation groups, Ann. Math. (2), 179, 611-658 (2014) · Zbl 1295.20027 |
| [11] | Humphries, S. P., Generation of special linear groups by transvections, J. Algebra, 99, 480-495 (1986) · Zbl 0587.20030 |
| [12] | Kowalsky, E., Explicit growth and expansion for \(S L_2\), Int. Math. Res. Not., 2013, 24, 5645-5708 (2013) · Zbl 1353.20036 |
| [13] | McLaughlin, J., Some groups generated by transvections, Arch. Math., 18, 364-368 (1967) · Zbl 0232.20084 |
| [14] | McLaughlin, J., Some subgroups of \(S L_n( \mathbb{F}_2)\), Ill. J. Math., 13, 108-115 (1969) · Zbl 0179.04901 |
| [15] | Pyber, L.; Szabó, E., Growth in finite simple groups of Lie type, J. Am. Math. Soc., 29, 95-146 (2016) · Zbl 1371.20010 |
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