Abstract
Let G = {text {SCl}}_n(q) be a quasisimple classical group with n large, and let x_1, ldots , x_k in G be random, where k ge q^C. We show that the diameter of the resulting Cayley graph is bounded by q^2 n^{O(1)} with probability 1 - o(1). In the particular case G = {text {SL}}_n(p) with p a prime of bounded size, we show that the same holds for k = 3.
Highlights
Let G be a group and S a symmetric (S = S−1) subset of G
Having a classification of finite simple groups (CFSG)-free method is valuable for transparency, but we think it is essential for attacking Babai’s conjecture
We proceed (Sect. 4) by showing that a given short word w evaluated at random elements x1, . . . , xk ∈ G almost surely has large support (Theorem 4.2). This is a kind of antithesis to step 1 of Pyber’s programme: all sufficiently short words in random generators will fail to have degree (1 − )n
Summary
Having a CFSG-free method is valuable for transparency, but we think it is essential for attacking Babai’s conjecture It is well-known that two random elements of SCln(q) almost surely generate the group: this is a result of Kantor and Lubotzky [30]. Xk ∈ G almost surely has large support (Theorem 4.2) This is a kind of antithesis to step 1 of Pyber’s programme: all sufficiently short words in random generators will fail to have degree (1 − )n. For each classical group we find a large normal set C, all of whose fibres over Gab are large (allowing us to ignore linear characters), and a small integer m such that for every g ∈ C the power gm has minimal degree in SCln(q) This completes the proof of Theorem 1.1. For independent interest and for motivation, these are presented in Appendix A
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