Fixed point ratios in actions of finite classical groups. II. (Q875077)

This is the second in a series of four papers on fixed point ratios for actions of classical groups [see \textit{T. C. Burness}, part I, ibid. 309, No. 1, 69-79 (2007; see the preceding review Zbl 1128.20003), for more details on the set-up]. The main theorem of these papers is the following: if \(G\) is a finite almost simple classical group and \(\Omega\) is a faithful transitive non-subspace \(G\)-set, then either \(\text{fpr}(x)\lesssim|x^G|^{-\frac 12}\) for all \(x\in G\) of prime order, or the pair \((G,\Omega)\) is one of a small number of known exceptions. The proof of this result first reduces to primitive actions and then considers the different possibilities for the stabiliser \(G_\omega\) of a point \(\omega\in\Omega\); in the paper under review the result is proved when \(G_\omega\) lies in a maximal subgroup from one of the Aschbacher families \(\mathcal C_i\), where \(4\leq i\leq 8\) [see \textit{M. Aschbacher}, Invent. Math. 76, 469-514 (1984; Zbl 0537.20023)]. Prior to the main proof, the author includes a section containing many preliminary results which may be of interest independent of their applications in these papers.