Fixed point ratios, character ratios, and Cayley graphs (Q2704359)

The paper offers an excellent survey of recent results on fixed point ratios in primitive permutation representations, and character ratios in irreducible complex representations of finite simple groups. Various important applications are also discussed.NEWLINENEWLINENEWLINELet \(G\) be a finite primitive permutation group of degree \(n\). The fixity \(\text{fix}(G)\) is defined to be the maximum number of points fixed by a nontrivial element in \(G\), and the fixed point ratio \(\text{rfix}(G)\) is \(\text{fix}(G)/n\). There are many results showing that \(\text{rfix}(G)\) is bounded away from \(1\) with some known exceptions. A recent result of the author and \textit{M. W. Liebeck} [J. Am. Math. Soc. 12, No. 2, 497-520 (1999; Zbl 0916.20003)] significantly refines the case of finite groups of Lie type. Namely, it is shown that there is \(\epsilon>0\) such that \(\text{rfix}(G)\leq q^{-\epsilon l}\), whenever \(G\) is a finite group of Lie type of Lie rank \(l\) and defined over \(\mathbb{F}_q\) and the primitive action is not a subspace action. This result allows the author and Liebeck to prove Cameron's conjecture on the base size of an almost simple primitive permutation group. Fixed point ratios results are also important in connection with the Guralnick-Thompson conjecture, posed in [\textit{R. M. Guralnick} and \textit{J. G. Thompson}, J. Algebra 131, No. 1, 303-341 (1990; Zbl 0713.20011)], which has now been proved thanks to works of Guralnick, Thompson, Aschbacher, Liebeck, Saxl, Frohardt, Magaard, and others.NEWLINENEWLINENEWLINENext the author discusses results concerning upper bounds for \(|\chi(x)/\chi(1)|\), where \(\chi\) runs over all nonprincipal irreducible complex characters of a finite simple group \(G\) and \(x\) runs over all nontrivial elements of \(G\). Such upper bounds have been obtained by Gluck for simple groups of Lie type and Roichman for alternating groups. The author then conjectures a very strong bound for these character ratios, which is similar to the aforementioned bound for fixed point ratios. This conjecture would lead to the best possible upper bound \(c\log|G|/\log|C|\) for the diameter of the Cayley graph \(\Gamma(G,C)\) (where the simple group \(G\) is generated by a single conjugacy class \(C\)), and a similar bound for the mixing time of random walks on \(\Gamma(G,C)\). For the theory of random walks on finite groups, see [\textit{P. Diaconis}, Group representations in probability and statistics, IMS Lecture Notes-Monograph Series, 11. Hayward, CA: Institute of Mathematical Statistics. vi, 198 p. (1988; Zbl 0695.60012)]. Whereas the bound for the mixing time cannot hold for arbitrary \(G,C\) (and the author proposes a conjecture giving the list of exceptions where the bound fails), he and \textit{M. W. Liebeck} have proved the bound for the diameter of Cayley graphs in the recent paper [Ann. Math. (2) 154, No. 2, 383-406 (2001; Zbl 1003.20014)]. This important results in turn leads to interesting consequences concerning random generation and covering numbers of finite simple groups, expanders, Waring-type theorems for finite simple groups, and subgroups of profinite groups.NEWLINENEWLINEFor the entire collection see [Zbl 0953.00031].