Grassmannian fixed point ratios
Let \(G\) be a group of permutations on the finite set \(\Omega\) of cardinality \(n\). If \(g\) is a nonidentity element of \(G\), let \(f(g)=|\Omega^g|/n\), where \(\Omega^g\) is the set of fixed points of \(g\). This has long been studied, particularly in the case of primitive permutation groups. If \(G\) is primitive, these numbers are typically very close to \(0\) unless we are considering natural actions. See [\textit{M. W. Liebeck} and \textit{A. Shalev}, J. Am. Math. Soc. 12, No.~2, 497-520 (1999; Zbl 0916.20003)] on asymptotic results for actions other than these natural ones. This paper provides a major contribution to the subject. These types of results have had many applications to many different problems in group theory -- in particular to probabilistic methods. In this paper, the authors study the case where \(G\) is a finite almost simple classical group (i.e., linear, unitary, symplectic or orthogonal) and \(\Omega\) is taken to be an orbit of nonsingular or totally singular subspaces for \(G\) in the natural module. They then give quite good estimates for \(f(g)\) in terms of the size of the field and dimensions of eigenspaces of \(g\). Let \(n\) be the dimension of the natural module and suppose the orbit consists of subspaces of dimension or codimension \(k\leq n/2\). They show that typically \(f(g)<(4000+150n)/q^{(n-6)/2}\) or \(g\) has an eigenspace of codimension \(d\) with \(dk<n/2\) and \(f(g)<q^{-dk}+2000/q^{(n-6)/2}\). For small values of \(n\) and \(q\), one can do a bit better, but asymptotically these results are quite sharp. One motivation for these results was a conjecture of the reviewer and \textit{J. G. Thompson} [J. Algebra 131, No.~1, 303-341 (1990; Zbl 0713.20011)] about monodromy groups of coverings of Riemann surfaces. The conjecture is that for a fixed genus \(g\), the nonabelian simple composition factors of the monodromy group of a cover \(\phi\colon X\to Y\) with \(X\) of genus \(g\) are either alternating groups or are in a finite set \(E(g)\). In a forthcoming paper [Ann. Math., II. Ser. (to appear)] the authors complete the proof of this conjecture using the results of this paper (as well as the work of several other articles).
scientific article; zbMATH DE number 1547909