Fixed point ratios for actions of finite classical groups, I

This is the first in a series of four papers on fixed point ratios in actions of finite classical groups. Our main result states that if G is a finite almost simple classical group and Ω is a faithful transitive non-subspace G-set then either fpr(x) < |x^G|^{-1/2} for all elements x ∈ G of prime order, or (G, Ω) is one of a small number of known exceptions. Here fpr(x) denotes the proportion of points in Ω which are fixed by x. In this introductory note we present our results and describe an application to the study of minimal bases for primitive permutation groups. A further application concerning monodromy groups of covers of Riemann surfaces is also outlined.