Derangements in Subspace Actions of Finite Classical Groups

TL;DR

This paper proves that in primitive subspace actions of finite classical groups, the proportion of derangements is bounded away from zero and approaches one as dimensions grow, with implications for group generation and algebraic maps.

Contribution

It establishes bounds and asymptotic behavior for derangements in subspace actions of finite classical groups, extending previous conjectures and results.

Findings

Proportion of derangements is bounded away from zero in primitive subspace actions.

As dimension and codimension increase, derangements approach totality.

Results have applications in probabilistic group generation and algebraic geometry over finite fields.

Abstract

This is the third in a series of papers in which we prove a conjecture of Boston and Shalev that the proportion of derangements (fixed point free elements) is bounded away from zero for transitive actions of finite simple groups on a set of size greater than one. This paper treats the case of primitive subspace actions. It is also shown that if the dimension and codimension of the subspace go to infinity, then the proportion of derangements goes to one. Similar results are proved for elements in finite classical groups in cosets of the simple group. The results in this paper have applications to probabilistic generation of finite simple groups and maps between varieties over finite fields.