Infinite transitivity of tame groups of automorphisms of affine spaces

Abstract

For positive integers $n$ and $m$, we study the actions of the groups of tame automorphisms of the $n$-dimensional affine spaces over finite fields on ordered subsets of $m$ points. Our primary interest lies in constructing tame automorphisms that take one ordered sequence to another and in proving upper and lower bounds on the maximal complexity of such automorphisms. Our preferred measure of complexity of an automorphism is the maximum of the degrees of the polynomials that define it and its inverse. Using methods and results from various branches of mathematics, including the theory of symmetric groups, affine geometry over finite fields, polynomial interpolation, combinatorics of projective spaces over fields, and polynomial automorphisms, we obtain a wide variety of qualitative and quantitative results.