Linearity and Nonlinearity of groups of polynomial automorphisms

TL;DR

This paper investigates the linearity and nonlinearity properties of groups of polynomial automorphisms of affine spaces, revealing dimension-dependent phenomena and subgroup structures over different fields.

Contribution

It demonstrates that the automorphism group of the plane is nonlinear over infinite fields and contains nonlinear subgroups, contrasting with higher dimensions where such subgroups are always nonlinear.

Findings

Automorphism group of K^2 is nonlinear over infinite fields.

Contains nonlinear FG subgroups when characteristic is zero.

Finite codimension subgroups of Aut K^3 are nonlinear, even over finite fields.

Abstract

Let $K$ be a field, and let $\Aut \,K^2$ be the group of polynomial automorphisms of $K^2$. If $K$ is infinite, this group is nonlinear. Moreover it contains nonlinear FG subgroups when $\ch\,K=0$. On the opposite, it contains some linear "finite codimension" subgroups. This phenomenon is specific to dimension two: it is also proved that "finite codimension" subgroups of $\Aut\,K^3$ are nonlinear, even for a finite field $K$.