Linearity and Nonlinearity of Groups of Polynomial Automorphisms of the Plane

Abstract

Given a field $K$, we investigate which subgroups of the group Aut${\mathbb{A}}_K^2$ of polynomial automorphisms of the plane are linear or not. The results are contrasted. The group Aut${\mathbb{A}}_K^2$ itself is nonlinear, except if $K$ is finite, but it contains some large "finite-codimensional" subgroups which are linear. This phenomenon is specific to dimension two: it is easy to prove that any "finite-codimensional" subgroup of Aut${\mathbb{A}}_K^3$ is nonlinear, even for a finite field $K$. When ch$K=0$, we also look at a similar questions for f.g. subgroups, and the results are again disparate. The group Aut${\mathbb{A}}_K^2$ has a one-related f.g. subgroup which is not linear. However, there is a large subgroup, of "co-dimension-three", which is locally linear but not linear. This paper is respectfully dedicated to the memory of Jacques Tits.