Linearity and Nonlinearity of Groups of Polynomial Automorphisms of the Plane

Abstract

Given a field K, we investigate which subgroups of the group Aut A 2 K of polynomial automorphisms of the plane are linear or not. The results are contrasted. The group Aut A 2 K 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 A 3 K 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 also contrasted. Some ''finite-codimensional'' subgroups are locally linear but not linear. This paper is respectfully dedicated to the memory of Jacques Tits