Abstract
Given a field K, we investigate which subgroups of the group AutAK2 of polynomial automorphisms of the plane are linear or not.The results are contrasted. The group AutAK2 itself is nonlinear, except if K is finite, but it contains some large subgroups, of “codimension-five” or more, which are linear. This phenomenon is specific to dimension two: it is easy to prove that any natural “finite-codimensional” subgroup of AutAK3 is nonlinear, even for a finite field K.When chK=0, we also look at a similar questions for f.g. subgroups, and the results are again disparate. The group AutAK2 has a one-related f.g. subgroup which is not linear. However, there is a large subgroup, of “codimension-three”, which is locally linear but not linear.
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