Linearity and Nonlinearity of Groups of Polynomial Automorphisms of the Plane

2023/03/14 by Mathieu, Olivier · 1 citation
#14R10 (Primary) 20F29 #20B27 #20E06 (Secondary) #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2303.07734

Abstract

Given a field K, we investigate which subgroups of the group Aut\mathbbA²_K of polynomial automorphisms of the plane are linear or not. The results are contrasted. The group Aut\mathbbA²_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\mathbbA³_K 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 Aut\mathbbA²_K 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.

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