Normal subgroups generated by a single polynomial automorphism

Abstract

We study criteria for deciding when the normal subgroup generated by a single special polynomial automorphism of $\mathbb{A}^n$ is as large as possible, namely, equal to the normal closure of the special linear group in the special automorphism group. In particular, we investigate m-triangular automorphisms, i.e., those that can be expressed as a product of affine automorphisms and m triangular automorphisms. Over a field of characteristic zero, we show that every nontrivial 4-triangular special automorphism generates the entire normal closure of the special linear group in the special tame subgroup, for any dimension n = 2. This generalizes a result of Furter and Lamy in dimension 2.

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