We give a family of polynomial automorphisms of the complex a ne plane whose generic length is 3 and degenerating in an automorphism of length 1 with surprisingly high degree.
We construct explicitly a family of proper subgroups of the tame automorphism group of affine three-space (in any characteristic) that are generated by the affine subgroup and a nonaffine tame automorphism. One important corollary is the titular result that settles negatively the open question (in characteristic zero) of whether the affine subgroup is a maximal subgroup of the tame automorphism group. We also prove that all groups of this family have the structure of an amalgamated free product of the affine group and a finite group over their intersection.
We provide explicit families of tame automorphisms of the complex affine three-space which degenerate to wild automorphisms. This shows that the tame subgroup of the group of polynomial automorphisms of C 3 is not closed, when the latter is seen as an infinite dimensional algebraic group.2010 Mathematics Subject Classification. 14R10.
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