A note on Automorphisms of the Affine Cremona Group

Let G be an ind-group and let U⊆G be a unipotent ind-subgroup. We prove that an abstract group automorphism θ:G→G maps U isomorphically onto a unipotent ind-subgroup of G, provided that θ fixes a closed torus T⊆G, which normalizes U and the action of T on U by conjugation fixes only the neutral element. As an application we generalize a result by Hanspeter Kraft and the author as follows: If an abstract group automorphism of the affine Cremona group G3 in dimension 3 fixes the subgroup of tame automorphisms TG3, then it also fixes a whole family of non-tame automorphisms (including the Nagata automorphism).