A note on automorphisms of the affine Cremona group

Abstract

Abstract. 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 G 3 in dimension 3 fixes the subgroup of tame automorphisms T G 3 , then it also fixes a whole family of non-tame automorphisms (including the Nagata automorphism).

Automorphisms of ℂ3Commuting with a ℂ+-Action

“…3 for an exact definition of Aut(C 3 / /u, Γ)). In contrast to the two-dimensional case (see (1)), the homomorphism p in the three-dimensional case is in general not surjective (see [Sta13, Proposition 1]).…”

Continuous automorphisms of Cremona groups

“…In dimension 2, the question has a positive answer if k is uncountable [9]. A partial generalization of [9] to higher dimensions has been obtained in [17], [22], [23].…”