A note on Automorphisms of the Affine Cremona Group
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).