On Automorphisms of the Affine Cremona Group

This is a joint work with Hanspeter Kraft.
We studied group automorphisms
of the group G_n of polynomial automorphisms of the affine space \C^n. Clearly,
conjugation by an element of G_n
defines a group automorphism of G_n,
and also a
field automorphism of \C yields a group automorphism of G_n. The result we found
is that these are all group automorphisms when we restrict to the subgroup of
tame automorphisms. More precisely,
we proved the following

Theorem. If \theta is a group automorphism of G_n then there exists g in G_n and a field
automorphism \tau of \C such that

\theta(f)=\tau(gfg^-1)

for all tame automorphisms f .
For n = 2 this result is due to Julie Deserti
(note that all automorphisms of \C^2 are
tame).
The aim of this talk is to give an idea
of the proof of the theorem mentioned
above.