Seminar Algebra and Geometry: Christian Urech (Universität Basel)

On automorphisms of the affine Cremona group as an abstract group and as an ind-group

The affine Cremona group $\mathcal{G}_n$ is the group of polynomial automorphisms of Aⁿ. Hanspeter Kraft and Immanuel Stampfli showed that every automorphism of $\mathcal{G}_n$ as a group is inner up to field automorphism if restricted to the subgroup of tame automorphisms. First I will sketch a proof of this theorem.

In a next step we only consider those automorphisms of $\mathcal{G}_n$ that also respect its additional algebraic structure as an ind-group. It turns out that these are exactly the inner automorphisms of $\mathcal{G}_n$. To prove this I will follow an idea recently presented by Belov-Kanel and Yu, which uses tame approximation.

These are results from my Master's thesis under the supervision of Hanspeter Kraft.

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