Continuous automorphisms of Cremona groups

TL;DR

This paper proves that automorphisms of Cremona groups that are also topological homeomorphisms are essentially inner, up to a field automorphism, extending to polynomial automorphism groups of affine spaces.

Contribution

It establishes a rigidity result for automorphisms of Cremona and polynomial automorphism groups under topological conditions, showing they are inner up to field automorphisms.

Findings

Automorphisms that are homeomorphisms are inner up to field automorphisms.

The result applies to Cremona groups of any rank.

Similar rigidity holds for polynomial automorphism groups of affine spaces.

Abstract

We show that if a group automorphism of a Cremona group of arbitrary rank is also a homeomorphism with respect to either the Zariski or the Euclidean topology, then it is inner up to a field automorphism of the base-field. Moreover, we show that a similar result holds if we consider groups of polynomial automorphisms of affine spaces instead of Cremona groups.