Topologies and structures of the Cremona groups

Abstract

We study the algebraic structure of the $n$-dimensional Cremona group and show that it is not an algebraic group of infinite dimension (ind-group) if $n\ge 2$. We describe the obstruction to this, which is of a topological nature. By contrast, we show the existence of a Euclidean topology on the Cremona group which extends that of its classical subgroups and makes it a topological group.

20 pages, to appear in Annals of Mathematics

References in corpus (1)

• On the Topologies on ind-Varieties and related Irreducibility Questions

Cited by in corpus (17)

• Dynamical degrees of birational transformations of projective surfaces
• Birational geometry of del Pezzo fibrations with terminal quotient singularities
• The annihilator of the Lefschetz motive
• Connected algebraic groups acting on three-dimensional Mori fibrations
• On the dynamical degrees of reflections on cubic fourfolds
• Three plots about the Cremona groups
• Automorphisms of surfaces over fields of positive characteristic
• The WYSIWYG compactification
• Birationally isotrivial fiber spaces
• The Abelianisation of the real Cremona group
• The Cremona group is compactly presentable
• On degenerations of plane Cremona transformations
• Conjugacy classes of special automorphisms of the affine spaces
• Tits-type alternative for certain groups acting on algebraic surfaces
• Maximal subgroups in the Cremona group
• Borel subgroups of the plane Cremona group
• The embeddings of the Heisenberg group into the Cremona group