Topologies and structures of the Cremona groups

TL;DR

This paper investigates the algebraic and topological structures of the Cremona group, revealing it is not an ind-group for dimensions two and higher, but admits a Euclidean topology making it a topological group.

Contribution

It demonstrates the non-ind-group nature of the Cremona group in higher dimensions and constructs a Euclidean topology that turns it into a topological group.

Findings

Cremona group is not an ind-group for n ≥ 2

Existence of a Euclidean topology on the Cremona group

This topology extends classical subgroup topologies

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.