Properties of the Cremona group endowed with the Euclidean topology

TL;DR

This paper studies the topological properties of the Cremona group with the Euclidean topology, revealing its non-metrisability, the behavior of sequences of bounded order, and the limitations on path lifting.

Contribution

It establishes new topological properties of the Cremona group, including the behavior of sequences of bounded order and the non-existence of certain subgroup convergences.

Findings

Sequences of bounded order converging to identity are constant.

Cremona groups do not contain non-trivial converging subgroup sequences.

Paths in Cremona groups generally do not lift or behave like morphisms.

Abstract

Consider a Cremona group endowed with the Euclidean topology introduced by Blanc and Furter. It makes it a Hausdorff topological group that is not locally compact nor metrisable. We show that any sequence of elements of the Cremona group of bounded order that converges to the identity is constant. We use this result to show that the Cremona groups do not contain any non-trivial sequence of subgroups converging to the identity. We also show that, in general, paths in a Cremona group do not lift and do not satisfy a property similar to the definition of morphisms to a Cremona group.