Properties of the Cremona group endowed with the Euclidean topology

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.