Topological simplicity of the Cremona groups

TL;DR

This paper proves that Cremona groups are topologically simple and path-connected under Zariski or Euclidean topology across all dimensions ≥ 2 over any infinite field, highlighting their fundamental topological properties.

Contribution

It establishes the topological simplicity and path-connectedness of Cremona groups in any dimension ≥ 2 over infinite fields, a significant advancement in understanding their structure.

Findings

Cremona groups are topologically simple under Zariski and Euclidean topology.

They are path-connected, with any two elements connected by an affine line.

The results hold in all dimensions ≥ 2 over any infinite field.

Abstract

The Cremona group is topologically simple when endowed with the Zariski or Euclidean topology, in any dimension $\ge 2$ and over any infinite field. Two elements are moreover always connected by an affine line, so the group is path-connected.