Three plots about the Cremona groups

TL;DR

This paper explores fundamental properties of Cremona groups, including subgroup compressibility, embeddability relations, and their connectedness, advancing understanding of their algebraic and geometric structure.

Contribution

It provides new insights into the structure of Cremona groups by analyzing subgroup compressibility, embeddability, and connectedness.

Findings

Finite subgroups are compressible within Cremona groups

Certain groups can be embedded into Cremona groups and vice versa

Cremona groups exhibit connectedness properties

Abstract

The first group of results of this paper concerns the compressibility of finite subgroups of the Cremona groups. The second concerns the embeddability of other groups in the Cremona groups and, conversely, the Cremona groups in other groups. The third concerns the connectedness of the Cremona groups.