A note on 3-subgroups in the space Cremona group

Abstract

We prove that a finite 3-group in the Cremona group ${\text{Cr}}_3(\mathbb{C})$ can be generated by at most 4 elements. This provides the last missing piece in bounding the ranks of finite p-subgroups in the space Cremona group.

Keywords:

• p-group
• 3-group
• Cremona group
• birational automorphism

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 14J45
• 14E07
• 20D15

Acknowledgements

The author thanks Yuri Prokhorov for reading the draft of the paper and many valuable comments, Alexandra Kuznetsova and Constantin Shramov for useful discussions.

Additional information

Funding

This work is supported by Russian Science Foundation under grant 18-11-00121.