A note on 3-subgroups in the space Cremona group

TL;DR

This paper proves that any finite 3-group in the Cremona group of rank 3 can be generated by at most four elements, completing the understanding of the structure of finite p-subgroups in this group.

Contribution

It establishes an upper bound of four generators for finite 3-groups in the Cremona group, filling a key gap in the classification of p-subgroups.

Findings

Finite 3-groups in the Cremona group can be generated by at most 4 elements.

Provides the last missing piece in bounding ranks of finite p-subgroups.

Advances understanding of the structure of the Cremona group.

Abstract

We prove that a finite $3$-group in the Cremona group $\mathrm{Cr}_3(\mathbb{C})$ 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.