Finitely generated subgroups of algebraic elements of plane Cremona groups are bounded

TL;DR

This paper proves that finitely generated subgroups of the plane Cremona group made up of algebraic elements have bounded degree, using results on 'decent' actions, and analyzes their degree growth.

Contribution

It establishes the boundedness of degrees for such subgroups and introduces a general framework involving 'decent' actions on infinite direct sums.

Findings

Finitely generated algebraic subgroups have bounded degree

Degree growth of subgroups can be explicitly described

Results apply to understanding Cremona group structure

Abstract

We prove that any finitely generated subgroup of the plane Cremona group consisting only of algebraic elements is of bounded degree. This follows from a more general result on `decent' actions on infinite direct sums. We apply our results to describe the degree growth of finitely generated subgroups of the plane Cremona group.