Finite subgroups of the birational automorphism group are 'almost' nilpotent of class at most two

TL;DR

This paper proves that the birational automorphism group of a variety over a field of characteristic zero is 'almost' nilpotent of class at most two, meaning finite subgroups contain large nilpotent subgroups of low class.

Contribution

It introduces the concept of nilpotently Jordan groups and establishes that the birational automorphism group fits this framework with class at most two.

Findings

Finite subgroups contain nilpotent subgroups of class at most two

The birational automorphism group is nilpotently Jordan of class at most two

Provides a new structural understanding of automorphism groups in algebraic geometry

Abstract

We call a group $G$ nilpotently Jordan of class at most $c$ $(c\in\mathbb{N})$ if there exists a constant $J\in\mathbb{Z}^+$ such that every finite subgroup $H\leqq G$ contains a nilpotent subgroup $K\leqq H$ of class at most $c$ and index at most $J$. We show that the birational automorphism group of a variety over a field of characteristic zero is nilpotently Jordan of class at most two.