# Finite subgroups of the birational automorphism group are `almost' nilpotent

## TL;DR

This paper proves that the birational automorphism group of a d-dimensional variety over a characteristic zero field has a structure where finite subgroups contain large nilpotent subgroups of bounded class, generalizing Jordan's theorem.

## Contribution

It introduces the concept of nilpotently Jordan groups and establishes that birational automorphism groups are nilpotently Jordan of class at most the variety's dimension.

## Key findings

- Finite subgroups contain large nilpotent subgroups of bounded class
- Birational automorphism groups are nilpotently Jordan of class ≤ d
- Generalizes Jordan's theorem to a broader class of groups

## 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 $d$ dimensional variety over a field of characteristic zero is nilpotently Jordan of class at most $d$.