Jordan property for Cremona groups

TL;DR

Under the assumption of a conjecture, the paper proves a uniform bound on the structure of finite subgroups of Cremona groups, establishing the Jordan property for these groups, including the three-dimensional case.

Contribution

It proves the Jordan property for Cremona groups of dimension three assuming the Borisov--Alexeev--Borisov conjecture, providing a uniform bound on subgroup indices.

Findings

Existence of a constant J(n) for finite subgroup structure

Cremona group Cr_3 satisfies the Jordan property

Provides bounds on abelian subgroup indices in birational automorphism groups

Abstract

Assuming Borisov--Alexeev--Borisov conjecture, we prove that there is a constant $J=J(n)$ such that for any rationally connected variety $X$ of dimension $n$ and any finite subgroup $G\subset Bir(X)$ there exists a normal abelian subgroup $A\subset G$ of index at most $J$. In particular, we obtain that the Cremona group $Cr_3=Bir(P^3)$ enjoys the Jordan property.