Tori in the Cremona groups

TL;DR

This paper classifies certain subgroups within Cremona groups, proves their normalizers are algebraic, and extends classical linearization results to disconnected groups, introducing new structural concepts like fusion theorems and Jordan decomposition.

Contribution

It provides a classification of subgroups in Cremona groups, establishes algebraic normalizers, and generalizes linearization results to disconnected groups, introducing fusion theorems and Jordan decomposition concepts.

Findings

Normalizers of classified subgroups are algebraic.

Fusion theorems for tori in Cremona groups are established.

New notions of Jordan decomposition and torsion primes are introduced.

Abstract

We classify up to conjugacy the subgroups of certain types in the full, in the affine, and in the special affine Cremona groups. We prove that the normalizers of these subgroups are algebraic. As an application, we obtain new results in the Linearization Problem generalizing to disconnected groups Bia\lynicki-Birula's results of 1966-67. We prove "fusion theorems" for $n$-dimensional tori in the affine and in the special affine Cremona groups of rank $n$. In the final section we introduce and discuss the notions of Jordan decomposition and torsion primes for the Cremona groups.