Tori in the Cremona groups

We classify up to conjugacy all subgroups of certain types in the full, affine and special affine Cremona groups and prove that the normalizers of these subgroups are algebraic. As an application, we obtain new results on the linearization problem by generalizing Białynicki-Birula's results of 1966–67 to disconnected groups. We prove fusion theorems for $n$-dimensional tori in the affine and special affine Cremona groups of rank $n$, and introduce and discuss the notions of Jordan decomposition and torsion primes for the Cremona groups.