A classical theorem of Jordan asserts that each finite subgroup of the complex general linear group $GL(n)$ is “almost commutative”: it contains a commutative normal subgroup with index bounded by an universal constant that depends only on $n$. We discuss an analogue of this property for the groups of birational (and biregular) automorphisms of complex algebraic varieties and the groups of diffeomorphisms of real manifolds.

This is a report on joint work with Tatiana Bandman (Bar-Ilan University).