Jordan groups

The concept of an abstract Jordan group, introduced by the author in 2010, was inspired by the classical Jordan theorem of 1878 and the Serre theorem of 2008. It turned out to be fruitful in the context of the classical directions of algebraic geometry and differential topology — studies of finite subgroups of groups of biregular or birational automorphisms of algebraic varieties, diffeomorphism groups of topological smooth manifolds, automorphism groups of complex spaces. The concept of Jordan group introduces a “social” aspect into these studies, which concernins the qualitative properties of all such subgroups together. The last few years have been marked by notable activity in this direction.