Flexible varieties

According to A. Borel and F. Kno,p an algebraic group cannot
act 3-transitively on an irreducible affine algebraic variety.
However the automorphism group of the affine space A^n, n \ge 2,
acts infinitely transitively i.e., m-transitively for any m \ge 1.
An affine algebraic variety X of dimension at least 2 is called
flexible if its special automorphism group
(that is a certain priviledged subgroup
of the full automorphism group Aut(X)) acts infinitely transitively
on the smooth locus of X.
We provide criteria of flexibility, numerous examples, and applications.

This is a report on a joint work with Ivan Arzhantsev, Hubert Flenner,
Shulim Kaliman, Karine Kuyumzhiyan, and Frank Kutzschebauch.