Infinite transitivity on affine varieties

Let $X$ be an affine variety of dimension at least 2. We define the group of special automorphisms $SAut(X)$ as the subgroup in $Aut(X)$ generated by one-parameter unipotent subgroups. We prove that if the group $SAut(X)$ acts on the smooth locus of $X$ transitively, then this action is infinitely transitive. Moreover, these two properties are equivalent to flexibility, that is the tangent space at every smooth point is generated by sections of locally nilpotent vector spaces. We sketch the proof of the main results describe wide classes of flexible varieties, explain why any flexible variety is unirational and discuss a related conjecture due to Bogomolov.
The talk is based on joint works with H. Flenner, S. Kaliman, F. Kutzschebauch, K. Kuyumzhiyan and M. Zaidenberg. Also I plan to discuss infinite transitivity on universal torsor over some complete smooth rational varieties. This is a joint project with A. Perepechko and H. Suess.