Flexible varieties and automorphism groups

Given an affine algebraic variety X of dimension at least 2, we let SAut (X) denote the special automorphism group of X i.e., the subgroup of the full automorphism group Aut (X) generated by all one-parameter unipotent subgroups. We show that if SAut (X) is transitive on the smooth locus of X then it is infinitely transitive on this locus. In turn, the transitivity is equivalent to the flexibility of X. The latter means that for every smooth point x of X the tangent space at x is spanned by the velocity vectors of one-parameter unipotent subgroups of Aut (X). We provide also different variations and applications.