Flexible varieties and automorphism groups

Given an irreducible affine algebraic variety X of dimension n\ge2 , we let \operatorname {SAut}(X) denote the special automorphism group of X , that is, the subgroup of the full automorphism group \operatorname{Aut}(X) generated by all one-parameter unipotent subgroups. We show that if \operatorname {SAut}(X) is transitive on the smooth locus X_{\mathrm{reg}} , then it is infinitely transitive on X_{\mathrm{reg}} . In turn, the transitivity is equivalent to the flexibility of X . The latter means that for every smooth point x\in X_{\mathrm{reg}} the tangent space T_{x}X is spanned by the velocity vectors at x of one-parameter unipotent subgroups of \operatorname{Aut}(X) . We also provide various modifications and applications.