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.Comment: Final version; to appear in Duke Math.
