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
X_reg then it is infinitely transitive on X_reg. In
turn, the transitivity is equivalent to the flexibility of X.
The latter means that for every smooth point x in X_reg the
tangent space of X at x is spanned by the velocity vectors at x of
one-parameter unipotent subgroups of Aut (X). We provide also
different variations and applications.