Flexible varieties and Automorphism groups

Given an affine algebraic variety $X$ of dimension $n\geqslant 2$, we let $\mathrm{SAut}(X)$ denote the special automorphism group of $X$ i.e., the subgroup of the full automorphism group $\mathrm{Aut}(X)$ generated by all one-parameter unipotent subgroups. We show that if $\mathrm{SAut}(X)$ 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 $T_x X$ is spanned by the velocity vectors at $x$ of one-parameter unipotent subgroups of $\mathrm{Aut}(X)$. We provide also different variations and applications.