The automorphism group of a flexible affine variety is infinitely transitive (joint work with H. Flenner, S. Kaliman, F. Kutzschebauch and M. Zaidenberg)

Given an affine algebraic variety $X$ of dimension $n\ge 2$, 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 $X_{\mathrm{reg}}$ 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 $x\in X$ the tangent space $T_xX$ is spanned by the velocity vectors at $x$ of one-parameter unipotent subgroups of $\mathrm{Aut}(X)$. So we obtain the result announced in the title. We also deduce different variations and applications.