Arkiv för Matematik

It is known that if the special automorphism group $SAut(X)$ of a quasiaffine variety X of dimension at least 2 acts transitively on X, then this action is infinitely transitive. In this paper we question whether this is the only possibility for the automorphism group $Aut(X)$ to act infinitely transitively on X. We show that this is the case, provided X admits a nontrivial ${\mathbb{G}}_a$- or ${\mathbb{G}}_m$-action. Moreover, 2-transitivity of the automorphism group implies infinite transitivity.