Abstract. Given an irreducible affine algebraic variety X of dimension n ≥ 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 ∈ X reg the tangent space T x X is spanned by the velocity vectors at x of one-parameter unipotent subgroups of Aut(X). We provide also various modifications and applications.

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It is an open question whether every normal affine surface V over C admits an effective action of a maximal torus T = C * n (n ≤ 2) such that any other effective C * -action is conjugate to a subtorus of T in Aut(V ). We prove that this holds indeed in the following cases: (a) the Makar-Limanov invariant ML(V ) = C is nontrivial, (b) V is a toric surface, (c) V = P 1 × P 1 \∆, where ∆ is the diagonal, and (d) V = P 2 \Q, where Q is a nonsingular quadric. In case (a) this generalizes a result of Bertin for smooth surfaces, whereas (b) was previously known for the case of the affine plane (Gutwirth [Gut]) and (d) is a result of Danilov-Gizatullin [DG] and Doebeli [Do].

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