A complex affine algebraic variety X of dimension at least two is called flexible if the subgroup SAut(X) of the automorphism group Aut(X) generated by all one-parameter unipotent subgroups acts m-transitively on the smooth locus of X for all positive integer m. It is known that any nondegenerate affine toric variety is flexible.

We prove that if such a toric variety X is smooth in codimension two then one can find a subgroup of SAut(X) generated by a finite number of one-parameter unipotent subgroups which has the same transitivity property. In fact, three such subgroups are sufficient when X is an affine space. Our proofs are based on the study of closures of some infinite dimensional groups of automorphisms of affine toric varieties in the ind-topology. This approach leads to natural constructions and questions on Lie algebras of such groups.

This is a joint work with Karine Kuyumzhiyan and Mikhail Zaidenberg.