Given a toric affine algebraic variety X and a collection of one-parameter unipotent subgroups U 1 , . . . , U s of Aut(X) which are normalized by the torus acting on X, we show that the group G generated by U 1 , . . . , U s verifies the Tits alternative, and, moreover, either is a unipotent algebraic group, or contains a nonabelian free subgroup. We deduce that if G is m-transitive for any positive integer m, then G contains a nonabelian free subgroup, and so, is of exponential growth.