Tits-type Alternative for Groups Acting on Toric Affine Varieties

Given a toric affine algebraic variety X and a collection of one-parameter unipotent subgroups U₁,… ,Uₛ of Aut(X), which are normalized by the torus acting on X, we show that the group G generated by U₁,… ,Uₛ verifies the following alternative of Tits type: either G is a unipotent algebraic group or it contains a non-abelian free subgroup. We deduce that if G is $2$-transitive on a G-orbit in X, then G contains a non-abelian free subgroup and so is of exponential growth.