  Given a toric affine algebraic variety $X$ and a collection of one-parameter
unipotent subgroups $U_1,\ldots,U_s$ of $\mathop{\rm Aut}(X)$ which are
normalized by the torus acting on $X$, we show that the group $G$ generated by
$U_1,\ldots,U_s$ 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.
