  The notion of $\Delta$-weakly mixing subsets is introduced for countable
torsion-free discrete group actions. It is shown that for a finitely generated
torsion-free discrete nilpotent group action, positive topological entropy
implies the existence of $\Delta$-weakly mixing subsets, and while there exists
a finitely generated torsion-free discrete solvable group action which has
positive topological entropy but without any $\Delta$-weakly mixing subsets.
