The polycyclic cyclic escape conjecture

A group $G$ has the cyclic escape property if, for every ergodic unitary representation $(V,\pi)$ of $G$ and every $v\in V$, there are infinite cyclic subgroups $C\leq G$ whose fixed-vector projections $P_Cv$ become arbitrarily small. The preceding result establishes this property for every infinite finitely generated nilpotent group.

Polycyclic cyclic escape conjecture. Every infinite finitely generated polycyclic group has the cyclic escape property.

This conjecture extends the nilpotent theorem to the broader class of finitely generated polycyclic groups. If true, it would imply directional expansiveness for every totally ergodic probability-measure-preserving action of such a group; the conjecture remains open in the supplied text.