Nilpotent Cantor actions

A nilpotent Cantor action is a minimal equicontinuous $\Phi \colon \Gamma \times \mathfrak {X} \to {X}$ on space $\mathfrak {X}$, where $\Gamma$ contains finitely-generated subgroup $\Gamma _0 \subset \Gamma$ of finite index. In this note, we show that these actions are distinguished among general actions: any effective finitely generated group space, which continuously orbit equivalent to action, must itself be action. As an application result, obtain new invariants under continuous equivalence.