Prime spectrum and dynamics for nilpotent Cantor actions

A minimal equicontinuous action by homeomorphisms of a discrete group $\mathrm{\Gamma }$ on a Cantor set $X$ is locally quasi-analytic, if each homeomorphism has a unique extension from small open sets to open sets of uniform diameter on $X$. A minimal action is stable, if the actions of $\mathrm{\Gamma }$ and of the closure of $\mathrm{\Gamma }$ in the group of homeomorphisms of $X$, are both locally quasi-analytic. When $\mathrm{\Gamma }$ is virtually nilpotent, we say that $\mathrm{\Phi }\text{\hspace{0.17em}⁣}:\mathrm{\Gamma }\times \mathfrak{X}\to \mathfrak{X}$ is a nilpotent Cantor action. We show that a nilpotent Cantor action with finite prime spectrum must be stable. We also prove there exist uncountably many distinct Cantor actions of the Heisenberg group, necessarily with infinite prime spectrum, which are not stable.