Prime spectrum and dynamics for nilpotent Cantor actions

Steven Hurder and Olga Lukina

Vol. 327 (2023), No. 1, 107–128

DOI: 10.2140/pjm.2023.327.107

Abstract

A minimal equicontinuous action by homeomorphisms of a discrete group $\Gamma$ on a Cantor set $\mathfrak{𝔛}$ is locally quasianalytic if each homeomorphism has a unique extension from small open sets to open sets of uniform diameter on $\mathfrak{𝔛}$. A minimal action is stable if the action on $\mathfrak{𝔛}$ of the closure of $\Gamma$ in the group of homeomorphisms of $\mathfrak{𝔛}$ is locally quasianalytic.

When $\Gamma$ is virtually nilpotent, we say that $\Phi :\Gamma \times \mathfrak{𝔛}\to \mathfrak{𝔛}$ 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.

Keywords

odometers, Cantor actions, profinite groups, Steinitz numbers, Heisenberg group

Mathematical Subject Classification

Primary: 20E18, 37B05, 37B45

Secondary: 57S10

Milestones

Received: 6 May 2023

Revised: 28 November 2023

Accepted: 6 December 2023

Published: 18 February 2024