Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Prime spectrum and dynamics for nilpotent Cantor actions

Steven Hurder, Olga Lukina

Abstract

A minimal equicontinuous action by homeomorphisms of a discrete group $Γ$ 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 $Γ$ and of the closure of $Γ$ in the group of homeomorphisms of $X$, are both locally quasi-analytic. When $Γ$ is virtually nilpotent, we say that $Φ\colon Γ\times \mathfrak{X} \to \mathfrak{X}$ 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.

Prime spectrum and dynamics for nilpotent Cantor actions

Abstract

A minimal equicontinuous action by homeomorphisms of a discrete group on a Cantor set is locally quasi-analytic, if each homeomorphism has a unique extension from small open sets to open sets of uniform diameter on . A minimal action is stable, if the actions of and of the closure of in the group of homeomorphisms of , are both locally quasi-analytic. When is virtually nilpotent, we say that 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.
Paper Structure (18 sections, 16 theorems, 44 equations)

This paper contains 18 sections, 16 theorems, 44 equations.

Table of Contents

  1. Introduction
  2. Cantor actions
  3. Basic concepts
  4. Profinite completion
  5. Algebraic Cantor actions
  6. Equivalence of Cantor actions
  7. Locally quasi-analytic
  8. Type and typeset for Cantor actions
  9. Type for profinite groups
  10. Nilpotent actions
  11. Noetherian groups
  12. Dynamics of Noetherian groups
  13. Basic examples
  14. Nilpotent actions with prescribed spectrum
  15. Basic components of the construction.
  16. ...and 3 more sections

Key Result

THEOREM 1.2

Let $({\mathfrak{X}},\Gamma,\Phi)$ be a nilpotent Cantor action. If the prime spectrum $\pi[{\mathfrak{X}},\Gamma,\Phi]$ is finite, then the action is stable.

Theorems & Definitions (32)

  • DEFINITION 1.1
  • THEOREM 1.2
  • THEOREM 1.3
  • THEOREM 1.4
  • COROLLARY 1.5
  • PROPOSITION 2.1
  • DEFINITION 2.2
  • DEFINITION 2.3
  • PROPOSITION 2.4
  • COROLLARY 2.5
  • ...and 22 more