[Paper Review] On the centralizers of minimal aperiodic actions on the Cantor set

María Isabel Cortéz, Samuel Petite|arXiv (Cornell University)|Jul 12, 2018

Mathematical Dynamics and Fractals17 references4 citations

TL;DR

This paper investigates the centralizers of minimal aperiodic actions on the Cantor set, proving that any countable residually finite group can be embedded as a subgroup of the centralizer of a minimal $χ$-action, and any countable group can be realized as a subgroup of the normalizer of a minimal aperiodic action of a free abelian group. The key contribution is showing that the centralizer of any minimal aperiodic $\Gamma$-action on the Cantor set is always a subgroup of the centralizer of some minimal $\mathbb{Z}$-action.

ABSTRACT

In this article we study the centralizer of a minimal aperiodic action of a countable group on the Cantor set (an aperiodic minimal Cantor system). We show that any countable residually finite group is the subgroup of the centralizer of some minimal $\mathbb Z$ action on the Cantor set, and that any countable group is the subgroup of the normalizer of a minimal aperiodic action of an abelian countable free group on the Cantor set. On the other hand we show that for any countable group $G$, the centralizer of any minimal aperiodic $G$-action on the Cantor set is a subgroup of the centralizer of a minimal $\mathbb Z$-action.

Motivation & Objective

• To understand the algebraic structure of centralizers of minimal aperiodic actions on the Cantor set, particularly in relation to group embeddings and dynamical constraints.
• To investigate whether arbitrary countable groups can be realized as subgroups of centralizers or normalizers of minimal aperiodic Cantor systems.
• To establish a hierarchy of centralizers by showing that all centralizers of minimal aperiodic $\Gamma$-actions are subgroups of centralizers of $\mathbb{Z}$-actions.
• To extend known results on full groups and outer automorphism groups by linking them to centralizer structures in minimal Cantor systems.

Proposed method

• Constructing generalized subshifts over the Cantor set using iterative, $\Gamma$-invariant word collections to ensure minimality and $\Gamma$-invariance.
• Using a recursive construction of word sets $B_n$ of increasing length $\ell_n$, preserving $\Gamma$-action and ensuring density of finite-orbit points.
• Applying a Lindenstrauss-Weiss-type argument to ensure that any point in the subshift can be approximated by images under shift and $\Gamma$-action.
• Proving faithfulness of the $\Gamma$-action on the resulting subshift by showing that only the identity element fixes a dense set of letters.
• Leveraging the fact that the centralizer of a minimal $\mathbb{Z}$-action can contain arbitrary countable residually finite groups via direct construction.
• Using the stability of the property 'being a subgroup of an automorphism group of an aperiodic minimal Cantor system' under direct products to embed direct sums into $\mathbb{Z}$-centralizers.

Experimental results

Research questions

• RQ1 Can every countable residually finite group be realized as a subgroup of the centralizer of a minimal $\mathbb{Z}$-action on the Cantor set?
• RQ2 Can any countable group be embedded as a subgroup of the normalizer of a minimal aperiodic action of a free abelian group on the Cantor set?
• RQ3 Is the centralizer of any minimal aperiodic $\Gamma$-action on the Cantor set always contained within the centralizer of some minimal $\mathbb{Z}$-action?
• RQ4 What are the structural limitations on centralizers of minimal aperiodic Cantor systems, especially in terms of residual finiteness and uncountability?
• RQ5 How do the automorphism groups of full groups relate to the centralizers and normalizers of minimal Cantor systems?

Key findings

• Any countable residually finite group $\Gamma$ can be embedded as a subgroup of the centralizer of a minimal $\mathbb{Z}$-action on the Cantor set.
• Any countable group $G$ can be realized as a subgroup of the normalizer of a minimal aperiodic action of a free abelian group on the Cantor set.
• The centralizer of any minimal aperiodic $\Gamma$-action on the Cantor set is isomorphic to a subgroup of the centralizer of some minimal $\mathbb{Z}$-action.
• The construction of the generalized subshift ensures that the $\Gamma$-action is faithful and that the set of points with finite $\Gamma$-orbits is dense in the subshift.
• The property of being a subgroup of an automorphism group of an aperiodic minimal Cantor system is preserved under direct products.
• The result extends prior work on full groups and outer automorphism groups by showing that centralizers of $\mathbb{Z}$-actions can realize complex algebraic structures, including non-residually finite groups via normalizer embeddings.