[Paper Review] On Virtual Conjugacy of Generalized Odometers

María Isabel Cortéz, Konstantin Medynets|arXiv (Cornell University)|Aug 7, 2015

Advanced Operator Algebra Research29 references3 citations

TL;DR

This paper establishes a rigidity theorem for continuous orbit equivalence in minimal equicontinuous systems, proving that such systems are virtually piecewise conjugate if and only if their defining groups are virtually isomorphic and preserve the finite-index subgroup structure. It further shows that the topological full group of a minimal equicontinuous system is amenable if and only if the acting group is amenable, extending Boyle’s flip-conjugacy theorem and providing a dynamical classification of restricted isomorphisms in generalized Bunce-Deddens C*-algebras.

ABSTRACT

The paper is focused on the study of continuous orbit equivalence for generalized odometers (profinite actions). We show that two generalized odometers are continuously orbit equivalent if and only if the acting groups have finite index subgroups (having the same index) whose actions are piecewise conjugate. This result extends M.~Boyle's flip-conjugacy theorem originally established for $\mathbb Z$-actions. As a corollary we obtain a dynamical classification of the restricted isomorphism between generalized Bunce-Deddens $C^*$-algebras. We also show that the full group associated with a generalized odometer is amenable if and only if the acting group is amenable.

Motivation & Objective

• To establish a dynamical rigidity theorem for continuous orbit equivalence in minimal equicontinuous systems.
• To characterize when two profinite actions are continuously orbit equivalent via virtual isomorphism of their defining groups.
• To extend Boyle’s flip-conjugacy theorem to higher-rank $Δ^{d}$-actions.
• To classify restricted isomorphisms of generalized Bunce-Deddens C*-algebras using dynamical invariants.
• To determine the amenability of the topological full group in terms of the acting group’s amenability.

Proposed method

• Representing equicontinuous systems as $G$-odometers using nested finite-index subgroups, not necessarily normal.
• Defining continuous orbit equivalence via homeomorphisms that preserve local conjugacy on clopen neighborhoods.
• Using the orbit cocycle $f(\gamma,x)$ to relate elements of the topological full group to group elements acting locally.
• Constructing the topological full group $[[G]]$ as the union of subgroups $[[G]]_n$ compatible with a nested partition structure.
• Proving that $[[G]]_n$ is isomorphic to a semidirect product $G_n^{[G:G_n]} \rtimes S_{[G:G_n]}$ for normal subgroups $G_n$, enabling inductive limit analysis.
• Applying group-theoretic results on amenability and embeddings to relate full group amenability to group amenability.

Experimental results

Research questions

• RQ1 When are two free minimal equicontinuous systems continuously orbit equivalent?
• RQ2 What group-theoretic condition ensures continuous orbit equivalence between two profinite actions?
• RQ3 How does continuous orbit equivalence relate to virtual conjugacy in $Δ^{d}$-actions?
• RQ4 What is the role of the topological full group in determining amenability of the system?
• RQ5 How does the structure of finite-index subgroups determine orbit equivalence in generalized odometers?

Key findings

• Two profinite actions $(X,G)$ and $(Y,H)$ are continuously orbit equivalent if and only if $G$ and $H$ are virtually isomorphic and the isomorphism preserves the structure of the finite-index subgroups defining the actions.
• Continuous orbit equivalence of minimal equicontinuous $\mathbb{Z}^d$-systems implies virtual piecewise conjugacy, extending Boyle’s flip-conjugacy theorem.
• The topological full group $[[G]]$ of a minimal equicontinuous system $(X,G)$ is amenable if and only if the group $G$ is amenable.
• For a free exact odometer defined by normal subgroups $\{G_n\}$, the topological full group $[[G]]$ is isomorphic to the inductive limit $\varinjlim(G_n^{[G:G_n]} \rtimes S_{[G:G_n]}, \tau_n)$.
• The topological full group of any free minimal equicontinuous Cantor system is isomorphic to a subgroup of the full group of an exact odometer, enabling amenability transfer.
• The restricted isomorphism class of generalized Bunce-Deddens $C^*$-algebras is completely classified by the virtual isomorphism type of the associated group and the preservation of finite-index subgroup structure.