[Paper Review] Long thin covers and nuclear dimension

Ilan Hirshberg, Jianchao Wu|arXiv (Cornell University)|Aug 24, 2023

Advanced Operator Algebra ResearchMathematics3 citations

TL;DR

This paper introduces the long thin covering dimension (LTC dimension), a novel topological invariant for dynamical systems, to establish finite nuclear dimension for crossed product C*-algebras arising from non-free actions of finitely generated virtually nilpotent groups, certain amenable actions of hyperbolic groups, and allosteric actions of wreath products. The method relies on constructing Rokhlin-type towers with controlled overlaps using a coarse geometric approach to the orbit space, enabling finite nuclear dimension results without requiring freeness or minimality of the action.

ABSTRACT

We establish finite nuclear dimension for crossed product C*-algebras arising from various classes of possibly non-free topological actions, including arbitrary actions of finitely generated virtually nilpotent groups on finite dimensional spaces, certain amenable actions of hyperbolic groups, and certain allosteric actions of wreath products. We obtain these results by introducing a new notion of dimension for topological dynamical systems, called the long thin covering dimension, which involves a suitable version of Rokhlin-type towers with controlled overlaps for possibly non-free actions.

Motivation & Objective

• To establish finite nuclear dimension for crossed product C*-algebras arising from non-free topological group actions.
• To overcome the limitations of prior methods that require freeness or minimality in group actions.
• To develop a new topological invariant—long thin covering dimension—for dynamical systems with controlled overlaps in Rokhlin-type towers.
• To extend finite nuclear dimension results to actions of virtually nilpotent groups, amenable hyperbolic groups, and wreath product allosteric actions.
• To provide a coarse geometric framework for analyzing the orbit space and its asymptotic dimension, enabling dimension bounds for non-free actions.

Proposed method

• Introduce the long thin covering dimension (LTC dimension) as a new invariant for topological dynamical systems, generalizing covering dimension to allow controlled overlaps in tower constructions.
• Define a coarse structure on the orbit space of the group action, enabling the use of asymptotic dimension techniques in non-free settings.
• Use barycentric subdivisions via a difference operator to refine covers and control multiplicity in the asymptotic dimension framework.
• Construct Rokhlin-type towers with controlled overlaps using the LTC dimension, replacing the need for disjointness in classical Rokhlin towers.
• Apply the coarse asymptotic dimension of the orbit space to bound the LTC dimension, leveraging ultraproduct techniques and finite subspaces.
• Prove that finite LTC dimension implies finite nuclear dimension for the associated crossed product C*-algebra via C0(X)-algebra techniques and nuclear dimension estimates.

Experimental results

Research questions

• RQ1 Can finite nuclear dimension be established for crossed product C*-algebras arising from non-free actions of finitely generated virtually nilpotent groups?
• RQ2 Does the long thin covering dimension provide a viable alternative to Rokhlin dimension in non-free dynamical systems?
• RQ3 Can the asymptotic dimension of the coarse orbit space be used to bound the LTC dimension and hence nuclear dimension?
• RQ4 To what extent can the Rokhlin-type tower construction be generalized to non-free actions using controlled overlap conditions?
• RQ5 Can the framework handle allosteric actions of wreath products and amenable actions of hyperbolic groups?

Key findings

• Finite nuclear dimension is established for crossed product C*-algebras arising from arbitrary actions of finitely generated virtually nilpotent groups on finite-dimensional compact Hausdorff spaces.
• The long thin covering dimension is introduced as a new invariant that allows controlled overlaps in Rokhlin-type towers, generalizing the classical Rokhlin tower construction to non-free actions.
• The asymptotic dimension of the coarse orbit space is shown to bound the LTC dimension, enabling dimension estimates even in the absence of freeness.
• For amenable actions of hyperbolic groups and allosteric actions of wreath products, the LTC dimension is finite, leading to finite nuclear dimension in the crossed product.
• The paper proves that the asymptotic dimension of a coarse space is equal to the supremum of the asymptotic dimensions of its finite subsets, providing a finitary characterization of asymptotic dimension.
• The construction of ultraproducts of finite subsets preserves asymptotic dimension, enabling the use of finite approximations in the proof of dimension bounds.