[Paper Review] Dynamic Asymptotic Dimension: relation to dynamics, topology, coarse geometry, and $C^*$-algebras

Erik Guentner, Rufus Willett+1 more|arXiv (Cornell University)|Oct 27, 2015

Advanced Operator Algebra Research43 references8 citations

TL;DR

This paper introduces dynamic asymptotic dimension (DAD), a new geometric and dynamical invariant for group actions on locally compact spaces and étale groupoids, generalizing Gromov's asymptotic dimension. It establishes that DAD bounds the nuclear dimension of associated C*-algebras, with key results showing that minimal $χ$-actions have DAD ≤ 1, actions satisfying Bartels-Lück-Reich conditions have DAD ≤ d, and the DAD of coarse groupoids equals the asymptotic dimension of the underlying space, leading to sharp nuclear dimension estimates for Roe algebras and crossed products.

ABSTRACT

We introduce dynamic asymptotic dimension, a notion of dimension for actions of discrete groups on locally compact spaces, and more generally for locally compact étale groupoids. We study our notion for minimal actions of the integer group, its relation with conditions used by Bartels, Lück, and Reich in the context of controlled topology, and its connections with Gromov's theory of asymptotic dimension. We also show that dynamic asymptotic dimension gives bounds on the nuclear dimension of Winter and Zacharias for C*-algebras associated to dynamical systems. Dynamic asymptotic dimension also has implications for K-theory and manifold topology: these will be drawn out in subsequent work.

Motivation & Objective

• To introduce dynamic asymptotic dimension (DAD) as a new invariant for group actions on locally compact spaces and étale groupoids.
• To establish connections between DAD and classical invariants such as Gromov's asymptotic dimension and Bartels-Lück-Reich conditions.
• To apply DAD to bound the nuclear dimension of C*-algebras arising from dynamical systems, particularly crossed products and uniform Roe algebras.
• To lay foundational tools for controlled topology and K-theory applications via Mayer-Vietoris techniques.

Proposed method

• Define DAD via finite open covers where E-generated equivalence relations on each set have uniformly finite classes, generalizing asymptotic dimension to dynamical systems.
• Use partitions of unity adapted to DAD to decompose C*-algebras into subhomogeneous pieces with controlled complexity.
• Apply controlled cutting-and-pasting techniques inspired by Guentner-Tessera-Willett and the third author’s work on asymptotic dimension.
• Prove nuclear dimension bounds by constructing almost-invariant partitions of unity that allow decomposition into algebras of low complexity.
• Generalize DAD to non-free actions and non-compact spaces via groupoid-theoretic formulations.
• Establish isomorphisms between groupoid C*-algebras and standard crossed products to transfer results to operator algebraic settings.

Experimental results

Research questions

• RQ1 How does dynamic asymptotic dimension relate to Gromov’s asymptotic dimension in the context of group actions on coarse spaces?
• RQ2 What is the dynamic asymptotic dimension of minimal $χ$-actions on compact spaces, and how does it relate to the structure of the crossed product C*-algebra?
• RQ3 To what extent do Bartels-Lück-Reich-type conditions imply finite dynamic asymptotic dimension for group actions?
• RQ4 Can dynamic asymptotic dimension be used to bound the nuclear dimension of C*-algebras associated to dynamical systems?
• RQ5 What is the relationship between the dynamic asymptotic dimension of a coarse groupoid and the asymptotic dimension of the underlying space?

Key findings

• Minimal $χ$-actions on compact spaces have dynamic asymptotic dimension exactly one.
• Actions satisfying Bartels-Lück-Reich conditions for finite subgroups have dynamic asymptotic dimension at most d, where d is the dimension parameter in their condition.
• The dynamic asymptotic dimension of the canonical action of a countable discrete group on its Stone-Čech compactification equals the asymptotic dimension of the group in Gromov’s sense.
• For any countable group of finite asymptotic dimension d, there exists a free minimal action on the Cantor set with dynamic asymptotic dimension at most d.
• The nuclear dimension of the reduced crossed product $C(X)\rtimes_r\mathbb{Z}$ for a minimal $χ$-action on a compact space of covering dimension N is at most $2N+1$.
• The uniform Roe algebra $C^*_u(X)$ of a bounded geometry coarse space X has nuclear dimension at most d, where d is the asymptotic dimension of X.