[Paper Review] On asymptotic property C

Trevor Davila|arXiv (Cornell University)|Nov 18, 2016

Advanced Operator Algebra Research9 references3 citations

TL;DR

This paper establishes that asymptotic property C is preserved under finite products and holds for countable restricted direct products of groups with finite asymptotic dimension. It introduces hyperbolic property C and weak hyperbolic property C as infinite-dimensional analogues, proving that weak hyperbolic property C implies straight finite decomposition complexity (sFDC), and thus Property A.

ABSTRACT

Asymptotic property C was introduced by Dranishnikov to study spaces with infinite asymptotic dimension. We show that asymptotic property C is preserved by infinite products. We also show that countable restricted direct products of countable groups with finite asymptotic dimension have asymptotic property C. Then we introduce hyperbolic property C, an infinite dimensional version of hyperbolic dimension.

Motivation & Objective

• To prove that asymptotic property C is preserved under finite direct products, resolving a question from [BBGRZ] and [BM].
• To show that countable restricted direct products of countable groups with finite asymptotic dimension possess asymptotic property C.
• To introduce hyperbolic property C and weak hyperbolic property C as infinite-dimensional analogues of hyperbolic dimension.
• To establish implications between asymptotic property C, hyperbolic property C, weak hyperbolic property C, sFDC, and Property A.
• To demonstrate that weak hyperbolic property C implies sFDC and hence Property A, advancing the understanding of large-scale geometric properties in infinite-dimensional metric spaces.

Proposed method

• Uses the definition of asymptotic property C: for any increasing sequence of radii $ R_0 \leq R_1 \leq \cdots $, there exists a finite cover by uniformly bounded $ R_i $-disjoint families.
• Applies the construction of finite products by combining $ R_i $-disjoint families from each factor space to form a cover of the product space.
• Applies the restricted direct product construction on countable groups with finite asymptotic dimension, using coarse equivalence of proper left-invariant metrics.
• Introduces hyperbolic property C as a cover condition requiring uniform large scale doubling, and weak hyperbolic property C as a weaker version.
• Employs the $ R $-decomposition relation $ \mathcal{X} \stackrel{R}{\longrightarrow} \mathcal{Y} $ to build chains of decompositions leading to uniformly bounded families.
• Applies Proposition 4.4 (sFDC for uniformly bounded finite asymptotic dimension families) and Proposition 4.5 from [NR] to show that weakly uniformly large scale doubling families have uniformly finite asymptotic dimension, enabling sFDC.

Experimental results

Research questions

• RQ1 Does asymptotic property C persist under finite direct products of metric spaces?
• RQ2 Can asymptotic property C be established for countable restricted direct products of countable groups with finite asymptotic dimension?
• RQ3 What infinite-dimensional generalizations of hyperbolic dimension can be defined such that they imply sFDC and Property A?
• RQ4 How do hyperbolic property C and weak hyperbolic property C relate to asymptotic property C and sFDC?
• RQ5 Does weak hyperbolic property C imply sFDC and thus Property A?

Key findings

• Asymptotic property C is preserved under finite direct products, confirming a conjecture from [BBGRZ] and [BM].
• Countable restricted direct products of countable groups with finite asymptotic dimension possess asymptotic property C, providing a partial affirmative answer to Question 3.2 in [Y].
• Hyperbolic property C is defined as a cover condition requiring uniformly bounded, $ R_i $-disjoint families that form a uniformly large scale doubling cover.
• Weak hyperbolic property C is defined analogously but with a weaker large scale doubling condition.
• Weak hyperbolic property C implies sFDC, as shown by constructing a decomposition chain ending in a uniformly bounded family with finite asymptotic dimension.
• Consequently, weak hyperbolic property C implies Property A, completing the chain: asymptotic property C $\Rightarrow$ hyperbolic property C $\Rightarrow$ weak hyperbolic property C $\Rightarrow$ sFDC $\Rightarrow$ Property A.