[Paper Review] Examples of metric spaces with asymptotic property $C$

Yan Wu, Jingming Zhu|arXiv (Cornell University)|Dec 3, 2019

Advanced Operator Algebra Research14 references4 citations

TL;DR

This paper constructs metric spaces with arbitrarily large transfinite asymptotic dimension and complementary-finite asymptotic dimension, specifically showing that for any $k \in \mathbb{N}$, there exist spaces with both dimensions equal to $\omega + k$, and a space with dimension $2\omega$. It further demonstrates that asymptotic dimension growth and transfinite asymptotic dimension are independent invariants by constructing examples where one grows faster than the other, even when the transfinite dimension is fixed.

ABSTRACT

We construct a class of metric spaces whose transfinite asymptotic dimension and complementary-finite asymptotic dimension are both $ω+k$ for any $k\in\mathbb{N}$, where $ω$ is the smallest infinite ordinal number and a metric space whose transfinite asymptotic dimension and complementary-finite asymptotic dimension are both $2ω$. Moreover, we study the relationship between asymptotic dimension growth, transfinite asymptotic dimension and finite decomposition complexity.

Motivation & Objective

• To construct metric spaces with transfinite asymptotic dimension $\omega + k$ for any $k \in \mathbb{N}$, extending prior results on infinite-dimensional metric spaces.
• To construct a metric space with transfinite asymptotic dimension $2\omega$, resolving a long-standing question about intermediate infinite dimensions.
• To demonstrate the independence of asymptotic dimension growth from transfinite asymptotic dimension and finite decomposition complexity by explicit counterexamples.
• To show that for any unbounded increasing function $g$, there exists a metric space $X_{\omega}(g)$ with $\text{trasdim}(X_{\omega}(g)) = \omega$ and $ad_{X_{\omega}(g)}(r) \geq g(r)$ for all $r > 0$.
• To establish that asymptotic dimension growth does not control transfinite asymptotic dimension or finite decomposition complexity, even when the latter is finite.

Proposed method

• The construction uses a coarse disjoint union of rescaled integer lattices $(2^{\tilde{g}(i)}\mathbb{Z})^{g(i)}$ to build $X_{\omega}(g)$, where $\tilde{g}(r)$ is chosen so that the asymptotic dimension of each component is $g(r)$.
• Transfinite asymptotic dimension is analyzed via the notion of $\mathcal{D}_\alpha$-decomposition, showing $X_{\omega}(g) \in \mathcal{D}_\omega$ but not in any $\mathcal{D}_n$ for finite $n$, implying $\text{trasdim}(X_{\omega}(g)) = \omega$.
• The space $Y_{2\omega}$ is constructed as a coarse disjoint union of scaled copies of $X_{\omega}(g)$, with dimensions carefully tuned to achieve $\text{trasdim}(Y_{2\omega}) = 2\omega$ and $\text{coasdim}(Y_{2\omega}) = 2\omega$.
• Asymptotic dimension growth is estimated by analyzing the growth rate of $ad_X(r)$, showing it can be made arbitrarily large while keeping $\text{trasdim}(X) = \omega$.
• Independence results are proven by comparing $ad_{X_{\omega}(g)}(r)$ and $ad_{Y_{2\omega}}(r)$, showing $ad_{X_{\omega}(g)}(r) > ad_{Y_{2\omega}}(r)$ for all $r > 0$ while $\text{trasdim}(Y_{2\omega}) = 2 \cdot \text{trasdim}(X_{\omega}(g))$.
• The proof relies on covering arguments in $\ell^\infty$-type spaces and the use of $r$-disjoint, uniformly bounded families to control dimension growth and transfinite decomposition.

Experimental results

Research questions

• RQ1 Can metric spaces be constructed with transfinite asymptotic dimension $\omega + k$ for any $k \in \mathbb{N}$?
• RQ2 Is there a metric space with transfinite asymptotic dimension $2\omega$, and if so, can it be constructed explicitly?
• RQ3 Does asymptotic dimension growth control transfinite asymptotic dimension, or are they independent invariants?
• RQ4 Can a metric space have arbitrarily large asymptotic dimension growth while maintaining transfinite asymptotic dimension $\omega$?
• RQ5 Is finite decomposition complexity independent of asymptotic dimension growth, even when transfinite dimension is fixed?

Key findings

• For any $k \in \mathbb{N}$, there exists a metric space $X_{\omega+k}$ such that both its transfinite asymptotic dimension and complementary-finite asymptotic dimension are exactly $\omega + k$.
• A metric space $Y_{2\omega}$ exists with $\text{trasdim}(Y_{2\omega}) = \text{coasdim}(Y_{2\omega}) = 2\omega$, providing the first explicit construction of such a space.
• For any unbounded increasing function $g: \mathbb{R}^+ \to \mathbb{Z}^+$, a metric space $X_{\omega}(g)$ exists with $\text{trasdim}(X_{\omega}(g)) = \omega$ and $ad_{X_{\omega}(g)}(r) \geq g(r)$ for all $r > 0$.
• There exists a metric space $X_{\omega}(g)$ such that $ad_{X_{\omega}(g)}(r) > ad_{Y_{2\omega}}(r)$ for all $r > 0$, despite $\text{trasdim}(Y_{2\omega}) = 2 \cdot \text{trasdim}(X_{\omega}(g))$, proving independence between asymptotic dimension growth and transfinite dimension.
• The space $X_{\omega}(g)$ belongs to $\mathcal{D}_\omega$ but not to any $\mathcal{D}_n$ for finite $n$, confirming $\text{trasdim}(X_{\omega}(g)) = \omega$ and showing that $\mathcal{D}_\omega$ captures the correct level of decomposition complexity.
• The results show that asymptotic dimension growth and finite decomposition complexity are independent invariants, as demonstrated by constructing examples where one grows rapidly while the other remains bounded or grows slowly.