论文标题
在粗空产品的渐近维度上
On the asymptotic dimension of products of coarse spaces
论文作者
论文摘要
我们证明，对于任何粗空间，$ x_1，\ dots，x_n $的渐近维度$ \ ge 1 $，产品$ x = x_1 \ times \ times \ dots \ times x_n $具有渐近尺寸$ \ ge n $。另一个结果指出，如果$ z $允许组$ \ mathbb z^n $的几乎免费操作，则一个finarity coare space $ z $具有$ \ mathrm {asdim}（z）\ ge n $。我们从以下组合结果（概括了大风的十六进制定理）来得出这些结果：对于离散盒的任何封面$ \数学f $ 1 $这样，以至于$ | \ {f \ in \ mathcal f：f \ cap b \ ne \ emptyset \} |> n $。
We prove that for any coarse spaces $X_1,\dots,X_n$ of asymptotic dimension $\ge 1$, the product $X=X_1\times\dots\times X_n$ has asymptotic dimension $\ge n$. Another result states that a finitary coare space $Z$ has $\mathrm{asdim}(Z)\ge n$ if $Z$ admits an almost free action of the group $\mathbb Z^n$. We deduce these results from the following combinatorial result (that generalized the the Hex Theorem of Gale): for any cover $\mathcal F$ of a discrete box $K=k_1\times \dots \times k_n$, either some set $F\in\mathcal F$ contains a chain connecting two opposite faces of $K$ or there exists a set $B\subset K$ of diameter $\le 1$ such that $|\{F\in\mathcal F:F\cap B\ne\emptyset\}|>n$.
