In this paper, we study the relationship between complementary-finite asymptotic dimension and transfinite asymptotic dimension. We prove that for every metric space X, coasdim$(X)\le \gamma$ implies trasdim$(X)\le \gamma$ for every countable ordinal number γ. Therefore, the metric space with complementary-finite asymptotic dimension has asymptotic property C. Furthermore, we construct an example of a metric space satisfying asymptotic property C and finite decomposition complexity with arbitrary high asymptotic dimension growth, which shows that asymptotic property C and finite decomposition complexity need not imply sub-exponential dimension growth in general.

中文翻译：

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渐近分解性质之间的关系

在本文中，我们研究了互补有限渐近维和超限渐近维之间的关系。我们证明对于每个度量空间X，coasdim$（X）\le \gamma$ 暗示trasdim$（X）\le \gamma$对于每个可数序数γ。因此，具有互补有限渐近维数的度量空间具有渐近性质C。此外，我们构造了一个满足渐近性质C和有限分解复杂度且具有任意高渐近维数增长的度量空间的例子，这表明渐近性质C和有限分解一般而言，复杂性不一定意味着次指数维增长。