自由群的满盒空间

摘要：

本文研究了满盒空间和它们之间的粗等价。我们分两部分来做。第一部分比较了不同生成元个数的自由群的满盒空间。特别地，自由群$F_k$的满盒空间不粗略地等价于自由群$F_d$的满盒空间，如果$d\ge 8k+10$。在第二部分中，我们比较了$\Box_f\mathbb{Z}^n$与$2$生成群的全盒空间。特别地，我们证明了$\mathbb{Z}^n$的满盒空间不粗等价于任何$2$生成群的满盒空间，如果$n\ge 3$ .

英文摘要：

In this paper we investigate full box spaces and coarse equivalences between them. We do this in two parts. In part one we compare the full box spaces of free groups on different numbers of generators. In particular the full box space of a free group $F_k$ is not coarsely equivalent to the full box space of a free group $F_d$, if $d\ge 8k+10$. In part two we compare $\Box_f\mathbb{Z}^n$ to the full box spaces of $2$-generated groups. In particular we prove that the full box space of $\mathbb{Z}^n$ is not coarsely equivalent to the full box space of any $2$-generated group, if $n\ge 3$.

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