Wreath product $\mathbb{Z} \wr \mathbb{Z}$ has infinite asymptotic dimension

Question

I want to know this question that wreath product $\mathbb{Z} \wr \mathbb{Z}$ has infinite asymptotic dimension. In my text , $\mathbb{Z} \wr \mathbb{Z}$ contains a subgroup isomorphic to $\mathbb{Z}^n$ for every $n \in \mathbb{N}$, but I cannot understand this fact . So , please advice to me.`

Answer 1

The restricted wreath product $A\wr K$ is defined to be

$$\underbrace{A\oplus A\oplus\cdots\oplus A}_K\rtimes K,$$

where $K$ acts on $\bigoplus A$ by permuting its coordinates. The direct sum of copies of $A$ indexed by $K$ is the same as the space of functions $K\to A$ with finite support. How? Any coordinate vector with cofinitely many zeros is interpretable as a function of the set of indices. In particular, coordinate vectors from $\Bbb Z^n$ are interpretable as functions that take an index from $\{1,\cdots,n\}$ and return the corresponding integer-valued coordinate. Obviously $\Bbb Z^n$ is isomorphic to a subgroup of $\Bbb Z^m$ for all $m\ge n$; this is no less obvious when $m$ is infinite! That is, maps $\{1,\cdots,n\}\to\Bbb Z$ are a subset of the maps $\Bbb Z\to\Bbb Z$ with finite support. Just take a function $f$ and define it to be $0$ outside of $\{1,\cdots,n\}$.