On the first-order genus of wreath products and their central extensions

We prove that groups of the form ${\mathbb{Z}}^m\mathrm{w}\mathrm{r}{\mathbb{Z}}^n$, where $m,n\in \mathbb{N}$, are regularly bi-interpretable with $\mathbb{Z}$ and therefore are first-order rigid: every finitely generated group elementarily equivalent to ${\mathbb{Z}}^m\mathrm{w}\mathrm{r}{\mathbb{Z}}^n$ is isomorphic to ${\mathbb{Z}}^m\mathrm{w}\mathrm{r}{\mathbb{Z}}^n$. On the other hand, we show that ${\mathbb{Z}}^2\mathrm{w}\mathrm{r}\mathbb{Z}$ admits $2^{{\mathrm{\aleph }}_0}$ elementarily equivalent, pairwise non-isomorphic central extensions with finite kernel.