We prove that groups of the form $\mathbb Z^m { \,\rm wr\,} \mathbb Z^n$, 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 $\mathbb Z^m { \,\rm wr\,} \mathbb Z^n$ is isomorphic to $\mathbb Z^m { \,\rm wr\,} \mathbb Z^n$. On the other hand, we show that $\mathbb Z^2 { \,\rm wr\,} \mathbb Z$ admits $2^{ \aleph_0}$ elementarily equivalent, pairwise non-isomorphic central extensions with finite kernel.
Categories: math.GR