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

TL;DR

This paper investigates the logical properties of certain wreath product groups, establishing their first-order rigidity and revealing a vast diversity of their central extensions with finite kernels.

Contribution

It proves that groups of the form Z^m wr Z^n are first-order rigid and bi-interpretable with Z, and shows the existence of many non-isomorphic central extensions of Z^2 wr Z with finite kernels.

Findings

Groups Z^m wr Z^n are first-order rigid and bi-interpretable with Z.

Z^2 wr Z admits 2^{aleph_0} non-isomorphic central extensions with finite kernel.

Abstract

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.