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

Olga Kharlampovich, Alexei Miasnikov, Denis Osin

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.

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

Abstract

We prove that groups of the form , where , are regularly bi-interpretable with and therefore are first-order rigid: every finitely generated group elementarily equivalent to is isomorphic to . On the other hand, we show that admits elementarily equivalent, pairwise non-isomorphic central extensions with finite kernel.
Paper Structure (16 sections, 42 theorems, 105 equations)

This paper contains 16 sections, 42 theorems, 105 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Interpretability and bi-interpretability
  4. Interpretability in $\mathbb N$ and $\mathbb Z$.
  5. Wreath products of free abelian groups
  6. Interpretability of $G$ and ${\mathbb{Z}}$ in each other
  7. Interpretability of $G$ in ${\mathbb{Z}}$
  8. Interpretability of ${\mathbb{Z}}$ in $G$
  9. Definability of ${\mathbb{Z}}$- and ${\mathbb{Z}} A$-exponentiations in $G$
  10. Definability of ${\mathbb{Z}}$-exponentiations in $G$
  11. Interpretation of the ${\mathbb{Z}} A$-action on $N$ in $G$
  12. Bi-interpretability of $G$ and ${\mathbb{Z}}$
  13. Condensed center-by-metabelian groups
  14. The space of finitely generated marked groups.
  15. The main construction
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $A$ and $B$ be free abelian groups of finite ranks. Then $B\,{\rm wr}\, A$ is regularly injectively bi-interpretable with $\mathbb Z$.

Theorems & Definitions (83)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Definition 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Corollary 1.7
  • Corollary 1.8
  • Definition 2.1
  • Definition 2.2
  • ...and 73 more