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Groups elementary equivalent to finitely generated free metabelian

Olga Kharlampovich, Alexei Miasnikov

TL;DR

The paper classifies all groups elementarily equivalent to a free metabelian group $G$ of finite rank by establishing a strong regular bi-interpretation with ${\mathbb{Z}}$ and describing non-standard models $G(\tilde{\mathbb{Z}})$. It develops a robust toolkit of absolute and regular interpretations, including ${\mathbb{Z}}$-exponentiation and ${\mathbb{Z}}[a_1^{\pm1},\dots,a_n^{\pm1}]$-actions on the metabelian structure, and shows that bases of $G$ are definable in a uniform way. The results imply that elementary equivalence classes correspond to non-standard exponentiations over ${\mathbb{Z}}$, and extend to $A$-metabelian exponential groups with a detailed module-theoretic picture for the commutator subgroup. This yields strong model-theoretic properties for $G$ (richness, elimination of imaginaries) and provides a comprehensive description of the structures elementarily equivalent to free metabelian groups, including regular and injective bi-interpretations with ${\mathbb{Z}}$ and related rings.

Abstract

We describe groups elementarily equivalent to a free metabelian group with n generators. We also explore an exponentiation that naturally occurs in metabelian groups.

Groups elementary equivalent to finitely generated free metabelian

TL;DR

The paper classifies all groups elementarily equivalent to a free metabelian group of finite rank by establishing a strong regular bi-interpretation with and describing non-standard models . It develops a robust toolkit of absolute and regular interpretations, including -exponentiation and -actions on the metabelian structure, and shows that bases of are definable in a uniform way. The results imply that elementary equivalence classes correspond to non-standard exponentiations over , and extend to -metabelian exponential groups with a detailed module-theoretic picture for the commutator subgroup. This yields strong model-theoretic properties for (richness, elimination of imaginaries) and provides a comprehensive description of the structures elementarily equivalent to free metabelian groups, including regular and injective bi-interpretations with and related rings.

Abstract

We describe groups elementarily equivalent to a free metabelian group with n generators. We also explore an exponentiation that naturally occurs in metabelian groups.
Paper Structure (13 sections, 30 theorems, 110 equations)

This paper contains 13 sections, 30 theorems, 110 equations.

Table of Contents

  1. Introduction
  2. Interpretability and bi-interpretability
  3. By-interpretation of a free metabelian group with $\mathbb Z$
  4. Preliminaries for metabelian groups
  5. Normal forms of elements
  6. ${\mathbb{Z}}$ is absolutely interpretable in $G$
  7. Interpretation of ${\mathbb{Z}}$-exponentiation in $G$
  8. Regular injective interpretation of ${\mathbb{Z}}$ in $G$
  9. Absolute and regular interpretations of ${\mathbb{Z}}[a_1,a_1^{-1}, \ldots,a_n,a_n^{-1}]$ in $G$
  10. Interpretation of ${\mathbb{Z}} \bar{G}$-module $G'$ in $G$
  11. Regular bi-interpretability of ${\mathbb{Z}}$ and $G$
  12. Groups elementarily equivalent to a free metabelian group
  13. $A$-metabelian groups

Key Result

Lemma 2.3

Let ${\mathbb{A}}=\langle A;L({\mathbb{A}})\rangle, {\mathbb{B}}=\langle B;L({\mathbb{B}})\rangle$ and ${\mathbb{C}}=\langle C;L({\mathbb{C}})\rangle$ be algebraic structures and $\Gamma,\Delta$ be codes as above. If ${\mathbb{A}}\stackrel{\Gamma}{\rightsquigarrow}{\mathbb{B}}$ and ${\mathbb{B}}\sta

Theorems & Definitions (58)

  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3: DM1
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Theorem 2.7: DM1MN
  • Definition 2.8
  • Theorem 2.9
  • Lemma 3.1
  • ...and 48 more