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Nonstandard free groups

Alexei Miasnikov, Andrey Nikolaev

TL;DR

The paper develops a unifying framework for nonstandard models of groups via interpretations in arithmetic, focusing on nonstandard free groups and their ultrapower realizations. It shows that interpreting a group $G$ in $\mathbb{Z}$ yields nonstandard models $G(\widetilde{\mathbb{Z}})$ that are elementarily equivalent to $G$, with finitely generated $G$ yielding models largely independent of the interpretation; for free groups, the nonstandard free group $\widetilde{F}$ arises from a nonstandard list superstructure that supports infinite products and enables defining nonstandard subgroups and presentations. The authors link ultrapowers to these nonstandard models, providing a robust description of ultrapowers $F^I/D$ (and more generally $G^I/D$ for groups interpretables in $\mathbb{N}$) as nonstandard versions $G(\widetilde{\mathbb{Z}})$, and derive structural results on centralizers, subgroups, and definable maps. The framework extends to a broad class of algebraic structures via interpretability in arithmetic, offering new tools for analyzing elementary theories, model substructure, and nonstandard presentations with potential applications to Malcev-type ultrapower questions.

Abstract

Interpretation of a structure $\mathbb A$ in $\mathbb B$ allows to produce structures elementarily equivalent to $\mathbb A$ given those elementarily equivalent to $\mathbb B$. In particular, interpretation of the free group in $\mathbb N$ enables us to introduce and study a family of elementary free groups, which we call nonstandard free groups. More generally, for a wide class of groups we introduce nonstandard models arising from interpretation in $\mathbb N$. We exploit interpretation to show that under mild assumptions, ultrapowers of a group can be viewed as nonstandard models of that group. This leads us to describe the structure of the ultrapowers in terms of structure of nonstandard models of natural numbers, offering insight into a longstanding question of Malcev. We also introduce fundamentals of nonstandard combinatorial group theory such as the notions of nonstandard subgroups, nonstandard normal subgroups, and nonstandard group presentations.

Nonstandard free groups

TL;DR

The paper develops a unifying framework for nonstandard models of groups via interpretations in arithmetic, focusing on nonstandard free groups and their ultrapower realizations. It shows that interpreting a group in yields nonstandard models that are elementarily equivalent to , with finitely generated yielding models largely independent of the interpretation; for free groups, the nonstandard free group arises from a nonstandard list superstructure that supports infinite products and enables defining nonstandard subgroups and presentations. The authors link ultrapowers to these nonstandard models, providing a robust description of ultrapowers (and more generally for groups interpretables in ) as nonstandard versions , and derive structural results on centralizers, subgroups, and definable maps. The framework extends to a broad class of algebraic structures via interpretability in arithmetic, offering new tools for analyzing elementary theories, model substructure, and nonstandard presentations with potential applications to Malcev-type ultrapower questions.

Abstract

Interpretation of a structure in allows to produce structures elementarily equivalent to given those elementarily equivalent to . In particular, interpretation of the free group in enables us to introduce and study a family of elementary free groups, which we call nonstandard free groups. More generally, for a wide class of groups we introduce nonstandard models arising from interpretation in . We exploit interpretation to show that under mild assumptions, ultrapowers of a group can be viewed as nonstandard models of that group. This leads us to describe the structure of the ultrapowers in terms of structure of nonstandard models of natural numbers, offering insight into a longstanding question of Malcev. We also introduce fundamentals of nonstandard combinatorial group theory such as the notions of nonstandard subgroups, nonstandard normal subgroups, and nonstandard group presentations.
Paper Structure (24 sections, 29 theorems, 44 equations, 1 figure)

This paper contains 24 sections, 29 theorems, 44 equations, 1 figure.

Table of Contents

  1. Motivation and main ideas
  2. First-order classification and elementary equivalence
  3. Ultrapowers and nonstandard groups
  4. Interpretability
  5. Interpretations
  6. Bi-interpretation
  7. Nonstandard tuples
  8. Nonstandard models of arithmetic
  9. List superstructure
  10. Interpretation of the list superstructure in arithmetic
  11. Nonstandard list superstructure
  12. Properties of nonstandard tuples
  13. Nonstandard tuples that are always there
  14. Nonstandard permutations
  15. Nonstandard sum, product, and exponentiation
  16. ...and 9 more sections

Key Result

Theorem 2.1

Let $\mathbb A$ be absolutely interpretable in $\mathbb B$ as $\mathbb A \cong \Gamma(\mathbb B)$. Then for any $\widetilde{\mathbb B}\equiv\mathbb B$ the algebraic structure $\Gamma(\widetilde{\mathbb B})$ is elementarily equivalent to $\mathbb A$.

Figures (1)

  • Figure 1: Nonstandard model of arithmetic.

Theorems & Definitions (61)

  • Definition 2.1
  • Theorem 2.1: Lemma 4.4(2), KharlampovichMyasnikovSohrabi:2021
  • Remark 2.1
  • Theorem 2.2: Uniqueness of nonstandard models, Daniyarova-Myasnikov:I
  • Definition 2.2
  • Theorem 2.3: Daniyarova-Myasnikov:I
  • Remark 2.2
  • Remark 3.1
  • Definition 3.1
  • Lemma 3.1
  • ...and 51 more