Nonstandard free groups

TL;DR

This paper introduces nonstandard free groups via interpretation in natural numbers, explores their properties, and connects ultrapowers of groups with nonstandard models, providing new insights into group theory and longstanding questions.

Contribution

It develops the theory of nonstandard free groups and models, linking ultrapowers to nonstandard models, and introduces foundational concepts in nonstandard combinatorial group theory.

Findings

Ultrapowers of groups can be viewed as nonstandard models under mild assumptions.

Interpretation in natural numbers produces elementary free groups, including nonstandard variants.

Provides new structural insights into ultrapowers and longstanding group theory 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.