Nonstandard free groups
Alexei Miasnikov, Andrey Nikolaev

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 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…
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Taxonomy
TopicsMathematical and Theoretical Analysis · Computability, Logic, AI Algorithms · Advanced Topology and Set Theory
