On the Isomorphism Relation for Omnigenous Locally Finite Groups

TL;DR

This paper proves that the class of all countable omnigenous locally finite groups is Borel complete, indicating it has the highest complexity of classification among countable structures.

Contribution

It establishes the Borel completeness of the class of all countable omnigenous locally finite groups, a significant complexity result in model theory.

Findings

The class of all countable omnigenous locally finite groups is Borel complete.

This class has the maximum Borel complexity among all countable structures.

The result generalizes the understanding of isomorphism relations in locally finite groups.

Abstract

The concept of an omnigenous locally finite group was introduced in [2] as a generalization of Hall's universal countable locally finite group. In this paper we show that the class of all countable omnigenous locally finite groups is Borel complete, hence it has the maximum Borel cardinality of isomorphism types among all countable structures. [2] M. Etedadialiabadi, S. Gao, F. Le Ma\^{i}tre, J. Melleray, Dense locally finite subgroups of automorphism groups of ultraextensive spaces, Adv. Math. 391 (2021), 107966.