Hall's universal group is a subgroup of the abstract commensurator of a free group

TL;DR

This paper proves that Hall's universal group, a countable locally finite group, can be embedded as a subgroup within the abstract commensurator of a finite-rank nonabelian free group, linking two important algebraic structures.

Contribution

It establishes that Hall's universal group is a subgroup of the abstract commensurator of a free group, revealing a new connection between locally finite groups and free groups.

Findings

Hall's universal group embeds into the abstract commensurator of a free group

The result bridges locally finite groups and free groups in algebraic structure

Provides new insights into the structure of the universal group and its relation to free groups

Abstract

P. Hall constructed a universal countable locally finite group U, determined up to isomorphism by two properties: every finite group C is a subgroup of U, and every embedding of C into U is conjugate in U. Every countable locally finite group is a subgroup of U. We prove that U is a subgroup of the abstract commensurator of a finite-rank nonabelian free group.