Preserving torsion orders when embedding into groups with `small' finite presentations

TL;DR

This paper surveys a construction that embeds any finitely presented group into a small presentation group while preserving torsion properties, leading to a universal torsion-free finitely presented group.

Contribution

It demonstrates that torsion orders are preserved under a specific embedding into groups with 8 generators and 26 relations, and constructs a universal torsion-free finitely presented group.

Findings

Embedding preserves torsion properties.

Existence of a universal torsion-free finitely presented group.

Construction uses Boone and Collins' method.

Abstract

We give a complete survey of a construction by Boone and Collins for embedding any finitely presented group into one with $8$ generators and $26$ relations. We show that this embedding preserves the set of orders of torsion elements, and in particular torsion-freeness. We combine this with the independent results of Belegradek and Chiodo to prove that there is an $8$-generator $26$-relator universal finitely presented torsion-free group (one into which all finitely presented torsion-free groups embed).