On torsion in finitely presented groups

TL;DR

This paper introduces a uniform method to construct torsion-free groups from recursive presentations, re-establishes the existence of a universal finitely presented torsion-free group, and analyzes the complexity of recognizing embeddability and torsion element orders.

Contribution

It provides a new uniform construction for torsion-free groups, re-derives the universal torsion-free group, and studies the complexity of embeddability and torsion element order recognition.

Findings

Constructs torsion-free groups from recursive presentations.

Re-establishes the universal finitely presented torsion-free group.

Shows embeddability recognition is $ ext{Pi}^0_2$-hard, $ ext{Sigma}^0_2$-hard, and in $ ext{Sigma}^0_3$.

Abstract

We give a uniform construction that, on input of a recursive presentation $P$ of a group, outputs a recursive presentation of a torsion-free group, isomorphic to $P$ whenever $P$ is itself torsion-free. We use this to re-obtain a known result, the existence of a universal finitely presented torsion-free group; one into which all finitely presented torsion-free groups embed. We apply our techniques to show that recognising embeddability of finitely presented groups is $\Pi^{0}_{2}$-hard, $\Sigma^{0}_{2}$-hard, and lies in $\Sigma^{0}_{3}$. We also show that the sets of orders of torsion elements of finitely presented groups are precisely the $\Sigma^{0}_{2}$ sets which are closed under taking factors.