Torsion, torsion length and finitely presented groups

TL;DR

This paper explores the properties of torsion length in finitely presented groups, demonstrating how certain constructions preserve torsion length and establishing the existence of hyperbolic groups with infinite torsion length.

Contribution

It introduces new constructions showing that every finitely presented group can be realized as a quotient of a hyperbolic group by torsion elements, and demonstrates the existence of hyperbolic groups with infinite torsion length.

Findings

Construction preserves torsion length in certain embeddings

Every finitely presented group is a quotient of a hyperbolic group by torsion subgroup

Existence of hyperbolic groups with infinite torsion length

Abstract

We show that a construction by Aanderaa and Cohen used in their proof of the Higman Embedding Theorem preserves torsion length. We give a new construction showing that every finitely presented group is the quotient of some $C'(1/6)$ finitely presented group by the subgroup generated by its torsion elements. We use these results to show there is a finitely presented group with infinite torsion length which is $C'(1/6)$, and thus word-hyperbolic and virtually torsion-free.