Torsion Subgroups of Groups with Quadratic Dehn Function

TL;DR

This paper constructs the first finitely presented groups with quadratic Dehn function that contain infinite torsion subgroups, establishing optimality and embedding properties related to Burnside groups.

Contribution

It provides the first examples of such groups with quadratic Dehn function containing infinite torsion subgroups and demonstrates embedding of certain Burnside groups.

Findings

Existence of finitely presented groups with quadratic Dehn function containing infinite torsion subgroups

Embedding of infinite free Burnside groups into these groups

Optimality of quadratic Dehn function for such constructions

Abstract

We construct the first examples of finitely presented groups with quadratic Dehn function containing a finitely generated infinite torsion subgroup. These examples are "optimal" in the sense that the Dehn function of any such finitely presented group must be at least quadratic. Moreover, we show that for any $n\geq2^{48}$ such that $n$ is either odd or divisible by $2^9$, any infinite free Burnside group with exponent $n$ is a quasi-isometrically embedded subgroup of a finitely presented group with quadratic Dehn function satisfying the Congruence Extension Property.