Algorithmic problems in groups with quadratic Dehn function

TL;DR

This paper constructs finitely presented groups with quadratic Dehn functions and explores their properties, including undecidability of the isomorphism problem and the existence of subgroups with faster-growing Dehn functions.

Contribution

It introduces new finitely presented groups with quadratic Dehn functions and demonstrates their complex algorithmic properties, such as undecidable isomorphism problems.

Findings

Isomorphism problem is undecidable in QD-groups

Existence of QD-groups with subgroups growing faster than any recursive function

A QD-group with undecidable conjugacy but decidable power conjugacy

Abstract

We construct and study finitely presented groups with quadratic Dehn function (QD-groups) and present the following applications of the method developed in our recent papers. (1) The isomorphism problem is undecidable in the class of QD-groups. (2) For every recursive function $f$, there is a QD-group $G$ containing a finitely presented subgroup $H$ whose Dehn function grows faster than $f$. (3) There exists a group with undecidable conjugacy problem but decidable power conjugacy problem; this group is QD.