The isoperimetric spectrum of finitely presented groups

TL;DR

This paper explores the isoperimetric spectrum of finitely presented groups, linking it to computational complexity conjectures and recent mathematical results, to better understand the possible growth rates of Dehn functions.

Contribution

It demonstrates how Olshanskii's recent result advances the understanding of the isoperimetric spectrum, contingent on the unresolved P=NP conjecture.

Findings

Complete description of the isoperimetric spectrum modulo P=NP

Connection between group theory and computational complexity

Implications for the growth rates of Dehn functions

Abstract

The isoperimeric spectrum consists of all real positive numbers $\alpha$ such that $O(n^\alpha)$ is the Dehn function of a finitely presented group. In this note we show how a recent result of Olshanskii completes the description of the isoperimetric spectrum modulo the celebrated Computer Science conjecture (and one of the seven Millennium Problems) $\mathbf{P=NP}$ and even a formally weaker conjecture.