The isoperimetric spectrum of finitely presented groups

Abstract

The isoperimeric spectrum consists of all real positive numbers $\alpha$ such that $n^{\alpha }$ is equivalent to 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) $\mathbf{P}=\mathbf{NP}$ and even a formally weaker conjecture.