Polynomially-bounded Dehn functions of groups

Abstract

It is well known that every subqadratic Dehn function is linear. A question by Bridson asked to describe the isoperimetric spectrum of groups, that is the set of all numbers $\alpha$ such that $n^{\alpha }$ is equivalent to the Dehn function of a finitely presented group. The goal of this paper is to give a description of the isoperimetric spectrum. Earlier a similar description was given by Sapir, Birget and Rips for the intersection of the isoperimetric spectrum with $[4,\infty ]$. Lowering the bound from 4 to 2 required significant new ideas and tools.