Polynomially-bounded Dehn functions of groups

TL;DR

This paper constructs finitely presented groups with Dehn functions growing as n^α for all computable α ≥ 2, filling a gap in understanding the possible polynomial and superpolynomial growth rates.

Contribution

It introduces new methods to realize Dehn functions of the form n^α for all computable α ≥ 2, expanding the known spectrum of Dehn function growth rates.

Findings

Constructed groups with Dehn functions n^α for all computable α ≥ 2

Established lower bounds for Dehn functions matching n^2

Extended the class of known Dehn function growth rates

Abstract

On the one hand, it is well known that the only subquadratic Dehn function of finitely presented groups is the linear one. On the other hand there is a huge class of Dehn functions $d(n)$ with growth at least $n^4$ (essentially all possible such Dehn functions) constructed in \cite{SBR} and based on the time functions of Turing machines and S-machines. The class of Dehn functions $n^{\alpha}$ with $\alpha\in (2; 4)$ remained more mysterious even though it has attracted quite a bit of attention (see, for example, \cite{BB}). We fill the gap obtaining Dehn functions of the form $n^{\alpha}$ (and much more) for all real $\alpha\ge 2$ computable in reasonable time, for example, $\alpha=\pi$ or $\alpha= e$, or $\alpha$ is any algebraic number. As in \cite{SBR}, we use S-machines but new tools and new way of proof are needed for the best possible lower bound $d(n)\ge n^2$.