Polynomially-bounded Dehn functions of groups

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 $d(n)\ge n^2$.