Polynomially-bounded Dehn functions of groups

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