On subgroup distortion in finitely presented groups

It is proved that every computable function $G\to \mathbb N=\{0,1,\dots\}$ on a group $G$ (with certain necessary restrictions) can be realized up to equivalence as a length function of elements by embedding $G$ in an appropriate finitely presented group. As an example, the length of $g^n$, the $n$th power of an element $g$ of a finitely presented group, can grow as $n^{\theta }$ for each computable $\theta \in (0,1]$. This answers a question of Gromov [2]. The main tool is a refined version of the Higman embedding established in this paper, which preserves the lengths of elements.