Geodesic
On subgroup distortion in finitely presented groups
Sbornik. Mathematics, Tome 188 (1997) no. 11, pp. 1617-1664

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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. 
@article{SM_1997_188_11_a2,
     author = {A. Yu. Ol'shanskii},
     title = {On subgroup distortion in finitely presented groups},
     journal = {Sbornik. Mathematics},
     pages = {1617--1664},
     publisher = {mathdoc},
     volume = {188},
     number = {11},
     year = {1997},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1997_188_11_a2/}
}
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A. Yu. Ol'shanskii. On subgroup distortion in finitely presented groups. Sbornik. Mathematics, Tome 188 (1997) no. 11, pp. 1617-1664. http://geodesic.mathdoc.fr/item/SM_1997_188_11_a2/