On subgroup distortion in finitely presented groups
Abstract
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 nth power of an element g of a finitely presented group, can grow as n^{\theta } for each computable \theta \in (0,1 \rbrack . 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.
- Publication:
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Sbornik: Mathematics
- Pub Date:
- December 1997
- DOI:
- 10.1070/SM1997v188n11ABEH000276
- Bibcode:
- 1997SbMat.188.1617O