Abstract
It is proved that every computable function 
on a group 
(with certain necessary restrictions) can be realized up to equivalence as a length function of elements by embedding in an appropriate finitely presented group. As an example, the length of 
, the 
th power of an element 
of a finitely presented group, can grow as 
for each computable 
. 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.