Are there any procedures which given a nonnegative nondecreasing function on the integers will construct a finitely generated group with the same growth up to the usual equivalence of growth functions? Can this at least be done for nice functions? For example Bergman gives an explicit construction of semigroups of any growth between quadratic and cubic. 


Also, is there an algorithmic way to do this if your function is recursive and given as input by a Turing machine?



How to construct a group with specified growth function
Up to equivalence there is only one exponential type of growth in which case the answer is trivial. In polynomially bounded growth the answer follows from various old theorems (notably of Wolf and Gromov): the possible growths are $n^d$ for $d$ non-negative integer and things are classified. What's remaining is intermediate growth, and the first examples with growth determined up to equivalence were obtained quite recently by Bartholdi and Erschler (Inventiones, [arxiv 1011.5266](http://arxiv.org/abs/1011.5266), 2012, see also their more recent [arxiv 1110.3650](http://arxiv.org/abs/1110.3650)). Although the growth of the first Grigorchuk group $\Gamma$ is not yet known up to equivalence, $\Gamma$ is used (in a permutational wreath product construction) in the construction of the Bartholdi-Erschler groups.


