<strong>Edit</strong> I edited the question. <hr> I think that the question http://mathoverflow.net/questions/151165/how-to-construct-a-group-and-the-corresponding-manifold-with-specified-growth-fu Is not an unreasonable question and I don't fully see why it is closed. The English is not optimal and no motivation is given, but it is not alone in that respect. The basic underlying question is reasonable (note also that the OP, based on previous questions, seems to come from formal language theory, not group theory). The basic question is whether there is some procedure (perhaps algorithmic) to construct groups with a prescribed growth function. Of course, constructing groups with intermediate growth is a difficult problem and I don't even think there is a procedure in general for semigroups (except one by Bergman for certain growth ranges). Work like http://arxiv.org/abs/1108.0262 is to some extent concerned with variations of this question. So I think it is reasonable to leave it open even if a complete answer seems out of reach.