Relatively Free Semigroups of Intermediate Growth

The problem of calculating the growth of a finitely generated (f.g.) semigroup satisfying the given system of identities is considered. Examples of relatively free semigroups having intermediate growth, new growth criteria, and constructions are given. The main ideas employed in the proofs are based on some new points of view on growth. Our method is also useful for constructing various examples of f.g. nilpotent and nil-semigroups having intermediate growth and arbitrarily small Gelfand–Kirillov superdimension.